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THE

OF

PRINCIPLES MATHEMATICS

BY

RUSSELL

BERTRAND

NEW

W-

W-

NORTON

YORK

"

Publishers

COMPANY,

INC

rights

All

:P R

i

isr X

E

ID

i

"

T

isr

G:R:EA.T

INTRODUCTION

TO

SECOND

EDITION

Principles of Mathematics

JL

it

was

which

In

in

some

interest that

it

have

therefore

to

say

in

what

represents

be

The

certain

thesis of the

modify.

This

thesis

of

none

for

if

first,that make

justifies,or which the

for

there

of

to

seems

fact I

subject.

Introduction, and

it expresses, have

to

me

shown

less

seen

a

new

feelings would find support

to

in

of

with

represented by

kinds

is of

that

as

logic.

be

accepted,

two

;

it

Cantor, of

account

on

led

to

Georg

These

:

logic,

is believed

formalists,

the

reasons

opposite

mathematicians

many

it shares

lasting

no

serious

mathematics

such

work,

difficult

mathematical

in

mathematics

logical basis

themselves

had

more

of two

difficulties than

felt it

rather

have

broadly speaking,

unsolved

traditionally

considered and

to

reason

any

logicis

and

mathematicians

who

master

such

certain

which

that

so

those

to

suspicion by

paradoxes are

and

justify,much

with

criticism

are,

if the

to

Aristotle,

unable

certain

are

tends

and

But

reasons

since

never

mathematics

that

pages,

first,unpopular, because

asked

been

it appear

is viewed

opposite

by Hilbert, and

intuitionists, led by Erouwer. The

for

These

I have

business,

being

they had

unsolved

lines the

was,

secondly that,

and

at

technique.

doubt.

which

one

their

resented

mathematical

influence

research

its

in this

which

opinions

following

which

philosophy

with

logicians

endeavour,

the

to

of

the

in

or

Such

erroneous.

logic are

be

I adhere

this

views.

consists

development

shall

respects subsequent

identical, is

to

but

forms.

new

amend

present

my

they

though

completely

attempt

of

problems

new

to

of

technique

others,

on

of

most

subjects

the

historical, and

stage in the

nothing,

fundamental

associated

to

expresses

is

the

some

and

taken

useless

longer

no

1903, and

years

while

;

solved,

have

possesses

respects

other

to

been

condition,

in

discussed, and

have

now a

widely

it seemed

altered

what

in

them

book

the

as

subsequent

ones

which

book,

the

greatly improved

circumstances,

that, in the

published

was

In

been

controversial

a

these

old

"

been

logichas

arisen,

remain

have

it treats

mathematical have

1900,

in

written

THE

our

interpretation of

formalist purposes

example

undefined,

in

but

we

the

may

ignore

sphere

mathematics

its older

of number,

asserting concerning

forms. it consists

them

by

is

such

no

means

new,

but

presented by Hilbert,

As in

leaving

axioms

as

the

integers

shall

make

Introduction

vi

possiblethe to

say,

that

we

they

propositions. That is except symbols 0, 1, 2,

arithmetical

usual

of the

deduction

meaning to our certain properties enumerated

assign any

do

not

are

to

have

.

the

in

.

axioms.

The

variables.

later regarded as symbols are, therefore, to be merely something integersmay be denned when 0 is given, but 0 is to having the assignedcharacteristics. Accordinglythe symbols 0, 1, 2, The do not represent one definite series,but any progressionwhatever. needed, not only for doing are formalists have forgottenthat numbers be

These

...

or

"London

system.

6,000,000

has

For

symbol

the

as propositions

Such

for counting.

but

sums,

"

"

inhabitants 0

may

There

cannot

be taken

"

"

to

be mean

12

were

"

Apostles

interpretedin their finite integer, any

false ; and thus every thereby making any infinitely ambiguous. The formalists are like number-symbol becomes look pretty that in making his watches who is so absorbed a watchmaker has therefore of tellingthe time, and he has forgotten their purpose of Hilbert's

without

omitted There

to

insert

any

is another

to

a

works. in difficulty

Hilbert

regardsexistence.

; accordingly,in

place

formalist

the

that

assumes

contradiction, there must

axioms

axioms

be of

if a set of axioms set of

some

that

does

objectswhich

himself

to methods

not

is

as

lead

satisfies the

establish existence

seeking to

instance, he devotes

and position,

theorems

of

proving the stood, him, existence," as usually underis an unnecessarily metaphysical concept, which should be replaced Here, again, he has by the precise concept of non-contradiction. that has There is no arithmetic limit to the uses. practical forgotten that Our might be invented. systems of non-contradictory axioms for the in axioms interested that lead to ordinary reasons being specially arithmetic lie outside arithmetic, and have to do with the applicationof to empiricalmaterial. This applicationitself forms no number part of but makes either logic it a which or arithmetic a prioriimpossible ; theory be "right. The cannot makes their connection logicaldefinition of numbers the with actual world of countable objects intelligible ; the formalist theory does not. The intuitionist theory, represented first by Brouwer and later by serious matter. ttis There is a more a associated with Weyl, philosophy the theory, which, for our it is we its ignore only ; bearing purposes, may that concerns on The essential point here is us. logicand mathematics the refusal to regard a propositionas either true or false unless some method exists of deciding the alternative. Brouwer denies the law of excluded middle where such exists. method no This destroys, for real numbers example, the proof that there are than more rational by producingan

self-consistency of his axioms.

numbers, and limit.

been

that, in the

series of real

Consequentlylarge parts thought

Associated

well with

established,are this

"

For

of

numbers, every progressionhas

analysis,which

rendered

for centuries

a

have

doubtful.

theory is the doctrine called finitism, which calls in question propositionsinvolvinginfinite collections or infinite the that series, on such ground This propositions are unverifiable.

Introduction is

doctrine

aspect of

an

thorough-going empiricism,and

have consequences seriously, recognizedby its advocates. if their number

we "

about

more

any generalstatement" collection defined by its

a

"

such

as

This would

mathematics,

of

only

not

questionable.Disastrous proving disproved,can

a

doctrine

only be

make

All

by

properties,not

that

than

must,

if taken

those

that

are a

infinite. If the finitist'sprincipleis admitted,

were

all its members.

not

destructive

make

not

must

even

Men, for example, though they form and practically empirically, justas impossibleto enumerate

finite class,are, as

Vll

men

are

actual

"

mortal

mention

of

clean sweep of all science and of all parts which the intuitionists consider

the

a

consequences,

however,

be

cannot

regarded as

is false ; and the finitist doctrine, if it is to be met by a complete theory of knowledge. I do

believe it to be true, but I think

short and

no

easy

refutation

of it is

possible. An

excellent and and

"

Treatise

full discussion of the

very

identical will be

logic

are

of Formal

found

Logic/1 pp, 57-200, of the

dispassionateexamination

in

where

arguments

matics question whether matheVol. Ill, of Jorgensen's the

reader

have

that

will find

been

a

adduced

against this thesis,with a conclusion which is, broadly speaking, the same as mine, namely that, while quite new grounds have been given in recent for to reduce mathematics to logic, of these refusing none years grounds is in any degreeconclusive. This bringsme to the definition of mathematics which forms the first of the sentence Principles."In this definition various changes are To begin with, the form is only one of many necessary. p impliesq forms that mathematical take. I logical was originally propositions may led to emphasise this form by the consideration of Geometry. It was clear "

"

"

that Euclidean

and

non-Euclidean

systems

alike must

be included

in pure

be regarded as mutually inconsistent ; we not mathematics, and must that assert not the axioms therefore, must, only imply the propositions, that the instances among when

axioms

led

are

me

to

true

and

lay

undue

truth-functions,and it is said that

"p

variables,"it would, of

therefore the propositionsare

and

more

q

course,

Such

is

only one implication,which than Next the others. important are propositionscontainingone or more to say that they are be more correct

stress

no

true.

on

:

be excused functions ; what is said, however, may the on prepositional and that not functions been had not were defined, propositional ground yet or mathematicians. yet familiar to logicians I come to a more serious matter, namely the statement that next "neither I except logicalconstants." q contains any constants p nor the the what for discussion to moment, as logicalconstants are. postpone, absence of non-logical this is the that known, present Assuming my point character of condition for the mathematical constants, though a necessary i s sufficient a condition. Of not this, a perhaps, the best proposition, of things in the world. examples are statements concerning the number at least three things in the world." There This is are Take, say : equivalentto : Thereexist objectsx, y, z, and properties #, such that y, -y", "

"

Introduction

viii

has the property y),and y but not i be enunciated in purelylogical can

but not y has the property q",x but not has the property #." This statement

z

x

of classes of classes of classes

proved to be true terms, and it can be logically of these there must, hi fact,be at least 4, even For

that

in

there would

case

be

if the universe did riot exist.

class,the null-class ;

one

:

classes of

two

is the only member which is null class ; and four classes of classes of classes, namely the one whose the the class of null one classes, is null,the one whose only member is the null class,and the one is the class whose only member only member

classes,namely, the class of

is the

which

of the two

sum

that

of classes,and

that

there

least three

at

are

sort is to

something of this that

It is true

logically prove of logic, very be expected; for logicaims at independence of

did not

if the world

cannot

classes,we From

members.

the

of the universe

the existence

fact, and empirical

of

But hi the lower types,that of individuals,

last.

of classes

that

the class whose

classes and

no

nature

is

an

empiricalfact.

would exist, logic-books

exist ; but

not

is not one logic,nor can it logic-books that has a rightto be in a logic-book. be inferred from any proposition is possiblewithout assuming In practice, a great deal of mathematics the existence of anything. All the elementary arithmetic of finite integers the

and

rational fractions

classes

of

which

constructed

be

can

but

;

integersbecomes impossible.This excludes analysis. To include them, we need the

"

that,it n

states

At the time

members.

I wrote

when

proved,but by

be

this could

real numbers

the

time

"

the

that

of

axiom

finite number, there is at least

is any

infinite

involves

whatever

of

the whole

n

of

existence

premissesof

of the

and

infinity," having

class

one

I supposed that Principles/*

Dr.

Whitehead

and

I

published convinced that the supposed PrincipiaMathematica," we had become proofwas fallacious. The above argument depends upon the doctrine of types, which, although "

it

hi

occurs

reached

the

a

crude

stage

form

of

in

Appendix

development

infinite classes cannot

be

existence-theorems

the

in

at

B of the

which

demonstrated

"

had Principles,"

it showed

What logically.

last

of

not

that the existence is said

as

yet of to

last

the paragraph chapter of the be valid : such to to me longer appears existence-theorems,with certain exceptions, are, I should now say, examples of propositions which can be enunciated hi logical terms, but can only be proved or disprovedby empiricalevidence. Another example is the multiplicativeaxiom, or its equivalent, Zermelo's axiom of selection. This asserts that, given a set of mutually exclusive classes,none of which is null,there is at least one class consisting "

of

"

Principles (pp. 497-8) no

representativefrom

one

not,

no

true, and it would

knows.

one

each

It is easy

class of the to

it is impossible to prove be false ; but it is also

that there

imagine that there

set.

Whether

this is true

universes hi which

it would

or

be

possibleuniverses in which impossible(at least,so I believe)to prove are

possibleuniverses hi which it would be false. I did not the necessityfor this axiom until a year after the Principleswas published. This book contains, in consequence, certain errors, for example the assertion,in "119 (p. 123),that the two definitions become "

are

no

of

aware

"

Introduction of

are infinity

equivalent,which

can

ix

only be proved if the multiplicative

is assumed.

akiom

Such

examples which might be multipliedindefinitely show that a the definition with which satisfy the proposition may Principles opens, be incapableof logical and yet may mathematical or proof or disproof. All mathematical included under propositionsare the definition (with certain minor emendations),but not all propositions that are included are In order that a propositionmay mathematical. belong to mathematics further have it must a it must be property: according to some and accordingto Carnap it must be tautological/' analytic." It is by "

"

"

"

"

"

means

no

easy

definition of this characteristic ; moreover, " that it is necessary to distinguish between analytic"

to

get

Carnap has shown "

and

exact

an

demonstrable," the

latter

being a somewhat is or is proposition

concept. not question analytic,"or demonstrable depends upon the apparatus of premisseswith which we criterion as to admissible logical begin. Unless, therefore,we have some whole the what to as are question logicalpropositionsbecomes premisses, considerable extent to a very arbitrary.This is a very unsatisfactory I and do it not But before anything more conclusion, as final. can accept be said on this subject,it is necessary to discuss the questionof logical constants," which play an essential part in the definition of mathematics of the in the firstsentence Principles." in regardto logical There are three questions constants : First,are there ? how defined ? such things in Second, are Third, do they occur they of logic? Of these questions, the firstand third are highly the propositions clearer by a little ambiguous, but their various meanings can be made And

whether

the

a

narrower "

"

41

"

"

discussion.

constants ? There is one of this question sense logical definite affirmative answer hi which we can givea perfectly : in the linguistic words there of or are or symbolic expression logical symbols propositions, the same contribution to the sigwhich play a constant nificanc part, i.e.,make for wherever of propositions Suck are, they occur. example,

First

"

Are there

:

"

"

"

"

"

"

"

1," 2," not," if-then," the null-class," 0," is that, when we analyse the propositionsin the written difficulty

or,"

The

and,"

.

.

.

find that they have no symbols occur, we in question. In some cases to the expressions constituents corresponding such

expression of which this is that

obvious fairly

the

perfect

: "

"

or

not

ardent

the most

even

is laid up

in heaven, and

Platonist would that

the

"

or's

"

suppose here on

archetype. But in the case of numbers Pythagoras,which began with arithmetical mysticism, influenced all subsequent philosophy and Numbers were mathematics more profoundlythan is generallyrealized. bodies numbers were like the : and immutable intelligible heavenly eternal, ; The last of these the key to the universe. the science of numbers was earth

imperfectcopies of

are

the celestial

this is far less obvious.

beliefs has to

the

which

misled

doctrines of

and

mathematicians

present day. mean

The

Consequently,to

nothing appears

as

a

the

Board

say

that

horrible form

of Education numbers

of atheism.

are

down

symbols

At the time

Introduction

x

I shared with Frege a belief in the Platonic Principles," peopled the timeless realm realityof numbers, which, hi my imagination, which I later with regret. of Being. It was abandoned a faith, comforting it. Something must now be said of the steps by which I was led to abandon In Chapter IV it is that of the word said Principles every again Whatever occurringin a sentence must have some meaning ; and in any true or false proposition, be an object of thought,or may occur may I wrote

when

"

the

"

"

"

"

"

or

be counted

can

as

I call

owe,

a

class, a relation,a chimsera,

is

sure

A

term.

a

a

moment,

else that

be

man,

...

number, mentioned, a

or can anything deny that such and such a thing is a term must always be false." This way of understanding language turned out have to be mistaken. That a word must the word, some meaning of course, being not gibberish,but one which has an intelligible use

be

to

; and

term

a

to

"

"

"

"

is not

always is that

true

which

it

occurs

is

no

this

there

the

word

as

contributes

but that is

;

applying

the

to

the

to

word

in isolation.

meaning of the

What

is in

sentence

different matter.

a

very first step in the process was the theory of descriptions. According " in the theory, proposition Scott is the author of Waverley/'

The to

if taken

true

constituent

"

the author of Waverley corresponding to : the analysisof the proposition is,roughly : Scott wrote Waverley, and whoever Scott wrote Waverley was ; or, more accurately: ''The prof unction wrote x is positional Waverley equivalent to x is Scott is true for all values of x." This theory swept advanced, away the contention for instance,by Meinong that there must, in the realm of Being, be such "

"

"

'

'

"

"

objects as about

the

golden "

them.

The

and

mountain

the

round

since

square,

round

we

can

talk

does not exist had always been a square What is it that does not proposition; for it was natural to ask exist ? and any possibleanswer had seemed to imply that, in some sense, there is such an objectas the round though tliis object has square, the odd property of not existing. The theory of descriptionsavoided "

difficult

"

"

this and The "

other difficulties. next

step

the abolition of classes.

was

PrincipiaMathematica,"

like

those

then-

for

uses

descriptions, are,

it is said in

our

"

:

This The

system,

step

was

taken

in

symbols for classes, incomplete symbols ;

defined, but they themselves not assumed are to mean Thus classes,so far as we introduce them, are merely linguistic conveniences,not genuine objects (Vol.I,pp. 71-2).

are

anything at all. symbolicor Seeing that

cardinal

numbers

"

"

had

been

defined

classes of classes, they conveniences." linguistic Thus, for as

merely symbolic or proposition 1-}~1=2," somewhat

the

following

...

"

also became

example,

where

"

Form

the

becomes simplified,

the

'

is not 6, and propositionalfunction a whatever x may be, x is a y is always equivalent to x is a or x is b ; form also the propositionalfunction is a y, and, whatever a x be, x is a y but is may not a is always equivalent to x is ".'"' Then, whatever be, the y may :

'

e

assertion

that

(fordifferent is not and

a

one

values

of these of

a

and

propositionalfunctions 6) is equivalent to

always false." Here the numbers similar analysiscan be applied to

1 and any

is not

the assertion 2 have

always false that the other

entirelydisappeared,

arithmetical proposition.

Introduction

xi *

Dr.

Whitehead,

this

at

stage, persuaded

to abandon me points of for them particlesof matter, substituting constructions of In the end, it seemed events. logical to composed result that none of the raw material of the world has smooth logical but that whatever to have such is properties, structed properties conappears

instants

space,

of time, and

in order artificially

apparently about

Occam's

entities which

that, when

they

to

example,

may

mentions

neither

follows

as

those

which let

these

time

which

end

that their

not

be

nor

instants.

any after it "

of instants

wholly later

a

let

Time

than

consists

of

instants," for

but

in

either

statement, It may, define

us

begins,but begin

other

any

true

x,

initial

as

"

in them.

event

than

consists is

may

define

us

later

wholly "

:

razor

time

Given

that statements

mean

numbers, or any of the other abolishes,are false,but only that they need

mentioned or

I do not

or

form is misleading,and linguistic in question are rightlyanalysed,the pseudo-entities

are

be not

them.

instants

shows

interpretationwhich found

to have

.pointsor

roughly, be interpreted its as contemporaries "

before

contemporaries

"

"

it ends

of

contemporaries of "

it

case

those

which

Then

the

x

x.

is true

if,given any contemporary of

and

;

event

among not

are

statement event

x, every

than wholly initial contemporary A similar process of interpretation is of x. some in regardto most, if not ail,purely logical constants. necessary Thus the question whether in the propositions constants occur logical of logicbecomes difficult than it seemed at first sight. It is, in more be given, can fact,a question to which, as thingsstand, no definite answer in definition of because there is no exact a proposition. occurring In the first place,no But something can be said. propositionof logic mention If Socrates is a man can object. The statement any particular some

later

is

#

"

"

"

and

ail

men

are

mortal, then Socrates

the above

logic; the logicalpropositionof which 4"

"

is mortal

propositionof is : case particular

is not is

a

a

word

"

from

which

the

99 has

has the property whatever x has the property of "p, and x, "p, y may property ip, then x has the property y", whatever

If

be." the

The

correct

here, disappears occurs property," of the statement if-then," or something proposition; but symbolic efforts to reduce After the utmost serving the same purpose, remains. shall find elements in the logicalcalculus, we of undefined the number "

left with

ourselves

incompatibility ; the function.

they

(By not

are

said

was

would

seem

"

absurd

to

what

them,

the

"

vt

or

say

"c

Principles."It

looks very

applies that generality is we

be treated

it to

will still be

as

are

to

be

true

a

at

that

the no

that

meant

What

substantial.

incompatibility ; and of

constituent

be

able

to

when

constants

general

as

part of

much

logicbecomes time

a

I

it

anything

say

part of the language,not

In this way,

speaksabout.

language linguisticthan I believed

propositionsis

equally to

therefore, if

must

of two

of these

true.) Neither about

is : one indispensable of a propositional

seem

of all values

truth

incompatibility

proposition. Logical constants, definite about

is the

other

"

the

both

earlier

(at least) which

two

wrote

except

more

the

logical

Introduction

xii constants

in the verbal

occur

"

is intended

"

Socrates

as

that

be true

it will not

but

of logical propositions, symbolicexpression of objects, constants names are these logical or

be.

to

easy except logic,or mathematics, is therefore by no means in relation to some given set of premisses. A logicalpremiss must have have be defined : it must complete can certain characteristics which no that it mentions particularthing or quality ; in the sense generality, definite set of logical Given a be true in virtue of its form. and it must define logic,in relation to them, as whatever they can premisses,we

define

To

enable

But

demonstrate.

to

us

it is hard

(1)

of its form

to

what

say

makes

; (2) it is difficult to

a

of

see any way propositiontrue in virtue of from is set a the premisses that resulting given system proving should wish of embracing everything that we complete, in the sense second As this regards to include among point, it logicalpropositions. a datum, as has been customary to accept current logicand mathematics be reconstructed. can and seek the fewest premissesfrom which this datum the have arisen But when doubts arise as concerning validityof they "

"

certain It

in

be

clear that there must

seems

than

this method

parts of mathematics, relation

to

a

leaves

some

us

of

way

in the lurch.

defining logicotherwise

particularlogicallanguage. The

fundamental

is indicated when is that which we logic,obviously, say of in virtue their form. The true that logical question of propositionsare since enter cannot in one in, which, proposition demonstrability every the premisses,might, in another from system, be system, is deduced If is the itself taken a as proposition premiss. complicated,this is be impossible. All the propositionsthat inconvenient, but it cannot in any admissible logicalsystem must demonstrable share with the are virtue the of in true of their form ; and all being property premisses propositionswhich are true in virtue of their form ought to be included in writers,for example Carnap in his Logical any adequate logic. Some of Syntax of Language," treat the whole problem as being more a matter I choice than In the above-mentioned believe it to be. can liguistic work, has of which admits two the logicallanguages,one Carnap multiplicative

characteristic of

"

and

axiom

the

It

to

seems

of formal event

clear account virtue to

the

which

of what

of its form." which

one

to

the

other

be decided

either do,

characterizes

include

must

them,

problem

as

these axioms

truth

logic

every

exclude

must

matter

a

that

me

while infinity,"

of

axiom

myself regard such

them,

or

by

does our

arbitrarychoice.

do not, have

logic,and

while

I confess,however, that

in the I

am

I cannot

not.

that

the

istic character-

in the

latter unable

is meant

former

logic give any

every to

by saying that a proposition is true in But this phrase,inadequate as it is,points,I think, be solved if an adequate definition of logic must "

is to be found. I of no

come

to finally

the

questionof

the

contradictions

and

the doctrine

considered mathematical types. Henri Poincare, who logic to be help in discovery,and therefore sterile, rejoicedin the contradictions : La logistiquen'est plus sterile ; elle engendre la contradiction ! "

Introduction All

that

mathematical

it evident to make that logicdid, however, was follow from premissespreviouslyaccepted by all logicians,

contradictions

dated

some

of mathematics.

innocent

however

the

In

from

the

Greek

Nor

were

the contradictions

all new

;

times.

"Principles," only

Forti's

Burali

xiii

concerning the

three

contradictions

greatest ordinal, the

greatest cardinal, and

mine

mentioned:

are

contradiction

concerning the

cerning con-

classes that

are

of themselves

members

is said as to (pp. 323, 366, and 101). What solutions be possible ignored, except Appendix B, on the theory may of types ; and this itself is only a rough sketch. The literature on the is vast, and the subject is stillcontroversial. contradictions The most the of treatment to is be known to in found me complete subject Carnap's he says Logical Syntax of Language (Kegan Paul, 1937). What the either difficult to to me refute that a on subject seems right or so refutation could not possiblybe attempted in a short space. I shall, therefore, confine myself to a few generalremarks. At first sight,the contradictions seem to be of three sorts : those that and those that may be suspected are mathematical, those that are logical, due trivial less trick. to of being Of the definitely some more or linguistic mathematical contradictions,those concerning the greatest ordinal and be taken as typical. the greatest cardinal may not

"

"

The

first of these, Burali

ordinal numbers will call N, from

0

up

in order

is the to

N

of

Forti's,is

magnitude

;

as

then

follows

:

Let

us

arrange

the last of these, which

But the number greatest of ordinals. We N+l, which is greater than N.

is

all we

of all ordinals cannot

escape

has no last term by suggestingthat the series of ordinal numbers ; for in this than series has that case itself ordinal number an equally greater any than number. of the series,i.e., ordinal term greater any The second contradiction, that concerning the greatest cardinal, has evident the need for some doctrine of of making peculiarly the merit the know number that of from elementary arithmetic types. We that is of at time class number n a combinations 2n, i.e.., a things any We that this has 2n sub-classes. of n terms can proposition prove And is infinite. remains Cantor true when n proved that 2n is always

greater than have

n.

supposed

Hence that

there

the

greatest possible number

can

class of

be

no

greatest cardinal.

Yet

containing everything would

terms.

Since, however,

the

one

would

have

the

number

of

of things,clearlyclasses of things tilingsexceeds the number this statement can not things. (1 will explainshortlywhat mean.) are in discussed is Of the obviouslylogical Chapter X ; contradictions, one invented in the linguistic by famous, that of the liar,was group, the most If he J the Greeks. am It is as follows : Suppose a man lying." says is lying,his statement is true, and therefore he is not lying; if he is not lying,then, when he says he is lying,he is lying. Thus either hypothesis impliesits contradictory. The logicaland mathematical contradictions, as might be expected, the linguistic not are : but reallydistinguishable group, according to

classes of

**

Introduction

xiv Ramsey*,

be solved

can

They

considerations.

means

or

depend

which

solutions

notions

by

somebody

possibleto

is

the

asserts

find

logicalconsiderations.

than

other

upon

as

linguistic

sense,

logicalgroup what

it logical,

not

are

broad

a

the

notions, such empirical

these

since

; and

be called,in

may

distinguishedfrom

are

they introduce

fact that

what

by

This

of the theory of types, which, as possiblea great simplification wholly to appear unplausible it emerges from Ramsey's discussion, ceases avoid the ad hoc hypothesis designed to artificial or a mere or renders

contradictions. technical

The a

of 'the

essence "

prepositionalfunction "

of

All values

is

*eif Socrates

if a

is

x

Socrates

man,

is

of contradiction

law

x

is is

substitute for "

are

; but

mortal"

a

law

the

man,

a

all values

mortal

a

merely

of types is

which

to legitimate man

a

theory of

(px

it is not

which expressions

"

of

true,

are "

x."

true, and we

cannot

contradiction

theory of types declares this latter set of words values of " x " in " gives rules a$ to permissible The

complications,but

this

For we

infer

,can

a

are

example:

infer is

Given

:

fchere

"if the

mortal."

be nonsense, "px" In the

and

to

detail

the

general principle is preciseform of one that has always been recognized. merely a more In the older conventional logic,it was customary to point out that such of words form virtue is triangular is neither true nor as false, but a no attempt was made to arrive at a definite set of rules for decidingwhether not significant.This the theory of types a given series of words was or was classes of things are achieves. Thus, for example I stated above that will is a "If is not This member of the class a x : a mean things." and is a proposition, then a proposition, propositi6n, "px "po. is not but a meaninglesscollection of symbols." There controversial still many are questions in mathematical logic, I have no which, hi the above pages, I have made attempt to solve. to which, in my mentioned only those matters as opinion,there has been since the time when the some fairlydefinite advance Principles was I written. Broadly speaking, stillthink this book is in the right where it disagreeswith what had been previouslyheld, but where it agrees with older theories it is apt to be wrong. The changes in philosophy which to me to be called for are seem of partly due to the technical advances mathematical have logic in the intervening thirty-fouryears, which the of and ideas simplified apparatus primitive propositions,and have there

difficulties and

are

"

"

"

'

'

'

'

'

'

"

swept away

many

apparent entities,such

Broadly,the result is an mediaeval

sense

of nominalism whether

by

means

outlook

of the word.

remains,

to

*

Foundations

or

not,

an can

classes,points, and

is less Platonic,or

far it is

mind,

my

completely soluble of mathematical

which

How

as

"

possibleto go

unsolved

only

be

less realist in the in the

Mathematics, Kegan Paul,

direction

question,but one which, adequately investigated

logic.

of

instants.

1931, p. 20 ff.

PREFACE.

has

flpHE present work -L

that

in

all pure of

terms

all its

that

mathematics

very small

a

of

objects. One

these, the proof

with concepts definable exclusively

deals

number

logicalconcepts, and

of fundamental

deducible

propositionsare

fundamental

main

two

from

a

is undertaken logical principles,

small

very

in Parts

of

number

VII. of this

II."

Volume, and will be established by strict symbolicreasoningin Volume demonstration

The

of this thesis

of and precision certainty the

As

thesis

is very

denied universally defend

to

theories I

have

its as

mistaken, all the

not

am

demonstrations

mathematical

recent

various

occasion

widely held

most

important stages

more

adverse

disprove.

untechnical

language as the

in

almost

this volume,

difficult to

most

present, in

to

is

against such

arose, or

capable.

are

mathematicians, and

among

parts, as

endeavoured

the possible,

I

I have undertaken,in by philosophers,

appeared

also

which

has, if

n.

deductions

as

the

which

by

thesis is established. The

other

object of the

explanation of as

indefinable.

myself that give

done

methods

"

the

endeavour

is the

logic

entities

the

which

which

the chief

forms

the

them as

which in the

necessary

a

process

Neptune, with

the

for telescope

confess,I have

see

and clearly,

it has

present

residue

that there must

there is

difficult

to

in

entity which

final

has

part of the undertaking. In failed to

of

entities than

the

perceiveany

others

or

the

of

a

is often

actuallyto perceive

"

of

discovery

search

the

inferred case

of

obtained

are

it analysis,

stage

the

kind

taste

resulted in the

been

clearly,

see

that

have

the indefinables

process

conducted.

part of philosophical

redness

that which

difference that the

case, a

be such

analogousto

with

flatter

and inquiry,

be

inquirymay

may

easier to know

most

"

by

accepts

I cannot

field of

vast

a

concerned,in order that the mind

primarilyas

mental

indicate

make

pineapple. Where,

;

mathematics

to

acquaintancewith

them

concepts which

than

more

I, is the

occupies Part

task,and purelyphilosophical

a

discussion of indefinables

The

of

is

of the

sample

a

work, which

fundamental

This

I have

this

"

with

is often I classes,

a

the must

the conditions concept fulfilling

Preface

xvi

Chapterx.

second

The

that

proves

failed to

hitherto

to

notion

the

for requisite

of

And

class.

something is amiss, but

discussed in

this is I have

what

discover.

volume,

I have

in which

the collaboration of Mr

secure

the contradiction

A.

had

the

great good fortune

N. Whitehead, will be addressed

it will contain chains of deductions, to mathematicians; exclusively of symboliclogicthrough Arithmetic,finite and the premisses from to Geometry,in an order similar to that adoptedin the present infinite,

volume

;

it will also contain

various

in which developments, original

the

Logic of Relations, of mathematical instrument has shown itselfa powerful investigation. The presentvolume, which may be regardedeither as a commentary introduction to, the second volume, is addressed in equal upon, or as an of Professor

method

to

measure

will be

more

the

Peano,

supplemented by

as

and philosopher

to the interesting

to

the

the mathematician

;

but

some

I should

one, others to the other.

parts advise

interested in SymbolicLogic, mathematicians,unless they are specially arises. to beginwith Part IV.,and onlyrefer to earlier partsas occasion I. Part : specially philosophical followingportionsare more (omittingChaptern.); Part II.,ChaptersXL, xv., xvi., xvn.; Part III.; Part V., Chapters XLL, Part IV., ""07, ChaptersXXVL, xxxi.; XXVIL, Part VI., ChaptersL., LI., ui.; Part VII., ChaptersLIIL, XLIL, XLIIL; which belong to Part I., uv., LV., LVIL, LVIII. ; and the two Appendices, and should be read in connection with it. Professor Frege's work, which for the most part unknown to me when anticipates largely my own, was the printing of the present work began ; I had seen his Grundgesetze der Arithmeti/c^ of his symbolism, I had but, owing to the great difficulty failed to grasp its importanceor to understand its contents. The only to his work, was to devote method, at so late a stage,of doing justice an Appendix to it; and in some pointsthe views contained in the in ""71,73, 74. Appendix differ from those in Chapter vi., especially On questions discussed in these sections, I discovered errors after passing

The

the sheets for the press ; these errors, of which the chief are the denial of the null-class, and the identification of a term with the class whose

only member treated

are

it is, so

are

rectified in

difficult that

opinions,and

regard any essentially hypotheses. A

few words

the

I

the

Appendices. The

subjects

feel little confidence

conclusions

which

may

in my present be advocated as

to originof the present work may serve show the importance of the questions discussed. About six years ago, I began an into the philosophy of Dynamics. I was investigation met is subject that, when a particle to several forces, by the difficulty as

to

Preface of

one

no

the

resultant

the

xvii

accelerations

component

actuallyoccurs,

of which acceleration,

but

only

they parts; such causation of illusory is as particulars by particulars at firstsight, affirmed, the law of by also that gravitation.It appeared in regardto absolute motion is insoluble the difficulty relational on a this fact

not

are

rendered

theoryof of the

From

space.

of principles

these two

led to

questionsI was

thence Geometry,

re-examination

a

the

philosophyof continuity the meaning of the discovering final outcome, as regardsthe

to

and thence,with a view to infinity, word any, to Symbolic Logic. The of Dynamics,is philosophy perhaps rather slender;the almost all the that is, problems of Dynamics appear to and

therefore outside the

and

scope

have interesting questions

very VI. and

of such had

work

a

be

to

as

actual

to applied

are

mathematical form

are objects

actual space

counted, or or

when

actual matter,

reasoningis appliedto

what

empirical,

me

present. Many

in omitted,especially

VII., as not relevant to my purpose, it may be well to explainat misunderstandings, When

the

of this

reason

or

Parts

which, for fear of this

stage.

Geometry and Dynamics when, in any other way,

the reasoning exists, employed

to which it is applied dependentupon the objects being that they are, but only objects upon their having certain In pure mathematics, actual objects in the world generalproperties. of existence will never be in question, but only hypothetical objects those which deduction whatever having general properties depends upon is being considered; and these will always be general properties

has

a

not

just those

in terms expressible

logicalconstants. mathematics, it

of the fundamental Thus

is not

when

or

space

actual space

concepts which

or

motion

actual

is

I have

called

spokenof

motion, as

we

in pure know them

in

that are spokenof,but any entitypossessing those abstract experience, of space or motion that are employed in the reasonings general properties of geometry or dynamics.The question whether these properties belong, as

a

matter

of

fact,to

actual

mathematics, and therefore a

space to

the

to be purelyempirical question,

or

actual motion, is irrelevant to pure

present work, being,in in the investigated

opinion, the or laboratory my

it is true, the discussions connected with pure observatory.Indirectly, mathematics have a very importantbearing questions, upon such empirical held by many, perhaps most, since mathematical are space and motion different and therefore necessarily to be self-contradictory, philosophers

actual space and motion, whereas, if the views advocated in the such self-contradictions are to be found in no following pages be valid, from

extra-mathematical considerations

But space and motion. of this kind have been almost whollyexcluded from

mathematical

the

presentwork.

Preface

xviii

in all its chief of philosophy, questions my position, I have acceptedfrom him G. E. Moore. is derived from Mr features, of propositions the non-existential nature (exceptsuch as happen to also of any knowing mind; assert existence)and their independence the pluralism which regardsthe world, both that of existents and of mutually that of entities,as composed of an infinite number independententities,with relations which are ultimate,and not of the whole which these of their terms reducible to adjectives or Before learningthese views from him, I found myself compose. whereas completelyunable to construct any philosophyof arithmetic, liberation from a large their acceptance brought about an immediate number of difficulties which I believe to be otherwise insuperable. The doctrines justmentioned in opinion,quite indispensable are, my of mathematics,as I hope to any even satisfactory tolerably philosophy the following leave it to my readers to But I must pages will show. these doctrines, and how far it judge how far the reasoningassumes are supports them. Formally,my premisses simplyassumed; but the fact that they allow mathematics to be true, which most current do not, is surelya powerfulargument in their favour. philosophies In Mathematics, is indeed evident, as to are my chief obligations, Cantor and Professor Peano. I If had become Georg acquainted of work with the Professor Frege, I should have sooner owed a fundamental

On

great deal

him, but

to

results which

he

I have

assisted

been

and criticisms,

he

also has

the

a

than

more

I

can

E. Johnson

much

to Mr

;

of

at independently

by

the

of Mr

A.

work, the suggestions, N.

Whitehead

greatlyimproved the passages. Many useful

and in the

G. E. Moore

many

every stage of my

express

proofs,and

my

large number

W.

owe

I

At

encouragement

generous

very

also to Mr

owe

of the book

arrived

alreadyestablished.

kindlyread

expressionof I

had

it is I

as

more

;

final hints

philosophical parts

besides the

generalposition

which underlies the whole. In the

acquire

an

many labour

endeavour

to

exhaustive

knowledgeof the literature. There

cover

importantworks with

so

which

wide

I

a

it field,

has been

to impossible are

doubtless

unacquainted; but where the thinkingand writingnecessarily absorbs so much time, such however regrettable, ignorance, not seems whollyavoidable. Many words wiU be found,in the course of discussion, to be defined in

am

of

apparentlydeparting widely from I must ask the reader to believe, departures, are senses

been

made

been

necessitated

with

great reluctance. In

mainly by

two

causes.

common never

Such usage. wanton, but have

philosophical matters,they have it often First, happensthat

Preface two

cognatenotions

two

names

both

are

to

for the one, but

convenient

be

none

between distinguish

to

xix

and that languagehas considered, for the other.

It is then

the two

commonly used

names

highly as

for the usual, the other for the hitherto nameless one synonyms, keeping The other cause arises from philosophical with sense. disagreement received views.

Where

are qualities commonly supposed inseparably which has the name regardedas separable,

two

but are here conjoined, to their combination will usually have to be restricted to one applied other. For example, or are commonly regardedas (1)true propositions I do, that what is true or false is not or false, as (2) mental. Holding, in

generalmental,I requirea

this

name

be scarcely

can

for the true

name

or

false

as

such,and

other than

In such a case, the proposition. As regards mathematical degreearbitrary.

usage is in no for establishing the existence-theorem in each terms,the necessity from departure

i.e.the

proofthat

there

many definitionswhich attached to the terms of

ordinal and complexnumbers. cardinal,

and

in

other cases, the

many

definition

such

doubt

no

as

done

more

than

to

But

a

by

the

these,

fact that it

in many

instances

be doubted whether

it may

to giveprecision

former of

derived from the class,

a

to the existence-theorem.

from usage, apparent departure

has been been

as

"

In the two

of abstraction, is mainly recommended principle leaves

"

question has led to appear widelydifferent from the notions usually in question.Instances of this are the definitions are

entitiesof the kind in

case

notion

had

which

of

more

hitherto

less vague.

or

so a work containing publishing many unsolved difficulties, my revealed near no apologyis,that investigation prospectof adequately the contradiction discussed in Chapter x., or of acquiring a resolving better insight of errors into the nature of classes. The repeated discovery

For

hi solutions which for

a

time

had

satisfiedme

caused these

problemsto satisfactory seemingly

onlyconcealed by any theories which a slightly longerreflection might have produced ; it seemed better, than to wait therefore, merelyto state the difficulties, of the truth of some almost certainly until I had become persuaded such

appear

My in

would

have

been

doctrine.

erroneous

their

as

thanks

Mr Secretary,

regardto

the

Syndicsof Wright, for

due to the

are

H.

T.

presentvolume

LONDON, December, 1902.

the

Press,and University

their kindness

and

to

courtesy

TABLE

OF

CONTENTS

JPAGE

INTRODUCTION

TO

SECOND

THE

EDITION

v ,

PART THE

I.

INDEFINABLES

OF

MATHEMATICS.

CHAPTER DEFINITION

1.

Definition

2.

The

3.

Pure

of

OF

L

PURE

MATHEMATICS.

mathematics

pure

3

mathematics

principlesof mathematics

are

only

uses

longer

no

few

a

controversial

3 ...

notions^

and

these

are

logical

constants 4. 5.

All

Asserts

formal

6.

And

7.

Which

8.

Mathematics

9.

Applied

employs

implications

with

Definition The

13.

.

without

types

exception

Q

of relations

hy

the

7 occurrence

of constants

which

logical

8

of mathematics

and

to

Symbolic logic

of

scope

indefinables

of

logic

8

........

II.

LOGIC.

symbolic logic

10

symbolic logic

10

of three

11

consists

14.

Definition

15.

Distinction

between

16.

Implication

indefinable

17.

Two

parts

'TJie Propoffitwnal

A.

Calculus. 13

indefinables

and

implication

and

formal

The

Disjunction

and

implication

14 ...

14 ten

in primitive propositions

primitive propositions

18.

19.

ten

4 .

5

SYMBOLIC

12.

premisses

5

CHAPTER

11.

twenty

.

is defined

mathematics

Relation

from

.

value

any

deals

not

formally

variables

have

may

are

10.

4

follows

mathematics

pure

negation

this calculus

15 .

.

16

defined .......

17

of Contents

Table

xxii

of Ckiw"t.

Calculus

The

B.

PAGE

20.

Three

21.

The

*8

indefinables

new

.........

22.

relation of an individual to its class functions ......... Prepositional

23.

The

24.

Two

25.

Relation to

calculus prepositional

26.

Identity

............

.........

28. 29. 30.

20

propositions primitive

........

new

21 .......

23 C.

27.

ld 20

of such that

notion

19 ......

Calculus

The

logicof relations essential to New primitive propositions Relative products Relations with assigneddomains

The

of Relations.

mathematics

23

.....

24

.........

25

..........

26

........

Peantf* SymbolicLogic.

D.

definitions philosophical

31.

Mathematical

#2.

Peano's indefinables

33.

Elementary definitions

34.

Peano's

and

26

......

27

..........

28 .........

propositions primitive

35.

Negation and disjunction

36.

Existence and

29

........

31

.........

the null-class

32

........

III.

CHAPTER IMPLICATION

AND

37.

Meaning

38.

Asserted and unassorted

IMPLICATION.

FORMAL

42.

implication propositions Inference does not requiretwo premisses Formal implication is to he interpreted extensionally The variable in a formal implication has an unrestricted field A formal implication is a singleprepositional not a function,

43.

Assertions

44.

Conditions that

45.

Formal

39.

of

....

33

....

^

34

.......

35

......

40. 41.

36

....

oftwo

.

36

.

relation 38

...........

......

39

......

a

term

in

an

involved implication

in rules of inference

CHAPTER PROPER

NAMES,

he varied

implication may

39

...

40

....

IV.

ADJECTIVES

AND

46.

Proper names,

47.

Terms

48.

51.

Things and concepts Conceptsas such and as terms Conceptualdiversity Meaning and the subject-predicate logic

52.

Verbs

53.

All verbs,except perhapsi", express relations Relations per se and relating relations

and verbs distinguished adjectives

VERBS. 42

....

43

............

44

..........

49.

.

50.

.45

.....

40

..........

and

47 47

......

truth ...........

54.

.....

55.

Relations

are

not

particularized by their

49 49

......

terms .....

50

Table

of

xxiii

Contents

CHAPTER

V.

DENOTING. PAGE

56.

Definition of

53

57.

Connection

54

58.

Denoting concepts obtained

59.

Extensional

60. 61.

Intensional account Illustrations

62.

The

denoting with subject-predicate propositions

of the

55

predicates

of all,every, any,

account

and

a

56

some

....

58

same

59

difference between in the

from

of

all,every,

lies in the objectsdenoted^ not

etc.

denotingthem

61 62

63.

The

notion

way of the and

definition

64.

The

notion

of the and

identity

65.

Summary

63 64

CHARTER

VI.

CLASSES. 66.

Combination

67.

Meaning

68.

Intensional and

60.

Distinctions

70.

The

class

71.

The

notion

72.

All

73.

There

74.

The

as

extensional and

one

by as

genesisof

classes

class

as

.68

.68

many

69

72

into all and men analyzable but there class-concepts,

one,

as

it has

except when

is one

and

73

...

from

is distinct

term,

the 76

each denote

Every, any,

76.

The

relation of

77. 78.

The

relation of inclusion between

The

contradiction

79.

Summary

a

some

term

one

object,but

an

ambiguous

one

classes

.78 79 80

of such Indefinability

VII.

can

be

FUNCTIONS. 82

that

fixed relation to

a

function

a

analyzed

fixed term into

a

is

asserted, a prepositional

variable

subjectand

a

constant

assertion

83. 84.

85.

83

in other cases is impossible analysis Variation of the concept in a proposition to classes functions Relation of prepositional A prepositional function is in general not analyzableinto

But this

84 86

88X

.....

and

77 77

.

PROPOSITIONAL

82.

.

to its class

CHAPTER

Where

null class

no

many a

66

67

.....

Peano

75.

81.

.

of and

null

are

class

80.

required standpoints

.67

overlooked

is not

men

extensional

of intensional and

of class

a

variable element

a

constant 88

of

Table

xxiv

Contents

VIII.

CHAPTEK

VARIABLE.

THE

IX.

CHAPTER

RELATIONS.

94.

Characteristics

9-5.

Relations

96.

The

97.

Logical

98.

A

99.

Relations

terms

and

domain

the

.96

themselves

to

domain

converse

logical product

sum,

relation

.95

relations

of

of

is not of

a

class

relation

a

of to

of

....

100.

Consequences

101.

Various

102.

An

103.

Variable

104.

The

105.

Other

106.

Summary

of

the

the

generalized

analogous

prepositional

contradiction

.

.

.99

X.

CONTRADICTION.

contradiction of

statements

98 .

terms

CHAPTER

THE

relations

of

product

99

couples its

a

relative

and

97

relation

102

argument

102

functions

arises

101

contradiction

from

are

in

general

treating

as

one

inadmissible

103 .

a

class

which

.

only

is

104

many

primd of

facie Part

possible I

solutions

appear

inadequate

.105 .

.

.

106

Table

of

Contents

PART

xxv

II.

NUMBER.

CHAPTER DEFINITION

OF

XL

CARDINAL

NUMBERS. PAGE

107.

Plan of Part II

108.

Mathematical

109.

Definition of

110.

to Objections

111.

Nominal

Ill

meaning of definition numbers by abstraction

Ill 112

this definition

114

definition of numbers

115

CHAPTER ADDITION

AND

XII. MULTIPLICATION.

113.

be considered at present Definition of arithmetical addition

114.

Dependenceupon

115.

Definition of

116.

Connection

112.

Only integersto

addition logical multiplication the

of classes

AND

FINITE

117.

Definition of finite and infinite

118.

Definition of a0 Definition of finite numbers

120.

Peano's indefinables and

exponentiation .

121. 122.

Peano

123.

Proof of Peano's

.110-

XIII. INFINITE.

121

by

mathematical

OF

FINITE

induction

.

.

.123

XIV. NUMBERS. 124

primitive propositions

independence of the latter

Mutual

.

.121

CHAPTER THEORY

118 119-

and of addition,multiplication

CHAPTER

119.

117 117

not finite numbers reallydefines progressions, primitive propositions

125 ....

125

127

Table

xxvi

Contents

of

XV.

CHAPTER

ADDITION

CLASSES.

OF

ADDITION

AND

TERMS

OF

PAGE

124.

mathematics

distinguished

fundamental

sense

Philosophyand

125.

Is there

126.

Numbers

127.

Numbers

more

a

must

129

that defined above ?

than

of number

be classes

129.

classes as many not of terms, but of unit classes One is to be asserted, Countingnot fundamental in arithmetic

130.

Numerical

131.

Addition

132.

A

128.

term

apply to

is

but indefinable,

135. 136.

Singleterms

133

the number

not

How

139.

A class as

be defined

part cannot

far

137 137

....

.

the numerical

.

.

one

is

an

aggregate

140.

Infinite

141.

Infinite unities,, if there

.

.

.

.

.140

conjunctionof its parts

141

.

141 ,

XVII. WHOLES.

be admitted

aggregatesmust

are

any,

are

143 unknown

Are all infinitewholes aggregates of terms Grounds in favour of this view

RATIOS 144.

Definition of ratio

145.

Ratios

146.

Fractions

are

one-one are

AND

to

us

...

.

.144

.

.146

?

146

XVIII. FRACTIONS. 149

relations

150

concerned

with relations of whole and part upon number, but upon magnitude of

.

148.

.

141

CHAPTER

147.

138

.

is falsification analysis

JNFINITE

143.

PART.

AND

CHAPTER

142.

135

be either

may

is distinct from

whole

A

138.

135

.

XVI.

.

137.

.

1

simpleor complex by logical priority Three kinds of relation of whole and part distinguished Two kinds of wholes distinguished and

Whole

132

.

133

numbers.

not generatesclasses primarily,

WHOLE

134.

.

and plurality conjunction of terms

CHAPTER

133.

.132

"

.

130

.131

"

Fractions depend, not Summary of Part II

.

.

150

divisibility151 152

Table

xxvii

Contents

of

PART

III

QUANTITY. CHAPTER THE

XIX.

MEANING

MAGNITUDE.

OF

PAGE

the relation of number

and

1-57

149.

Previous views

150.

Quantity

151.

157.

Meaning quantity theories of equality to be examined possible is of number of parts Equality not identity relation of quantities Equalityis not an unanalysable of magnitude Equalityis sameness Every particular magnitude is simple of abstraction The principle

Io8.

Summary

167

Note

168

152. 153. 154. 155. 156.

fundamental

not

of

on

in mathematics

.

.

.

158

......

15S)

magnitude and

Three

159

....

160 .

1(52

.

.

104 164 16f"

........

XX.

CHAPTER THE

RANGE

belongto

159.

does Divisibility

160.

Distance

161.

Differential coefficients

162.

A

163.

quantity

not

all

173

but divisible,

never

magnitude is

be

may

NUMBERS

magnitude of divisibility .

XXL :

MEASUREMENT. 176

164.

Definition of measurement

165. 166.

holdingall magnitudesto Intrinsic measurability

167.

Of divisibilities

168.

And

grounds for

be measurable

.

176

.

-

Measure

170.

Distance-theories

171.

Extensive and

177 178 179

of distances

169.

173

174

MAGNITUDES

EXPRESSING

AS

a

unanalysable CHAPTER

Possible

170

quantities

171

magnitude is

Every

QUANTITY.

OF

of distance

and

measure

aiidstretch-theories

intensive

181

of stretch

magnitudes

of

geometry .

.

.

181

....

.

.

.

.182

XXIL

CHARTER ZERO. 172.

Difficultiesas to

184

zero

173.

Meinong'stheory

174.

Zero

175.

Zero

distance

176.

Zero

as

177.

Zero and

178.

Every

.184 185

minimum

as

a

null

as

identity

186

.....-..-

186

segment

186

negation

kind

of

zero

magnitude is in

a

sense

indefinable

....

187

xxviii

Table

Contents

of

XXIII.

CHAPTER THE

INFINITY,

CONTINUITY.

AND

INFINITESIMAL,

PAGE

179.

Problems

180.

Statement

181.

Three

182.

Of which

183.

And

184.

Which

185.

Provisional

186.

Summary

of

not specially infinity quantitative of the problem in regardto quantity

188 188

.

antinomies

the

189

the antitheses

depend upon

of mathematical

nse

are

both

to be

sense

of

of finitude

axiom

an

.

190

.

.

induction

rejected continuity

192 192

.

.

.

.

.

.

.193

.

of Part II I

194

PART

IV,

ORDER.

CHAPTER THE

XXIV.

GENESIS

OF

SERIES,

187-

Importanceof

188.

Between

189.

Generation

190.

By transitive asymmetrical relations By distances relations By triangular between relations By asymmetricalrelations And by separationof couples

191. 192. 193. 194.

and

order

199

separationof couples of order

by

one-one

199

relations

CHAPTER THE 195.

What

is order?

196.

Three

theories of between

197.

First

203 204 204 205 205

XXV. OF

ORDER.

207 207

theory

208

relation is not between its terms

198.

A

199.

Second

200.

There

201.

Reasons

202.

MEANING

200

theoryof

210

between

.211

relations appear to be ultimate triangular for rejecting the second theory

.

.

.

theoryof between to be rejected Meaning of separationof couples

204.

Reduction

205.

This

.

.

.213

!

Third

203.

211

.

*

.

.

*

to

transitive asymmetrical relations

215

is formal reductio.il

216

......

206.

But

is the

207.

The

second the

reason

way

meaning

213 214

why separationleads to order .....* generatingseries is alone fundamental,and gives

216

of

of order

j|

^

2ig

Table

Contents

of

CHAFfER

xxix

XXVI.

ASYMMETRICAL

RELATIONS. PAGE

Classification

208.

209.

Symmetrical

210.

lleflexiveness

211.

Kelative

212.

Are

213.

Monadistic

214. 215.

216.

of relations

regards symmetry

transit!

218

veuess

.

21 i)

principleof

the

abstraction

219

position

relations

220

reducible

to

? predications

221

theory of relations Reasons for rejectingthis theory Monistic it for rejectingtheory and the reasons Order requires that relations should be ultimate

CHAPTER DIFFERENCE

217.

Kant

218.

Meaning

219.

Difference

220.

In the

221.

And

222.

Right

223.

Difference

difference

on

of

220

OF

DIFFERENCE

AND

sense

of

222 224

SIGN.

227

........

228

sense

sign

228

-

229

of finite numbers

cases

of

of

of difference

222

XXVII.

SENSE

OF

.

229

magnitudes

and

231

left of

sign

asymmetrical

THE

of

difference

from

arises

sense

among

232 .

DIFFERENCE

XXVIII. OPEN

BETWEEN

difference between

224,

What

is the

225.

Finite

closed

22(5.

Series

generated by triangularrelations

227.

Four-term

228.

Closed

open

and

closed

AND

CLOSED

series ?

are

as

have

PROGRESSIONS

an

236

arbitraryfirst term

Definition

230.

All finite arithmetic

appliesto

231.

Definition

of ordinal

numbers

232.

Definition

of "wth"

233.

Positive

238 ....

XXIX.

AND

229.

and

234

237

such

CHAPTER

of

SERIES.

234

series

relations

series

transitive

relations

CHAPTER ON

and

relations

transitive and

as

ORDINAL

NUMBERS. 239

progressions

negative ordinals

every

progression

240 ....

242 243 244

Table

xxx

of

Content*

XXX.

CHAPTER

NUMBER.

OF

THEORY

DEDEKIXD'S

PAGE

ideas principal

234.

Dedekind's

235.

of Representation

236.

The

notion of

237.

The

chain of

238.

Generalized

239.

Definition of

240. 241.

Definition of cardinals Dedekind's proofof mathematical

242.

Objectionsto his definition of ordinals

243.

And

a

245

.

245

system

chain

246k

element

24(5

an

246

induction

of mathematical

form a

a

"

247

singlyinfinite system

247 248

induction

248 249

of cardinals

XXXI.

CHAPTER DISTANCE. Distance

245.

Definition of distance

240.

Measurement

247.

In most

not

252

essential to order

244.

253

of distances

254

.........

the existence series.,

is doubtful

of distances

254

....

'

248.

Summary

of Part IV

255

PAKT INFINITY

AND

CONTINUITY.

CHAPTER THE infinitesimal and

250.

The

principles supposedcontradictions

251.

Correlation of series

252.

Independentseries and

253.

Likeness

254.

Functions

255.

Functions

XXXII.

CORRELATION

The

249.

V.

space

are

OF no

SERIES.

longer required in

a

statement

of

259 of

have been resolved infinity

.

.

.

260

260 series

by

correlation

262

of relations

262

263 of

a

variable whose

values form

a

series

264

....

256.

Functions

257-

Complete series

which

are

defined

by

formulae

267 ^

269

Table

of

CHAPTER

xxxi

Contents

XXXIH.

REAL

NUMBERS. PAGE

258.

Real numbers

259.

Segments of rationals

260.

of segments Properties

261.

Coherent

not limits of series of rationals

are

classes in

a

270

....

271 272

series

274

Note

274

CHAFfER AND

LIMITS

XXXIV.

IRRATIONAL

NUMBERS.

limit

262.

Definition of

263.

of limits Elementaryproperties

264.

An

265.

Dedekind's

266.

Defects in Dedekind's

267.

to Objections

268.

Weierstrass's

282

269.

Cantor's

283

270.

Real numbers

a

arithmetical

theoryof

276 277

irrationals is indispensable .

.

277

.

theoryof irrationals his

axiom

theoryof

of

278 279

continuity

irrationals

280

theory theory are

segments of rationals

XXXV.

CHAPTER CANTOR'S

FIRST

285 ......

CONTINUITY.

OF

DEFINITION

Cantor

287

271.

The

272.

Cohesion

288

273.

Perfection

290

274.

Defect in Cantor's definition of

275.

The

arithmetical

is due theoryof continuity

ORDINAL 276.

Continuityis a purelyordinal

277-

Cantor's ordinal definition of

Only

ordinal notions

occur

special grounds

293

.

XXXVI. CONTINUITY. 296

notion

continuity

296

in this definition

298

Infinite classes of

281.

integerscan be arrangedin of Segments generalcompact series Segmentsdefined by fundamental series

282.

Two

280.

.

.

291

without

not be assumed

existence of limits must

.

perfection

CHAPTER

278. 279.

to

a

continuous

series

.

300

......

compact series may compact

be combined

to

form

298 299

a

series which .

.

is not .

.

303

XXXVIL

CHAPTER

CARDINALS.

TRANSFJNITE

PAGE

transfinite ordinals

283.

Transfinite cardinals differ widelyfrom

284.

Definitionof cardinals

285.

286.

of cardinals Properties and exponentiation Addition,multiplication

307

287.

The smallest trausfinite cardinal

309

288.

Other transfinite cardinals

289.

Finite and

304

...

.304 306

er0

310

transfiriite cardinals

form

a

singleseries

by relation to

greater and less

311

XXXVIII.

CHAPTER TRANSFINITE

ORDINALS.

CHAPTER THE 303. 304.

305.

XXXIX.

INFINITESIMAL

CALCULUS.

The infinitesimalhas been usually supposedessential to the calculus Definition of a continuous function Definition of the derivativeof a function

The infinitesimal is not implied in this definition "307. Definition of the definite integral

.

320 328

306.

#08.

Neither the infinite nor

THE 309.

INFINITESIMAL

329

ANTD

.

330

XL. THE

IMPROPER

INFINITE.

A

311.

precisedefinitionof the infinitesimalis seldom given Definitionof the infinitesimaland the improperinfinite Instances of the infinitesimal

312.

No

310.

329

....

the infinitesimalis involved in this definition

CHAPTER

32-5

313.

infinitesimal segments in compact series Orders of infinity and infinitesimality

314.

Summary

.

.

.

.

.

.

331 331 332 334 33,5 ^37

Table

Contents

of

CHAPTER PHILOSOPHICAL

xxxm

XLI.

ARGUMENTS

THE

CONCERNING

INFINITESIMAL. PAGE

philosophical opinionsillustrated by

Current

Cohen

#38

....

bases the calculus upon infinitesimal^ Space and motion are here irrelevant Who

317318.

Cohen

310.

And

320.

regardsthe doctrine

of limit**as

#J8 331)

insufficient for the calculus

quantitative supposes limits to he essentially To involve infinitesimal difference

330

.

340

....

341

.......

321.

And

to introduce

322.

He

323.

Consecutive

324.

Cohen's

a

of

meaning

new

identifies the inexten^ive

numbers

with

equality

the intensive

supposed

are

be

to

change to be

are

3*2-5.

sense Philosophical

32(".

The

327.

Zeno

328.

The

argument

329.

The

and objectionable

330.

Extensional

of

is

continuum

of

344

XLIL THE

OF

here

continuitynot

in

question

,

.

34("

.

units .

.

34("

.

:U7

dichotomy

348

........

the innocent

kind

of endless regress a

348

.

.

.

whole

341)

....

Achilles and The

the tortoise

350 350

arrow

333.

Change

334.

The

does

involve

not

of the

argument

a

change

332

continuity

.

333

.

35:3

Historical retrospect Positive doctrine of the infinite

339.

Proof that there

340.

The

341.

A

342.

Whole

343.

No

paradox of a

are

THE

OF

338.

INFINITE. 333 35("

infinite classes

Tristram

part may

337

Shandy

338

be similar

359

part and formal implication

and

immediate

predecessorof

o"

or

regardsthe number

344.

Difficulty

345. 34(5.

Cantor's first proof that there is His second proof

347-

Every class

348.

But

349.

Resultingcontradictions

as

this is

Summary

.

XLIII.

PHILOSOPHY

THE

and

....

consists of elements

continuum

whole

351

........

of

of Cantor's doctrine

Summary

of

state

measure

CHAPTER

350.

CONTINUUM.

composed of mutuallyexternal

intensional definition of

and

332.

837.

344

\Veierstrass

and

331.

The

continuous

rejected

PHILOSOPHY

THE

33"".

342

required for ,

views

CHAFFER

333.

341

has

more

sub-classes

in impossible

of Part

V

o0

.

3("0 .

of all terms, no

.

.

,

.

objects,or proposition^

301 3(52

.

greatest number

....

3("3

3("4 than

terms .....

certain

.

cases

.......

3("H

3("("

3"7 SS8

xxxiv

Table

of

Contents

VI.

PART SPACE.

XLIV.

CHAPTER AND

DIMENSIONS

NUMBERS.

COMPLEX

PAGE

371

353.

Retrospect

352.

Geometry

353.

Non-Euclidean

354.

Definition of dimensions

355.

Remarks

356.

The

of series of two

is the science

on

or

more

372

.

372

"

374

.........

375

the definition

359. 360.

Remarks

358.

.

geometry

purelylogical and universal algebra Complex of number Algebraical generalization Definition of complex numbers

357.

dimensions

definition of dimensions

370

is

376

numbers

on

the definition

XLV.

CHAFfEB PROJECTIVE Recent

377 378 379

GEOMETRY.

threefold

381

#"3.

scrutinyof geometricalprinciples and metrical geometry Protective, descriptive and lines Projectivepoints straight

364.

Definition of the

365.

Harmonic ranges Involutions

385

367.

Projectivegenerationof order

386

368.

Mobiusnets

388

369.

Protectiveorder presupposedin assigningirrational coordinates

370.

Anharmonic

371.

372.

Assignment of coordinates to any pointin space and Euclidean geometry Comparison of projective

373.

The

361. 362.

366.

384 384

389 .

ratio

390 390

....

391

....

of duality principle

DESCRIPTIVE 374.

Distinction

375.

Method

376.

Method

392

XLVI.

between

of Pasch

and

GEOMETRY.

and descriptive projective geometry

.

.

.

Peano

393 394

employingserial relations Mutual independenceof axioms Logicaldefinition of the class of descriptive spaces .

378.

381 382

plane

CHAPTER

377.

....

.

.

...

"

.

395 396

....

397

Table

xxxv

of Contents

PAGE

,379, Parts of

397

380.

lines straight Definition of the plane

,398

381.

Solid geometry

382.

to Euclidean Descriptive geometry applies elliptic space

383.

Ideal elements

400

384.

400

38.5.

Ideal points Ideal lines

380.

Ideal

387-

The

399 and

but hyperbolic,

not

399

401

planes

removal

of

402 a

suitable selection of

pointsrenders

a

projective space

descriptive

403

CHAPTER

XLVII.

METRICAL 388.

Metrical geometry presupposes

mi

Errors in Euclid

390.

is not Superposition

391.

Errors in Euclid

392.

Axioms

393.

Stretches

394.

Order

395.

Geometries

390.

In

GEOMETRY. or descriptive projective geometry

404

.

404 a

405

valid method

406

(continued)

407

of distance

as

most

408

from resulting which spaces,

distance alone

derive the

409

......

distance

line from straight

magnitude

of

divisibility can

be

.410 .

.

instead

used

of 411

distance 397. 398.

Meaning of magnitudeof divisibility of making distance independent of Difficulty

399.

Theoretical

meaning of

411

stretch

.413 .

.

.

414

measurement

400.

Definition of

414

401.

Axioms

415

402.

An

angle concerningangles

403.

stretch of rayp, not Areas and volumes

404.

Right and left

angleis

a

class of

points

410

....

417 417

CHAPTER RELATION

a

OF

METRICAL

DESCRIPTIVE

XLVIII. TO

AND

PROJECTIVE

GEOMETRY. 419

408.

Non-quantitative geometry has no metrical presuppositions Historicaldevelopmentof non-quantitative geometry theoryof distance Non-quantitative In descriptive geometry

409.

And

410.

Geometrical

426

411.

New

427

405.

400. 407.

....

in

projective geometry theoryof imaginary point-pairs projective theoryof distance

.

.

420 421

423 425

Table

xxxvi

of

CHAPTER OF

DEFINITIONS

Contents

XLIX. SPACES.

VARIOUS

PAGE

412. 41 3. 414. 415.

All kinds of spaces are purelylogical of dimensions three Definition of projective spaces

definable in

CONTINUITY

THE 416.

The

417.

The

418.

An

.

.

.

430 ....

432

spaces of three dimensions Definition of Clifford'sspaces of two dimensions Definition of Euclidean

CHAPTER

429

terms

....

4-34

L.

SPACE.

OF

of a projective continuity space metrical o f a continuity space enables us to dispensewith axiom of continuity

437 488 of the postulate

the

440

circle 419. 420. 42 L

422.

? priorto pointsand induction premisses Empirical There is no reason to desire our premissesto is of not an Space points, a unity aggregate

440

Is space

CHAPTER LOGICAL

ARGUMENTS

441

be self-evident

441 .

.

442

LI. AGAINST

POINTS.

423.

Absolute and relative position

44-5

424.

Lotze's arguments against absolute Lotze's theoryof relations

440

425.

position

......

440

.

420.

The

427.

Lot/e's three kinds of

subject-predicate theoryof propositions Being of indiscernibles identity

428.

Argument

from the

421).

Points

not active

430.

Argument

431.

are

CHAPTER KANT'S The

4*3.3.

Summary

434,

Mathematical

present \\ork

is

44i) 451 452

from the necessary truths of geometry Points do not implyone another

432.

448

THEORY

454 ....

454

LII. OF

diametrically opposed to

SPACE. Kant

450

....

of Kant's

theory element reasoningrequiresno extra-logical

435.

Kant's mathematical

430.

Summary

of Part VI

antinomies

450 .

.

.

457 458 401

Talk

of Contents

PART MATTER

xxxvii

VII.

AND

MOTION.

LIIL

CHAFIER MATTER.

PAGE

437-

Dynamics is here

438.

Matter is not

439.

Matter

440.

Relations of matter

as

considered

as

a

branch

of pure mathematics

405 .

.

405

implied by space

466

substance f

441.

Definition of matter

467

to space and time

of

in terms

constants logical

408 ....

LIV.

CHAPTER MOTION. 442.

Definition of

443.

ITiere is no

469

change

such

thingas

a

state of

471

change

471 472

involves existence

444.

Change

44-5,

of Occupation

446.

Definition of motion

472

447.

There is no state of motion

473

a

placeat

time

a

LV.

CHAPTER CAUSALITY. 448. 440. 450. 451.

452.

The

theoryof dynamics descriptive Causation of particulars by particulars Cause and effectare not temporally contiguous ? Is there any causation of particulars by particulars Generalized form of causality

....

CHAPTER DEFINITION

OF

A

474 47*5 476

477 478

LVL DYNAMICAL

WORLD.

453.

Kinematical motions

480

454.

Kinetic motions

480

Table

xxxviii

of

Content*

LVII.

CHAPTER

MOTION.

OF

LA\V'S

NEKTON'S

I"A"K

45.5. 4o"J.

and acceleration

Force The

4-57. The

482

fiction*

are

law* of inertia

*^~

second law of motion

^M

third law

4JW

458.

The

4oi).

Summary

4("0.

in Causality

of Newtouiai

48o

principles

i

48("

dynamics

487

Accelerations as caused hy particular* 4(51 is an // prioritruth 4(\-2. No part of the laws of motion .

LVIII.

CHAPTER AND

ABSOLUTE Newton

4(54.

(irounds

4"fo.

Neumann's

RKLATIVB

48J)

for al^olute motion

400

theory

490

4W5.

Streinty/s theory

4(J7.

Mr

4(J8.

Absolute rotiitiou is ";tilla

4"9.

Mach's

491

Macaulay'stheory replyto

491

change of

relation

492

HERTZ'S of Hertz's

470.

Summary

471.

Hertz's innovations pure

492

New ton

LIX.

CHAPTER

DYNAMICS. 494

system fundamental

not

are

from

the point of view

of

mathematics

472.

Principles

473.

of Principle

474.

Summary

the

49.5

to Hertz

common

of equality

and cause

Newton and

49f;

effect

49(5

of the uork

4")7

APPENDIX THE

MOTION.

his critics

and

4""3.

488

....

LOGICAL

AND

ARITHMETICAL

A. DOCTRINES

OF

KREGE.

47^.

Principal pointsin Frege'sdoctrines

.501

47"".

Meaning and indication

.502

477.

Truth-values

478.

Criticism

and

judgment

.502 ^Q.J

479.

Are assumptionsproper

480.

Functions

481.

Begriffand Gege,nstand

names

for the true

or

*.

the false? .

.

504 -05

.507

XXXIX

Table

Content*

of

PACE

482.

Recapitulation

48:5.

Can

484.

Ranges

485.

Definition

of

of

theory

prepositional

functions

508 ....

made

be

concepts

logical

?

subjects

510 ......

510 and

of

of

relation

512

f ........

48(5.

Reasons

487.

A

488.

Possible

481).

Recapitulation

490.

The

491.

Classes

having

492.

Theory

of

493.

Implication

494.

Definition

for

class

exteiisional

an

which

view

only

has

of

member

one

classes is

513

distinct

its

from

only

member

513 .

theories

to

of

subject

of

for

account

theories

already

proposition

a

only

this

514

discussed he

may

tact

515

plural

5 US

member

one

517 ........

495.

496*.

Frege's

518

types and of

symbolic

cardinal of

theory

Kerry's

518

numbers

519

series of

criticisms

logic

520

Frege

520

APPENDIX

THE

of

497-

Statement

498.

Numbers

499.

Are

500.

Contradiction

and

the

of

DOCTRINE

TYPES.

523

concepts

arising propositions

OF

doctrine

propositions

prepositional

B.

from than

as

individuals the

525

types

question

propositions

?

526 whether

there

are

more

classes

527

529

I.

PAET

THE

INDEFINABLES

OF

MATHEMATICS.

CHAPTER

DEFINITION

variables,the

term

a

the

to

in

the

to

these, mathematics

propositionswhich unusual.

of

it

definition

use

The

that

sense,

is

simple,from

the

respect

one

included

a

that

those

which,

in

the

we

past,

have

philosophicalstrife.

time

and

motion,

and

of

addition of

the

in

what some-

be

capable of object of the

whatever

those

has, in the definition,,

our

marks

by

seek to

to

its

The

pass

which

nature

mathematical

will be

arrive

from

we

shall

find

concerned

in all the

number,

inference

the

to

the

those

in

many

;

of

the

to

certainty

at

be

in

premisses.

differ

be

therefore

philosophical the complex

able, thanks

shall

of

precise implied in a

"

from

we

involved

will

indemonstrable

shall to

method

called

discussions

our

solution

been

be

may

rather

unconsciously,are Our

term.

which

the

the

to

the

is included

less

or

more

of

questions with capable of an exact

"of

most

that

be

but signification,

themselves,

mathematician*

the

In

doubt,

is, no

possesses

uncommon

which,

few

involved

other vaguely distinguished from professesto be, not an arbitrary decision to

demonstrable

not

be

constituent

a

that,

such

though

an

say,

relation

of truth.

will

shown

usuallycalled philosophical. We

are

of

to

it

mathematics,

ordinary employment and of analysis, our problem

one

be

will

It

of

the

of

in

ideas

the

is not

parts, nevertheless, appear

is

commonly

word

analysisof

in

else

definition

common

a

pure

may

all

are

of

form.

mathematics

which justification

whatever

above

the

which

as

more

q contains

nor

notion

notions

considers, namely the notion

"

is

further

the

form

or

logicalconstants

member,

a

notion

of

p

the

following: Implication,the

it is

a

justificationa to provide. present work been regarded as pure past, mathematics

the

neither

And

propositionsof

Its various

exact

studies.

propositionsof propositionscontaining one

such

iisea

above

The

of

which

of

notion

general

that

all

of

propositions,and

two

relation, and

in

and

class

are

q

tenns

class

a

of

notion

2,

and

p

MATHEMATICS.

except logical constants.

definable

notions

is the

in the

same

constants

any

of

where

implies q"

"p

PURE

OF

Mathematics

PURE

1.

I

that

labours

regard

and

the

But

to

among

problems

traditional of

uncertainty of infinity, space,

itself,are

all

questions

of Mathematics Indefinables

The

4

[CHAP,i

itself demonstrable professing will be given an answer with mathematical which, however, certainty consists in reducingthe above problemsto problemsin pure logic,

to

which,in the presentwork, an

answer

"

which last will not be found The

3.

solved satisfactorily

Philosophyof

obscure and

Although it

has

Mathematics

been

follows.

hitherto

unprogressivethe other branches of as

agreed that generally

was

in what

as true, philosophers disputed

mathematics

is in

as

troversial, con-

philosophy. some

sense

mathematical

to what

propositions really : althoughsomething was true, no two peoplewere agreedas to what it was that was and if was known, no one knew true, something what it was that was So long,however, as this was known. doubtful, it could hardlybe said that any certain and exact knowledgewas to be in obtained mathematics. We that idealists have find, accordingly, tended more and more to regardall mathematics as dealingwith mere while have held m athematical to be empiricists appearance, everything to some truth about which they had nothing to exact approximation tell us. This state of things, it must be confessed, was thoroughly asks of Mathematics unsatisfactory. does it mean ? Philosophy : What Mathematics in the past was unable to answer, and Philosophyanswered by introducingthe totallyirrelevant notion of mind. But now meant

Mathematics

is able to

answer,

so

far at least

to

as

reduce

the whole

of its

to certain fundamental propositions notions of logic. At this t he discussion be resumed must point, by Philosophy.I shall endeavour to indicate what the fundamental notions are to

length that the A

others

no

in

occur

mathematics,and

difficultiesinvolved philosophical

complete treatment

Logic,which 4

There

was,

and

yet

the

of analysis

of these difficultieswould

will not

of mathematics.

in the

involved, prove at to point out briefly

be found

until

very

It seemed

orthodox

involve

these a

notions.

treatise

on

in the

following pages. a special in the principles lately, difficulty

plainthat mathematics

accounts

of

deduction

consists of

deductions,

largelyor wholly only the Aristotelian doctrines of SymbolicLogic, mathematical at or reasoning, any rate requiredsuch artificialforms of statement that they could not be practically applied.In this fact lay the strengthof the Kantian view, which asserted that mathematical reasoningis not formal, strictly but always uses i.e. the a priori intuitions, of space and knowledge time. Thanks to the progress of SymbolicLogic,especially treated as Professor this Peano, by part of the Kantian philosophy is now capable

to inapplicable existingmathematics. syllogistic theory,but also the modem either theoretically were inadequateto

ol

of

a

final and

deduction

irrevocable refutation. and

ten

(e.g. implicationis

a

formally deduced; and be defined

in

terms

other

were

Not

By

premissesof

the a

helpof

ten

principles

general logicalnature be strictly can and

all mathematics relation"),

all the entities that in mathematics occur of those that occur in the above

can

twenty premisses

Definition of Pure

2-6]

Mathematics

includes not this statement. Mathematics and but also Geometry, Euclidean Analysis,

only

In

Dynamics,and an indefinite number of other their infancy.The fact that all Mathematics of

greatestdiscoveries of

the

the remainder established,

our

of the

and

Arithmetic

non-Euclidean,rational

studies stillunborn is

and

age;

5

or

SymbolicLogic

is

this fact has

when

of mathematics principles

in one

been

consists in

of Symbolic analysis Logicitself. 5. The is deduction generaldoctrine that all mathematics by from was logicalprinciples logicalprinciples stronglyadvocated by that axioms Leibniz, who urged constantly ought to be proved and the

all

that

few

except a

fundamental

notions

ought

But

defined.

be

to

necessity owing partlyto a faultylogic, partlyto belief in the logical led into hopeless of Euclidean Geometry,he was in the endeavour errors known to carry out in detail a view which, in its general outline,is now do The actual propositions of Euclid,for example, to be correct*. not of logic follow from the principles alone ; and the perception of this fact led Kant in the theory of knowledge. But since to his innovations the growth of non -Euclidean Geometry, it has appeared that pure has

mathematics

no

with

concern

question whether

the

the

axioms

and

of Euclid hold of actual propositions space or not: this is a question for applied decision is possible, mathematics,to be decided,"so far as

any

by experimentand observation.

asserts is merely pure mathematics Le. follow from the Euclidean axioms propositions

that the Euclidean it asserts has

"

space which

implication: any

an

and

also such

such

other

mathematics, the Euclidean true:

What

such

Thus, properties.

and

non-Euclidean

nothingis affirmed

in each

has

of proposition

a

pure thus

properties

dealt with

as

in pure

equally propositions in, belongto experimental when theybelongto are

All implications.

to what

variables in

such

Geometries

except the like live exists, actually space we or science,not to mathematics empirical ; arise from givingto appliedmathematics, they

as

and

one

mathematics

of

more

or

constant

some

the

value

value of the enablingus, and consequent instead of to assert both hypothesis variable,actually We assert always in mathematics merely the implication. asserting the satisfying

that if

and hypothesis,

certain

a

assertion p then some ,

entities#, ?/, 2, but we do not assert either p ...

a

between

relation

the

for that

of any set of other assertion q is true of those entities ;

is true

of any

entitya.\

of our q separately assertions p and j, which or

or

entities. I

shall

We

assert

calljbrmal

implication, 6.

Mathematical

fact that

they assert The

variables.

which

Logic has *

On

this

not are propositions only characterized by the but also by the fact that theycontain implications,

notion to

of the variable is

deal,and

in the

one

of the

presentwork

a

most

difficultwith

satisfactory theory

cf. Couturat, La Logiquede Leibniz,Paris,1901. subject,

of Mathematics Indefinable^

The

6

[CHAP.I

spiteof much discussion,will hardlybe found. variables it plainthat there are For the present,I onlywish to make where at first sightthey might in all mathematical even propositions, to be absent. seem Elementary Arithmetic might be thought to form variables nor to an exception:1 + 1=2 appears neither to contain in But matter of fact,as will be shown assert an a as implication. is: "If x is one and Part II, the true meaning of this proposition y

in

its nature,

to

as

and

is one,

y, then

differs from

x

oc

and

And

two."

are

y

this

both contains variables and asserts an implication.We proposition that the words any in all mathematical propositions, shall find always, or

soine

occur

;

and these words

are

the marks

of

variable and

a

a

formal

-in the form: expressed proposition Any unit and any other unit ai^e two units." The typical is of the form """(#,#, """"") of mathematics implies^r(oc^?/, #,...), where whatever values x^ "/, z, have"; ""("r, may #, #, ...) and for every .set of values of #, y, ",..., are propositions. ^Or, y, ",...)" that is is alwaystrue, nor It is not asserted that "j" -fy alwaystrue, yet is true, but merelythat,in all cases, when ""is false as much as when (f"

Thus implication.

the above

proposition may

be

"

...

T|Tfollows from it. distinction between

The

obscured of

by

mathematical

parameters

in

as

a

usage. sense

some

variable and

is somewhat

constant

a

It is customary, for

constants,but

example,to speak

this is

a

usage

which

reject.A constant is to be somethingabsolutely Thus which there is no ambiguitywhatever. definite, 1, ", concerning we

shall have

to

Socrates,are constants; and so are man, and the human race, and considered future, collectively. tion, implicaProposition, past,present but etc. are constants a class, some proposition, ; any proposition, 3, ",

TT,

not are constants, for these phrasesdo not denote one proposition, definite object. And called parameters are thus what are simply for variables. the Take, -f-c 0, considered equation ax -f-Iry example, the in line Here to a we as a equation straight plane. say that x and y But unless we while a, ", c are constants. are variables, are dealing with one line,say the line from a particular absolutely particular point in London to a particular point in Cambridge,our #, 6, c are not t

=

definite

And

in

numbers, but stand for any numbers, and Geometry

nobody

does

deal with

actual

always discuss any line. The point is that couples"r, y into classes of classes,each class couplesthat class to

from 7.

It

is

restricted to

supposed to stand

have

a

fixed relation

class,a, /;, c customary in mathematics ako

certain

stand

for numbers,

they are

certain

numbers

vary, and

are

are

we

thus also variables.

lines ; particular collect the

being defined

as

those

triad (",i, c). But therefore properly variables.

to

one

variables regard our classes: in Arithmetic,for instance,they

for numbers.

But

we

various

to

this

only means

that

as are

if they some that they satisfy formula,i.e. the hypothesis the formula. is what is really This, then, implies

Definition of Pure

6-8] and asserted,

in this

not

is

it proposition

variables should be numbers: are

Mathematics

7

longernecessary that our holds equally when they implication

the

no

" and y Thus, for example,the proposition x

so.

implies(x + y)* "

op +

and Plato*

Socrates

holds

%xy + y^

if for equally

numbers

are

and y

x

tute substi-

we

both

and consequent, in this case, hypothesis will be false,but the implication will still be true. in every Thus of when fullystated,the variables haMe proposition pure mathematics, unrestricted field: conceivable an tuted absolutely entitymay be substiany of our variables without impairingthe truth of our for any one proposition. We understand why the constants in mathematics 8. now can are in constants the sense The defined above. to be restricted to logical in constants of into variables leads a transforming proposition process and givesus, as it were, the formal is called generalization, to what is interested exclusively of a proposition. Mathematics in types essence of propositions onlyconstants be proposed, ; if a proposition p containing of its terms we imagineothers to be successively and for a certain one the result will in general be sometimes true and sometimes substituted, false. Thus, for example,we have "Socrates is a man"; here we turn Socrates into a variable, and consider "x is a man."" Some hypotheses "x is a Greek,"insure the truth of "x is a man"; to %) for example, as thus "x is a Greek ""r is a man," and this holds for all values of implies But is not one of pure mathematics, the statement because it depends x. We nature of Greek and man. may, however*,vary upon the particular :

"

too, and

these

then

of pure contained

and

b

and classes,

are

m"

and those involved in the notion So

variable,our

long as any can proposition

it is the possible,

term

in

be

our

have

of formal

and generalized; to do

a

in

it.

",

proposition

the constants

doss^

with implications be turned

propositioncan

business of mathematics

of deduction which

is contained

a

"x is a b" Here at last we implies three variables and mathematics, containing

variables. a

a

a"

is an

"x

obtain: If

long as

so

If there

are

into

this is several

differ onlyas to the

meaning of the symbols, that so capableof several is to form the class of the proper course, mathematically, interpretations, meanings which may attach to the symbols,and to assert that the that the symbolsbelong formula in question follows from the hypothesis to the class in question.In this way, symbolswhich stood for constants chains

identical become symbolically propositions

become

transformed

into

and variables,

of classes to which consisting are generalization

so

the

*

may not

Whenever It is necessary

be

two to

done)so easily

numbers.

many instances

sets of terms

suppose that the

constants

old constants

frequentthat

mathematician, and innumerable work.

new

have

will

belong. occur

will be

mutual

arithmetical addition and above

formula

remains

substituted,

are

at

Cases

to

once

given in

of such

the

every

present

relations of the

same

defined (as multiplication

when significant

x

and

y

arc

of Mathematics Indefinabks

The

8

type,the

pointsin

relations of

mutual

branch

Euclidean

a

complexnumbers;

those of the

of pure

of

in every branch

relations

of

are

a

example,the

of the

type as

same

to decide whether

deal,

to

mathematics, with any class of entities whose mutual specified type; thus the class,as well as the particular

and the onlytrue constants are becomes a variable, considered, of relation Now a type types of relations and what theyinvolve. term

formal

the is to

class of relations characterized by the above a discussion,

in this

mean,

a

set of entities havingthe

other

some

as

its variables

we ought Speakinggenerally,

relations.

type of mutual

same

or

planeare

For

planegeometry,considered

hence

mathematics, ought not

pointsor complexnumbers

are

will applyto both.

of deduction

form

same

[CHAP,i

of identity

the

the

possiblein regard to

deductions

various

will appear more as class;and hence a type of relations, fullyhereafter,if not alreadyevident,is always a class definable in

members

of

terms as

of the

We

constants*. logical

class of relations defined

a

Thus

by

therefore define

a

type of relations

property definable in terms

some

of

alone.

constants logical

9.

may

mathematics

pure

contain

must

no

indefinable^

except

indemonstrable no or logicalconstants,and consequently premisses, constants but such as are concerned exclusively with logical propositions, and with variables. It is precisely this that distinguishes pure from In applied mathematics, results which have been appliedmathematics. shown by pure mathematics to the to follow from some as hypothesis variable are actually asserted of some constant the hypothesis satisfying in

question. Thus terms which were variables become constant, and a new premiss is alwaysrequired, namely: this particular entitysatisfies the hypothesis in question. Thus for exampleEuclidean Geometry, as a branch of pure mathematics,consists whollyof propositions having the "S hypothesis that exists is the

is

Euclidean

a

space." If

this enables Euclidean,"

us

we

go

on

"The

to:

Euclidean consequentsof all the hypothetical constituting

where

the

variable

S is

space

to assert of the space that exists

the constant

Geqmetry,

But replaced by by this stepwe pass from pure to appliedmathematics. 10. The of mathematics connection with logic, accordingto the above account, is exceedingly close. The fact that all mathematical constants and that all the premisses are of mathematics constants, logical now

concerned

are

with

have philosophers fact is

that,when,

actual spac'e.

I believe, the precise these,gives, statement meant

in

that mathematics asserting

is

a,

of what

priori.

The

the apparatus of logichas been accepted, all follows. The logical themselves are constants necessarily

mathematics

once

onlyby enumeration* for theyare so fundamental that all properties by which the class of them might be defined presuppose

to be

the

*

defined

One-one, many-one,

with which

we

transitive, symmetrical,are

shall be often concerned.

instances

of types of relations

Definition

8-10]

of

terms

some

logical

of is

following

and

premisses

together Thus

with

belong

will

implies

constants

but

;

both

for

while

that

recognizing

logical

and adhere

to

propositions

reader

will

to

define

and

logic,

certain

,"

desire

the

for

rather

me

r,"

the will

relation

a

variables

only

leads

tradition

is

and

containing

of

implies

p

But

mathematics

marks.

principle

the

the

of

these

"implication

as

of

variables,

have

then

r,

such

identify

propositions

respect

implies

logical

definition

consequences

e.g.

as

with

with

containing

mathematics.

to

might of

the

made

together

as

mathematics, q

be

may

above

themselves

others,

not

all

of

the from

to

the

belong

to

sciences. what

From

has

present

work

has

follows

from

symbolic

what

the

are

be

will

And

necessary

branch

of

to

in

fulfil

first to

give

mathematics.

two

logic,

of

an

as

outline This

a

first,

secondly

and

logic

following all,

the

objects,

symbolic

of the

said,

been

now

principles

pursued

I.

Part be

but

we

class

distinction,

above

q while

logic

usage, the

as

of and

it

the

be

will

mathematics,

implications

premisses

premisses

p

to

to

either

the

mathematics,

to

adhere

the

"if

syllogism, belong

of

desired,

fulfil

not

the

mathematics

exclusively

consists formal

assert

such of

some

do

Mathematics

which

of

of

concerned

but

which

logic,

is

discovering

of

distinction

premisses are

9

method

symbolic

distinction

the

variables

(" 1).

above

the

The

a

which

with

mathematics

if

of

consists

propositions

constants

of

chapters. but

Mathematics

practically,

analysis

arbitrary,

Logic

other

But

the

is

the

very

follows. all

class.

constants

business

logic

the

Pure

of

will

Symbolic occupy

The

itself.

Parts,

that

discover,

to

preliminary of

show

to

while to

a

Logic the

first

the

mathematics

all

as

second

critical considered

following

far of

the

that

perceive

as

possible,

these

objects

belongs analysis,

simply chapter.

it

to

will

as

a

II.

CHAPTER

LOGIC,

SYMBOLIC

SYMBOLIC

11.

study

the

is

synonyms" The word

Formal

or

shall

I

Logic "

the

of

these

use

general types

various

Mjmbolicdesignatesthe subjectby

accidental

an

terms

as

deduction,

of

characteristic,

employment of mathematical symbols,here as elsewhere,is merely The irrelevant convenience. syllogismin all its figures a theoretically be the whole subject if all belongs to Symbolic Logic, and would the scholastic tradition supposed. It is deduction as were syllogistic, inferences that modern from the Symbolic recognitionof asyllogistic for the

Leibniz

Logic, from

has

onward,

the motive

derived

to

Since

progress.

of Thought (1854), the subject has publicationof Boole's Laws able been pursued with a certain vigour,and has attained to a very consideralmost technical development*. Nevertheless, the subjectachieved either to philosophyor to other branches of mathematics, nothing of utility methods of Professor transformed until it was Peanof. by the new essential to every not onlyabsolutely Symbolic Logic has now become philosophicallogician,but also necessary for the comprehension of and even mathematics for the successful practiceof certain generally,

the

of

branches

judged by from

mathematics. those

who

acquiring it;

the

How

have

useful

it is in

experienced the

its theoretical

functions

only

practicecan

increase be

must

of

be

derived

power

brieflyset forth

in

present chapter*.

*

By

in the

far the

three

1890, 1891, t

See

\

was

In

de

will be

account

of the

non-Peanesque methods

Schroder, Vorkmngen

fiber die Algebra der

follows

have

;

even

been

Mathematiquw, Turin, 1895, with de

Mathemaliques, Vol.

quoted

originallythe what

regards relations considered

of

; also Revw

Fonnulaire

which

complete

will be found

Logik^ Leipzig,

189-5.

Formulaire

later years the

most

volumes

as

Rimta

the

F. di

and

cases

suggested to

me

by

his

I

1

The

on.

subsequent editions (1900). The ilevue

will be referred due

are

where

vn,

so

Matematka, outlines

main

in those

189-5

No.

to

Professor

depart from

works.

to

in

editions of

de

Mathtmatiques,

as

R. d. "L

Peano, except as problems

his views, the

11-13]

SymbolicLogic

11

concerned with inference in Symbolic Logic is essentially and is various of mathefrom b ranches matics general*, distinguished special mainly by its generality.Neither mathematics nor symbolic but relations as (say)temporalpriority, logicwill study such special mathematics with the class of relations possessing will deal explicitly the formal propertiesof temporal prioritypropertieswhich are summed the formal properties in the notion of continuity!.And up 12.

"

of

relation may be defined as those that can in terms be expressed of logical constants, or againas those which, while they are preserved, permit our relation to be varied without invalidating any inference in a

which

the said relation is

symboliclogic,in the

regardedin

narrower

what inferences investigate

the

lightof

which

sense

a

valuable.

But

convenient, will not

is

relations in respectof continuous possible this investigation (Le.relations generatingcontinuous series); belongs to mathematics, but is still too specialfor symboliclogic.What is the generalrules by which inferences symboliclogicdoes investigate and it are made, requiresa classificationof relations or propositions in far The these notions. so as only generalrules introduce particular notions which of symboliclogic, particular appear in the propositions of these notions,are the logical and all others definable in terms constants. is not great: The number constants of indefinable logical it appears, in fact,to be eightor nine. These notions alone form the of the whole of mathematics : no others,except such subject-matter of the original definable in terms as are anywhere eightor nine,occur the technical in Arithmetic, Geometry, or rational For Dynamics. studyof SymbolicLogic,it is convenient to take as a singleindefinable is the notion of a formal implication, Le. of huch propositions x as whose a man impliesx is a mortal,for all values* of **""" propositions (x\ ("r)implies^ (j?)for all values of x* where "f" generaltype is : "f" of this notion The analysis ^ (x\ for all values of #, are propositions. but is not of the of formal implication subject, belongsto the principles requiredfor its formal development. In addition to this notion-,we between : Implication propositions requireas indefinables the following are

"

"

"

the containingvariables,

not

is

member,

a

By

the

notion

of these

means

relation

of

of siiclithoiythe

notions,all the

a

term

to

notion

a

class of which

it

and -truth. of relation,

of symbolic logiccan propositions

be

.stated. The

13.

calculus of relations.

subjectof Symbolic Logic consists of and the calculus of classes, propositions,

Between

which parallelism, *

I may deduction. "or

a

mere

t See

the arises

there

first two, as

follows

:

In

that I do not well say at once induction is called What appears to as

method

of

making plausible guesses

below, Part V, Chap, xxxvr.

is, within any

three

parts,the

the

calculus of

limits,a

certain

the symbolicexpression,

and inference l"etween distinguish d eduction either t"e to me disguised

12

of Mathematics Indefinables

The

9

letters may

be

classes

as interpreted

of inclusion in the

one

case

be

may

as

or

[CHAP,n

and propositions,

by that replaced

the relation

of formal

implication

of the syllogism, if Thus, for example,in the principle and a is contained in ", b in c, then a is contained in c; a, b9 c be classes, and a implies but if #, ^ c be propositions, b, b implies c. c, then a implies A great deal has been made and in the later editions of of this duality, the Formulaire,Peano to its precision appears to have sacrificedlogical *. But, as a matter of fact,there are many ways in which preservation in the

the

other.

calculus

of

differs from that of classes. Consider, propositions for example, the following If and r : are propositions, p implies jp, q, then r? is true ; but This or r, proposition q p implies q or p implies its correlative is false, and a is contained namely: If a, b" c are classes, in b or c, then a is contained in b or a is contained in c" For example, all either men but are not all men Englishpeopleare or nor women, yet all women. The fact is that the duality holds for propositions asserting of a variable term that it belongsto a class, i.e. such propositions as "x is a man," providedthat the implication involved be formal, i.e.one "

"

which

holds

for

at proposition

entities that

all values of

we

concerned

are

To genuinepropositions.

that,for all values of man"

or

woman" "

is a

x

But

x.

all,beingneither true

"r,

"

is

man

"

or

for all values of

x.

But man" Thus

is itself not

a

such

but calculus, propositional

with

illustration: "

a

a woman." implies x either implies"^ is a

woman

man"

a

false ; and it is not

the above

is

"

is

with

nor

in the

continue #"

"a?

a

woman

either

it is false that

It is true

implies x "

"

x

is

a

is

man

a

or

for all values of #, or implies the implication which involved,

is alwaysone

of the two, is not formal,since it does not hold for all values of 07, beingnot of the two. The symbolicaffinity one alwaysthe same of the propositional and the class logic is,in fact,somethingof a snare, and we have to decide which of the two we to make fundamental. are

Mr

McColl, in

view

that

inclusion does not

importantseries of papers f, has contended for the and implication fundamental than propositionsare more and classes; and in this opinionI But he agree with him. appear

an

to

me

to

realize

the adequately

distinction between

and such as contain a real variable : thus he is led genuine propositions to speak of propositions sometimes true and sometimes as which false, of course is impossible with a genuine proposition. As the distinction involved

further.

*

is of I shall dwell on it before very greatimportance, A proposition, we may say, is anythingthat is true

On the points where Lecture 21.

the

dualitybreaks down,

cf.

proceeding or

that is

Schroder,op. tit,,Vol

'

n

'

t Of. "The Calculus of EquivalentStatements," Proceeding*of the London Mrthtmatical Society, Vol. ix and subsequent volumes ; "Symbolic Reasoning/' Mind, Jan 1880, Oct. 1897, and Jan. 1800 ; -La Logique Symboliqueet ses

Applications

tiibltotheqmdu Congre*International de Philosophic, Vol. m (Paris, 1901) future quote the of the above Congress proceedings by the title

Cougrto.

I shall in

13,

14]

Symbolic Logic such expression

false. An

for it is neither true

"

as

is

x

false. If

nor

becomes whatever,the expression form

schematic And we

when not

are

have

we

we

now

there is

standingfor

way as the result is no

"

is a

x

under

x

giveto

we

x

: it proposition

a

of

the

whole

any

tion, proposi-

a

value

constant

is thus

it

as

any

a

were

a

class of

man

:

the variable is absorbed

integral signin of

function

longera

one

a

Peano

x.

appears in this way as apparent^since the " is a man x upon the variable ; whereas in for different values of the I shall

calls real*.

is therefore not

propositions. of x? or is a mortal for all values implies but a asserting singleimplication, a class of implications ; in the letter a genuine which,though x appears, proposition, say

real variable

no

"

man

a

13

in the

there

that the so integral, a variable which distinguishes does not depend proposition there are different propositions "

and variable,

are

one

the variable is what

real

more

or

of

definite

where speakof propositions exclusively

real variable: where

kind

same

of the variables the

Peano

there is

and variables,

no

for all

involved I shall a expression proposition, call the expression a propositioned function.The study of genuine fundamental than that of classes ; more is,in my opinion, propositions but the study of prepositional functions appears to be strictly on a and indeed scarcely therefrom. distinguishable par with that of classes, first like fundamental McColl, at more as Peano, regardedpropositions considered tions funcbut he, even than classes, more definitely, propositional From this criticism, Schroder is exempt: rather than propositions. and pointsout their deals with genuinepropositions, his second volume values

is

formal differencesfrom classes.

Calculus. Propositional

The

A.

calculus is characterized by the fact that propositional have as hypothesis and as consequentthe assertion of all its propositions plies is of the form "p imthe hypothesis material implication. a Usually, to the assertion that the letters p? etc.,which ("16) is equivalent Thus the consequents in the consequent which occur are propositions. of all propositions. functions which true consist of propositional are the letters It is importantto observe that,though employedare symbols the variables are given and the consequentsare true when for variables, these values must be genuinepropositions* values which are propositions, The functions. not propositional hypothesis "p is a propositionis 14.

The

"

satisfiedif for p Socrates is a man"

not "

or

of x.""

put if

"

x

is

put

we

Shortly, may in this calculus letters single

all values

by

we

variables "

in

the

which propositions

we

case,

the

is to

that

a

man," but

"a? is

are

say,

calculus asserts *

F.

a

man

say that the

1901, p.

it is satisfied if is

are

2.

put

a x implies represented propositions

but variables, where

we

mortal for

the

do

not

contain

hypothesesof

satisfied.

the

of Mathematics Indefinable^

The

L4 15.

calculus

Our

studies

the

relation

This be relation must propositions. )f formal implication, which holds

"vhen the

impliesthe

one

is also implication studied :

do

we

only certain of

not

in

consider

of

between implication from the relation distinguished between functions prepositional

other for all values

involved

this

of the

calculus.

variable.

calculus,but

is not

functions in prepositional

definite prepositional functions which

our

[CHAP,n

How

far formal

Formal

explicitly

general,but

in the sitions propois definable in implication occur

of

terms

it as be implication simply,or material implication may difficult which will be in discussed a question, Chapter in. the difference is between the two, an illustration will explain.

called,is What The

fifth proposition of Euclid

is true,

is the

so

This is a

follows from

the fourth: if the fourth

fifth,while if the fifth is false, so

is the

fourth.

of material

for both propositions absolute are implication, their for the dependent meaning upon assigningof a value to a variable. But each of them states a formal implication.The fourth states that if x and certain conditions, fulfilling y be triangles then x and y are triangles certain other conditions, and that fulfilling this implication holds for all values of x and y ; and the fifth states that if x is an isosceles triangle, has the anglesat the base equal. The x formal implication involved in each of these two propositions is quite differentthingfrom the material implication a holdingbetween the propositions as wholes; both notions are requiredin the propositional but it is the study of material calculus, which specially implication this subject, for formal implication distinguishes occurs throughoutthe case

constants, not

whole

of mathematics.

It has been kinds of where

in treatises on logic, to confound the customary, and*oftento be really implication, the formal considering

the

material

kind

only was

apparentlyinvolved.

when

it is said that "Socrates is a man, Socrates isfeltas a variable: he is a would

man

have done

well.

example,

therefore Socrates is a mortal,"

type of humanity,and

any other

For

two

kind

one

feels that

If,instead of

which therefore, impliesthe truth of hypothesis and consequent,we put "Socrates is a man impliesSocrates is a mortal,"it appears at once that we may substitute not onlyanother but other man, entity whatever,in the any placeof Socrates. Thus althoughwhat*is explicitly stated,in such a case, is a material implication, what is meant is a formal implicationand ; effort is needed to confine our some to material implication. imagination 16. A definitionof is quiteimpossible. implication If p implies then if q" truth p is true q is true, i,e. p^ implies j'struth ; also if q is false p is false, i.e.g^sfalsehood implies falsehood*. Thus /s truth and falsehood give us merelynew implications, not a definition of implication. *

The

statements truth,

reader

is recommended

formal,ie

to

as

observe that the maiix implications in these

"p impliesq" forrmlh, implies "p's truth implies ?'s while the subordinate implications material are are

SymbolicLogic

15-17]

15

If p implies g, then both are false or both true, or p is false and q true ; to have q false and p true,and it is necessary to have it is impossible q true or p false*. In fact,the assertion that q is true or p false turns be

to "p implies strictly equivalent means gr";but as equivalence this stillleaves implication mutual fundamental,and not implication, of disjunction. the other hand, is definable in terms on Disjunction, of implication, shall shortlysee. definable in terms It follows as we that of any two propositions from the above equivalence there must be which implies the other,that falsepropositions one implyall propositions, But these are and true propositions are impliedby all propositions. of the d eal results to be demonstrated; premisses our subject exclusively out

to

with

rules of inference.

It

be

may

observed be

that, although implicationis indefinable,

defined.

and Every propositionimpliesitself, whatever is not a proposition impliesnothing. Hence to say "p is a to is p"\ and this equivalence proposition" equivalent saying"p implies the mathematical As of be used to define propositions. sense may that current is widelydifferent from definition among philosophers,

propositioncan

it may

well to

be

observe

that, in

the

mathematical

sense,

a

new

defined when it is stated to be function is said to prepositional and be function to imply to (i.e. impliedby) a prepositional equivalent indefinable or has been defined in which has either been accepted as be

terms

of

indefinables.

The

definition

of

entities

which

are

not

functions is derived from such as are in ways which will prepositional in connection with classes and relations. be explained indefinables We then, in the prepositional calculus, no 17. require, of kinds the two implicationremembering,however, that formal except remains to be undertaken. is a complex notion,whose analysis implication we As requirecertain indemonstrable regardsour two indefinables, have not succeeded in reducingto less which hitherto I propositions, there indemonstrables must than ten. Some be; and some propositions, be of the number, since no demonstration must such as the syllogism, But concerning is possible without them. others,it may be doubted whether they are indemonstrable or merely undemonstrated; and it of supposingan axiom and should be observed that the method false, which has of this been found assumption, deducingthe consequences is here not universally admirable in such cases as the axiom of parallels, of deduction;and if they axioms are principles available. For all our are true, the consequences which appear to follow from the employment will not really follow,so that arguments from of an opposite principle of the axiom here subjectto special of are the supposition falsity an of indemonstrable be the number fallacies. Thus propositions may of them and in regardto some I know of capableof further reduction, "

*

I may as be considered

well state as

once

for all that the

mutuallyexclusive

unless

alternatives

expresslysaid

of to

a

be

will disjunction so.

never

The

16

no

of Mathematics Indefinable^

them groundsfor regarding

hitherto

18.

indemonstrable

except that theyhave

undemonstrated.

remained The

as

[CHAP,n

axioms

ten

are

the

following.(1)

If p

impliesq, then be, "JP implies "?"

implies"?*;in other words,whatever p and q may is a proposition. (") If p implies p; in other words, "?,then p implies whatever q, then q impliesanythingis a proposition.(3) If p implies whatever is implied by anythingis a proposition. implies q; in other words, be dropped,and the may (4) A true hypothesisin an implication of formal symbolic incapable consequentasserted. This is a principle the essential limitations of formalism a statement, and illustrating Before later at I which shall return a stage. proceeding point to of assertion two the define desirable is to it joint propositions, further, is definition This their called is what highly artificial, logical or product. mathematical and philoand illustratesthe great distinction between sophical if definitions. It is as follows: If p implies q implies jp, then, g, if t hat that of and product p q) means p implies q implies pq (thelogical their joint In other words, ifp and q are propositions, r, then r is true. is true which is assertion is equivalent to sayingthat every proposition it. We that the second implies such that the first implies cannot, with p

"

formal

correctness, state

our

definition

in

this

shorter

form, for the

the is already propositions'"

logical productof hypothesis "p q are and "j is a proposition.We state the can now "^" is a proposition1'' of inference, six main principles to each of which, owing to its importance, and

"

all except the last will be found in of the subject.(5) If p implies Peano^s accounts p and q implies y, This and is called then pq implies simplification,asserts merelythat p. the jointassertion of two propositions impliesthe assertion of the first a

is to

name

be

given; of

these

This r. (6) If /; implies r, then /; implies q and q implies called the syllogism. (7) If q impliesq and r impliesr, and

of the two. will be

This is the principle of r. implies r, then pq implies have a productof three propositions; the hypothesis, we but this can of course of the product of two. be defined by means The that q implies states that if p implies follows r principle ?*, then from the jointassertion of p and q. For example:"If I call on so-andI shall be admitted" so, then if she is at home implies"If I call on so-and-so and she is at home, I shall be admitted.'1 (8) If p implies that q implies r. p and q implies r, then /; implies 5, then, if pq implies This is the converse of the preceding and is called principle, exportation^. The previousillustrationreversed will illustratethis principle.(9) If p impliesq and p implies/-, then p impliesqr: in other words, a if p

that implies In importation.

*

q

Note that the implications denoted by if and then,in these axioms, are

formal, by impliesare material. t (7) and (8) cannot from the definition of the logical (I think) be deduced product,because they are requiredfor passingfrom "If p is a proposition, then 'q is a proposition* then etc." impliesetc." to "If p and q are propositions, while those denoted

17-19]

SymbolicLogic

which implies each proposition

17

of two

propositions impliesthem both. If composition.(10) p implies p and This is called 'p impliesq* implies p* implies q, then q implies p. of reduction; it has less self-evidence than the previous the principle but is equivalent to many that are self-evident. principles, propositions it to these,because it is explicitly I prefer concerned,like its predecessors, with implication, and has the same kind of logical character as they If we remember is have. that "/?implies to "q or not-/?," q" equivalent is true; for can we easilyconvince ourselves that the above principle to "p or the denial of *q or not"*j?impliesq* implies p" is equivalent p" i.e. to "/? or *p and not q" i.e.to p. But this way of persuading of reduction is true involves many ourselves that the principle logical which have been be not and cannot demonstrated, principles yet demonstrated is except by reduction or some equivalent.The principle useful in connection its help,by with negation. Without especially firstnine of the we can means principles, prove the law of contradiction ; that p implies if and be can we propositions, not-not-p; that p q prove, and to not-py; to "j implies not-/?" "/? impliesnot-y"is equivalent that p impliesthat that "/" implies q" implies"not-" impliesnot-/?"; is equivalent and that that not-/? to "/?implies not-/?"; not-/?implies/?; to "not-not-/? impliesnot-y." But we tfp impliesnot-y"is equivalent without reduction cannot or some (so far at least as equivalent prove This

is called the

of principle

"

I have

be true (thelaw of able to discover)that p or not-/?must is equivalent to the negation middle);that every proposition

been

excluded

that not-not-/? that "not-yimplies implies/?; proposition; that not-/?"" y"; p" implies "not-p implies implies"/?implies /?, or that of Each these assumptionsis "/? impliesq" implies""/ or not-/?." and may, if we choose,be subof reduction, stituted to the principle equivalent of them excluded for it. Some middle and double especially have self-evidence. But when we negation appear to have far more and in of define how to we seen implication, disjunction negation terms and for shall see formal that the supposedsimplicity vanishes, that, is than reduction at simpler purposes any of the possible any rate, I retain it among For this reason alternatives. premissesin my obvious usual and more to more superficially propositions. preference is 19. defined a ddition follows: as or Disjunction logical "/;or q" is equivalent It is easy to persuade to ""/? impliesq* impliesq? ourselves of this equivalence, by remembering that a false proposition impliesevery other; for if p is false,/? does imply 9, and therefore, if "p implies But this argument q" implies 5, it follows that q is true. which have not yet been demonstrated,and is again uses principles From this merelydesignedto elucidate the definition by anticipation. definition, by the help of reduction,we can prove that "/? or q" is deducible from the equivalentto "q or /?." An alternative definition, above, is: "Any proposition impliedby/? and impliedby q is true,"or, of

some

other

"

"

in other

*' together *' and (qimplies implys, whatever words," "p implies Hence

be."

may

s

to equivalent "r

[CHAP,n

of Mathematics Indefinabks

The

18

the

proceedto

we

assertion

that

impliesall

negation:not-p is i.e. that propositions,

whatever

may

the

p r" r" "_p implies implies implies

definition of

r

From

be*.

this

middle

and laws of contradiction and prove the of logical double negation,and establish all the formal properties commutative and distributive the associative, and addition multiplication

point we

excluded

can

"

the

Thus

laws.

logicof

will Philosophers

is now propositions

objectto

the

complete. definitions of

above

and disjunction

notions is something by negationon the ground that what we mean the their definitions meanings, assignas quitedistinct from what matter the of in definitions stated a the as and that are, equivalences indications as to the way in not mere propositions, fact,significant Such an objection which symbolsare goingto be used. is,I think,wellfounded, if the above account is advocated as givingthe true philosophic where But of the matter. a purelyformal purpose is to be analysis side in which a certain notion appears on one served,any equivalence the advantageof the other will do for a definition. And but not on formal developmentis that it prohaving before our minds a strictly vides definite shape than in a more the data for philosophical analysis of formal logic, Criticism of the procedure would be otherwise possible. these

will be best therefore,

brought to

an

end.

The

B. 20.

In "

this calculus there in

has been

until the presentbrief account postponed

fact,two

in the way difficulties

seem

of

Calculus are

of

Classes.

very much

fewer

sufficient but "

primitive propositions

new

there

much

are

greater

of the ideas embedded non-symbolicexposition

far as possible, willbe postponed as symbolism. These difficulties, I shall try to make to later chapters.For the present, an exposition which is to be as straightforward and simpleas possible. The calculus of classes as fundamental by regarding may be developed in

our

the notion

of

doss, and also the relation of

class. This method

is

a

member

of

a

class to its

Professor Peano, and is perhaps more correct than a different method philosophrcally which, for formal purposes, I have found more convenient. In this method stilltake as we *

adoptedby

The

principlethat false propositions imply all propositionssolves Lewis logical paradox in Mind, N. S. No. 31 (1894). The assertion made in that and paradox is that, if p, q, r be propositions, q implies r, while p impliesthat then be must not-r, false,on the supposedground that 'q implies r" and q implies p not-r are incompatible.But in virtue of our definition of negation., if q implies will hold : the two ever q be false both these implications together, in fact, whatr be, are equivalentto not-gr. Thus the only inference proposition may warranted is that if p be true, q must be false, by Lewis Carroll's premisses i.e. that which ; and this is the conclusion, iiot-# p implies would oddlyenough, common sense have drawn in the particular which he discusses. case

Carroll's

'

ff

"

19-22]

SymbolicLogic

fundamental of

the relation

individual to

an

to the human

a

function and

this,we the

(which,following Peano, I shall denote by e)

class to which it

which

race

In addition to

is

i.e.the belongs,

of siich that.

of

man.

prepositional

a

It is these three

notions

be said in

that

explanation

of them.

The

21. whole

and

insistence

the distinction between

on

part between

classes is due

the whole

importanceto

technical

to mathematics. applications

and

Socrates is a

indefinables the notion

as

characterize the class-calculus.Somethingmust of each

relation of Socrates

expressed by sayingthat

take

notion

19

in all

except between

of

Peano, and

to

developmentand

is of very great the whole of the

In the scholastic doctrine of the

the previoussymboliclogic,

in the work

and the relation of

"

Frege*.

The

relations

two

the relation of individual to

and species

confounded,

are

distinction is the that

syllogism,

same

as

of

that

speciesto

genus, between the relation of Socrates to the class of Greeks and the relation of Greeks to men. On the philosophical of this distincnature tion I shall enlarge when I come to deal critically with the nature of

classes;for the present it is enough to observe that the relation of whole and part is transitive, while e is not so : we have Socrates is a and men a are a class,but not Socrates is a class. It is to be man, observed

that

the

predicate by

or

is

man

and

Socrates

all

be

and

from distinguished

it is to be defined: thus relation

considered

men

I shall return

man.

that

which

The class-concept.

a

between

class must

to

The

22.

-

"

shall find

fundamental

next

class-concept a

class,while

regardedas holding

not between collectively,

is

x

an

to

reason

a? where

Socrates

a

is

a

constant

doubt.

is that of

notion

functions Although propositional there each defined as they are notion

be

are

point in Chapter vi. Peano holds containingonly a singlevariable are

functions prepositional in expression the form we

must

men

this

capableof

class;but this view

e

the

a

function. propositional

in the calculus of propositions it occurs, so that the general occur

is not

duce required. But in the class-calculus it is necessary to introthe generalnotion explicitly. Peano, does not requireit,owing to and assumptionthat the form "x is an a" is generalfor one variable,

his that

extensions

variables.

of the

But

avoid

must

we

introduce the notion of

form

same

available for any number of assumption,and must therefore

are

this

function. We may explain propositional (but not define) this notion as follows ; "frx is a propositional function if,for determinate when is given. x every value of x, $x is a proposition, Thus is a man is a propositional x function. In any proposition, ever howwhich contains no real variables, we complicated, imagineone may of the terms, not a verb or adjective, be other to terms : instead replaced by u

of

"

*

p. 2.

Socrates

a

"

is

"

a

man

we

may

put

See his flegrifffsschrift^ Halle,1870, and

"

Plato

is a

man/'

"

the number

52

Grundge"etzeder Arithmetik,Jena.,1893,

is

Indefnablesof Mathematics

The

20

a

and

man/'

except

the

to

as

Thus

on*.

so

variable

one

man

a

variable,so far "

y

The

false for others.

and

is

impliescc

man

a

they

as

assertingthat

are

is a

other

no

for all values

of

I know

mortal"; but

functions prepositional

of the

such implications,

all express

to me,

term,

proof the variable

values

some

it is true

where

instances known

for

variable

propositions.A

of all such

"

is

cr

for the

Putting or

term.

the type expresses function in generalwill be true positional "

all agreeing propositions

get successive

we

[CHAP, n

no

priori

for all values of

true

are

as

for

reason

a

the variable.

brings me

This

23.

render

which

a

function fa prepositional the latter

indeed

equation "

values of x may consider all the form values these class,and a

true

which

are

such that

$x

a

be

x an

and

"

In

is true.

of of

roots

of the former

class may function. propositional in fact

values

like the

are

case particular

a

are

The

that.

of such

notion

the

to

we

general,

denned

all

as

There is,however, some satisfying in this statement, though I have not been able to limitation required some the limitation is. This results from a certain what discover precisely I shall discuss at length at a later stage (Chap. x). which contradiction for definingelectsin this way are, that we requireto provide The reasons

the

terms

null-class,which

for the which to

some

the

defining a class

our

relation

6,

that

and

by relations,i.e. all the terms the relation R are to form a class,and such

define

terms

has

other

prevents

we

term

a

to

be

to

have

which

classes

as

wish

able other

to

requiresomewhat

cases

functions. complicatedpropositional

notions, we require regard to these three fundamental first asserts that if x belongs to the The primitive propositions. With

24. two

function fa, then fa is true. a propositional satisfying for all The second asserts that if fa and fa are equivalentpropositions values of ^ then the class of #'s such that fa is true is identical with the class of a?'ssuch that fa is true. here, is Identity,which occurs defined as follows : x is identical with y if y belongsto every class to which other words, if for tr belongs, on x is aw" implies"y is a u all values of u. With regard to the primitiveproposition it is to itself, class of terms

"

be observed Two

that

it decides

class concepts need

in favour

not

axiom with

is to hold, it must

classes.

terms, not

problem

with

We

must

not

be

of

extensional

an

be identical when

andfeatherless bipedare by and b etween 1 and prime integer man

"

no

3.

These

concerned

with

any concept denoting that purposes, this is quite essential. combinations as to how can many

no

more

speak

so:

are

even

are

and class-concept,

are

be of these that

of classes.

their extensions

identical,and

means

view

if

our

in

dealing assemblage of matical assemblage. For matheConsider, for example, the be formed of a given set we

the

are

to

actual

*

Verbs and adjectivesoccurringas such are distinguishedby the fact that, if be taken as variable,the resulting function is only a proposition for some values of the variable, i.e.for such as are verbs or adjectives See iv.

they

respectively.

Chap.

SymbolicLogic

22-25] taken any

of terms

contained

common

number

time, i.e.as

a

how

to

classes are many have the same tension, ex-

given class. If distinct classes may this problem becomes utterlyindeterminate. usage would regarda class as determined when in

a

And

certainly

all its terms

are

extensional view of

The

given. SymbolicLogicand Arithmetic

in some classes, form, is thus essential to and its is expressed in the mathematics, necessity

to

the

But

axiom.

above

at

21

axiom

itself is not

at least it need not

;

to employed until we come employed,if we choose to distinguish

be

equalityof classes,which is defined as mutual inclusion,from the of individuals. Formally, the two are totally distinct : identity identity of a and b is defined by the equivalence is defined as above,equality of the

"

"

is

so

an

and

a

deduced

part of

logical productof the A

the

new

"

of propositions

is

x

is

k

a

an

implies x

by

define we Similarly of class (not-a). negation a

If is true.

a

6),and the

or

", i.e.the class of

for all values of

u

are

class of o?s such that the

logical productand its logical classes, productis the "

is a

is

easily logical product

The

the

of the classes of "

class-calculus

b is the

and "x

a

is introduced

to each "

"

classes (a

classes. If fcis a class of

x.

the

calculus. prepositional

classes a and

two

of two

sum logical

idea

for all values of

those of the

from

common

or

"

is a b

x

of

Most

25.

"

The

u.

of

sum

class of

a

longing be-

class of terms

terms

such

x

"

that

u

is the class

sum logical

in every class in which every class of the class Jc is which "u is contained,i.e.the class of terms x such that,if " u is a k" implies is contained

contained And

we

in c" for all values of ?/, then, for all values of c, x is " is an x say that a class a is contained in a class 6 when

"

implies x we

is

b"

a

define the

may

for all values of

productand

In

x.

of

sum

a

like

Another propositions.

what is called the existence of very importantnotion is what existence means which must not be supposedto mean A

it has at least

class is said to exist when is

follows:

as

is

a

existent

an "

is true provided x proposition we

may

be

must

class

a "

an

giveto

a

It must

x.

exists when

the

is true,i.e. when

It is

is

a"

an

the

in

all such

tion defini-

when

only

any value

implied proposition of

function propositional of the propositions

are

clearlythe obtained

in

manner

from

Consider, for example,the

calculus. positional

formal

A

x.

form

"

x

is

false. are propositions

understand

class-calculus

that the

of all

sum logical

not

importantto

a

philosophy.

in

and

word

a

"

it whatever alwaysimplies

be understood

not genuineproposition,

a

class when

class

a

A

term.

one

"

a

above

the

with

manner

class of

c.

a

those

which in

positions pro-

the

prohave

syllogism.We

r" imply "/" implies?-." q" and "5 implies "p implies

Now

put

is

"x

definite have some a," "# is a 6," "a? is a r" for/?, 5, r, where x must that then find decide but We is it not necessary to what value. value, is an a implies is a 6, and x is a b x x if,for the value of x in question, an

x implies

is

a

irrelevant,we

c,

then

may

x

is

an

a

vary #, and

impliesx thus

we

is

a

c.

find that

Since

the value of

if

is contained

a

x

is

in b9

and b in c, then

applyingthis if fallacies are

be

of Mathematics Indefinables

The

22

is contained in

a

process it is necessary to avoided. to be successfully examine

instructive to

between

SchrcSder and

point upon

a

r" is implies

"pq propositions, r" or q implies

equivalent to

that

admits

McColl

Mr

this connection

In which

Schroder

McColl*.

Mr

But in class-syllqgism. caution, employ the utmost

is the

This

c.

[CHAP,n

it will

disputehas

a

arisen

asserts that

the the

ifp"q, r are r disjunction "p implies disjunction impliesthe

for the diverThe gence reason implication. other, but denies the converse of and plication immaterial propositions is, that Schroder is thinking

functions and thinkingof prepositional the truth of the principle As regardspropositions, formal implication. made plainby the followingconsiderations. If pq implies may be easily of them which is false implies the one either p or q be false, r, r, then,if But if all both be because false propositions true, propositions. imply plies and and is and therefore is therefore r true, true, p impliesr q impq are impliedby every proposition. r, because true propositions least of the in any at Thus one propositions case, p and q must imply r. (This is not a proof,but an elucidation.)But Mr McColl and r to be the objects: Supposep and q to be mutuallycontradictory, null proposition, then pq implies r but neither r. Her,e p nor q implies with and functions formal we areimplication.A dealing prepositional function is said to be null when it is false for all values of propositional the class of x\ satisfying the function is called the null-class, x ; and while Mr

being in

fact

McColl

class of

a

is

no

Either

terms.

Peano, I shall denote following

by

A.

Now

the let

function

our

r

be

or

the

class,

replaced by A,

and our q by not-"""#, where fovis any propositional function. p by "f"x, Then pq is false for all values of x9 and therefore implies But it is A. our

in

not

generalthe

case

that

$x

is always nor false,

yet that not-"^ris always

false;hence neither

Thus the above formula can A. alwaysimplies only the in in the calculus class-calculus trulyinterpreted : propositional it is false. This may be easily rendered obvious by the following considerations: Let $x, ^rjr, functions. Then yx be three propositional for all values of #, that either "f" $x -fyximplies yx" implies, x implies But it does not imply that either "f"ximplies yx or ^x implies yx. yx for all values of ac9 or tyx implies for all values of x. The disjunction yx is what I shall call a -variable disjunction, constant to as one: opposed a be

"

.

that

is,in

alternative is true, in others the other, whereas constant a there is one of the alternatives (thoughit is not disjunction stated which) that is alwaystrue. Wherever disjunctions in regard occur some

cases

one

in

functions, propositional theywill onlybe transformable into statements in the class-calculus in cases where the disjunction is constant. This is is both importantin itself and instructive in its a pointwhich bearings. of stating Another the matter is this: In the proposition: If way to

*

Schroder^ Algebra

cfer

Logik, Vol.

EquivalentStatements/'fifth

paper, Proa

of n, pp. 258-9; McColl,, "Calculus Land. Math. Soc. Vol. xxvzn, 182. p.

25-27]

Symbolic Logic

23

TJTXimpliesxp, then either $x implies^x or tyx implies%", the indicated by if and then is formal,while the subordinate implication $x

.

are material; hence implications

lead to the inclusion of formal

formal

the

same

laws of

class in

one

and negation addition, multiplication, tautology

regardsclassesand

as

that

states

do not implications another,which results only from

implication.

The are

the subordinate

made

change is

no

The law of tautology propositions. when a class or proposition is added to or

multiplied by itself. A new feature of the class-calculusis the null-class, class or havingno terms. This may be defined as the class of terms that the class which to does not exist (in the sense as belong every class, defined class A

above),as the class

in every class, the as is such that the prepositional function "# is a A" is false

which

which

is contained

for all values of x, or as the class of a?s fycwhich is false for all values of

function

shown easily

satisfying any prepositional All these definitions are

x.

to be

equivalent. important pointsarise in connection with the theory of identity.We have alreadydefined two terms as identical when the second belongsto every class to which the first belongs. It is easy to is transitive show that this definition is symmetrical, and that identity 26.

Some

reflexive (i.e. if

and

whatever

x

and z ; and and y, y and z are identical, x so are is identical with x). Diversityis defined as the

x

be, x

may

be any term, it is necessary onlymember is x : this may be

negation of identity.If from

x

the

class whose

class of terms

which

which distinction, discovered

was

the class of

even

which

are

the

to

return

it at

pointto

be

C.

The

a

later

as

stage.

be identified with the number

the

sum

Calculus

calculus of relations is

Thus

2, and

of 1 and " is not to be identified

the speaking, philosophically considered in Chapter vi.

with B.

defined

necessityfor this

The

x.

distinguish

to

from purelyformal considerations, primarily

by Peano ; I shall primesis not to

In what,

The

identical with

are

results

the class of numbers

27.

x

of a

difference consists,is

a

Relations.

more

modern

subjectthan

the

in Although a few hints for it are to first developed De Morgan*, the subject was by C. S. Peircef A careful shall find in the course of mathematical shows (aswe analysis reasoning of the presentwork) that types of relations are the true subject-matter this fact ; hence the however bad phraseology a discussed, may disguise immediate bearing on mathematics than logicof relations has a more calculus

of

be found

classes.

.

*

Phil.

Camb.

Relations." t See

x,

"On

the

Syllogism,No.

iv, and

on

in

the

Logic

Cf. ib. Vol. ix, p. 104; also his Formal Logic (London, 1847),p. 50. Journal the Algebra of Logic, American articles on

his especially

Mathematics, Vols. methods

VoL

Tram.

in

and

vn.

The

Schroder, op. cit.,Vol.

subjectis treated in.

at

length by

of

of

C. S. Peiree's

of Mathematics Indefinables

The

24

[CHAP,n

correct and adequate and any theoretically propositions, is Peirce by its means. of mathematical truths onlypossible expression

that of classes or

Schroder

and

great importanceof

realized the

have

the

but subject,

Peano, but on the their methods, being based, not on unfortunately from Boole, are so older SymbolicLogic derived (with modifications) which to the be of applications ought cumbrous and difficultthat most made

I do

not

relation

as essentially

formulae

the fact that

from presentdiscuss)

at

not

for

they regard a requiringelaborate

thus couples,

of

class

a

of summation

old

sufferstechnically (whetherphilosophically

method

SymbolicLogic,their or

In addition to the defects of the

not feasible. practically

are

dealingwith singlerelations.

from I think, probablyunconsciously, derived,

This

view

is

: it error philosophical l has propositionsess always been customary to suppose with ultimate than class-propositions propositions, (or subject-predicate and this has led which class-propositions are confounded), habitually

a

relational

to

desire to treat relations

a

be,

it

friend Mr

derived from

my treatment

formal

powerfulas

or

engineof

an

led to

was

treatment, whether

This

of relations.

correct

Moore*, that I

G. E.

not, is

this may

However

belief,which oppositephilosophical

the

from certainly

was

of classes.

kind

a

as

far certainly

more

different

a

sophically philo-

more

and

convenient

I

far

more

f.

in actual mathematics discovery

function we relation, express by xRy the prepositional We a primitive x has the relation R to y? (i.e. indemonstrable) require for the is that effect to a proposition all values of x proposition xRy 28.

If R

be

We

then

a

"

and y.

which

terms

class of

R

be

the relation R

to

the paternity,

jK; and

also

consider

to

other,which

or

the

The

the

fathers and

class of

I call the

class of terms

which

to

Thus

I call the class of relata.

referents will be

have

We

term

some

respect to has the relation J?,which

children.

classes: following

consider the

to

with referents

term

some

have

have

relata will be

classes corresponding

the

if

with

classes of terms : so-and-so's children, terms or or respectto particular the children of Londoners,afford illustrations. The

intensional view relations may

that two

same x

there

in the

was

determinate relation R

by

the

the

#,

of

case

when

equivalentto and

the

advocated

leads to the result

beingidentical. relations J?,R are said to be equalor equivalent, to have the or and is impliedby xRy for all values extension,when xRy implies and y. But there is no need here of a primitive as proposition,

Two

of

of relations here

have

*

same

in order classes,

extension

logicalsum the

See his article "On

t See

logical

sum

my

the Nature

articles hi JK. d. M.

obtain

to

product of

or

equivalentto R\ if K

without

is determinate.

i.e. by the assertion of

this is identical with

relations

extension

be of

some

or or

a

We

the

which

relation may

replacea

class of relations

of all such

relations ;

product of the

equivalentto

R.

Here

class of we

Judgment/* Mind, N. S. No. 30. 2 and subsequentnumbers.

Vol. vn, No.

is

use

SymbolicLogic

27-29]

25

of two classes, which results from the primitive identity proposition of classes, to identity to establish the identity of two relations procedurewhich could not have been appliedto classes themselves

the as a

"

without A has

a

that

vicious circle.

a

in regardto proposition primitive i.e.that, if R

converse,

is

xRy

be any

relations is that every relation

there is a relation R? such relation,

for all values of

to yKx equivalent

and y*

x

Following

of R by R. Greater and less, Schroder,I shall denote the converse after,implying and impliedby, are mutually converse relations. With such as some relations, identity, diversity, equality,

"before and

the inequality,

relations with

called

are

is the

converse

same

the

as

symmetrical.When

relation: such original is incompatible converse

the

the

in such cases I call the as relation, as original greaterand less, in intermediate cases, not-symmetrical. asymmetrical ; The most importantof the primitive in this subject is propositions that between any two terms there is a relation not holdingbetween any other terms. two This is analogous that an^ term is to the principle the only member of some class; but whereas that could be proved, extensional view this principle, far as I can the of classes, so owing to is incapable of proof. In this point,the extensional view of discover, relations has an advantage;but the advantageappears to me to be relations are considered outweighed by other considerations. When the above principle it may seem to doubt whether intensionally, possible is true at all. It will,however, be generally admitted that,of any two function is true which is not true of a certain terms, some prepositional If this be admitted,the above principle given different pairof terms. the logical follows by considering productof all the relations that hold be Thus the above first pair of terms. between our principle may which is equivalent to it: If xRy implies replacedby the following, xRy\ whatever R may be,so long as R is a relation,then x and x\ identical. introduces a But this principle respectively y and y are hitherto from which we have been exempt, namely a difficulty logical variable with a restricted field; for unless R is a relation, xRy is not a take and thus #, it would seem, cannot at all, true or false, proposition

relation

values,but only such as are relations. of this pointat a later stage.

all

that are assumptionsrequired and that the logical relation, productof

29. is

a

productof is

relations must

be

relation.

a

R^ S is the relation which

relations there

two

is a simultaneously)

a

relation S.

term

Thus

is the relative to

y to which

her

x

has

the relation of

the

a

relation

a

class of relations

(Le.the

a

maternal mother

relative

Also

relation. The

relative

between

the relation R

productof father and grandsonis

holds

negationof

the

Other

assertion of all of them

to the discussion

I shall return

productof

and

x

which

and

to grandfather ;

that

productof

of

a

the relative

z

whenever

has to his

two

z

the

grandson

mother grandpaternal

mother

and father;

of Mathematics Indefinabks

The

26

is the to grandchild grandparent

of

that

these

as product,

relative

parent. The

commutative, and does

productis a notion of it leads obey the law of tautology,

relative

and grandchild, the relative

so

relations R

of two

sum

when, if y be any other y has to z found no occasion

consider also what

and S, which

they call

holds between

x

and z9

has to y the relation jR, x notion,which I have complicated

whatever, either

term

This

the relation S.

or

of relations : the square of relation of grandparentand

to powers

Schroder

Peirce and

on.

show, is not in general

instances

child is the

parent and

relation of

the

relative productof parent and

generalobey the law of tautology.The very great importance. Since it does not

in

not

[CHAP,n

is

a

onlyin order to has a of and multiplication. This duality preserve the duality addition is considered as an independent when the subject certain technical charm in relation to it is when considered but mathematics branch of solely ; in questionappears devoid of of mathematics, the duality the principles all philosophical importance. I know, only two far as other Mathematics 30. so requires, material that is the one implication a relation, propositions, primitive is to a class to which it belongs) the other that e (the relation of a term without relation*. We can now a developthe whole of mathematics in the logic further assumptions indefinables. Certain propositions or of relations deserve to be mentioned, since they are important,and it of formal proof. If ", v might be doubted whether theywere capable be any two

employ,and

to

there is classes,

a

is introduced

which

relation R

the

assertion of which

between

to the assertion that x belongs to u any two terms x and y is equivalent and y to v. If u be any class which is not null,there is a relation which If all its terms have to it,and which holds for no other pairsof terms.

R

be

and relation,

any

with

respectto

u

R, there

any class contained in the class of referents is a relation which has u for the class of its

and referents,

is

the

where it holds,but

same

domain

the

R

as as

to equivalent

with class

synonymous

developmentof the

D. So

much

of

has

a

taken

questionas as

*

There

the

and indefinable,

is

a

of mathematics

above

brief

of the

outline

in regard to difficulty

E.g. F. 1901, p. ({;

f.

of

Symbolic

to discuss his work

I have

notions

of

Logic is explicitly,

from departed

mm

symboliclogic to be as indemonstrable, propositions to some extent insisted!* arbitrary.But it is which

has

result.

SymbolicLogic.

are

of the

this

discussed primitiveproposition,

94 below. t

(I use

From this pointonwards, of referents.) : special types of relations are

Peano^s

to which

is,as Professor Peano

restricted domain.

more

desirable inspired by Peano, that it seems justifying by criticism the pointsin which The

class; this relation is

is technical subject

and special branches considered,

31.

throughoutthat

R

1807, Part I, pp. 62-3.

in

"" 5"%

29-32]

SymbolicLogic establish all the mutual

importantto and of logic,

27

relations of the

simplernotions

the consequence to examine of takingvarious notions as It indefinable. is necessary to realize that definition, in mathematics, hi does not mean, of as an philosophy, analysis the idea to be defined

into constituent

ideas.

This

notion, in any

concepts,whereas in mathematics

it is

case, is

to possible

to onlyapplicable

which

define terms

concepts*.Thus also many notions are defined by symbolic since they are logicwhich are not capableof philosophical definition, and Mathematical definition consists in pointing simple unanalyzable. out a fixed relation to a fixed term, of which one term onlyis capable: this term of the fixed relation and the fixed is then defined by means term. The definition point in which this differs from philosophical be elucidated that the definition does mathematical by the remark may not pointout the term in question, and that onlywhat may be called there are. all the terms reveals which it is among philosophical insight This is due to the fact that the term is defined by a concept which denotes it unambiguously, not by actually mentioningthe term denoted. is meant What well the different ways of denoting, as as by denoting, be acceptedas primitive must ideas in any symboliclogicf: in this not in any degreearbitrary. respect,the order adoptedseems not

are

32.

For

the

sake

of Professor Peano's he

has

abandoned

of

let definiteness,

of the expositions the

attempt

to

us

now

examine

subject.In his later

some

one

J expositions

certain ideas and distinguish clearly

probablybecause of the realization that any propositions primitive, such distinction is largely arbitrary.But the distinction appears useful, and as showing that a certain set as introducing greaterdefiniteness, ideas and propositions of primitive sufficient;so far from being are in every possible it ought rather to be made I shall, abandoned, way. that therefore,in what follows,expound one of his earlier expositions, of 1897". Peano the following: starts are notions with which The primitive as

Class,the relation of the notion. of

a

an

individual to

class of which

a

it is

member,

a

where contain the both propositions implication of the simultaneous affirmation implication,

term,

i.e. formal variables,

same

the propositions,

two

From

notion

of

and the negation of definition,

with the division of notions,together

a

position. pro-

complex all symboliclogicby

these

to deduce into parts,Peano professes proposition Let us examine of certain primitive means propositions.

a

the deduction

in outline. We

may

observe,to begin with, that the simultaneous

might seem, at propositions idea. For althoughthis primitive of tzvo

the

to

simultaneous

*

See

t

F. 1901

first can

affirmation of any

not enough to take as a sight, extended, by successive steps, of propositions, finite number

be

t See

Chap. iv. and R. d. M.

Vol. vn, No.

1

affirmation

(1000).

"

F.

Chap.

v.

1897, Part

I,

The

28

easier to define than

much be

assertion of

simultaneous

a

that

is a ft"

a

If this

implies p.

but it

affirm

(see " 34, (3)). If propositions

that of two

might almost

have

false. We

defined in

be

well be taken

as

"

affirmation is the assertion

of the class are holds,all propositions

be at least must one true; if it fails, can productof two propositions logical manner;

able to

any class,finite or infinite. But class of propositions, oddly enough, is

their simultaneous class of propositions,

"p

be

requireto

we

[CHAP,n

of propositions

the

all simultaneously the

is wanted;

all that

this is not

yet

of Mathematics Indefinables

highlyartificial

a

since indefinable,

as

of the

that the

seen

no

definition.

We property can be provedby means may combined material that formal and a re implication observe,also, by be whereas to Peano into one kept they idea, ought separate. primitive

further

Before

33.

definitions.

some "

Peano propositions, givingany primitive

is an

x

"

means

a x

(1)

If

a

is

"

class,

a

If

and y is an a" (%) is an a implies that x is

and

a

6."

a

x

b

If

and

are

y

proceedsto

""V

is to

"

classes, every

are

mean

is

a

a

b

"

acceptformal implication

we

unobjectionable ; but it may well be held that the relation of inclusion between classes is simpler than This is a formal implication, and should not be defined by its means. for discussion. A difficult question, which formal I reserve subsequent of a whole class of material the assertion be to implication appears The introduced at this point arises from implications. complication the nature of the variable,a pointwhich Peano, though he has done ciently its importance, to show appears not to have himself suffivery much considered. The notion of one variable a containing proposition he takes which such is another as proposition, primitive, implying complex,and should therefore be separatedinto its constituents ; from this separation the simultaneous arises the necessity of considering affirmation of a whole class of propositions before interpreting such is is t hat We 6." next a a x as x an a implies come (3) proposition to a perfectly worthless definition, which has been since abandoned*. as

a

notion,this primitive

definition

seems

"

This is the definition of such that.

told,are

The

#\s such

that

x

is

an

a,

we

are

onlygivesthe meaning of such is an Now a? of the type x placedbefore a proposition it is often necessary to consider an x such that some is proposition true of it,where this proposition is not of the form a? Peano holds x is an (thoughhe does not lay it down as an axiom) that every proposition containingonly one variable is reducible to the form "x is an at." But we shall see (Chap,x) that at least one such proposition is not reducible to this form. in any case, the onlyutility of Mich that And to

mean

the class

a.

But this

that when

"

"

is to

effect the

effected already

but idea, *

one

reduction,which without it.

1, p. 25

be

assumed

to disengage from clearly

of the criticisms of

Vol. vir, No.

therefore

fact is that such that contains

which it is not easy

In consequence

t R. d. M.

The

cannot

; F.

Padoa, R.

d. M.

to a

be

tive primi-

other ideas.

Vol. vr, p. 112.

1901, p. 21, " 2, Prop. 4. 0, Note.

SymbolicLogic

32-34]

29

In order to grasp the meaning of such that,it is necessary to observe, and mathematicians Peano first of all,that what generallycall one a containing proposition

conjunctionof

the

constancy

of form

variable is

if the variable is apparent, really, class of propositions defined by some

certain

a

while if the variable is real,so

;

function,there propositional

is not

that

have

we

a

all,but

at proposition merely of certain a proposition a type. of the angles of a triangle is two rightangles," The sum for example, of a variable, becomes : Let x be a triangle when stated by means ; then of of the is the sum two angles x rightangles. This expresses the all in the o f propositions which it is said of particular conjunction the sum of their anglesis definite entities that if they are triangles, two function,where the variable is rightangles. But a propositional of a certain form, not all such proposireal,represents tions any proposition function,an (see""59-62). There is, for each propositional which indefinable relation between propositions and entities, be may all have the the that same form, expressedby saying propositions but different entities enter into them. It is this that givesrise to functions. Given, for example,a constant relation and propositional constant between the propoa term, there is a one-one correspondence sitions the said various have that relation to the said terms asserting which in these propositions. It is occur term, and the various terms for the comprehension of such that. this notion which is requisite Let be a variable whose values form the class a, and let f (x) be a onex for all values of x within valued function of x which is a true proposition a

of any representation

kind of schematic

"

the terms Then the class a, and which is false for all other values of x. such thatf(x)is a true proposition. This of a are the class of terms that

the

but

appearance

of values

number

such that.

of

it must

is fundamental

What

function. propositional

a

But

alwaysbe remembered of having one f(x} satisfied by a proposition is fallacious: f (x) is not a propositionat all, x

of gives'an explanation

is the relation of

the various to of given form terms entering propositions values of this i nto them variable the or as arguments severally ; the propositional relation is equally function requiredfor interpreting but is itself ultimate and inexplicable. f(x) and the notion such that, definition of the We the logicalproduct,or to next come (4) various

part,of

common

part consists of Here

of such

membership of 34.

The

x

b be two

such that

x

their classes, is

an

a

and

common x

is

a

b.

a

it is pointsout (loc.cit.), necessary to extend the where asserts our beyond the case proposition since it is onlyby means of the definition that class, that

part is

common

class of terms

and

a

Padoa

as already,

meaning the

the

If

classes.

two

shown

remainder less

to

of

be

the

a

class.

important,and may some primitivepropositions, appear to be

are propositions

precedingthe primitive be passed over. Of the with the concerned merely

definitions

of Mathematics Indefinables

The

30 and symbolism, the

others,on

of what real properties

to express any

"

impliesitself." proposition

every

which

of

to define

(%) propositions.

we

ought to have product,for a logical

the definition of the

genuineaxiom, which is very necessary by an generalized perhapsbe somewhat of a given form : propositions satisfying or

given relations

more

'In Section

axioms

axiom

which

defines the

are

common

as

different axioms,

give

order

It is

to terms

that is relevant.

productof K.

in

a

So

one

state

axiom

"

(3)

We

have

next

two

far

this of K

their logical that,if a, b be classes,

is contained

and

the

as

These

appear

as

ab

might be symbolismshows, that symbolisms they

have intrinsically

which

in this

case

be

if K

:

is involved. we

a

at

if K

Hence

representthe

none,

at

or

least

the class of classes,

belongingto

it becomes definition,

it is indifferent whether

in b.

of the defects of most

consists of all terms

K

With

of the terms

was

axioms

because,as

differentfrom ba. an

to

as

really onlyone, and appear distinct onlybecause Peano part of a part of two classes instead of the common

contained

product,"6, is

logical product

it is a indefinable, reasoning. But it might

definition of class.

the

classof classes. These two

to

class of classes; for when

concerningthe terms the terms having one e.g. more or given terms form a class.1" evaded by usinga generalized wholly

one

B, above, the axiom

of the

form

to

of

usingself-implication that the product been stated,as ought also

infinite class of classes. If class is taken

an

There law

axiom

be extended to the

it cannot onlytwo classes,

stated for

the

in

of

the

have

classesis a class. This

of two

of

Next

to equivalent

is

evadingthis axiom, adopted above, except the method identity, way

no

seems

symbolized ;

"every class is contained

is

axioms

Peano's

to itself." This is equivalent

is

importance. highlogical

of

contrary,are first of

The

(1)

not

n [CHAP,

once

has

every class that evident that no two oijly

terms,

none

logical belongs

a

logical productof K

order and

by

", ab

by ba, since the order exists only in the symbols,not in what is axiom with symbolized.It is to be observed that the corresponding that to of assertion the simultaneous class of is, regard propositions a of the class ; and this is pei^haps propositions impliesany proposition the best form of the axiom. is not Nevertheless,though an axiom it is of connecting to have a means required, necessary, here as elsewhere, the case where we from start class of classes or of propositions of a or or

relations with

the

case

where

the class results from

enumeration

of its

Thus

althoughno order is involved in the productof a class of there is an order in the productof two definite proposipropositions, tions it is significant and to assert that the products and p, 5, pq qp are B ut this be of the axioms with which equivalent. can provedby means we began the calculus of propositions ("IB). It is to be observed that this proof is prior to the proofthat the class whose terms are p and q is

terms.

identical with the class whose two

forms

of

terms

are

q and

p,

(4) We

both primitivepropositions. syllogism, The

have

next

first asserts

SymbolicLogic

35]

34,

and that,if a, 5, c be classes, a

5

;

the second

b in c, then

merits to have

class from

is contained

asserts that if a,

is contained

a

a

in

31

",and # is an a, then x is and a is contained in 6, b, c be classes, It is

in

the greatest of Peano^s the relation of the individual to its clearly distinguished c.

one

of

the relation of inclusion between

fundamental exceedingly

the former

:

classes. The

relation is the

difference is

and simplest

most

the latter a essential of all relations,

relation derived from complicated

implication.It logical

the

in Barbara assertion

has two that

assertion that are

results from

distinction that the

confounded forms,usually is

Socrates

Greeks

are

a

and

men,

stated by Peano's axioms.

definition of

wfyat is

the firstform

results from

meant

and

man,

:

the

therefore

mortal,the other

therefore mortal.

It is to be observed

by

one

the axiom

generaland

more

second form

cannot

These

two

the

forms

that,in virtue of the

being contained in another, and that,if /?, q, r be propositions, This r. productof p and q implies class

that q implies r, then the p implies substituted axiom is now by Peano for the first form it is

syllogism

the time-honoured

one

be deduced

from

of the

the

syllogism*:

said form.

The

instead of appliedto propositions This principle is transitive. asserts that implication is,of course, classes, have next a principle the very lifeof all chains of reasoning. (5) We calls composition:this asserts that if a is of reasoningwhich Peano of the

when syllogism,

also in c, then

contained

in b and

of both.

Statingthis

it is contained

with principle

regardto

in the

common

part

it asserts propositions,

implieseach of two others,then it impliestheir proposition which was product; and this is the principle jointassertion or logical above. called composition until we advance successfully this point, From 35. we requirethe in the of is edition the This Formulalre of idea taken, are we negation. and is defined i dea, as new a disjunction primitive considering, by its it is of course of the negationof a proposition, means. By means easy class for is a not-a is equivalentto of a : to define the negation x x axiom the But effect a? that is to is not an not-a we a requirean effect Peano the is also and that not-not-a another to a. a class, gives and ab is contained in c, and x third axiom, namely: If a, i, c be classes, in the form : If p, is an a but not a c, then x is not a b. This is simpler and and jp,q together imply r, q is true while r is q, r be propositions, then 5 is false. This would be stillfurther improvedby beingput false, and q implies in the form : If q9 r are propositions, r, then not-r implies Peano form which obtains as a deduction. By dealingwith ; a not-gr before classes or prepositional as we functions,it -ispossible, propositions all avoid and to replace to idea, treatingnegationas a primitive saw, of reduction. axioms respecting negationby the principle of We next to the definition of the disjunction or sum come logical has this subjectPeano times classes. On two changed his many that if

a

"

"

*

See

e.g. F.

1901, Part I, " 1, Prop. 3.

3

"

(p.10).

The

32

the edition

procedure.In negationof p.

find

19), we

which

consists of all terms contains

be observed that

a

belong to

definition

Either

b.

and b

n [CHAP, "

b

or

is defined

not-i, i.e.as the class of

In later editions

F. 1901, (e.g.

contains

b*

or

and

a

It is to logically unobjectionable.

seems

and classes,

are

class which

any

the

as

less artificialdefinition, namely: "a

somewhat

a

and

not-a

and not-J.

both not-a

not

are

"

considering,a

are

we

logical productof

the

which

terms

of Mathematics Indefinables

that

it remains

a

questionfor

logicwhether there is not a quitedifferent notion of the philosophical Jones.*" I shall consider of individuals, or as disjunction e.g. "Brown will remembered It be in Chapterv. this question that,when we begin is defined before negation disjunction by the calculus of propositions, ; to with the above definition (that of 1897), it is plainly take necessary first. negation class

notions of the null-class and the existence of In the edition of

dealt with.

next

are

null

connected

The

36.

it is contained

when

of

definition

class

one

in

every

being contained

a

1897,

class is defined

a

class. When

remember

we

in another

b

is

("#

a as

the "

an

a

implies"x is a b" for all values of x\ we see that we are to regard the implication as holdingfor all values,and not onlyfor those values This is a pointupon which Peano is an is not for which x really a. he has his whether made it. mind on and I doubt If the explicit, up when i s it hold would not to x were really an #, givea only implication is false for all values for which this hypothesis definition of the null-class, of

has

Peano

by

know

I do not

x.

since abandoned

of formal

means

whether

it is for this

the

reason

or

definition of the

for

some

other that

inclusion of classes

between implication

inclusion of classes appeal^ to be now definition which Peano jias sometimes

functions : the prepositional indefinable. Another regardedas favoured

(e.g.F. 1895, Errata,

116) is, that the null-class is the product of any class into its negation a definition to which similar remarks apply. In R. d. M. vn, No. 1 ("3, Prop.1-0), the null-classis defined as the class of those terms p.

"

belongto every class,i.e. the class of terms x such that "a is a for all values of a. There are of course class implies x is an a no in trying such terms x ; and there is a grave logical to interpret difficulty class which has no extension. This pointis one to which a extensionally I shall return in Chaptervi. this pointonward, Peano^s logic From proceeds by a smooth development.

that

^

"

"

But

ultimate

in

respectit is stilldefective : it does

one

relational

not propositions

not

recognizeas

asserting membershipof

a

class.

For this reason, the definitions of a function* and of other essentially But this defect is easily relational notions are defective. remedied by

applying,in

the

Formidaire

the

*

to

manner

explainedabove,

the

of principles

the

logicof relations*)".

E.g. F. 1001, Part I, " 10, Props.1.

+ See my

article " Sur la

0. 01 (p.33). -R. d. logiquedes relations/'

M.

Vol.

vir, 21 (1901).

CHAPTER

IMPLICATION

IN

37.

AND

III.

FORMAL

IMPLICATION.

the

and precedingchapterI endeavoured to present,briefly all the data, in the shape of formallyfundamental ideas uncritically, and propositions, that pure mathematics requires. In subsequentParts all the data by giving definitions of the I shall show that these are various mathematical the various continuity, concepts number, infinity, "

geometry, and motion. giveindications,as best I can, of

spaces

the

of

analysisof

the

data,and

In the

the

remainder

of Part

I, I shall

philosophical problemsarisingin

of the directions

in which

I

imagine these

will be elicited notions problemsto be probablysoluble. Some logical to logic, not commonly which, though they seem are quitefundamental discussed in works on the subject and thus problems no longerclothed ; in mathematical symbolism will be presentedfor the consideration of logicians. philosophical kinds of implication, the material and the formal, were found to Two be essential to every kind of deduction. In the present chapterI wish and distinguish these two kinds,and to discuss some methods to examine of attemptingto analyzethe second of them, In the discussion of inference,it is common to permit the intrusion of a element, and to consider our acquisitionof new psychological But it is plainthat where infer one we validly knowledgeby its means. from another, we do so in virtue of a relation which holds proposition whether the two between we perceiveit or not : the mind, propositions in inference as common in fact,is as purelyreceptive sense supposes it to it of sensible objects.The relation in virtue of which be in perception I call material is possible for us validlyto infer is what implication. be a vicious circle to define this have alreadyseen that it would We is true, then another is true, relation as meaning that if one proposition for if and then alreadyinvolve implication.The relation holds,in fact,

when

it does

hold,without

involved. propositions But in developing the we

were

the truth any reference to

consequences

led to conclusions

which

of

our

do not

or

falsehood

assumptions as by any means

to

agree

of the

cation, impliwith

of Mathematics Indefinable

The

34

[CHAP,m

for we found that any implication, commonly held concerning and any true proposition is false proposition impliesevery proposition Thus propositions are formallylike a set by every proposition. implied is like the inch or two, and implication of lengthseach of which is one such It relation "equalto or lessthan" among lengths. would certainly 4" can be deduced "2 2 maintained that from + not be commonly "Socrates is a man,""or that both are impliedby "Socrates is a triangle." is chiefly due, I think,to But the reluctance to admit such implications which is a much with formal implication, familiar more preoccupation where the before material mind, as a rule,even notion, and is really mentioned. from is inferences In "Socrates is what explicitly implication who vexed the is a man,""it is customary not to consider the philosopher Athenians,but to regardSocrates merelyas a symbol,capableof being in favour of by any other man ; and only a vulgarprejudice replaced stands in the Socrates by a number, a true propositions way of replacing what

is

=

plum-pudding.Nevertheless,wherever, as in Euclid, one is deduced from another,material implication is proposition particular

table,or

a

involved, though as instance particular constant

value

a

of

implication may be regardedas obtained by givingsome implication,

rule the material some

formal

to the variable

or

a

variables involved in the said formal

And although,while relations are stillregardedwith the implication. it is natural to doubt whether any such caused by unfamiliarity, awe is to be found, yet, in virtue of the general relation as implication there must laid down in Section C of the preceding principles chapter, be a relation holdingbetween nothingexcept propositions, and holding either the first is false or the of which between any two propositions Of the various second true. these equivalentrelations satisfying and if such a notion seems conditions,one is to be called,implication^ that does not suffice to prove that it is illusory. unfamiliar, this point,it is necessary to consider a very difficult 38. At logical problem,namely,the distinction between a proposition actually and a proposition considered merely as a complex concept. asserted, of our One indemonstrable principles was, it will be remembered, that if the hypothesis in an implication is true, it may be dropped,and the This asserted. it was consequent observed,eludes formal principle, to a certain failure of formalism in general.The statement, and points is employedwhenever a proposition is said to be proved; for principle what happensis,in all such cases, that the proposition is shown to be true some Another form in which the principle impliedby proposition. is constantly the employed is the substitution of a constant, satisfying in the consequent of a formal hypothesis, If "f"x implication. implies tyx for all values of #, and if a is a constant assert we can "/"#, satisfying droppingthe true hypothesis ijra, ever "j"a.This occurs, for example,whenof those of rules inference which the any employ hypothesis that the variables involved are propositions, are appliedto particular

37-39]

and Implication

Formal

Implication

35

The principle in question propositions. is,therefore, quitevital

to

any

kind of demonstration. The

of this principle is broughtout by a consideration independence Carroll's puzzle, What the Tortoise said to Achilles*." The of inference which we accepted lead to the proposition principles that,if and be then propositions, p togetherwith "p impliesq implies q p q. it might be thought that this would enable us to assert At firstsight, q is and But the true in that shows provided implies puzzle question p q. this is not the case, and that,until we have some shall new we principle, and more onlybe led into an endless regress of more complicated implications, the without ever at assertion of q. We need, in fact, arriving the notion of therefore, which is quitedifferent from the notion of implies, of Lewis

"

"

and holds between between and

"

a

different entities.

verb and

a

verbal noun,

.^s

In grammar,

between,say,

the distinction is that

greater than B" In the first of these,a proposition is "A

is

beinggreaterthan B" asserted,whereas in the second it is merely considered. But actually these are psychological terms, whereas the difference which I desire to logical.It is plainthat,if I may be allowed to express is genuinely the word the proposition assertion in a non-psychological use sense, though it does not assert p or q. "p impliesq asserts an implication, The p and the q which the enter into this proposition not strictly are the the i f which at least, they same as are separate propositions, p or q is: How The does differ by being true. a are question proposition true from what it would be as an entityif it were not true ? It actually alike are entities of a kind,but is plainthat true and false propositions that true propositions have a qualitynot belongingto false ones, a be called being sense, qualitywhich, in a non-psychological may in forming a consistent theory asserted.' Yet there are grave difficulties this point, for if assertion in any way no on changed a proposition, be unasserted could be in any context which can proposition possibly But true, since when asserted it would become a different proposition. false ; for in this is plainly q" p and q are not asserted,and p implies however, we must yet theymay be true. Leaving this puzzleto logic, "

"

insist that there is

a

difference of

some

kind between

an

asserted and

an

state a relation we therefore, thus which and which can only hold between asserted propositions, the hypothesis differs from implication.Wherever therefore occurs, to and the conclusion asserted by itself. This seems may be dropped, be the first step in answeringLewis CarrolPs puzzle. have inference must It is commonly said that an 39. premisses that two it is and more and a conclusion, or premisses held,apparently,

unasserted

are

f. proposition

necessary, if not

to

When

all

we

say

inferences, yet

out, at first sight, by obvious facts : every *

t

to

most.

This

view

is borne

for example,is held syllogism,

Mind, N. S. Vol. iv, p. 278. has a special symbol to denote assertion. Frege (loc.cit.)

The

36 to

have

of Mathematics Indefinables

[CHAP,m

premisses.Now such a theory greatlycomplicatesthe since it renders it a relation which may have any implication,

two

relation of

with respectto all but one of them, of terms, and is symmetrical This with respectto that one (theconclusion). but not symmetrical b ecause is,however, unnecessary, first, every simultaneous complication

number

is itself propositions

and single proposition, it is always because,by the rule which we called exportation, secondly, as to exhibit an holdingbetween single explicitly implication possible To take the first point first: if k be a class of propositions, propositions. asserted by the single of the class k are all the propositions if of x implies#, then x is a A: implies #, propositionfor all values x? or, in more ordinarylanguage, every Jc is true." And as regards of premisses the number to be finite, the second point,which assumes if q be a proposition, to "p impliesthat q r" is equivalent, pq implies hold explicitly between impliesr," in which latter form the implications Hence to be a we safelyhold implication singlepropositions. may number not a relation of an arbitrary relation between two propositions, to a single conclusion. of premisses which is a far more to formal implication, difficult 40. I come now In order to avoid the general notion notion than material implication. the of of propositional discussion a function,let us beginby particular mortal of #." is is values u# all for a a man instance,say impliesx is equivalent This proposition mortal all men is to are every man mortal11 and it seems is mortal.1' But highly doubtful "any man it is the same whether It is also connected with a purely proposition. intensional proposition in which man is asserted to be a complex notion of which mortal is a constituent, but this proposition is quitedistinct from the one we are Indeed,such intensional propositions discussing. not alwayspresentwhere one class is included in another : in general, are either class may be defined by various different predicates, and it is by that of the smaller class should no means necessary every predicate contain every predicate of the larger class as a factor. Indeed,it may are very well happen that both predicates philosophically simple: thus colour and existent yet the class of colours is appear to be both simple, assertion of

a

of

number

a

'

*

"

"

66

"

"

part of the

class of existents.

is in the predicates, and

main

I shall not

It may be is x a mortal implies 41.

terms,or onlyof

intensional view, derived from matics, SymbolicLogic and to Mathe-

consider it further at

present.

doubted, to begin with, whether "a? is a man is to be regardedas asserted strictly of all possible such terms as are men. Peano,thoughhe is not explicit, "

appears to hold the latter view. to be significant, and becomes a The

The

irrelevant to

"

But mere

in this case, the definition of x :

ceases hypothesis x

is to

mean

any

then becomes a mere hypothesis assertion concerning the of the meaning symbolar, and the whole of what is asserted concerning the matter dealt with by our symbol is put into the conclusion. The man.

39-41]

and Implication

Formal

Implication

37

The conclusion says : x is mortal, premisssays : x is to mean any man. But the implication is merelyconcerningthe symbolism: since any man is mortal, if x denotes any man, x is mortal. Thus formal implication, this view, has whollydisappeared, on leavingus the proposition "any man

is mortar1

as

with proposition

expressingthe whole variable.

It would

of what

is relevant

in

the

only remain to examine to explainthis proposition any possible the variable and formal implication. without reintroducing proposition be confessed that some It must avoided by this grave difficultiesare view. Consider,for example,the simultaneous assertion of all the class Jc: this is not expressed of some propositions by "c# is a i7 implies all values for For of x? it this does not express x as stands, proposition what is meant, since,if x be not a proposition, is a "" cannot x imply of x must be confined to propositions, the range of variability x ; hence unless we This prefix(as above, " 39) the hypothesis x impliesx? the remark applies to all generally, calculus, throughout propositional where the conclusion is represented cases by a singleletter: unless the the implication letter does actually asserted will representa proposition, since onlypropositions be false, be implied. The point is that, if x can for all values of x which be our itself is a proposition x variable, are for other This what the but not values. makes it plain propositions, within limitations are to which our variable is subject: it must vary only the range of values for which the two sides of the principal implication the is not other w hen in variable the two words, sides, are propositions, functions. If this replaced by a constant, must be genuinepropositional is not observed,fallaciesquickly restriction beginto appear. It should be which noticed that there may be any number of subordinate implications that their terms should be propositions : it is onlyof the do not require that this is required.Take, for example,the first principal implication of inference : If p implies principle g, then p implies q. This holds for if either is not a whether not ; or equally p and q be propositions to be a proposition, "p impliesq"" becomes false,but does not cease In fact,in virtue of the definition of a proposition, our proposition. states that "p implies function,I.e.that principle q" is a propositional But if we it is a propositionfor all values of p and q. apply the of obtain this to so as 'p implies proposition, principle importationto which is with formula have a only true q" we qj together p, implies in order to make it true universally, when p and q are propositions : we it by the hypothesis must q" In this preface "p implies p and q implies of the the restriction on the variability cases, if not in all, way, in many variable can be removed productof ; thus,in the assertion of the logical "if x implies ''x is a W the formula class of propositions, x^ then a tion. and allows x to vary without restricx* appears unobjectionable, implies and the conclusion in the premiss Here the subordinate implications is formal. material : onlythe principal implication are the

a

"

man

is

now

mortal," and

if

"

"

"

Returningnow restriction is

no

And to

x "

men

for #,

a

or

is

a? implies

is a

order to insure

our

man

a

in required

mortal,"it is plainthat havinga genuine proposition.

plainthat, althoughwe mightrestrict the values of in the proposition to be done althoughthis seems

mortal,"yet there is no

are

man

a

x

[CHAP,m

it is

is proposition

be

"

to

and

men,

all

a

of Mathematics Indefinables

The

38

concerned,why

not,

"

x

is

should

we

is

"

a

man

reason, so

far

so

restrict

when always,

a

as

the truth of

our

Whether

x.

our x

is substituted

constant

for implying, proposition

mortal."

unless

And

it is

we

" that value of #, the proposition is x admit the hypothesis equallyin the cases

shall find it

to deal satisfactorily with the impossible null-class or with null prepositional functions. We must, therefore, allow our truth the of formal is thereby wherever our implication #, unimpaired,to take all values without exception;and where any is required, restriction on the implication is not to be variability the formal until said has restriction been removed by being regardedas be whenever x satisfies""#, as prefixed hypothesis.(If-tyx a proposition where and if ^x, whenever it is a profunction, ""# is a prepositional position, then is not formal implies a "^ implies #r" implication, %#, but that -fyx "f"x implies %#" is a formal implication.) implies 42. It is to be observed that "x is a man impliesx is a mortal is not a relation of two propositional functions,but is itself a single function the prepositional having elegantproperty of being always For "x true. is a man" is, as it stands,not a proposition at all, and does not imply anything; and we must not first vary our in x is a man," and then "a independently vary it in "x is a mortal," for this would lead to the proposition that everythingis a man" is a implies everything mortal," which, though true, is not what was This proposition meant. would have to be expressed, if the language of variables were retained,by two variables, is a man as x implies is mortal." a But this formula too is unsatisfactory, for its natural y meaningwould be : "If anythingis a man, then everything is a mortal." The pointto be emphasizedis,of course, that our "r, though variable, be the same must both sides of the implication, on and this requires that we should not obtain our formal implication first by varying(say)

where

we false,

"

"

"

"

"

Socrates

in

"

Socrates

but

that

man

Socrates implies

as

a

Thus

class of

assert

towards

in

should

our

a

start

is a

man," and from

then

in

"

Socrates is

a

mortal,"

the whole

"Socrates is a proposition in this proposition asserts a class of implications, implication

mortal,"and vary Socrates

formal

at all. We singleimplication do not, in a word, have one plication ima but rather a variable implication. containing variable, We

a

have we

whole.

a

not

we

is

no implications, one

of which

contains

a

and variable,

that every member of this class is true. This is a firststep the analysis of the mathematical notion of the variable.

But, it may be asked,how the proposition Socrates is

conies

"

a

man

it that

Socrates

may be varied Socrates is mortal"? In implies

and Implication

41-44]

virtue of the fact that time

Formal

Implication

39

are propositions

impliedby all others, we but in this philosopher"; of Socrates is sadly restricted. This alas,the variability proposition, show that formal implication to seems involves something over and the relation of implication, above and that some additional relation have

is

"Socrates

hold where

must

a

impliesSocrates is a

man

term

be varied.

In

in question,it is the case natural to say that what is involved is the relation of inclusion l"etween the classes men and mortals the very relation which was to !"e defined and explained But this view is too simple by our formal implication. a

can

"

all

meet

is therefore not

and

A requiredin any case. of cases, though still not all cases, can be dealt with notion of what I shall call assertions. This notion must be now to

cases,

number

larger by the briefly

leavingits critical discussion to Chaptervn. explained, It

43.

and subject

has

always been customary to predicate ; but this division has

It is true

verb.

the

talk about thus

that

gracefulconcession

a

copula,but

to it.

We

paid

the verb

may

only one somethingwhich

and subject)

I shall call the

is

a

omittingthe

made

by loose respectthan is

more

verb,which

The

man.

is

remains with the assertion propositions, is neither true being robbed of its subject,

a

man""

is the but

;

but

mark distinguishing the

as

divided

be

may

assertion

false. In

nor

of assertion often occurs,

be

(the which something subject,

the

"Socrates

of

the notion

the defect of is sometimes

deserves far

is said about

Thus

assertion.

into Socrates and

propositionsinto

that every proposition say, broadly, may in several ways, into a term way, some

in

divided,some

divide

itself,

discussions, logical

the word

proposition

obtain separate consideration. Consider, for of indiscernibles : " If *r and^ example,the best statement of the identity

is used for it,it does not be any

diverse

two

assertion

some entities,

holds

of

which

x

does

not

be y? But for the word assertion^which would ordinarily would which this statement is one commonly replacedby proposition, said: be it "Socrates a was sopher, philomight pass unchallenged.Again,

hold

of

the

and

same

of Plato."

is true

Such

requirethe

statements

assertion and a subject, in order that into an proposition be said to be affirmed of be somethingidentical which can there may two subjects. into subject and We 44. now see how, where the analysis can there is a in which to distinguish assertion is legitimate, implications

of analysis

term

ways

which

of

a

making

decide between the two

others in which

be varied from

can

and the distinction may be suggested, said that It be there is a them. may

assertions "is

a

man""

the

whole

"Socrates. is proposition

Socrates

and

and

"is

holds,so does the other.

when

one

an

this is not the

a

relation

to

between

mortal,""in virtue of which, Or

again,we

Socrates implies

him, and

say that Neither of these theories

assertion about

questionholds of all terms.

man

a

we

Two

case.

shall have

analyzethe may is a mortal" into the

assertion

the replaces

in

above

Indefinables of Mathematics

The

40

is of x analysis implications ; but "

class of material

a

carries the the

from

analysis one

that difficulty

both

that

relation of assertions involved

the

essential to

into

is true

theory suffers

first

"

is a mortal

of the two

whichever

The

step further.

x implies

man

a

[CHAP,m

it is

assertions

it is otherwise

irrelevant though subject, second what on theory appears objectionable subjectwe of Socrates is the that a man the ground analysis implies suggested The in seems Socrates is a mortar' possible. scarcely proposition the terms being"Socrates consists of two terms and a relation, question of the

be made

should

same

The

choose.

"

is

and

man"

a

is

"Socrates

be

one

than

graver

at any therefore,

rate for the

a

e.g. "Socrates it is affirmed that

Here

This

after A?

Or better

another.

have

he must

relation between

the

Take propositions. father."

is

a

is married

because

this

is to

both

that

say

has

Socrates

take still,

"

A

the

"

subject

one

of

the notion

a

A

the

Such

instances

function,and prepositional

classes is relational

the

had

both

a

relation,

impliesB

assertions

the subjects;

have propositions

and BC"1

one

is before B

which, by the way, is quitedifferent from subjects, have

of

impliedSocrates

in which implication,

formal

This

assertions.

irreducible nature

at least) concerningdifferent (superficially

avoid

an

is asserted.

againstthe former view; I shall adopt the former view,and regard present,

from

results

insufficient. This

a

assertion,the

above that the relation of inclusion between

remarked

We

subjectand

that

from

as derived implication

a

that when

seem

of the relation which

of the terms

seems objection

formal

mortal"; and it would

analyzedinto

is relational proposition must subject

a

is

are

only

to

A

as

way and B

saying that they make

notion

of

it an

plainthat assertion,

the notion of dass" and that the latter is more are of formal implication. I shall not not adequateto explainall cases fundamental

than

be

point now, as it will subsequentportionsof the present work. this

enlargeupon

abundantlyillustrated

in

of importantto realize that, accordingto the above analysis notion the of is and indefinable ultimate. implication, every term which holds of every term, and therefore is one A formal implication of formal implication. be explained If a and b by means every cannot be classes, of "57 is an we can a explain every a is a 6" by means here is a derivative and impliesx is a 6"; but the every which occurs notion of t he It seems subsequentnotion,presupposing every term. It is

formal

"

to

be

formal

the

essence

of what

that reasoninggenerally,

every term truths are 45.

very

;

and

may some

unless the notion

be

called

formal truth, and

of

assertion is affirmed to hold

of

a

of every term

is

admitted, formal

impossible.

The

fundamental

importanceof

formal

is brought implication that is in the consideration involved it all the rules of inference. by This shows that we cannot hope whollyto define it in terms of material out

but implication,

that

some

further element

or

elements must

be involved.

We

according

point

in

which

it

is

is

which we

an

have

is,

and

dropped fact

that

must

be

and

often

it is

one

To we

the

a

address

we

of

the

question

formal

of

constituents

to

which

a

subjects, the

same

to

is

legitimate,

regard

a

Logic,

Book

perceive

to

it

implied

is

given

to

requires,

propositions.

To

its

n

a

this

theory

task

(p. 227).

as

raises

defence,

a

we

cerning con-

given

Where

possible,

This

cases,

made

another

imply

it, wherever

certain

a

simple

assertion,

subjects.

or

I, Chap,

of

is, in

fixed

concerned.

II, Part

implication,

formal

involved

for

and

:

ourselves.

*

all,

deduction;

implication

affirmed

subject

assertions

of

the

at

formal

perceive

a

be

that

case

that

rule.

may

introduced

any

as

implication

implications

agreed the

just

material

every

in

or

as

by

rule

the

of

rule the

dropping

the

if

here

process. a

by

of

by

that

by

implied

implication,

rule

endless

an

remains

guaranteed

present

the

inference. of

logical problems,

formidable

analysis

subject

between

relation

in

material

concerning holds,

implication some

of

said

it

respect

inference,

the

warranted

implication,

consequently

of

rules

propositions

certain

assertion

and

discussion

class

the

of

affirm

principle

But

is not

affirmation

all

of

class

and

easy,

the

our

the

is

said,

fixed

of up

sum

as

certain

connected a

affirm

rule

premiss.

a

that

to

into

on

non-formal a

rule

a

also

merely

not

asserted.

implication

the

more

or

class; and the

just

is

the

imply

perceived,

simply

immediately by

does

so

as

again

to

and

the

closely

being

need

implication

implies

implication

rule

our

of

then

and

and

it is

asserting

instance,

an

any

hold,

rule

our

the

shall

we

rule,

that

instance

premiss

inference,

apply

to

41

required

premiss, order

a

to

this

of

an

if

premiss:

true

the

rule

actually

simply

is

rule;

course,

does

inference This

of

In

the

instance

an

fact

from

go

can

we

The

of

is not

Bradley*;

true

a

have

to

particular

a

Mr

down.

necessary

instance

by

dropping

breaks

formalism

formally

case

of

principle

the

with

Implication

proceeds

emphasized

been

has

in

inference

the

which

to

Formal

that,

however,

observe,

may

This

and

Implication

45]

44,

formal due

to

many

thorough must

now

IV.

CHAPTER

belonging

what

to

in

of grammar,

my

opinion, is capable of throwing

grammatical

a

as

The

study light on

more

commonly supposed by philosophers.

distinction

be

cannot

assumed uncritically

to

a

evidence of

source

a

is

far

discussed

is prima difference,yet t)ie one genuine philosophical often he onost of the other, and usefullyemployed may

correspondto fade

questionsare to philosophicalgrammar.

called

be

may

questions than philosophical Although

be

present chapter,certain

the

IN

46.

VERBS.

AND

ADJECTIVES,

NAMES,

PROPER

be

it must

discovery. Moreover,

admitted, I think, that

a meaning perfectly occurringin a sentence must have some less fixed or meaninglesssound could not be employed in the more of our The which correctness sophical philolanguage employs words. way in analysisof a propositionmay therefore be usefullychecked by the exercise of assigning the meaning of each word in the sentence to to me seems expressingthe proposition. On the whole, grammar to a correct nearer bring us much logicthan the current opinions of and in what follows, grammar, philosophers master, ; though not our will yet be taken our as guide*.

word

every

Of

:

the

parts of speech,three

specially important: substantives,

are

and verbs. adjectives, substantives,some Among are from verbs, o r as human, or sequence adjectives humanity not speakingof an (I am etymologicalderivation,but of

Others, such

as

derivative,but obtain call

is

a

or

names,

proper

form

even

in

fact

is, as

a

substantives.

of words, classification, not

all or adjectives predicates in which

shall

we

notions

that

see,

but

which

would

grammar

time,

space,

primarily as

appear

and

human

of

derived from a

follows. logicalone.)

matter, What

ideas

;

from

are

not

wish

we

I shall

to

therefore

capable of being such,

are

call them

The

substantives.

and

humanity denote precisely being employed respectively according to .

the

same

concept,

the

kind

of relation in which

of

a

*

these

words

propositionin which 'Hie

excellence

of

it

grammar

this

concept stands The

occurs.

as

a

guide

r

inflexions,i.e.

to

the

degree

of

analysiseffected

to

the other

distinction

is

constituents

which

proportionalto by the language

we

the

require paucity

considered.

of

47]

46,

is not

be

identical

And

general

we

may

an

assertion

the

which

proposition,is "of

43

it is the

distinction

between

proper

between the dicated require, or objects inIn every in Chapter in, names. proposition, as we saw analysisinto something asserted and something about

names

make

adjective:

or

that

such

by

Verbs

substantive grammatical distinction between single concept may, according,to circumstances,

one

substantive

either

Adjectives, and

the

with

since adjective,

And

Names,

Proper

we

is

made.

always,

A

least

at

there

analysis(where

rather

when

name,

proper

according

to

of

one

it the

in

occurs

a

possibleways

several),the

subject that the proposition is or some proposition is about, and not what the said about the other subject. Adjectives and verbs, on hand, be are capable of occurring in propositions in which they cannot regarded as subject, but only as parts of the assertion. Adjectives I intend which Are distinguished by capacity for denoting- a term in technical be discussed in to sense to Verbs a use v. Chapter kind of are connection, exceedingly hard distinguishedby a special and of truth which to tinguish falsehood, in virtue define, with they disasserted from unasserted an an proposition one, e.g. "Caesar subordinate

are

constituent

"

"the

from

died"

of

I shall

amplified,and

with

begin

is familiar

Philosophy

47.

less

or

I wish

"what*.

I

equivalent:

substance

and

point

to

now

cognate

issues

between

monism

^empiricism,and

between

those

the

subject

with

is concerned

.all truth

theory

the

of

or

Whatever

false

widest

from

A

a

man,

-else that

the

can

it is essential

counted

can

it

the

words

fact

that

a

number,

one,

has

being,

sure

to

be

a

term

not

the of

general

on

in

i.e. is in

is

sense.

some or

use

The

third

the

deny

to

This, shall

I

entity.

while

true

any

term.

a

and

; and

part,

But

doctrine

occur

class,a relation,a chimaera,

a

and that

of account.

call

is one,

term

every

in

or

any

out

unit, individual, term

every

is

as

I

the

deny

whole

philosophicalvocabulary.

the

in

that

who

to

be

important,

bearings

Its

must always be false. thing is a term be thought that a word of might perhaps

such

those in

predicate, and

Idealism

present question.

variable.

the

be

mentioned,

be

between

wholly they are, of or thought, object may

fact

moment,

regard

the

to

me

subject is

and

maintain

to

to

appears The

will be left

an

the

emphasize

derived

"could

general

subject and adjective, this

exists,all depend,

word

with

synonymous

first two

It

and

of

and

monadism,

who

of

as

proposition,or

is the

in

nature

be

may

and

only because

here

philosophy,important

And

be

now

distinctions, all

of

distinctions

distinctions.

what

adopt

we

is treated

number

AS

must

between

set

brieflywhat

out

the

certain

a

substantive

attribute,

since

then,

distinctions

distinction

the

mean,

concerning these

or

These

the

with

"truth

upon

^

Caesar.

names.

proper

more

death

anything that

such

a

be

of

any *

This

great last

use.

pair of

Such terms

a

such

extreme

view, however,

is due

to

Mr

owing

Bradley.

generality to

certain

of Mathematics Indefinables

The

44

[CHAP*iv

A term doctrines,would be erroneous. is,, wide-spread philosophical of all the properties in fact,possessed commonly assignedto substances substantives. Every term, to begin with, is a logical : it is, subject or is itself that one. for example,the subjectof the proposition Again

and indestructible. What is immutable change can be conceived in it which would not term

every

is,it is,and

term

a

destroyits identity Another mark which term*. it another and make belongsto termswith themselves and numerical diversity from all is numerical identity the source and diversity of unity are other terms f. Numerical identity of and the terms thus admission and plurality monism. destroys ; many And undeniable that every constituent of every proposition it seems can contains than less two be counted as one, and that no proposition no

is,therefore,a useful word, since it marks dissent well as because, in many various philosophies, as statements, we Term

constituents. from

speakof

wish to 48.

term

any

or

Among

term.

some

terms, it is

kinds, which

two to distinguish possible

thingsand concepts. The former are the respectively by proper names, the latter those indicated by all other

I shall call

indicated Here is

proper

names

usual, and

to be understood

are

thingsalso

a

wider

somewhat

understood

be

to

are

in

as

terms-

words. than

sense

embracingall particular

commonly called be distinguished^ two kinds at least must things.Among concepts, again, The and those indicated by verbs. namely those indicated by adjectives former kind will often be called predicates latter or ; the class-concepts notion are alwaysor almost alwaysrelations. (In intransitive verbs,the asserts a definite relation expressed by the verb is complex,and usually and many pointsand instants,

to

indefinite relatum,as in

an

In more

The

"

other entities not

Smith

breathes."")

in one or we largeclass of propositions, agreed,it is possible, a subjectand an assertion about the subject. ways, to distinguish a

assertion

assertions

always contain

must

except in this respect,,

verb, but

a

properties.In a relational proposition, say "A is greaterthan B? we may regardA as the subject,, and is greaterthan B" as the assertion, A is B as the subject and or greater than" as the assertion. There are thus,in the case proposed,, of analyzingthe proposition two assertion. into subjectand ways have

to

appear

no

universal

"

"

Where

a

relation has

there will be

than

more

two

there propositions,

some

*

The

notion of from

notion a

which

t On

of

a

than

more

term

is

here

two

ways

only a set

conceptin his article "On Mr

1900-1901. Society, J This propositionmeans

Part VII that the relation

"

A

making singleway :

forth

G. E. Moore's

as

of

is

a

the Nature

notion,however,it differsin

see identity,

terms,

some

in "A the

analysis.But

these

modification

now^,"

is here

the

are

of Mr

of

G.

Judgment/' Mind, importantrespects.

in

subjectE.

N.

Moore's

S. No.

30,

article in the Proceedingsof the Aristotelian

is in this

expressedis not

place at

this time."

reducible to

a

two-term

It will be

shown

relation.

in

and Proper Names, Adjectives,

47-49]

such propositions, predicate "

humanity

human," is In

belongsto "

Socrates

about

is

a

former, the

humanity is

proposition those

in

notion

which

terms, however

proposition

to equivalent

it is

a

which

is

distinct proposition. human

when

occurs

numerous,

Socrates

"

expressed by it

45

occurs

it is called

but

in the

not

indicates

that

the terms in

occur

a

of

a

sition propo-

be

about which the proposition is. regardedas subjects of characteristic the of a proposition terms that any one of a be other replacedby any may entitywithout our ceasingto have and may

It is

that

The

being that in the latter case, This propositionis about this notion. not I a shall a thing. concept, speak of

as

them

is

humanity; but

human," the

different way from humanity,the difference

in

Socrates is human."

Socrates,"which

assertion

an

"

as

Verbs

Thus shall say that "Socrates is human" we is a proposition. having only one term; of the remaining components of proposition is the verb, the other is a predicate. one the proposition, With the sense

a

which is has

in this

we no at all proposition, longerhave a proposition if we by somethingother than a predicate.Predicates, other than verbs,which occur in propositions then, are concepts, having Socrates term is because Socrates or one a subject. can only thing, otherwise than as term in a proposition Soerates is not occur never : capableof that curious twofold use which is involved in human and states of mind, humanity. Points, instants,bits of matter, particular existents the above in and particular are generally, things sense, and which do not exist,for example,the pointsin a terms are so many of a novel. All classes, non-Euclidean space and the pseudo-existents it would seem, as numbers, men, terms, spaces, etc.,when taken as single are things; but this is a pointfor Chaptervi. from other terms Predicates are distinguished by a number of very which chief among is their connection with what properties, interesting I shall call denoting.One predicatealwaysgivesrise to a host of and humanity,which cognate notions : thus in addition to human have we some a man, man, man, only differ grammatically, any man, of be all which d istinct all to men*, one genuinely appear every man, of various notions vital these is The from another. study absolutely it is of mathematics and of them that account to any philosophy on ; is important. the theoryof predicates It might be thought that a distinction ought to be made 49.

human replace

between

a

concept

is and

such

pairsas "this as proposition e.g.,

I

use

all men

as

a

concept used

being,human

is one"

difficultieswill *

and

such

as

and'l

envelopus

in "1 if

we

men

are

mortal."

as

a

term,

humanity, is

a

number."

allow such

I shall Thus

with

a

But

the human

I shall say

all ft

a

tricable inex-

There

view.

alwaysuse

between, in such

one

i.e. as nearlysynonymous collective,

therefrom by beingmany and not one. differing to m yself confining every for the distributive sense.

mortal,"not "all

and

is,

race, but

collectively, every

man

is

of Mathematics Indefinables

The

46 of course,

and grammaticaldifference,

a

as

regardsrelations.

as

a

relate two

itself said

is

more

to

have

while

;

in

or predicate

a

I wish

and

to

not

urge in the

has adjective

1, in which

the

is

the concept is

case,

There

one

not

a

yet

must propositions

makes adjective

as

In

is, therefore* what

and

1

term

while

addition

in

one

this statement, one as either it has become

In

the

to

the second

be false,since

a

fact

denotes

there hypothesis,

and subject,

the

one

that

suppose

there self-contradictory ; or

in this latter

But

For

is that

to one

the

sitions propo-

maintain as

term

about proposition

is therefore

some

concept

a

be

must

concerningone as term, and we shall still have as concerningone as adjective propositions opposed to all such

to

hence

term;

a

is supposition

concept

term.

a

asserted

or

grammaticaldifference. But

term.

as

into

other difference between which

1

from

made

been

a

second

relation.

of the terms.

intrinsic nature

case

firstdenotes

term

a

is,that the difference lies solelyin external relations,

adjectivediffered

as

difference

a

concept in questionis used

the

a

in accounting for the difficulty

no

to corresponds

of predicated actually

terms

or

first case, the

In the

concept,that is,it

this

[CHAP,iv

reallyabout

;

one one

which could not be adjectives of substantives without change meaning, all propositions such adjectives them into turn (sincethey would necessarily concerning and so would would be false, the proposition that all substantives) such propositions into false,since this itself turns the adjectives are substantives. But this state of thingsis self-contradictory. The above argument proves that we were rightin sayingthat terms embrace with the possible in a proposition, that can occur everything exceptionof complexesof terms -of the kind denoted by any and cognate in a proposition, For if A words*. occurs then, in this statement, have A is the subject we justseen that, if A is ever not the ; and it is exactly A which is not subject the same and numerically subject, and is subjectin another. Thus the theory that in one proposition attributes or ideal things, there are adjectives whatever they may or or term.

as

made

short, if there

called,which

be

in

are

some

less self-identical, than true and

to

reduced easily

be

differ from

those

in

occur

a

50. which

fact

Two

be may fact that two

terms, even

are

any

less self-subsistent, substantial, substantives, appears to be whollyerroneous, Terms contradiction. which are concepts a but not, not in respectof self-subsistence, way

lass

that, in certain

true

is different in

or subjects

terms

them

as

false

or

of relations

terms, another

they propositions,

indefinable way

an

concepts have, iijaddition

belongsto

which

to

which

manner

in which

manner

which

of the

in virtue

as

were

into

from

the

occur.

the

to

numerical

diversity diversity characterized by the

specialkind

called

of

conceptual. This may be in which the conceptsoccur otherwise than propositions all other respects, the two propositions if,in are identical, *

See the next

chapter.

Proper Names,

49-52] yet differ

in

virtue

of

the

and Adjectives,

fact that

the

Verbs

47

concepts which

in

occur

diverse. Conceptualdiversity conceptually impliesnumerical but the converse does not hold, since not all diversity, implication its name are terms is the as concepts. Numerical diversity, implies, and of is less important to source plurality, conceptual diversity But the whole possibility mathematics. of making different assertions set of terms depends about a given term or diversity, upon conceptual in generallogic. which is therefore fundamental 51. It is interesting and not unimportantto examine very briefly the connection of the above doctrine of adjectives with certain traditional the nature views on of propositions. It is customary to regard all and as a as having a subject i.e. predicate, propositions having an and a general immediate this, it attached to of concept by way description. of the theoryin question This is,of course, an account which will strike indication its adherents as extremely crude; but it will serve for a. general view to be discussed. This doctrine develops of the by internal logical of Mr into the all that words stand for theory Bradley's Logic, necessity ideas having what he calls meaning,and that in every judgment there of the judgment,which is not an idea is a something, the true subject is a and does not have meaning. To have meaning,it seems to me, of and psychological notion elements. confusedlycompounded logical that they are all have meaning, in the simplesense Words symbols But a proposition, stand for somethingother than themselves. which it does not itself contain words: unless it happens to be linguistic, Thus contains the entities indicated by words. meaning, in the sense is irrelevant to logic. But such concepts in which words have meaning, have meaning in another sense to speak, : they are, so a man as symbolic have the because I call which in their own nature, logical they property in a proposition occurs (e.g. denoting.That is to say, when a man is not about the concept in the street "),the proposition "I met a man actual biped denoted but about something quitedifferent, some a man, by the concept. Thus conceptsof this kind have meaning in a nonthis is a man," in this sense, when And we sense. psychological say in which a concept is in some sense we are making a proposition

them

are

"

attached

to

what

is not

a

concept. But

indicated by John entity

the

contends*

;

and

even

among

when

does not have

concepts,it is

meaning is meaning,as

stood, thus underMr

Bradley

onlythose that denote

that

largelydue, I believe,to the notion in propositions, which in turn is due to the notion that that words occur mental and are to be identifiedwith cognitions. are essentially propositions be pursuedno further in must But these topicsof generalphilosophy have

The

meaning.

confusion is

this work. 52. it is

It remains

to

from distinguished *

discuss the

verb, and

In the adjective.

Logic,Book

to

find marks

regardto

verbs

I,Chap, i, "" 17, 18 (pp.58-60).

by

which

also,there

is

to corresponding

grammaticalform

twofold

a

of Mathematics Indefinables

The

48

(the various inflexions

difference in

a

of this form

of

be left out

may

account),and

"Felton

between

is that

distinction The present participle. Buckingham"and "Killingno murder."

verb

as

(in English)the

indicated by the infinitive or

there is the verbal noun,

merely

it has

which

is the verb in the form

external relations. There

[CHAP,iv

killed

By analyzingthis difference,

and function of the verb will appear. to beginwith,that the conceptwhich It is plain,

the nature

is the very

noun

that which

as

same

verb.

as

occurs

in the verbal

occurs

This

results from

of every the previous argument,that every constituent of beingmade be capable of self-contradiction,

must, proposition a logical subject. have already to kill" we the concept expressed by

pain

on

If

"killsdoes

"

say Utts

we

made

verb

as

also

as

say that subject.Thus

a

as

very verb which

the

is : What question

subject.The

grammaticalform

the difference of

expressed by

difference is

same

cannot

we

be made

occur

can

the

mean

and subject,

a

ktib cannot

the word occurs

not

logical And

?

it

plainthat the difference must be one in external relations. But the verb, in regardto verbs,there is a further point. By transforming into a verbal noun, the whole proposition in a proposition, it occurs as no longerasserted,and no subject, logical be turned into a single can in itselftruth or falsehood. But here too, there seems longercontaining which results of maintainingthat the logical subject to be no possibility

is

is

different

a

is asserted." In that case,

Caesar

is true

which

subject. The logical

mere

has

Caesar

falsehood

difficultto

substitute

I do

a

the

verbal

subjectof

some

"the

assert

not

that

(as the its

the truth

the

to have

appears be inherent not

an

to be appears which is lost as verb,

the

here become

in the very nature know how to deal

be made

cannot

This

of truth and

is

of Caesar's

ultimate

an soon

as

we

depend

does not

proposition," is

presentin

the contradiction which

inevitable.

may

truth

own

propositionin question

This proposition.

entitywhich

case

of

at any rate, is latter,

died is a for if I say "Caesar element which Caesar did die,and an

avoided,of

of

it analysis,

correct "

There

Thus died" has disappeared.

have been

death

falsehood of Caesar's death"

is lost when

other

"the

be that the death

form; grammatical

"Caesar

I do

and

is asserted

"the

other contains

quite plainthat died."

"Caesar

noun,

or

What be

falsehood

differs from

*

assertion, given by the

of

is made

it is

Yet

case.

to

seems or

and

falsity belongsto

nor

if this is the

it is true,

where

to equivalent

notion

upon

case

died

Caesar

"

must

seem,

or

way

some

But

element.

an

how

see

in the other never

as

in the

death"

here

answer

:

died"

it is the death of Caesar

neither truth

yet

died" in

Caesar

"

ask

we

answer

it would

external relation to truth

an

be),whereas or

false;and

or

the

died"?

"Caesar proposition

in the

"Caesar proposition.

will illustratethis point. If

death of Caesar"

a

the

entityfrom

a

falsehood,is one most

to

logical subject,

which difficulty,

The satisfactorily.

was

seems

with

obvious

to

which course

and Proper Names, Adjectives,

52-54]

Verbs

49

be to say that the difference between an asserted and an is not but psychological. In the sense logical, proposition would

unasserted which

in

be asserted, this is doubtless true. But there of is another sense assertion,very difficultto bringclearly before the and in mind, yet quite undeniable, which only true propositionsare and false propositions asserted. True alike are in some sense entities,,

propositions may

false

and

in

capableof being logicalsubjects;but when and a happens to be true, it has a farther quality, proposition over which it shares with false propositions, above that and it is this further is which I what mean quality as by assertion in a logical opposed to The of nature sense. a psychological truth,however, belongsno more of mathematics of everything to the principles than to the principles else. I therefore leave this questionto the logicians with the above are

some

sense

brief indication of 53. we

plainthat, if

difficulty.

be asked whether

It may

concerned

are

a

with, is

we

rightin

were

proposition having only express

relation in

a

verb,expresses

a

a

holdingthat

the

ordinarysense. distinguished by just this

are propositions a Nevertheless,

relation

relation

between

Socrates

or

sense logical

It

not.

seems *

is human

Socrates

"

term-,the is in this

one

the

everythingthat,in

is

a

cannot proposition

fact, subject-predicate

In

non-relational

and

humanity

character.

is

certainly

and it is very difficultto conceive the proposition as implied, expressing is relation at all. We it that no a relation, although perhapssay may

from other relations in that it does not permititself distinguished to be regardedas an assertion concerningeither of its terms indifferently, the referent. A similar remark may but onlyas an assertion concerning without "A which holds of every term is,*" apply to the proposition exception.The is here is quitedifferent from the is in "Socrates is be regardedas complex,and as reallypredicating human Y"; it may verb in a proposition Being of A. In this way, the true logical may be know hard relation. But it is to so always regardedas assertinga exactlywhat is meant by relation that the whole questionis in danger of becomingpurelyverbal. it is

54. noun,

if all expressed,

be

may

difference between

a

constituents of this B. difference,

reconstitute

a

be said that

notion we

when

we

say

"A

which

ought,in

difference has to A

to

be

relation

verbal

as

the as relations,

relating. actually

differs from

J?."

The

analyzeit,appear to be onlyA, do not thus placedside by side, constituents,

proposition actuallyrelates is analysis

a

"A proposition

The proposition.

the

are

held

if we proposition, these

Yet

verbs

relation in itself and

example,the

Consider, for

verb,as actual verb and

of the

twofold nature

The

has the

and

A no

difference which

B, whereas

connection

to analysis,

with

mention

the A the

occurs

in

the

difference after and

B.

It may

relations which

and B, relations which are expressed by is and from relations consist in the These is different from B?

of Mathematics Indefinable^

The

50

is referent and

A

fact that

with

relatum

B

[CHAP,iv

respectto difference. But

list of terms, not in fact, is essentially a unity,and when proposition, is still merelya

relatum, B" difference, "A, referent,

A proposition. of constituents enumeration the unity,no will has destroyed "analysis when used The verb, embodies the as a verb, restore the proposition. from the verb conand is thus of the distinguishable sidered proposition, unity clear how know I not to do of account a give as a term, though of the distinction. the precise nature whether the generalconcept difference be doubted It may 55. a

"A proposition

all in the

at

occurs

differsfrom B?

whether

or

there

is

difference of A and j#,and another specific difference not rather a specific affirmed in "A differs from B and of C and D, which are respectively "

which

there

and

as

are

many

instances

the

this way, instances

In

D."

C differsfrom

"

there

as

of class-concept

a

pairsof different terms

are

;

be said,in Platonic

may

this

of difference. As

nature

becomes difference

phrase,to partakeof the quite vital in the theory of

point is

point out

first of all,I must well to dwell upon it. And I intend to consider the bare differs from B" that in "A

numerical

difference in virtue of which

it may relations,

this

that

or

be

first try the

us

difference from particular

a

the no

difference in

two, not

respect.

that hypothesis with compoundedof differencetogether Let

they are

every

of difference itself is

relation

distinction

be

can

made

concerned,we

between

in qualities

different associated

difference is

complex notion, some special quality distinguishing other particular difference. So far as a

to

are

different cases

different

a

;

but there

But

cases.

that

suppose

since

to be

are cases

are

associated by their terms, the qualitymust be primarily distinguished be not a relation, with the terms, not with difference. If the quality it connection with the difference of A special from bare difference, render distinguishable and

can

have

was

to

irrelevant.

it becomes between two

and 5,

A

the

On and

over

have

terms

any latter not

two

of two

other

hand,

difference, we difference and relations, any

other

others,of which

pair of the

generalrelation of difference itself holds holds that

when

two

held with either the denial is to

be^said for and

Against the we

processes

relation

new

to hold that

the difference, specific

This

terms.

view

is

a

first holds

that

the

between

and

B, while the

A

abstract

differ they have, corresponding to

notion

or

the affirmation of the

are

led into

will

of

it may differences, specific

see

each

endless process.

an

in this

a

other.

Let

us

see

them. against

their differences from differencesdiffer, thus

it

relation of difference, and unique and unanalyzable specific Either of these views by any other pairof terms. may be

this fact,a not shared what

terms

a

shall have a

J?,which

if it fails in this

if it be

above

holdingbetween

combination second

and

no

be

other must

Those

who

proofthat differencesdo

urged that, if and differ,

also

objectto not

endless

differ. But

in

55]

54,

and Proper Names, Adjectives,

presentwork, it will be maintained that there

the

the notion

of

Verbs are

no

51

contradictions

endless process is not to peculiar an to unless it arises in the analysis be objected of the actual meaning of a In the present case, the of implications, not proposition. process is one to

analysis ; it must Against the notion of

one

and

A

and infinity,

therefore be that

that

regardedas

harmless.

the abstract relation of difference holds

the argument derived from the analysis 5, we of " A differs from B? which gave rise to the present discussion. It is to be observed that the hypothesis which combines the generaland the

between

have

difference must specific the

suppose that there are two distinct propositions, the general, the other ,the specific difference. Thus affirming if

one

there

be

cannot

generaldifference between A and B, this mediating impossible.And we saw that the attempt to avoid in the meaning of A differsfrom B^ analysis by including a

is also hypothesis the failure of

"

the relations of difference to A

and

B

vain.

was

This

attempt,in fact,

endless process of the inadmissible kind ; for we include the relations of the said relations to A and B and

leads to

an

shall have to and difference,

in this

and

continually increasing complexitywe are supposed the of This our only analyzing original proposition. meaning establishes of a point namely,that argument very great importance, so

on,

to

be

when

relation holds between

a

the terms, and

two

terms, the relations of the relation

of these relations to the relation and

the terms, and

to so

the though all impliedby the proposition affirming infinitum, of of the this relation,form no part original proposition. meaning ad

on

of A

argument does

the above

But to

B

be

cannot

suffice to prove that the relation abstract difference ; it remains tenable that, as not

suggestedto begin with, the true solution lies in regardingevery cannot as having a kind of unity which analysis proposition preserve, element and which is lost even as an though it be mentioned by analysis has but the This view doubtless its own in the proposition. difficulties, was

view

that

no

two

pairsof

terms

have

can

the

relation both contains

for the difficulty

and fails to solve the

difficulties of its own

same

sake of which

if the difference of A and B be absolutely For, even to A and B, stillthe three terms A, B" difference of A from J?, peculiar it

invented.

was

do not A

and B

did

reconstitute the

and difference did.

differ, they

the most

"J proposition

would

general way

And

it

stillhave

to

in which two

seems

differsfrom

more

than

if differences

plainthat,even somethingin common. have somethingin common can Hence if no giventerm.

have

terms

B? any

But

two by both having a givenrelation to a it follows that no two terms relation, pairsof terms can have Ithe same and hence different differences will not have anythingin common, can

is

be in any definable *

with

The

above

sense

instances of difference*.

argument appears

numerically diverse

instances

to

I

conclude,then, that

of universals prove that Mr Moore's theory in his paper on Identity(Proceeding*of the

B"

from

the

same

from

D"

numerically differs

C

"

be

reasons,

but

strictly

are

We verb.

the

it

though only,

must

logical

occurs

in

are

which

the

arising

this

subject

in

which

instance all

cases

it

has

its

where

lead

sketch

the

at

of

of

have

and

sight,

to

in

classes

at

be

such

somewhat

way every of

sum

which,

its

in

a

discussed. verbs

and

discussions

to

Chapter

to

vn

connected

are

verbs.

involve deal

the

to

of

nature

speaking,

functions to

instances,

the

chapters,

adjectives,

necessary first

independently

thoroughly

the

two

next

of

it

from

and

fully

regarded when

logical problems,

to

be

be

proposition,

a

distinct

single

a

but

Owing of

one

prepositional

relates,

adjectives,

terms

it

into

may

occurrence.

the

Broadly

been

like

not,

to

in

actually

and

proposition,

a

considered

their

general

seem,

it

may

is

for

with

a

It

length

from

remote

mathematics.

Society, to

of

prepositional

might of

principles

Aristotelian an

that

reason

points

verbs.

with

sense

renders

consideration

while

future

relates

proceed,

the

adjectives,

with

the

of

cases

a

in

do

deserve

given

call

word,

of

proposition,

every

shall

the

same

instances,

verb,

changed

relation

of

which

would

shall

bare

the

be

verb,

as

Verbs

these

now

I

I

in

discussion

One

but

;

D

the

adjective,

of

can

noun,

logical

is the

the

unity

a

connected

topics

the

actually

logic,

out

all

verb

All

adjectives,

every

occurs

it

proposition

and

occur.

our

the

terms

logical subject.

a

which

in it

in

constituents.

Having

kind

one

verbal

it relates.

has

on

a

noun

which identical

treatise

into

when

verbal

proposition

in

the

of

they in

differs

and

have

not

like

which,

concept

into

verb

a

elicited

points

being

made

verb,

Every

terms

but

of

main a

for

held,

which

iy

precisely

be do

in

is

C

relations

propositions

is

saw,

without

verb

its

subject,

as

the

as

relation;

a

we

be

also

occur

concept. as

verb,

may

turning

by

up

proposition

a

the

and

"A

between

must

relations;

all

in

proposition

affirmed

doctrine

this

other

same

sum

The

in

occur

the

now

may

all

of

true

to

And

the

[CHAP.

difference,

relation

the

as

in

E

of

relation

general

the

is

and

A

between

affirmed

relation

the

Mathematics

of

Indefinables

The

52

1900

"

universal, it

occurs.

1901) at

must

any

not

be

rate,, must

applied be

to

all

actually

concepts. and

The

numerically

relation the

of same

CHAPTER

V.

DENOTING.

THE

56.

obscured

been a

But

discuss. the

denoting,like by

undue

an

denote,when

we

for

symbols

of

hitherto

in which

sense

as

notion

fact that

of

employment to a logical relation of which

virtue

It is this

bottom

the

at

of

sense

out itself,

is not

of

which

the

is here

to

and

terms, in

some

such

This

terms.

lies

notion

question.

of

substance, of the

subject-

and

things

ideas,

ments, develop-

various

fact

mistaken, while the fundamental

me

hardly

is

grown,

discussed

ever

to

concept

a

in

These

words

able, by

are

is not

denote logically

perception.

have

we

which

is

I wish

that

sense

opposition between

immediate

they

the

concepts

some

all theories

of

main, appear

of

between

denotingwhich

predicatelogic,and discursive thought and in the

point or describe,or employ

we

and concepts inherently

(I think)

psychology. There

designatea thing

concepts,to

such

of

logic,has

of

notions

is possible"that description

is due

"

admixture

concepts ; this,however,

the

of the

most

in

its

logicalpurity. ,

concept denotes when,

A

ahoyt the

is not

with

peculiarway is not but

about

about

the

If I say

odd

which

not

almost

but

or

what

"Times"

nor

wife.

and

walk

not

a

tailor and

Again,

the

a

could

the

we

streets,

I met

bank-account

"any proposition

plainly,true ; yet the concept numbers It is only particular even. or

addition

even,

be

not

and

if there

Of the

even.

were,

it is

any

is mortal such

a

People

commas.

will notice

die, and as

the

yet

we

often

assert

should

be

following:"Died

finite finite

that

plainthat

are

it

concept any number," "

propositionsthat contain the phrase "any number wish to speak of the concept,we have to indicate the

inverted

or

entity, any

these, another

to

a

was

"

is not, in

is either odd

be odd

with

man

does

logic-books.What

the

certain

a

man," the proposition

a

is

even

odd

I met

"

in

connected

term

a

all the

false. If italics

or

of

actual

drunken

the proposition proposition,

a

concept which

a

limbo

"

there

even;

number, could

a

is neither

or

is

shadowy

is odd

number11

this

:

public-houseand

number

concept.

concept, an

a

in

occurs

concept, but

man

lives in the

thing,not a

a

if it

that

is

man

to surprised at

his

"

axe

fact

by

mortal;

find in the

residence

of

of Indefinables

The

54

Mathematics

[CHAP,v "

Camelot, Gladstone Road, Upper Tooting,on the 18th of June 19 , Man, eldest son of Death and Sin." Man, in fact,does not die ; hence about is mortal if " man man, were, as it appears to be, a proposition the is The fact about false. be is, men it would proposition simply ; "

"

here

and

The

conceptdenotes. of

of the

symbolism,and

denoting.

The

concept men, but about what this of identity, of classes, whole theory of definition,

notion

it is of its difficulties, The

57.

about

is not

again,it

is

variable

is

wrapped

fundamental

a

denoting may

notion

be

quiteessential to

of

notion

the

clear

as

be obtained

in

up of

the

theory of

and, in spite logic, about it as possible. by a kind of logical

from subject-predicate more or propositions, genesis upon which it seems those in which of propositions less dependent. The are one simplest and there than is otherwise as a term, occurs onlyone term of predicate is asserted. Such which the predicate in question be propositions may Instances A is,A is one, called subject-predicate are : propositions. A is human. might also be called classConceptswhich are predicates shall find it necessary because they give rise to classes, but we concepts, and the between words to distinguish predicate class-concept. Propositions and of the subject-predicate are impliedby other propotype alwaysimply sitions of the type which asserts that an individual belongsto a class. A is a unit, Thus the above instances are equivalent to : A is an entity, These new A is a man. not identical with the previous are propositions since To beginwith,?" is now different form. theyhave an entirely ones, the onlyconcept not used as a term. is neither shall find, A man, we a

concept nor

namely a

man

a

term, but

of those

which

a

certain kind

And

human.

are

the

is quitedifferent from his relation to "

is human

of certain terms,

of combination relation

of Socrates

to

humanity;indeed "Socrates

is cdrrect,to be not, in the judgment of relation between Socrates and humanity,

be

must

held,if

usual sense, a since this view would most

make

the above

human

occur

view

as

term

in "Socrates

is human."

It is,of course, undeniable that a relation to humanity is impliedby ** Socrates is human," namely the relation expressed by " Socrates has

humanity"; and proposition.But it is importantto

this relation

conversely impliesthe subject-predicat the two propositions and be clearly can distinguished, the theoryof classesthat this should be done. Thus in the of we have, case three types of propositions every predicate, which implyone Socrates has another,namely, Socrates* is human," humanity,"and "Socrates is a*man." The first contains a term and the second two terms and a relation (the second term being a predicate, identical with the predicateof the first while the third proposition)*, contains a term, a relation, and what I shall call a disjunction (a term which will be explained The if at differs little, shortly)f. class-concept "

*

Of.

" 4i).

t There fc

Socrates

"

are

two

is a-man"

allied propositionsexpressed by the "Socrates is-a man." The above

ami

same

words, namely

remarks

apply to

the

56-58]

or

sum

while the class, predicate, as opposed to of all the terms which have conjunction

relation which

The

55

the

all,from the

Denoting

in the second

occurs

the the

is class-concept,

given predicate.

has humanity)is type (Socrates

characterized completely by the fact that "itimpliesand

is

impliedby

with onlyone term, in which the other term of proposition has become a predicate.A class is a certain combination is akin to a predicate, and the terms class-concept closely forms the class are

determined

by the

certain sense, the simplest are, type of the simplest type of proposition.

in

58.

There

is,connected

with

of

terms, a

whose

bination com-

Predicates class-concept.

since they occur concepts,

a

a

the relation

in

a predicate, great variety of far as so it is they are distinct, for distinguish. Starting, example,with hinnnn^ we have

every

closelyallied concepts, which, in importantto

all men, the first are

men,

man,

except

man,

the

wan,

any

human

race, of which

twofold,a denotingconceptand

the notions also,less closely analogous,

have

we

every

an

and

man"

"a

all

objectdenoted

;

"some

man," which

again denote objects*other than themselves. This vast with every predicate be borne in mind, and muht be made endeavour must to o f an givean analysis all the above notions. for the present,it is the property of denoting, But rather than the various denotingconcepts that we are concerned with. apparatus connected

The

combination

complexitythan

of

their

is constituents,

said many things. But be what by analogymay

form

of greater concepts,

new

subjectupon

a

the

logichave to

to form

conceptsas* such

called

which

mathematics, since the upon

nature

justthis point. Six words,

are

any, a, same these words

should

these

:

as

on

such,

the

philosophyof in

occurrence are

the

words

daily life, aR, every,

it is essential that reasoning, from another; but one sharplydistinguished and is almost whollyneglected difficulties, by

For

the.

and

of constant

of mathematics

also characteristic

writers

and of the variable turns

of number

both

of terms

complex terms, is a subject onlythe scantiest discussion.

old and new, giveus logicians, Nevertheless,the subjectis of vital importanceto

upon

which

combination

be

subjectbristleswith -f\ logicians It is plain, to beginwith, that

the

of

correctness

a

one phrasecontaining

of the above

former ; but in future, unless the contrary is indicated by a hyphen*i|rotherwise, of Socrates the latter will always be in question. The former expresses the identity with an ambiguous individual ; the latter expresses a relation of Socrates to the tntm. class-concept *

and

I shall

be framed t On

with the

"Abstrahiren

the

use

and plural,

also

word cases

in object

of

a

wider

as

Jw"th singular term, to cover The fact that a word can man/*

than

sense

ambiguity,such

"a

problems. C-f. " 47. meaning than term raises grave logical made by Meinong, are indefinite article,some good remarks Mid Phyxiotogieder und fur P"yMogie Vergleichen/'Zeiltchrift a

wider

Sinnexoryane, Vol.

xxiv,

p. 63.

v [CHAP,

of Mathematics Indefinables

The

56

always denotes. It will be convenient, for the present I shall call from a predicate : a to distinguish class-concept discussion, though the distinction is and man human a class-concept, a predicate, guished distinas perhapsonly verbal. The characteristic of a class-concept, is that "a? is a u* is a prepositional in general, from terms It must be held that function when, and onlywhen, u is a class-concept. hut do not have a false proposition, we is not a class-concept, when u This value we at all,whatever giveto x. may simplyno proposition for the null-class, enables us to distinguish a belongingto class-concept from a term which is of the above form are false, which all propositions of the at all,for which there are no not a class-concept propositions is not a term Also it makes it plainthat a class-concept above form. if the in the propositionx is a u? for u has a restricted variability A denotingphrase, formula is to remain a proposition. we may now say, of the six words above consists alwaysof a class-concept precededby one words

six

"

or

of one of them. synonym The questionwhich first meets

some

regard to denoting is this : Is there one or are way of denotingsix different kinds of objects, the ways of denotingdifferent ? And in the latter case, is the object denoted the same in all six cases, or does the objectdiffer as well as the it will be first this question, In order to answer way of denotingit ? necessary to explainthe differences between the six,words in question. Here it will be convenient the to beginwith, since this to omit the word 59,

word is in from

a

which

In

from different position

they are where

cases

number

the

in

us

and others,

is liable to limitations

exempt. the

of terms, it

indicate the various

class defined

has only a finite class-concept is possible the class-concept to omit wholly,and objectsdenoted by enumeratingthe terms and

by

a

them of and or or as the case It will be. connecting by means may isolate to of a help our part problem if we first consider this case, the lack of i n although subtlety languagerenders it difficult to grasp the difference between objects form of words. indicated by the same Let us begin by two terms considering only,say Brown and Jones. The denoted objects by all,every, any, a and some* are respectively involved in the following five propositions.(1) Brown and Jones are of Miss

two

Miss

Smith

Smith's

(3) if it was

;

lover; (4) if it Brown

or

only

two

suitors

Jones

was

(5) Miss

;

forms

one

;

(2) Brown

Brown

or

of Miss Smith

Jones Smith's

I maintain propositions,

involved.

distinctions, some

*

I intend

to

The

between distinguish

the distinction of all and avoid

circumlocution.

every is also

a

Jones

you

and a

suitors,it or

a

must

Jones.

Brown

or

to

very ardent have been

Although Jones, are

that five differentcombinations

of which in

are

a way not of straining usage.

ttome

paving court

are

met, it was

will many Brown and Jones and

of words, Brown

involved in these are

and

rather

subtle, may

warranted Both

are

be

by language; necessary

to

59]

58,

Denoting the

broughtout by is Brown

and

Jones

considerations. following who

neverthelessit is not

The

composed of Brown and Jones are a genuine combination which,as

is characteristic of classes. In the chapter, is asserted is true

to Miss

payingcourt

the combination

Thus

first case: second

the

first all

concerns

Smith

by

concerned

case

is

in the next

the on proposition, the Jones severally;

and

is payingcourt is not

all of

i.e. each distributively,

the sake of

and

see

which of Brown

(I think)identical with,"Brown

and Jones

indicated

shall

we

second

of Brown

is equivalent proposition to, though not is

true of either separately;

two

Jones,the kind of combination

contrary, what

In the firstproposition, it

two, and this is not

are

the whole

two, for this is onlyone. with

57

the

them

same

Smith."

to Miss

here

as

in the

while collectively,

the

of them. For every one first a numerical conjunction, or

distinction, we may rise to gives number, the second a prepositional conjunction, since the proposition in which it occurs is equivalent of to a conjunction propositions. (It should be observed that the conjunctionof propositions call the

it

since

in

questionis of a whollydifferent kind from any of the combinations of the which is we are kind called considering, being in fact The combined logical are product. propositions qua propositions,

the not

qua The

terms.) third proposition givesthe kind

defined.

There

of

conjunction by

is

which any

is some

about this notion,which seems half-way difficulty further be This notion and conjunction a disjunction. may explainedas follows. Let a and b be two different propositions, each of which Then the disjunction c. impliesa third proposition "a b" implies let a and b be propositions Now or c. assigningthe combination is to two then there different subjects, same a predicate of the two to which the givenpredicate subjects may be assignedso that the resulting is equivalent to the disjunctiona or 5." proposition

between

a

"

Thus and "

"

" if you met suppose we have if you met Jones,you met

if you

and The

we

met

Brown

regardthis

combination

or as

if you

very ardent lover,1" infer Hence we very ardent lover."

Brown, you

met

Jones, you

met

a

met

very ardent lover," Brown Jones, etc." or a

if you met and Jones here indicated is the

to equivalent

of Brown

a

"

same

as

that

It differsfrom a disjunction by the fact by either of them. both ; but in that it impliesand is impliedby a statement concerning fails. The more some complicatedinstances,this mutual implication of combination method is,in fact,different from that indicated by both, I shall call it the and is also different from both forms of disjunction. is given by (4): this variable conjunction.The firstform of disjunction is the form which I shall denote by a suitor. Here, althoughit must

indicated

have been Brown nor

yet that it

to equivalent or

or

it must

must

have

been

true

Jones.

that

it must

Thus

the

have

been

Brown,

is proposition been

not

Brown

"it must have of propositions disjunction of in fact,is not capable The proposition, been Jones."

the

have

Jones, it is not

statement

either

except in

the

as

or disjunction

a

is,whichever

of the two

which

is increased

of terms

was

rapidly

beyond two,

the number

of terms

is

variable term, that fix upon, it does not denote this term, we This form accordingly I other of them.

denotes disjunction

of

this form

Thus

form

a

it

Brown,

not

was

Brown,"

inadmissible when theoretically

becomes

infinite.

it

"if

the number

intolerable when

becomes

[CHAP,v

of propositions, conjunction

a

as

form: very roundabout if it was not Jones, it was

Jones, and and

of Mathematics Indefinables

The

58

terms

a

or and yet it does denote one shalf the second form of disj unction callthe variable disjunction. Finally, the call what I shall constant since is givenby (5). This is disjunction,

is undecided.

That

is to

is proposition

our

say,

of propositions, namely disjunction Jones." she will marry denotes a disjunction either

nor

to equivalent

now

will marry

it may

them, though

these five combinations

that

terms, while the

is neither

which

nor

one

or

the

denote

distinct.

are

yieldneither

terms

The first only combinations of terms. others yieldsomethingabsolutely peculiar,

of relations.

is,at least if the

there

combination

combinations

The

many. the use

terms, effected without

combinations

are

Correspondingto form

combined

terms

definite concept,which denotes the various terms perfectly combined

in the

distinctions in

enumerated,as 60.

above, but

When

are

a

In this way,

enumeration. enumeration I leave

a

or

by

a

the class

given,it must

be

terms

a

But

shall see

meaning of such

AU

a\

to

the

be

or

a

held

not

that

the

is to say, any it belongsto the

That

not

givenotherwise

be

are

certain class.

otherwise

than than

by by

given questionwhich,for the present, of givinga collection by a possibility

in Part V.

phrasesas begin with,denotes a

is

given.

can

can

is highlyimportant, since class-concept

the

also

class,a

is a class-concept,

undetermined.

collections, as we

as

each

of the combination

combined of

are

collection

be

to

the terms

be decided whether

collectionof

Whether

terms

defined

class-conceptis

a

belongingto it can beingproposed,

class.

the

where

case

a

a

of

this,let us repeat explain

To

manner. specified

various terms term

a

and concepts,but strictly

yields many

our

Brown,

two, and

of the

one

all the five combinations

Thus

It is to be observed

some

of

one particular

one. particular

Smith

Miss

"

will marry

She

denoted,but the alternative

is

Jones

is denoted,,or

here either Brown

given. The

it enables

For

the

us

to deal with infinite

I wish present,

all

a\ every ", any a, an numerical conjunction ;

concept all {fa is

to examine

a. a, and some it is definite as

definite single perfectly concept,which denotes the terms of a taken all together.The terms have a number, which may thus be regarded, so taken if we choose, as of the since a is it determinate for class-concept, property any given the it on all the still denotes class-concept. Every a, contrary,though "X denotes them in a different way, I.e.severally instead of collectively. denotes Any a onlyone a9 but it is whollyirrelevant which it denotes, and what is said will be equally true whichever it may be. Moreover, soon

as

a

a

59-61]

Denoting

59

variable a, that is,whatever particular a we may fasten is certain that it does not a denote that one any upon, ; and yet of that i s true which is one of true denotes a An proposition a any any a. variable disjunction that is to say, a proposition : which holds of an a denotes

a

any

a

be false

concerningeach particular fl, so that it is not reducible to of For a disjunction propositions. example,a point lies between any and other it but would not be true of any one point; point any point that it lay between any point and any other point, particular since there would be many pairsof pointsbetween which it did not lie. to This bringsus finally some This denotes a, the constant disjunction. of term the class but the it denotes may be any term term justone a, may

of the class. Thus that there

mean

"

was

a

"

any moment

first moment

the exact

means

does not follow any moment in time,while " a moment

moment

some

opposite, namely,that

*"

would

precedes has

every moment

predecessors. In the

61.

of

case

a

class a which

.

-

has

a

finite number

illustratethese various notions

a"n" say #1, #2, a*, (1) Ml a?s denotes al and 0% and we

can

and

...

of terms

as

follows

"

:

o".

and denotes a". (") Every a denotes a^ and denotes a2 and has the meaning or or (3) Any a denotes a^ or 0% or "", where ...

...

that it is irrelevant which

we

take.

denotes ^

or

a^

(4) that

no

An one

take any

one

in

particular

in

particular.

Some

(5)

or

denotes

.a

a^

or

...

be

must

where

#",

taken,just

on

0% or

the

...

or

has the

in all ans

as

denotes

or

is taken,but

irrelevant which

not

we

meaning must

denotes o^, where

or

contrarysome

not

it is

a particular

one

be taken.

must

the nature

and

of the various ways of combiningterms properties of vital importance of mathematics, it may be well to the principles illustratetheir properties by the following importantexamples. As

are

to

(a)

Let

and

The

be

a

class,and b

cases

a

a

class of classes.

to b from

We

are

obtain

then

various combinations

and every do not, in this case, introduce follows. as

AU

some.

six

a

relations of possible

in all six a

a

of any,

anythingnew.

to 6,in other words,the (1) Any a belongsto any class belonging class a is wholly contained in the common product of part or logical the various classes belongingto b. (2) Any a belongsto a i, i.e. the class a is contained in any of class which contains all the 6's,or, is contained in the logical sum

all the Vs.

(3) Any in which

a

belongsto

some

the class a is contained.

the second

arises from

5, i.e.there is The

a

whereas belongs,

the fact that here

a

6,and

before it

was

belongingto by this

difference between there is

onlydecided that different o*s might belongto different Zfs.

a

class

one

case

b to which

every

a

and every

belongedto

(4?) An in

Indefinables of Mathematics

The

60

with

common

6,i.e.whatever b

any

take,it has

we

part

a

a.

to a 5,i.e.there is a b which has a partin common belongs to some 6." to This is equivalent some (oran) a belongs a. to to any 6,i.e.there is an a which belongs (6) Some a -belongs

An

(5)

a

"

with

the

belongsto

a

[CHAP,v

partof all the #s, or

common

These

all the

are

($)

a

part.

common

that arise here.

cases

to compare the above considered,

series of real numbers;

a, b be two arise.

cases

and all the 6'shave

of the type as instructive, showing the generality

It is

relations here Let

a

(1) Any

then six

following.

analogous precisely

any 6,or, the seriesa is contained among

is less than

a

with the

case

of

lessthan every b. (2) Any a is less than

numbers

i,or, whatever

a

a

take,there

we

numbers which is greater, or, the series a is contained among of the series b. It does not follow that some term a (variable)

is

b

a

less than of

term

than all the dfs. the series b is greater

6,or, there is a term of b which is (3) Any a is less than some than allthe cts. This case is not to be confounded with ("). greater (4) An a is less than any 5, i.e.whatever b we take,there is an a

which is lessthan it.

(5)

An

a

is less than

a

is less than

such that the

any Some

a

all the "'s. This whereas here In

the

6.

merelydenies

This

an

a

and

that any

b

a

is

a

b.

greaterthan

(6)

to find i, i.e. it is possible

a

is lessthan any not

was

",i.e.there

is an

impliedin (4),where

a

which is lessthan

the

a

was

variable,

it is constant.

actual mathematics

this case,

have

the compelled

distinction

between the variable and the constant where

mathematics

have

But in other cases, disjunction. obtained sway, the distinction has been

not

and the mathematicians, have not investigated as was natural, neglected; the logical nature of the disjunctive notions which theyemployed. I shall giveone other instance, it brings in the difference as (y) between any and evety, which has not been relevant in the previous cases.

Let

a

and

relationsbetween of their terms.

b be two them

classes of

arise from

classes;then twenty different

different combinations

of the terms

The

technical terms will be useful. If a be following its logical consists of all terms belonging sum a classof classes, to any such Le. all terms that there is an a to which theybelong, while a, its

common

partof all the dfs. (1) Any

a

all terms

consists of logical product term

is contained in the

(%) Any

term

is contained in the

of any

We a

belongs

to every

logical productof of any

sum logical

a

to belonging

have then the

cases. following Le. the 6, sum logical

of

b.

to belongs

of b.

every ", Le. to the

a

b, i.e.the logical sum

of

a

62]

61,

Denoting

(3) Any

sum logical

contains the

Any

(4) a

of any

term

of

of

term

which

(6) Any contains

b which

A

(7) (8)

.

a

A

A

(9) has

any

a

and

the

A

(10) (11)

of b have

productof

(16)

productsof

productof

an

every 6, Le. the

a

is

a

a

and any

has

a

part

a

b with which

belongsto

a

of

sum logical

part. 6, Le. givenany 5, an

any

a

a

can

a

and

part,

common

a

the logical b,~Le. sums

a

of every

of

a

of b.

sum logical

of every

term

belongsto every ft,Le. the logical logical productof b. belongsto a b,i.e.the logical product

a

in the

a

belongsto

term

of b have

a

a

of every

a

common

part.

belongsto

A

the term

which

any

of any

b has

a

J, Le. there

is

every 6,Le. the

belongsto a

belongsto logical productof b. of some a belongsto

term

some

a

term

is contained.

of every a of b have and the logical sum

with

(19)

J,Le. there is a

common

(orsome) term

Some

common

i, le. any class of

some

belongsto

a

of every

in the

and

a

a

(18)

with

of

(orsome)

A

(17)

an

a

the logical productof A

b, i.e.there

some

b, Le. any class of

a

to belongs

a

it has

term

is contained

of b in which

a

of

term

(15) Any

of

an

is contained

a

an

part.

(14) Any

in

of

common

a

is

common.

term

(13) Any

a

sum logical of any a belongs to

tenn

A

", le. there

a

of b.

with which

(12)

of

to belongs

a

logical productof b have

be found

an

part.

of any

term

A

is

a.

belongsto any

a

the

part in

a

(or an) a belongsto

some

common

term

6,i.e.there

of b.

sum

of any

term

with

in common

to (oran) a belongs

class belonging to

one

term

class of b have

of

term

every

productof b.

some

in the

J, Le. -there is a b which

some

to (or an) a belongs

some

of

term

is contained

belongsto

a

a.

which is contained in the

(5) Any

61

a

any

6, Le.

the

logical

part.

common

every

a

logical

6, Le.

b9 Le.

any

a

there is

has

some

a

part term

part.

common

belongsto any 6, i.e.any b has a part in with the common logical productof a. above The examplesshow that,althoughit may often happen that there is a mutual (which has not always been stated)of implication and #, or concerning"my propositions concerningsome corresponding mutual Thus other such there is no and every yet in cases implication. distinct, the five notions discussed in the presentchapterare genuinely (20)

A

term

of every

a

,

and to confound 62.

etc.

are

may

lead to

definite fallacies. perfectly

there are discussion that, whether denoted of denoting or by all men" every not, the objects to say therefore distinct. It seems legitimate certainly

It appears

different ways y

them

from

the above

of Mathematics Indefinables

The

62

the

all

in

same

There

cases.

the

with

connected

and objects,

difference lies in the

that the whole

[CHAP,v

that

denotingitself is difficult problems

however, many

are,

as regardsthe especially subject,

of

nature

the

with the class of men, denoted. AU men, which I shall identify objects it is plural, to be an althoughgrammatically seems unambiguousobject, we : questionis not so simple may doubt whether an ambiguous objectis unambiguouslydenoted,or a definite "I met Consider again the proposition objectambiguouslydenoted. that It is quite certain,and is impliedby this proposition, man." a in definite the technical man what I met was : an unambiguousperfectly is the is here which adopted, proposition expressedby I met language

in the other

But

the

cases

"

the

But

man."

some

actual

whom

man

I met

forms

part of the

no

denoted by some man. specially the concrete event which happenedis not asserted in the proposition. is class that of of concrete What is Asserted merely one some a

and in question, proposition Thus

place. The

took

events

who

is not

whole human

existed

is involved

race

in my

not existed

will exist had

assertion

:

been

going would have been different. Or, proposition if I substitute for intensional language, to put the same pointin more to the individual whom man applicable any of the other class-concepts I had the honour to meet, my i s proposition changed,althoughthe denoted as before. What individual in questionis justas much this must not be regardedas actually man denoting proves is,that some Smith and actuallydenotingBrown, and so on : the whole procession of human beingsthroughoutthe ages is alwaysrelevant to every proposition in which and what denoted is is some man essentially occurs, if any

ever

or

or

exist,the purport of my

to

each

not

is

man

separateman,

but

evident in the

case

a

kind

of combination

of all

men.

This

of every, any, and a.. There is,then, a definite something,different in each of the five cases, which must, in but is characterized as a set of terms combined a sense, be an object, more

in

a

certain a

man,

any

objectthat concept is 63.

which

way, man

or

some

somethingis man\

and

denoted

concerned are propositions used as denoting.

It

remains

to

discuss the

all men, every man, this very paradoxical which the corresponding

by

it is with

in

notion

been

of

the.

This

notion

has

symbolically emphasizedby Peano, very great advantageto his calculus; but here it is to be discussed philosophically. The use and the theoryof definition are dependent of identity upon this notion, has thus the very highest which philosophical importance. is correctly The word the,in the singular, employedonlyin relation of which there is only one to a class-concept instance. We speak of the

with

King* the

time); and term

in such

by means

words.

Prime

of

a

Minister,and cases

there is a

which concept,

at the present (understanding method of denoting definite one single is not givenus by any of our other five so

on

It is owing to this notion that mathematics

can

givedefinitions

62-64]

Denoting which

of terms

not

are

difference between

concepts

mathematical

which possibility

a

"

63

and

illustrates the

definition. philosophical

Every

is the

and thus every term, onlyinstance of same class-concept, is capableof definition, theoretically, providedwe have not adopted

term

system in which

a

the said term

paradox,

puzzlingto

curious

is

of

one

indefinables.

our

It is

a

the

retically, theosymbolicmind, that definitions, irrelevant nothingbut statements of symbolicabbreviations, and inserted onlyfor practical to the reasoning while yet, convenience, in the developmentof a subject, amount a very large theyalwaysrequire and often of thought, of the greatest achievements of embody some This fact to be seems analysis. by the theoryof denoting. explained be to An object without the our mind, present knowing any concept may is ike instance ; and the discovery of which the said object of such a improvement in notation. The reason why this concept is not a mere are

to

appears becomes

be the"

is

case

that, as

soon

the definition is

as

the

found, it

actual wholly unnecessary reasoningto remember objectdefined,since onlyconcepts are relevant to our deductions. In of discovery, the definition is seen to be true^because the the moment in our objectto be defined was already thoughts; but as part of our since what the reasoning reasoningit is not true,but merelysymbolic, it should that is deal not that with but merely that object, requires it should deal with the object denoted by the definition. to

the

actual definitions of mathematics, what is defined is a class and the notion of- the does not then explicitly But of entities, appear. In

most

in this

even

case, what

conditions; for a

term

or

a

is

as class,

defined really shall

we

and

of terms conjunction

alwaysrelevant

is the class

in the next

see

never

the notion of

concept. Thus

a

in definitions; and

certain satisfying i s chapter, always

observe

that generally the adequacyof conceptsto deal with thingsis whollydependentupon which this notion gives. term the unambiguousdenotingof a single the nature with is of of identity connection 64. The denoting rather serious problems. important,and helps,I think,to solve some is or is not a relation, and even whether whether identity The question

the is

there is such

a

may

For, it may be easy to answer. relation,since, where it is trulyasserted,

concept at all,is not

said,identitycannot we

we

be

a

for a relation* onlyone term, whereas two terms are required be anythingat all : indeed identity, an objector may urge, cannot

have

And

terms

two

are plainly

is it identical with?

might attempt to two

terms

are

remove

and identical,

not

Nevertheless

But

some

then

terms

respectwhen shall have

to

to

cannot

be

must identity

from identity

identical in

term

one

be, for what

something. We

and say relations,

theyhave hold

a

that

givenrelation

either that there is

we giventerm. that the of the givenrelation, the or two between cases strict identity in the sense of havinga givenrelation to a given have identity two cases term ; but the latter view leads to an endless process of the illegitimate

to

a

of Mathematics Ividefinables

The

64 kind.

of

terms

two terms

are

necessary. not be

but these need not

the

to as difficulty

the

by a sheer denial that two different There must alwaysbe a referent and a relatum, is affirmed, distinct; and where identity theyare be met

relation must

a

admitted,and

be

must identity

Thus

[CHAR y

so*. is it

questionarises: Why

the

But

is answered question

identity?This

by

ever

the

while

worth

theoryof

to

affirm

denoting.If

King,"we assert an identity ; the reason making is,that in the one case the actual while in the other a denotingconcept takes its place. term occurs, form a I ignorethe fact that Edwards (For purposes of discussion, form a class having only one term. and that seventh Edwards class, Often VII is practically, a proper name.) Edward though not formally, is and itself not term the mentioned,as two denotingconceptsoccur, his generasurvivor is the last of the in the proposition presentPope tion.*" its of with is When t he assertion term itself, a identity given, is outside and made the is never futile, logicthough true, perfectly is at once books ; but where denotingconceptsare introduced, identity In this case, of course, there is involved, to be significant. seen though not asserted, a relation of the denoting concept to the term, or of the other. each But in such the is which occurs two denotingconceptsto is the

Edward VII say why this assertion is worth "

we

"

does propositions

itselfstate

not

this further

but relation,

states

pure

*f". identity To

65.

sum

When

up.

six words

allyevery, any^ is,as a rule,not proposition but together,

about

a

a,

by class-concept, preceded

some,

about the

the, occurs

in

conceptformed

one

of the

the proposition,

a

of the two

words

objectquitedifferent from this,in generalnot This may be seen a a or complex of terms. by in which such conceptspccur the fact that propositions in general are false concerningthe concepts themselves. At the same time, it is to consider and make about the conceptsthemselves, possible propositions but these are not the natural propositions to make in employing the concepts. Any number is odd or even is a perfectly natural proposition, whereas "Any number is a variable conj unction is a proposition discussion. In such cases, we say that the onlyto be made in a logical conceptin questiondenotes. We decided that denotingis a perfectly an

all,but

conceptat

term

"

"

"

*

relationsof terms

On

to themselves, ". inf.Chap,ix, " 95. is terribly and ambiguous, great care is necessary in order not to confound its various meanings. We have (1)the sense in which, it asserts Being,as in "A is" ; (2)the sense of identity; in "A is human" (3)the sense of predication,

t The

word

is

;

(4)the

sense

of

"

A

is

addition to these there

"

a^-man are

(cf.p. 54, note),which

less common

relation of assertions is meant, that rise to formal implication. Doubtless

a

occurred to

me.

On

the

"

is very

like

identity.In happy,"where it exists, gives

to be good is to be in fact^ relation, which,where there are further meanings which

uses,

as

have

not

meaningsof is,cf. De Morgan,Formal Logic,pp. 49, 50.

64,

Denoting

65]

65

in all six cases, and that it is the nature the same definite relation, of and the the denoted object the denotingconcept which distinguishes the

discussed at

We

cases.

denoted

objectsin the

of terms.

In

length the

some

a

five full

and

nature

in which

cases

the

these

differences

objectsare

of

binations com-

it would discussion,

be necessary also to the actual meaningsof these concepts,as

denotingconcepts: of the objects nature theydenote,have not been discussed opposed that there would be anythingfurther to But I do not know above. say discussed the^and showed that this notion this topic* Finally, we on

discuss the

to the

is essential to

what

mathematics

calls

as definition,

of uniquelydetermining term a possibility actual

use

of

though identity,

not

its

by

means

meaning,was

well of

as

to

the

concepts; the

also found

to

depend

of denoting a singleterm. this point we From can upon this way the advance to the discussion of classes, therebycontinuing development

of the

R.

topicsconnected

with

adjectives.

VI.

CHAPTER

CLASSES.

bring clearlybefore

To

66,

distinguishthis

to

of

one

the

philosophy. Apart the

concept, -ask the of

subtle

It has

been

that

of

held

been

admirable

a

these

pure

lair.

It

concepts,

general

terms

and

definition in

be

there to

a

class

and

to

of

question the point other

the

Symbolic general the

be

point

view

if

hand, of

enumeration

Logic

we

its terms,

of

view

*

as

of

La

and

objects

de

would is to

extension this

be

classes.

denoted

will

not

Thus

It

Paris, 1901,

is

the

given

the

to

vicious

a

terms

circle;

unavoidable.

class is defined allow

us

owing

to

extent

this

as

in

must

this to

by

deal,

to

classes

and

p. 387,

or

but

intensional

an

our

by concepts,

is essential.

Leibmx,

our

pure,

method

given,

to

extent

some

are

we

predicates

are

attach

involve

as

Symbolic

which

attempt

course

But

intension

with

terms

which

really were

that

predicates attaching

this

infinite

intension

Lo^ique

of

for

extension

take

does, with

regarded of

others,

no

of

its

only

pure

regions not

his

in

can

justified.

between

should

when

cannot

class

be

classes

and

predicates

We the

terms,

definite

many

as

of

hence On

be

others.

no

be

must

will

of

have

Couturat,

if there

and

would

the

that

points, stand-

Mathematics

while M.

intermediate

essential

composed

class

a

is

apparatus

Symbolic Logic ;

must

Philosophers

former.

on

follows.

distinguishtwo

positions intermediate

it is in these

I

x.

the

pedantry

that

subject

Chapter

intension.

statement

fundamental

this

in what

extension*

of

view, his

are

extension,

for

in

of

in

allied, is

mathematical

very

fundamental,

more

standpoint

fact, there

should

of

speciallywith the Leibniz, states roundly

the

concerned

as

idle

logic,to

on

that

in

found

deal

points

its

and

latter

and

has

in

a

required

as

be

to

works

is

it is

of

problems

discussed

regard

to

not

which

to

doss

is

by class, and

is meant

notions

that

be

to

discriminations

on

two

of

Logic

nicety

the

to

from

up

matter

and

and

extension

work

built

only

care

customary,

usually regarded

be

fact

reader, therefore,

somewhat

has

the

contradiction

the

what

important

from

utmost

of

"account

and

difficult

most

all the

from

notion

mind

the

con-

66-68]

Classes

67

sideration that the

is of such great importance. In theoryof denoting have to specify the present chapterwe the precisedegree in which extension and intension respectively enter into the definition and employment

of classes;and reader to remember as

well

I must ask the throughoutthe discussion, is said has to be to infinite applicable

that whatever

to finiteclasses.

as

When

denoted by a concept, objectis unambiguously I shall the concept) speakof the conceptas a concept(orsometimes,loosely, as of the objectin question. Thus it will be the necessary to distinguish of class class from a classa concept a -concept. We agreedto call man but its does usual in man not, concept, employment,denote anything. and all men On the other hand, men (whichI shall regardas synonyms)do denote,and I shall contend that what theydenote is the class composed of all men. is the class-concept, Thus man men (the concept)is the and men (the objectdenoted by the conceptmen) concept of the class, is It the class. doubt and at first, to use no are confosing, class-concept in different class but distinctions ate so concept of a senses; many that In of unavoidable. some required straining languageseems the phraseology of the preceding we chapter, may say that a class is a numerical This is the thesis which is to be conjunctionof terms. 67.

an

established.

regardedclasses as derived from assertions, i.e. as all the entities satisfying left some assertion,whose form was I shall discuss this view critically in the next chapter; whollyvague. for the present,we may confine ourselves to classes as they are derived from predicates, leavingopen the questionwhether every assertion is kind of genesis We to a predication. equivalent may, then, imaginea of classes, throughthe successive stagesindicated by the typical propositions Socrates has humanity," Socrates is a Socrates is human," the last Of these propositions, Socrates is one men." man," among contains the class as a constituent ; but only,we should say, explicitly givesrise to the other three equivalent proposition every subject-predicate be sometimes it can and thus every predicate (provided propositions, of classes from trulypredicated) givesrise to a class. This is the genesis 68.

In

-Chaptern

we

"

"

"

"

the intensions! On

standpoint,

the other hand, when

they call a it is equivalent name, to regard involved is finite,

deal with what

mathematicians

some or manifold,aggregate,Menge, ensemble, where the number of terms especially common, the object as in question (whichis in fact a class) of of its terms, and as consisting possibly

that

case

is the class. Here

but relevant, this word are

a

terms

connected

stands for

and Brown class,

of classes.

a

it is not

by

numerical is a singly

defined

by the

tion enumera-

singleterm, which and denotingthat predicates

the word

a

and,

in the

Thus conjunction. class. This

sense

Brown

in are

in which

and Jones

is the extensional

genesis

Indefinables of Mathematics

The

68

best formal

The

69. Peano*.

is .that of

of classes in existence

treatment

this treatment

in

But

[CHAP*vi

,of distinctions of great

number

a

philosophical importanceare overlooked. Peano, not I think quite the relation identifiesthe class with the class-concept ; thus consciously, For him, is a. of an individual to its class is,for him, expressed by is said to belongto term in which 2 is a number is a proposition a the class number. Nevertheless,he identifies the equalityof classes, "

"

having the

consists in their

which

is

which

terms, with

same

the class is

when quite illegitimate

In order class-concept.

to

that perceive

and

man

identity "

a

ceeding pro-

regardedas

the

bipedare featlierkas

not quite unnecessary to take a hen and deprivethe poor bird of its feathers. Or, to take a less complex instance,it is plainthat

it is identical,

prime

even

identical with

is not

integernext

after1.

when

Tims

we

admit that two classes the class with the class-concept, must we identify without be identical. it is plainthat Nevertheless, being equal may is involved, when two class-concepts for we say are equal,some identity Thus there is some that they have the same terms. objectwhich is this object, identical when two class-concepts are eqiial positively ; and the plucked it would seem, is more called the class. Neglecting properly the of class is featherless would the same hen, as bipeds, say, every one the class of

men

;

the class of

after 1. Thus we next integers or class-concept, regard Socrates "

of

individual to

an

consequences

a

primes is the

even

must

not "

is

a

class of which

man

same

as

class of

the

the class with the identify the relation as expressing

it is

a

member.

This

has two

which prevent the philosophical (tobe established presently) in Peano^s formalism. The firstconsequence points is no such thingas the null-class, though there are null

acceptanceof certain is,that there

The class-concepts.

second is,that

a

class

having only one

term

is to

I should identified, contraryto Peano\s usage, with that one term. his notation in consequence or propose, however,to alter his practice of either of these points rather I should ; regardthem as proofsthat far as notation goes, with SymbolicLogic ought to concern itself, as

be

not

rather class-concepts 70.

than

with classes.

A

class,we have seen, is neither a, predicatenor a for different concept, and different class-concepts predicates may to the

same

from

the whole

one,

while the

class. A

class

also,in

one

sense

classspond corre-

is distinct at least,

composed of its terms, for the latter is onlyand essentially former,where it has many terms, is,as we shall see later,

the very kind of object of which many is to be asserted. The distinction of a class as many from a class as a whole is often made by language: and time and points, t he the navy and the soldiers, instants, space army and

the sailors, the Cabinet and the Cabinet Ministers,all illustrate the distinction. The notion of a whole,in the sense of a pure aggregate *

Neglecting Frege,who

is discussed in the

Appendix.

69-71]

Classes

is here

which

is,we relevant,

of the

notion

though terms treated

class

shall find,not

belong

the

to

the

where alwaysapplicable

applies(see Chapter x).

as

many be said to

may

69

the class,

In

such

class must

cases,

be

not

itself a

arises singlelogical subject*.But this case never where class can be generatedby a predicate.Thus we a for the may dismiss this from our In a class as many, minds. complication present the component terms, though they have some kind of unity,have less than is requiredfor a whole. They have, in fact,just so much unity to make them many, and not enough to preventthem from is required as A further reason for distinguishing from wholes remaining many. as

classes as many as

be

as

many, of

in

is that

class is

"

of its own

one

Class

71.

class as

a

classes

"

are

one

be

may

one

among whereas class-concept"),

a

of itself

of the terms

one

classes ""

(the

extensions!

complexwhole

a

valent equinever

can

constituents. be

may

defined

either

intensionally.

or extensionally

is to say, we may define the kind of object the which is a class, or of concept which denotes a class: this is the precise meaning of

That kind the

But oppositionof extension and intension in this connection. although the generalnotion can be defined in this two-fold manner, classes, particular except when they happen to be finite,can only be denoted defined intensionally, i.e.as the objects cepts. by such and such conI believe this distinction to be purelypsychological : logically, to infinite the extensional definition appears to be equallyapplicable would if we cut short but practically, to attempt it,Death were classes, laudable endeavour before it had attained its goal. Logically, our I will begin to be on extension and intension seem a therefore, par. with

extensional view.

the

When it is

a

defined class is regarded as called naturally

more

this name,

as

of definite terms.

number

other

numbers,

B"

"

or

A

since

the terms

the

or

by

mean

adopt objects is

what

any other enumeration of the actual mention

than

1

be asserted.

can

they result

be

A

pointedout

of plurality

terms

propositionshave

not

and

is not one

allowed

the

presuppose

numbers. presupposed

is

the

in the

no

for.

togetherwith where case particular

method A

many

There

exists of

the a

avoidingit,

is grammatically, collection,

subjectwhen logical

subject,but

not

and:

numbers

since

Collections do terms

simplyfrom

which, grammatical difficulty must

and.

connected

are

of the collection themselves

terms

such

and B and C?

collection is defined

The

they could only presuppose

*

moment

that It would seem by representsa fundamental way of combiningterms, and that just is essential if anythingis to result of which a way of combination

terms, and

this

I shall for the

prejudgethe questionwhether trulyclasses or not. By a collection I

by it are conveyedby "A and

and

collection.

of its terms,

the enumeration

it will not

denoted

the

a

by

a

number

is asserted of it :

subjects. See. end

of

" 74.

of Mathematics Indefinables

The

70

[CHAP,vi

whereas A and B, A and B and C, etc. are essentially plural. singular, fact (to be discussed arises from the logical This grammaticaldifficulty in generalforms a whole which is is many that whatever presently) it is,therefore,not removable by a better choice of technical one; terms.

In brought into prominenceby Bolzano*. order to understand what infinity is,he says, we must go back to one in order to reach an of our of the -simplest understanding, conceptions the word that we are to use to denote it. This is agreement concerning which underlies the conjunction the conception and, which, however,if in is it is to stand out as clearly as required, many cases, both by the I believe to be best purposes of mathematics and by those of philosophy, of certain things,1 or by the words i 'A system (Inbegrrjf) expressed of certain parts.' But must add that every *a whole consisting we objectA can be combined in a system with any others arbitrary or 5, C, Z), (speakingstill more alreadyforms a system correctly) of itself less which some more or f importanttruth can be enunciated, by in fact J, J5, C, Z", providedonly that each of the presentations of the propositions in so far as none a or different object, represents

of and

notion

The

was

"

...,

,

...

is the

'A

same

to speakof

have

we

a

above

The

B?

is the

'A

same

CJ 'A is the

as

if e.g. A is the same as J5,then it is system of the thingsA and B*

For

is true.

as

passage,

found

necessary.

it does not make of

from

collection

a

collection

it does not appear Secondly, is not practically applicable

this is connected with the second

we

have to consider is the

on

the

if any, of difference,

hand, and from

one

But

let

the whole

first examine

us

point,

of the notion

of intensional definition nor

mention

the other.

on

unreasonable certainly

as

and Thirdly,

any

class. What

a

D? etc.,

as

several distinctions which it is,neglects First and foremost,it does not distinguish

good

the many from the whole which theyform. of enumeration to observe that the method to infinitesystems.

same

a

class

of the

formed

further the

notion

of and.

Anything of which a finite number would be commonly said to be many, alwaysof the form "A and B and C each

and

one

are

the

same

A3 and ...."

To

much

to

only to u

A

not

and

be A

all different. To as

to

other than 0 and and

or

it

many, ..."

say that A ...

are

be

said,are

J, 5, C, seems "

...

are

to amount

Al and A* and

all different seems

symbols:

be asserted

can

might

Here

say that A is one is not of the form

say that A9 B, C, condition as regardsthe

1

to amount

be held

that

that diversity is implied so meaningless, by and, and stated. specially

need

a

A

term

"

it should

is

A

which

is

one

may

be

regarded as

*

Paradoxien des UnendRchen, Leipzig, 1854 t i.e. the combination of A with B3 6',D, ...

a

of

case particular

(2nd ed.,Berlin,1889)," a system.

alreadyforms

3.

a

Classes

71] collection, namely as

collection of

a

71 Thus

term.

one

collection

every

is many

which

presupposes many collectionswhich are each one : A and collections of some presupposes A and presupposes B. Conversely term those one which are presuppose many, namely complex: thus differs from B" is one, hut "A and B. presupposes A and difference B

there is not

But

symmetry in

this

respect,for

the ultimate

tions presupposi-

of

anythingare alwayssimpleterms. Every pair of terms, without exception,can indicated

manner

At

and

B

by

two.

are

A

and

A

and

B, and B

be

be

may

in the

combined

if neither A

be many, then nor conceivable entities, any

any

B

objectsof thought,they may be pointsor numbers or possible false propositions events or or people,in short anythingthat counted.

A

teaspoon and

dimensional to be

be

space,

be

number

3,

two. certainly

are

A

placedon

observed

the

Thus

or

a

true

be

can

and

chimaera

or

four-

a

restriction whatever

no

is

and B, exceptthat neither is to be many. It should and B need not exist,but must, like anythingthat

that A

mentioned,have Being. The

distinction of Being and existence a nd is well illustrated important, by the process of counting. What be counted be something, must and must can be, though it certainly

can

is

need

by

Thus

what

no

of possessed

demand

we

should be

be

means

of the terms

entity. be questionmay now this mean anythingmore

That of B?

our

asked

than

by of juxtaposition

What

:

the

is meant

A

and

B ?

A

with

B?

is,does it contain any element over and above that of A Is and besides A, a separate concept,which occurs

either

there

answer

suppose,

the first place,and,

In objections.

are

and that To

B?

might

we

concept,for if it were, it would have to be then be a relation between A and B; A and B would be

cannot of

kind

some

of

of existence. privilege collection is merelythat each

further

an

The Does

the

a

new

be one, not two. Moreover, if there are two concepts,there are two, and no third Thus and would mediatingconcept seems necessary to make them two. at or proposition,

least

meaningless.But

seem

with,it seems the word

idea

seems

and,

a

it is difficultto maintain

rash to hold that any do not seem to be we

word

is

this

theory.

begin

To

meaningless.When

utteringmere

to the word. correspond

to

would

prepositional concept,and

we

idle breath,but

Again some

use some

kind of combination

impliedby the fact that A and B are two, which is not true of either separately. can When we replace we say "A and B are yellow,11 cannot this but is A and B the proposition is yellow"; by yellow" to be

seems

"

"

be done

Thus

it

for

"

and B

A

best to

seems

combination, not which

would

be

called addition to

terms, and

a

are

two^

regardand

and relation,

;

on

as

not

the

contrary,A

is

one

This

of individual*.

is

one.

a definite unique kind expressing

combining A and

B

into

a

of

whole,

will in future be

uniquekind of combination to observe It is important only appliesto numbers in consequence one.

and B

that it of thdr

applies being

of Mathematics Indefinables

The

72

present,1 and

the

for

Thus

terms.

meaningless. As regards what is meant from what indistinguishable That A

and B

propositionsA is # u of the same propositions class whose only terms a "

A)

"

If

is

E

"

form

be

u

a

B

conceptof

;

conceptdenotes the terms

and B" "

are

the

are

those terms

but class,

from the concept of the class. of and, however, does not enter into the

are

bined com-

distinct

and class-concept

the

The

all other

all its " is the

"

this

in

the

true, but of which

then false,

and

class of which

of which class-concept

a

are

u

a

is

1

by and, it is conjunction.

numerical

a

and

1

indicated

concept of

the

*

are

A

are

by

certain way, and " A a " Thus A and B in justthat way.

combined

B

from

before called

we

onlymembers.

the

are

two, and

are

the combination

is denoted

IB is what

and

is,A

by

2

[CHAP,vr

notion

class,for

is

singleterm

a

conjunction.If

be

u

a

and class-concept,

a

meaning of

class,although it is not

a

numerical

a

only one

of the proposition conceptdenotinga single

be true, then "all "V is a " all its is a concept. Thus term, and this term is the class of which essential to a class is not the notion of and, but the being what seems

form

"

is

x

a

u"

"

denoted

conceptof

by some

a

class. This

bringsus

to

the intensional

of classes.

view

agreed in the precedingchapter that there are not but onlydifferent kinds of denotingconcepts different ways of denoting, different kinds of denoted have and correspondingly objects.We which constitutes a class ; we discussed the kind of denoted object have to consider the kind of denoting now concept. The consideration of classes which results from denotingconcepts is more and that in two generalthan the extensional consideration, respects. In the first place it allows, what the other practically excludes,the admission of infinite classes ; in the second place it these introduces the null concept of a class. But, before discussing there is a matters, purelylogicalpoint of some importanceto be We

72.

examined. If

u

be

a

is the concept"all class-concept, and

all constituents,

relation to t/, and to to

no

beginwith,that a

very

w,

"

common

is it

or

new

into two analyzable concept,defined by a certain

complex than

more

all ?/s

"

is synonymous

of the

use

a

itself?

u

with

plural.Our

M'S"

u

u\n

the indication of the

common

a

concept,namely to to

relation.

fact that

All

they both

"

men

have

a

may at least

observe,

according

questionis5then, as has certainly some

meaning of the plural. The word all meaning, but it seems highlydoubtful whether "

We

and

"

it

means

all numbers

certain

relation

definite than

more "

to

the

to

have a

in

class-

and

man number respectively. But it is very difficult isolate any further element of aU-ness which both share,unless we

take

as

It would

this element seem,

the

mere

then, that "all

fact that both iTs" is not

are

conceptsof classes.

into validly analyzable

all

71-73]

Classes

and u, and that The

in language,

remark

this

will

case

73

as

in

others,is a misleading

some

guide. every, any, some, a, and the. It might perhapsbe thought that a class ought to be considered, not merely as a numerical conjunctionof terms, but as a numerical conjunctiondenoted by the concept of a class. This complication, same

however, would a

a

is inadmissible in

numerical

view

our

conjunctionconsidered

when

Peano's

preserve

onlyterm it is the class is identified with the

is easy to grasp when which

but class-concept, evident

no

a

distinction which that

useful purpose, except to term and the class single whose

serve

distinction between

applyto

as

of

classes. It is

denoted

is either

so entity else is a complex of considered, or with the denoted and the objectdenoted is denoting together object ; what class. mean we a by plainly With regardto infinite classes, say the class of numbers, it is to be the observed that concept all numbers, though not itself infinitely denotes an infinitely yet complex, complex object.This is the inmost to deal of our with secret An infinity. power infinitely complex there be such, can not be concept,though certainly may manipulated the notion intelligence to ; but infinite collections, by the human owing be manipulatedwithout introducing of denoting, can any concepts of infinite complexity. Throughout the discussions of infinity in later

the

as

same

of the

Parts

not

presentwork,

air of

there is an it is forgotten, to

this remark

magic which

difficulties are

Great

with generally

concept as

idea of

the

nothing,and

associated

nothing. It is

that in

the

which interpretation

whether

in

class which

quite necessary

consider

makes

discussed

contradictions

the null-class is the it is

causes

borne

in mind

:

if

the results obtained

the

to

with

the

null-class,and

plainthat there

is such

nothingis

a

In

something. undoubtedlycapable it true a pointwhich givesrise to Plato's Sophist. In Symbolic Logic has no terms at all ; and symbolically some

fact,the proposition "nothing is an

be

doubtful.

seem

73.

of

should

sense

not

introduce

is nothing"" "

some

contradictions

such which

notion.

We

have

to

can

be

naturallyarise

avoided. that a in the first place, to realize, It is necessary denote althoughit does not denote anything. This occurs are

in propositions

which

the said

concept occurs,

and

concept may when which

there not

are

false. Or rather, are propositions of a denoting concept the explanation the above is a first step towards It is which denotes nothing. not, However, an adequate explanation. animals "chimaeras the or are for proposition Consider, example, These propositions appear even primes other than % are numbers." concerned with the not that they are to be true, and it would seem denotingconcepts,but with what these concepts denote; yet that is for the concepts in questiondo not denote anything. impossible,

about

the said

concept,but all

such

"

"

Indefinabksof Mathematics

The

74

[CHAP,vi

and that SymbolicLogic says that these conceptsdenote the null-class, in questionassert that the null-class is contained in the propositions certain other classes. But

extensional strictly

the

with

of classes

view

propounded above, a class which has no terms fails to be anythingat cannot subsist when all : what is merelyand solely a collection of terms all the

terms

Thus

removed.

are

of classes,or interpretation

else find

either

must

we

method

a

find

different

a

dispensingwith

of

null-class.

the

imperfectdefinition of a concept which denotes,but follows. All denoting be amended does not denote anything,may as and a is a classconcepts,as we saw, are derived from class-concepts; The denoting function. conceptwhen "# is an a" is a prepositional conceptsassociated with a will not denote anythingwhen and only The

when

above

"#

is

definition of

a

which

is false for all values of

#.

This

is

a

complete

anything; and all aV and that class-concept,

denotingconcept which

a

does not

shall say that a is a null null concept of a class. Thus for

in this case is

a"

an

we

denote

"

a

system such

as

Peano's, in

called classes are

technical difficulties really class-concepts, need not arise ; but for us a genuinelogical problemremains. The proposition chimaeras are animals interpreted may be easily of chimaera is formal as a by means meaning x implication, implies is an animal for all values of x? But in dealingwith classes we x have been assuming that propositions containingdH or any or every, to fonnal implications, were though equivalent yet distinct from them, and involved ideas requiring in the case Now treatment. independent of chimaeras,it is substitute the to intensional view, according easy pure to which what is really in stated is a relation of predicates : in the case animal is part of the definition of the adjective questionthe adjective what

are

"

"

"

chimerical to

(if we

denote the

ourselves to

word, contraryto usage, of chimaeras). But here again it is defining predicate

fairly plainthat chimaeras

allow

we

are

dealingwith

animals,but

a

which impliesthat proposition

propositionindeed,in the is not even present case, the implication By a negation reciprocal. we can givea kind of extensional interpretation : nothingis denoted chimaera which is a not denoted by But this is a very by an animal. roundabout O n it to the correct most interpretation. whole, seems while retaining the various other rejectthe propositionaltogether, that would be equivalent chimaeras. to it if there were propositions who have experienced of the nullBy symboliclogicians, the utility will this be felt view. class, as But I am not at present a reactionary what should be done in the logical the discussing calculus,where established practice the best,but what is the philosophical appears to me truth concerningthe null-class. We shall say, then, that, of the bundle of normallyequivalent of logical interpretations symbolic formulae,the class of interpretations considered in the present chapter, are

is not

this

use

the

same

"

73]

Classes

which

are

with null

75

fail where dependentupon actual classes, the that on there is no class-concepts, ground

We

we

are

concerned

actual null-class.

reconsider the

proposition may "nothingis not nothing"" true,and yet,unless carefully plainly of proposition handled,a source now

"

a

hopelessantinomies. apparently denotes

nothing.

The

denoted

i.e. it is not

Nothingis a denoting concept,which

conceptwhich

by

denotes is of

course

not

nothing,

itself. The

which looks so paraproposition doxical the Nothing, denotingconcept,is not nothing, i.e.is not what itselfdenotes. But it by no means follows this that there is an actual null-class: onlythe null class-concept from means

no

and the null But

than

more

conceptof

this :

class are

a

to be admitted.

has to difficulty

be met.

The

equalityof classlike all relationswhich are reflexive, and transitive, concepts, symmetrical, indicates an underlying i.e,it indicates that every class-concept identity, term a relation which all equalclass-concepts has to some also have to the term in term that questionbeing different for differentsets of but the for the various members of a single same equalclass-concepts, Now for all class-concepts which set of equalclass-concepts. not are class ; but where are we null,this term is found in the corresponding To this question several answers ? to find it for null class-concepts may of For be given,any be adopted. which may know what we now a now

new

a

"

class is,and

the class of all null

term may therefore adopt as our functions of all null prepositional or class-concepts we

and but genuineclasses, classes, have to

the

what

same

relation.

is elsewhere

to

shall be we class-concepts,

If be

we

to

either of them

then

called

a

These

are

not

null-

all null

class-concepts wish to have an entity analogous but to null class, corresponding

it is necessary (asin counting is identical for equalclass-concepts

wherever forced,

which to introduce a term classes) to substitute everywherethe class

of

class-concepts equal to a given for the class corresponding The to that class-concept. class-concept remains to the class corresponding fundamental, class-concept logically in The but need not be actually our null-class, symbolism. employed irrational in it to Arithmetic: in fact,is in some an ways analogous and if other classes, be interpreted the same cannot as on principles substitute must wish to givean analogous we elsewhere, we interpretation for classes other correlated

more

entities complicated

objectof

classes. The

technical; but failure

to

such

understand

"

a

the

in the

presentcase, certain

will be mainly procedure extricab will lead to inprocedure

of the symbolism. A very interpretation constantlyin Mathematics, for analogousprocedureoccurs closely of number; and so far as I know, example with every generalization either by no interpreted singlecase in which it occurs has been rightly So many instances will meet "us or by mathematicians. philosophers in the course of the presentwork that it is unnecessary to lingerlonger must the point at present. Only one misunderstanding over possible difficultiesin the

be

[CHAP,vi

of Mattiematics Indefinables

The

76

guardedagainst.No

vicious circle is involved

in the above

of doss

is first laid

generalnotion

of the null-class; for the

account

down,

is

existence,is then

not symbolically, by the notion of a class of equalclass-concepts, replaced philosophically* to what corresponds to and is found, in this new form, to be applicable which is i s class what not since now a null class-concepts, corresponds and classes of equal class-concepts classes wmpliciter null. Between

found

involve what

to

there is a

is called

breaks down

which correlation,

one-one

in the sole case

of the

null-class corresponds to which no ; and this class-concepts, whole for the fact is the reason complication. in the philosophyof which is A 74. question very fundamental fashion. Is be discussed in a more Arithmetic must now or lesspreliminary has many terms to be regarded or as itselfone a classwhich many? Taking the class as equivalent simplyto the numerical conjunction A and B it C and etc.,"1 seems and plainthat it is many ; yet it is quitenecessary class of null

"

that

should be able to count

we

speakof

a

class.

Thus

classeswould

each,and

one

be

to

seem

do

we

in

one

habitually and

sense

one

in another.

many

is

There class

as

certain

a

one,

e.g*.,all

class consists of

a

classes as

the temptationto identify men

more

the human

and than

be

can

and

many

the

wherever Nevertheless,

race.

term, it

one

class as

proved that

no

such

if it denotes a class concept of a class, any concept of the class which it denotes.

A identificationis permissible. as

one,

is not

the

same

as

is to say, classes of all rational animals,which denotes the human race one as term, is different from men, which denotes men, i.e. the identical with men, human But if the human race as race were many. That

follow that whatever

it would

the above

and to

denotes the

difference would

be

must

one

impossible.We

denote

the

other,

might be tempted

infer that Peano^

between a term and a class of which distinction, onlymember, must be maintained,at least when the is a class*. But it is more correct,I think,to infer an question

the said term in

term

is the

distinction between

ultimate

class

a

as

and

many

a

class

hold that the many are onlymany, and are not also one. be identified with the whole composedof the terms one may in the i.e.) But

where

of men,

case

can

there

we

the class

avoid

now

as

the

one

will be the human

as

The

one,

to

class

as

of the

class,

race.

contradiction

is

always to be feared, made a logicalsubject?

be something that cannot I do not myselfsee any way of eliciting a precisecontradiction in this the of concepts,we were In case case. dealingwith what was plainly the in with a complexessentially one ; entity presentcase, we are dealing into of units. In such "A and B are a capable analysis as proposition two," there is no logical subject:the assertion is not about J, nor

*

This

conclusion

Archivfttr*y*t.Phil

is

actuallydrawn by Frege See Appendix. iy p. 444.

from

an

analogousargument:

73-76] about

B,

Classes about

nor

and B.

about A

the whole

Thus

and only composed of both,but strictly that assertions are not necessarily seem be about many subjects ; and this removes

it would

but single subjects,

about

77

the contradiction which of

may

in the

arose,

of

case

which

We

said of the these

Are

denoted by a objects one objects

Socrates,nor

not

conclude

that

every one I think

is

possibilit the im-

no

every man, some neither? Grammar

or

many or the natural view,

Plato,nor

any

is true

in every case, but is neither 1 nor 2 nor

number

of the in

it is easy to conclude that any number but at first sightcontradictory, proposition

yet know

I do not

conceptevery

man.

distributive impartial

an

any other

whence

ambiguityin any, and more number." There one some

man,

is,which one ? Certainly objection other particular Can we person. As well might we conclude that

is denoted?

one

is to be

and any num. all as treats them

man,

denoted,which in fact

Any

unless

arise.

is denoted

one

manner.

them

what ask,as suggested by the above discussion,

may

But to this

one.

about

be feared does not

to

was

75.

a

concepts,from

turned making .theywere This here contradiction the subjects. absent, impossibility being

into

assertions

number, particular

is not

number,

one

any

from an really resulting expressed by any number is not correctly in this subject which are, however, puzzles **

to solve.

how

in regardto the nature of the whole remains logicaldifficulty composed of all the terms of a class. Two propositions appear selfevident : (1)Two wholes composed of different terms must be different ; It follows (2) A whole composed of one term onlyis that one term. is term that class that the whole composed of a classconsidered as one A

considered

composed firstof

as

one

term, and

is therefore identical with

of the class ; but

of the terms

however, is

difficult. The

not

first of

whole

this result contradicts the

The supposedself-evident principles.

our

the

in this case,

answer

is onlyuniversally principles

our

wholes

simple. A given than two if it has more whole is capable, beinganalyzedin a so constituents, of ways; and the resulting long as analysis plurality true

when

all the terms

composingour

two

are

parts,of

be different for different .ways Of analyzing.This proves that different sets of constituents may constitute of our difficulty. the same whole, and thus disposes

is not

pushedas

76. of which

far

as

will possible,

Somethingmust it is

a

be said

member, and

as

to the relation of

as

a

term

to

to the various allied relations.

the allied relations is to be called e, and is to be fundamental which of them extent optional Logic. But it is to some

in

a

class

One

of

Symbolic

we

take

as

fundamental. symbolically the fundamental Logically,

relation is that of

in Socrates is human expressed in that the Chapteriv, is peculiar "

" "

a

relatum

and predicate, subject

relation which, cannot

be

as

we

as regarded

saw a

in term

of Mathematics Indefinabks

The

80

[CHAP,vi

of themselves, predicates, negative though,by introducing predicated it will be found that there

One

of themselves. predicable

are

which many instances of predicates at least of these,namelypredicability,

justas

are

is not negative : predicability, as propertyof beinga predicate, of itself. But the most i.e.it is a predicate is evident, is predicable, instances are negative : thus common non-humanityis non-human, and or

the

so

on.

The

which predicates

therefore, onlya selection from among suppose that they form whether let us examine

themselves

of predicable and predicates,

not

are

are,

it is natural to

predicate.But if so, defining belongsto the class or defining predicate

class havinga

a

this

for that is of itself, it is not predicable class, if it is not predicable the characteristic property of the class. But of itself, then it does not belongto the class whose defining predicate On the other hand, if it it is,which is contraryto the hypothesis. If it belongsto the

not.

does not

belongto

i.e. it of itself, predicable

is

one

it is not

it is.then defining predicate

the classwhose

of those

that predicates

not

are

dicable pre-

and therefore it does belong to the class whose themselves, Hence from it is againcontraryto the hypothesis. predicate defining of

"

either

we hypothesi^

this contradiction

merelyas

it

deduce

can

its

I shall contradictory.

for the

in

return

to

introduced

present,I have

Chapter x; in distinguishing is likely to no subtlety

showing that

be

excessive.

lengthydiscussion. A class, up the above somewhat in extension; it is either to be interpreted agreed,is essentially To

79. we

sum

term, single

a

when not

or

which

is indicated

by the word and. But practically, though extensional method this purely can theoretically, onlybe applied terms

connected

are

to finiteclasses. All as

of terms

that kind of combination

the

whether classes,

finite or

be obtained can infinite, denoted by the plurals of class-concepts objects men, numbers, "

kinds of we two Startingwith predicates, distinguished is human and "Socrates has typifiedby "Socrates proposition, human the second as humanity,"of which the first uses predicate, of a relation. These two term classes of propositions, as a though not so relevant to Mathematics their are as very importantlogically, from human, we distinguished derivatives. Starting (1)the class-concept etc. points,

"

which

man,

differs

if at all,from human; (2) slightly,

denotingconceptsaU

men,

every man,

any

man,

a

man

the

and

some

various man

;

denoted by these concepts, of which the one denoted by (3)the objects called the class as aU men was was (theconcept) many^ so that all men called the conceptof the class; (4) the class as one, i.e.the human race. had

We

(1)

a

the above

upon has

also

"

classificationof

about Socrates,dependent propositions

and distinctions,

Socrates is-a man"

humanity";(")

with approximately parallel

is nearly, if not

"Socrates

identical with quite,

is a-man"

expresses

"

them: Socrates

between identity

Clashes

79]

78,

and

Socrates

men,"

among of

a

It work

which

appeared largely

logically may

has

be

terms,

no

throughout with

more

regarded

which

belongs

class

the

denoted

terms

individual

an

takes

requires, class,

Socrates

u

of

relation

the

proposition

a

(4)

men;

of

one

is

fiction,

a

that,

class-concepts

fundamental

as

our

for main

the

general

the

any

conclusion

null

are

symbolic classes

intension,

principles

alone

agreed

of

expresses relation

of

the

that

null-

class-concepts. must

treatment

and

one

plurality

the

possibility

there

though

is

Socrates to

which

We

many.

although and

as

**

owing

race,"

and,

as

(3)

;

difficulties

class,

not

man

a

human

the

to

one,

by

raises

its

to

as

81

extension

Mathematics

and

are

this

;

in

the

present

chapter.

YII.

CHAPTER

FUNCTIONS.

PKOPOSITIONAL

80. "the

IN

kind

of

the

notion.

not

defined of

did

It

to, it would

by

often

is

a

necessity is

this

general

The

with

connection

and

of

formal

implication.

such

to

be

that

is

the

and difficulties, with

a

capable

might

that

proposition the tf's such further objections,to be class

obtained

as

and

extension

that

be the

so-and-so11

and

that

remains

JfZo, where

reduce if

this we

R

is

what

be

thought,

an

is

tJiat must a

a

in

a

define

of

the

to

be

by

the

sight,

notion

class a?

is

"x

should

we

a*

an

from

Apart

is not

substitute

the

for

in

cedure pro-

afs such

"

that

class,

if Peano^s

necessary,

a

so-and-so," the

prepositional

emendation often

be

given relation must

is full

put forward

are

first

at

the

are

a

formally

that

formal

proposition to

ask

it is

such

x

purely

such

with

that

"

so-and-so

this

when

advocate

to

scope

connection

subject

The

obtained an

its

such that so-and-so," which class-concept"# from the predicate such that so-and-so

genuine

because

and

in

truth.

the

permissible,that

regarded as rather, "being

But

such

Thus

its

immediately, it is to be observed that is the genuine class,taken

whereas

many,

be

necessary

for

as

is

x

noticed

from

class-concept.

the

is to

#

assertions

present chapter

used, in fact, to

Peano

;

the

In

I intend

such

definition

of

which

in their

"

or

tion explana-

already explained

been

examined, critically be to investigated.

confidence

of

notion

has

assertion

an

doctrines

limited

very

The

may

of

object

an

The

subject-predicateproposition. to be sought in the theory of

notion

legitimacy are

classes

but

extension

class

a

class

that.

such

the

as

of

notion

restrict, the

recognize

to

of

purposes

subject-predicate the

to

as

indicate

to

for

from

greatly

necessary

of

means

class, and

a

view

our

made

was

derived

as

affect

not

if adhered

itself; but

called

be

to

endeavour

an

considered

were

This

propositions.

is

that

object

classes

discussion

and

preceding chapter

the

the form

be,

the

and u

x

answer

a

is

function

has

put

a

is

:

a

being

containing x. the point

propositions as We

term.

without must

form

made

such

given *

an

been

before a

latter

"

be

cannot

using

such

such

that

that each

;

81]

80,

Functions Propositioned

83

of its terms, and no other terms, have the relation R to a. To take from life the children of Israel : defined class examples are a daily by certain relation to Israel, and the class can a be the defined as only such that

terms

who

to

theyhave

which, and

or

in terms define,

different values of these relation

a

x.

Then

such that is defined

which when

entitywhich

the

type

"x

employed. already

the relation which

as

a satisfying

we may go further: given a class a, we of a, the class of propositions is an a" for *x It is plainthat there is a relation which each

any

of proposition

of

But

has to the x propositions in questionis determinate

relation J2. is

of

Such that is roughlyequivalent

representsthe generalnotion

function. propositional cannot

this relation.

is

is

And

is

a

a?

an

it,and that the Let

given.

referent with

a

the

in

occurs

But

holds between

"

x

is

of

the notion

onlybe

itselfcan and

a*

an

R

respectto

here

relation R

call the

us

x

for all

values of #, and does not hold between any other pairs of terms. Here which is chiefly such that againappears. The point importantin these remarks

been

have

defined. a

of propositional functions. indefinability

is the

admitted,the generalnotion of Every relation which is many-one,

given referent has onlyone function

is that

question.But

the function is

in the is presupposed

without

a

and symbolism,

a

we

type can

x

is

onlyanswer

a? if

an "

we

ask

is defined

what

In

the

assertion

of

case

relation in

a

by.means

togetherwith

of it function

a

of this

is said to

term

in

involved

of propositions

are propositions

in which all propositions

identified with

is easily

notion

generaldefinition of

and here the notion to be defined reappears. the indefinable element involved Can 81. be

the

be defined

cannot

functions alreadyoccur. propositional the

functions

by the proposition,

vicious circle: for in the above

**

these

I.e.every relation for which relatum,defines a function : the relatum

of the referent which

where

one-valued

When

type,

be

**

a

;

tions funcpropositional

the

notion

assertion made

of every

or an a givenassertion, concerning containing proposition is I far to The so can as only alternative, see, acceptthe every term ? function itself as indefinable, and for generalnotion of a propositional is certainly the best ; but philosophically formal purposes this course of analysis, and we have to the notion appears at first sightcapable this examine whether or not appearance is deceptive. in We verbs,in Chapteriv, that when a proposition saw discussing these constituents is completely analyzedinto its simpleconstituents, of reconstitute it. A less completeanalysis not taken togetherdo and assertion has also been considered; and into subject propositions and the proposition. A subject less this analysis to destroy does much do not, it is true, constitute a an assertion,if simplyjuxtaposed, asserted of the the assertion is actually but as soon as proposition; that the proposition subject, reappears. The assertion is.everything the verb omitted: is when the subject of the proposition remains

Indefinables of Mathematics

The

84 remains

the other terms from

the

of the

into

turned

is not

of legitimacy

noun

at

or

;

which distinguishes relation a relating proposition

relation abstractlyconsidered.

same

verbal

a

indefinable intricate relation to

retains that curious

the verb

rate

any

asserted verb,and

an

[CHAP,vn

of

this notion

be every proposition

is the

It

is

which

assertion

now

scope and examined.

be

to

regardedas an concerning any term limitations necessary as to the form of the and the way in which the term enters into it? proposition In some into subject simple cases, it is obvious that the analysis Can

occurringin

and

it, or

assertion

is

are

legitimate.In "Socrates

distinguishSocrates should

admit

Plato

something that

and

that the unhesitatingly

Aristotle.

or

assertion

Thus

this assertion, and containing is represented number by x "

assertion is to

must

appear as suffer" contains the

a

man,"

we

him;

be

class of

a

plainly

can

about

thing may

same

consider

can

we

is

is asserted

we

said about

propositions a typical

this will be the class of which is

It is to be observed

man."

a

assertion,not

as

term

assertion,but

same

thus

:

used

as

"

that the

be

to

man

a

term, and

this

does not belongto the class considered. In the case proposition of propositions assertinga fixed relation to a fixed term, the analysis seems than a yard long, for example, equallyundeniable. To be more is a perfectly definite assertion,and we consider the class of may in which this assertion is made, which will be represented propositions function "a? is more by the propositional than a yard long." In such phrase^as "snakes which are more than a yard long,"the assertion referred to a variable appeai-s very plainly;for it is here explicitly not asserted of subject, definite subject.Thus if R be a fixed any one relation and

a

fixed

a

term, ...#"

is

a

definite perfectly

assertion.

(I placedots before the J?, to indicate the placewhere

the

be inserted in order to make a whether a relational proposition can

be doubted

must

the relatum. in the

of

case

82.

More

a

part, I hold that this

difficult questionsmust

has

is

a

now

cerning con-

except

questionis

be

considered.

impliesSocrates

man

or

better

is

a

Is

such

mortal,"or

variable, we

impliesSocrates has a father,"an assertion concerning It is quitecertain that,if we replace Socrates obtain a propositional function ; in fact,the truth not ?

formal implication, corresponding which Now

be done

can

this

of this function for all values of the variable is what at

assertion

an

wife

a

Socrates

by

regardedas

subject

have discussed relations *.

propositionas "Socrates

"Socrates

be

subject-predicate propositions ; but

postponeduntil we a

For my

It may proposition.)

does not,

is asserted in the

might be thought relation between a two functions. propositional if intention, possible, to explain functions propositional as

first sight,assert it

by means

was

our

of assertions;

hence,if *

intention

our

See

"

06.

can

be earned

out, the

81,

82]

above

Functions Proportional

must propositions

be assertions

85

Socrates. concerning

however, a very great difficulty in so regarding them. to be obtained

a

that,in

the restoring

in the two

what

Of this

mortal."

In

the proposition,

where places

matter

not

assertion

An

was

a

is a implies...

man

is,

proposition by simplyomittingone of the terms B ut when we omit Socrates, proposition. obtain we

the

in occurring "...is

from

There

term

this formula of necessity

choose,but it must

we

however,no requisite,

should

term

same

dots indicate the

be substituted It does

term.

a

be identical in both

whatever

trace

it is essential

places.

in the would-be

appears all mention of the term to be since appear, inserted is necessarily omitted. When is inserted to stand for an x the variable, the identity of the term to be inserted is indicated by the

and assertion,

trace

no

of the repetition available.

can

letterx

;

but in the assertional form

method

such

no

is

And

it seems yet,at firstsight, very hard to deny that the in questiontells us a fact aboid Socrates, and that the same proposition

about

fact is true

Plato

or

a

that "Plato

is,in

other,the

some

sense

or

is of Socrates. proposition

would

be that the

the number

or plum-pudding

undeniable certainly

is

a

same

The

impliesPlato

man

function of Plato

natural

as

It is

2.

is

mortal"

a

previous

our

of this statement interpretation

relation as the has to Plato the same proposition other has to Socrates. But this requires should regard the that we function of its relation in question definable by means as prepositional function to the variable. Such a view,however, requires a propositional than the one more are we considering.If we represent complicated "x is a man x is a mortal" implies by fa, the view in questionmaintains definite that $ocis the term havingto x the relation R, where R is some

The

relation.

and y,

of

x

R

to x?

has far to

of this view is as follows

formal statement is identical with

"y

than what greatercomplexity

propositions may

in the

without function, and

a

For

all values

to "y has the relation equivalent since it not do as an explanation,

it

have

to

was

explain.It

would

seem

certain constancyof form,

a

pressed ex-

a given propositional they are into a the to propositions beingpossible analyze

instances

fact that its

:

is

$x*

It is evident that this will

follow that

constant

one

variable factor.

Such

a

view

of

is curious

and

difficult:

tions, constancy of form, in all other cases, is reducible to constancy of relanotion the in but the constancy involved here is presupposed in the therefore be explained and cannot of constancy of relation, usual way. The same

I think, will result conclusion,

from

the

case

of two

is a xRy" variable. It seems while x and y are independently constant relation, variables: function of two independent evident that this is a propositional of the in the notion of the class of all propositions there is no difficulty of all those members form xRy. This class is involved or at least

variables.

The

instance simplest

of this

case

is

where

R

"

the class that

are

true

are

involved" in the

notion

of the

classes of

of Mathematics IndefiTiabks

The

86

[CHAP,vn

tingly respectto J??and these classes are unhesitaand masters in such words as parents and children, admitted and wives,and innumerable other instances from servants, husbands and conclusions, notions such as premisses also in logical as dailylife, such All notions depend upon the class and so on. and effects, causes relata with

referents and

of

by zRy" where propositions typified

variable.

it is very difficultto

Yet

"assertion R

regardxRy

while

x

and

into analyzable

as

are

y

the

that this and y" for the very sufficientreason i.e. its direction from x to y9 of the relation,

x concerning

the destroys

view

is constant

R

sense

symmetricalwith respectto Given a and y, such as "the relation R holds between x and y? x distinct propositions are relation and its terms, in fact,two possible. it becomes if we take R itself to be an assertion, Thus an ambiguous to avoid ambiguity,we assertion: in supplyingthe term**, if we are We relatum. decide which is referent and which must quite may before was as ; but explained regard .Ry as an assertion, legitimately We here y has become constant. may then go on to vary "/, considering the class of assertions ...Ryfor different values of y\ but this process is indicated by the does not seem to be identical with that which function xRy. of x and y hi the propositional variability independent the element the of variation an Moreover, suggestedprocess requires in an assertion, namely of y in Ry, and this is in itself a new and

leavingus with

assertion

some

.

which

is

.

.

.

.

difficultnotion. curious

A

pointarises,in

this connection,from

consideration,

the

often essential in actual Mathematics, of a relation of a term Consider the propositional function xRx, where R is a constant Such

functions

requiredin considering, e.g.^the

are

of self-made men; or for which it is equalto

a

itself.

relation.

class of suicides

or

the values of the variable again,in considering a

which certain function of itself,

It Mathematics. necessary in ordinary this case, that the proposition contains it is analyzed into

to

term

x

and

an

often

be

evident,in exceedingly

seems an

may

element

assertion R.

which Thus

is lost when

here

again,the

function must be admitted as fundamental. propositional 83. A difficultpointarises as to the variation of the concept in a of the type aRb, Consider,for example,all propositions proposition. where

a

seems

and 6

and

no "

is

moreover,

fixed terms, and doubt that the

and legitimate,

are

R

is

a

variable relation.

"relation class-concept

There

between

a

that there is a

class ; but this corresponding of such propositional functions as aBb9 which, i n actual frequently required Mathematics,as, for example,

admission

countingthe

given require,an are

are

to

reason

the requires in

6

number

of many-one relations whose referents and relata classes. But if our variable is to have, as we normally

unrestricted field,it is necessary to substitute the profunction " R is a relation implies aRb? In this proposition positional the implication involved is material,not formal. If the implication were

83]

82,

Functions Prepositional

formal,the proposition would

be

87

function of R, but would be to the (necessarily equivalent "All relations hold false)proposition: between a and 6." Generally "aRb have some such proposition we as not

a

implies "f"(R)providedR is a relation," and we wish to turn this into a I f implication. "f" for all values of jff, (R) is a proposition our objectis effected by substituting "If R is a relation1 implies'aHbf then ^ (R)." Here R can take all values*, and the ifand then is a formal while the implies is a material implication. implication, If "f" (R) is not a propositional function,but is a proposition onlywhen R satisfies-f(Jff), where ^(R) is a propositional function impliedby "R is a relation" for all values of R, then our formal If be put in the form can implication 'R is a relation' implies aRb, then, for all values of R, ^r(R) implies ""(#)""where both the subordinate implications material. As regards are the material implication R is a relation impliesaRb? this is always a whereas aRb is only a proposition proposition, when R is a relation. The new function will onlybe true when R is a relation propositional

formal

"

"

"

which

"

does hold between is false and

'

and b

a

:

when

R

the consequent is not

is false; implication

when

R

is

is not a

the relation,

so proposition,

relation which

a

a

does

not

cedent ante-

that

the

hold between

a and ", the antecedent is true and the consequent false,so that again the implication is false ; onlywhen both are true is the implication true.

Thus

in

the class of relations holdingbetween and b, the a defining is to define them the values satisfy formallycorrect course as ing-"R is a relation implies aRb an which, though it contains a implication i s variable, not formal,but material, beingsatisfiedby some onlyof the real values of R. The variable R in it is,in Peano's language, possible ""

"

and not

apparent.

for involved is: If $x is only a proposition generalprinciple values of #, then some $x impliesfox:impliesfoe is a proposition for aU values of x^ and is true when and onlywhen for is true. (The both involved In for" are material.) some implications cases, "foe implies is "JS will be equivalent to some function as simpler -^.r(such propositional The

"

a

relation Such

appears an

"

a even

"

**

*

which may then be substituted for itf. in the above instance), function as "R is a relation impliesaRb" propositional

less capablethan

assertion about jK,since

into R and previousinstances of analysis should have to assigna meaning to a..J)? "

we

by a anything,not necessarily relation. There is here,however, a suggestionof an entitywhich has be It may not yet 'been considered, namely the couplewith sense. and yet such phrasesas there is any such entity, whether doubted where

the blank

*

It is necessary R is not a relation. t A

false3is type

in

be filled by

space may

to

assignsome

meaning (otherthan

for every value false,being what is denoted

function,though propositional not

itself true

which question/*

or

is not

itselfa

proposition.

a

to proposition)

of the

a

Kb

when

variable it is true

by "any

or

of the proposition

The

88

[CHAP,vn

to show that its rejection holdingfrom a to 5" seem would lead to paradoxes. This point,however, belongsto the theory and will be resumed in Chapterix ("98). of relations, From functions what has been said,it appears that prepositional It be accepted follows that formal implication ultimate data. must as and the inclusion of classes cannot be generally of a by means explained relation between assertions, function although,where a prepositional asserts a fixed relation to a fixed term, the analysis into subjectand and not unimportant. assertion is legitimate It onlyremains to say a few words concerning 84. the derivation functions. of classes from prepositional When consider the *r'ssuch we that ""#,where is a prepositional function,we are introducing a "f"a? notion of which,in the calculus of propositions^ a only very shadowyuse

"R

is

Indefinable^ of Mathematics

relation

a

is made

I

"

the

mean

of truth.

notion

We

are

considering, among

all the

of the type c^r,those that are true: the corresponding propositions values of x givethe class defined by the function $x. It must be held,I think,that every propositional function which is not null defines a class, which is denoted by "#'s such that ""#." There is thus will always a concept of the class,and the class-concept corresponding be the singular,x such that ""#." But it indeed the may be doubted with contradiction which I ended the precedingchaptergivesreason for whether is there of such classes. doubting alwaysa defining predicate the from contradiction in t his Apart question, point might appear to be verbal: it such that an x merely being "f"x" might be said,may always be taken to be a predicate.But in view of our all contradiction, remarks on this subjectmust be viewed with caution. This subject, will be in resumed however, Chapter x. 85. It is to be observed that, accordingto the theory of profunctions here advocated,the "f" in fat is not a separate and positional distinguishable : it lives in the of the form ""#,and entity propositions cannot survive analysis. I am highlydoubtful whether such a view does "

"

"

"

not

lead to

has

the merit

a

but contradiction, of

enablingus

it

appears to be forced upon us, and it to avoid a contradiction arising from the

opposite view. If "f" were a there would be a distinguishable entity, propositionasserting which we may denote by ""(0) ; there "" of itself, would also be a proposition not-""(""), denying""("f").In this proposition we regard "]6as variable; we thus obtain a propositional may function. The

questionarises: Can

the assertion in this propositional function be asserted of itself? The assertion is of self, non-assertibility hence if it can be asserted of itself, it and if it

cannot,

This

contradiction is avoided

part of

by

the

recognitionthat

cannot, it the

can.

functional

function is not an independent propositional entity. As the contradiction in questionis closely analogousto the other,concerning not predicable predicates of themselves, we may hope that a similar a

solution will

apply there

also.

CHAPTER

THE

86.

discussions

THE

of the

nature

VIIL

VARIABLE.

of

precedingchapter elicited

the

variable

apparatus of

no

;

dispensewith the consideration of in a propositionwhile the other variable

is

perhaps the

the

elements

or

one

remain

of all notions

also one of the most difficultto understand. certainly if not the deed, belongsto the present chapter.

theory as to the is previous discussions, while

the

but

of

must

Thus

term.

the

the

the

in .which (f" the

$x denotes

x

denoting which, have

Let

87.

us

observe,

"

Any "

u

not

We

I shall

is

that

us

of

other

any

propositions this of

stancy con-

class

a

the

general

the

former,

set

any

the

propositionof the x

notions

where

of

variable.

any

and

x.

oi

This

theory, objectionablethat 1

it forth

more

in detail.

oi explicitmention will : formal implication cussed instance an alreadydisthe

to

denoting, where

is

"^r,

any ""#, different values of

least

that

recur

term, and

by is the

x

in Mathematics

occur

with

now

begin with,

to

is

a

a

class and

b

a

class

have *

a

say

by

is the difficulties,

required. Let

connection

of classes.

given term

any term, a certain form will contain that

constant

is denoted

may

than

our

Taking

resultingfrom propositions

imagine.

any, same, etc., need that is express all in

of

terms

is denoted

what

what

We

occurs.

is full of

able to

been

latter.

to

admit,

I

class

in

functions, the prepositional presupposedin the notion of the

are

a

replacedby

The

defined

be

propositionsof

variable,is

class of

in addition

Thus

be

fundamental

more

can

of the

function,is prepositional

form

is

latter

in terms

class of

#,

form

constant

class,for

of any

member

following. When

be called constancy of form, and may The idea. notion be taken as a primitive

the former

not

the

;

attempt,

what

propositionsof

notion

variable,which

a

have

of form of

outline

in

The

results from

the

that term proposition, may unchanged. remaining terms are

obtained

so

in

term

as

occurs

of

nature

The

unchanged.

it is

The

to

us

elements

more

mathematical distinctively

most

enables

assertions of

varying

mental funda-

the

belongsto any b V implies*a? is a a

equivalentto

is if

"

"

*#

is

an

a1

impliesthai

of Mathematics Indefinables

The

90 "

Any

belongsto

a

6" is

a

is a b9 say u, such that "

Any

belongsto

a

that

*

is

j?

is

x

*a? is an

"

'there implies

a?

w^ "*;

a

"there is a ",say u, such

to equivalent

b" is

some

'x implies

a*

an

to equivalent

[CHAP,vm

is

u"

a

w;

remainingrelations considered in Chapterv. The constitute definitions of questionarises : How far do these equivalences these notions involved in the symbolism any, a, some, and how far are

and

so

for

on

the

itself? the characteristic notion standpoint, of stating it is the method Moreover generaltheorems, the from intensions! different propositions something

of Mathematics.

which

alwaysmean

to which

the to

as logicians

such

meaning of meaning of

the

assertion equivalent

an

therefore,I shall -not

any

man

about

the

conceptman,

are

not

about the

further.

argue

That

as

"

name

the

That

is different from

or

men

self-evident truth

a

John

about propositions

that

all

assertion about

an

to reduce them.

Bradley endeavour

Mr

confess,to be

I must

me,

the formal

variable is,from

The

appears the fact

evident

as

John.

This

point,

variable characterizes

admitted, though it is not generally generally metic, to be presentin elementaryArithmetic. ElementaryArithperceived characterized fact that the numbers is as by the taught to children, is the it to in answer sum are constants; occurring any schoolboy"^ But the fact obtainable without propositions concerningany number. that this is the case can only be proved by the help of propositions Arithmetic about any number, and thus we are led from schoolboy's to the Arithmetic and proves general letters for numbers which uses will be

Mathematics

theorems. be

at

seen

is from childhood^s enemy may very differentthis subject in such works as those of Dedekindf and StolzJ. Now

How once

the difference consists

simplyin this,that We being constants.

variables instead of n, not

concerning3

essential to absolutely nature

or

4

any

or

any

other

our now

numbers prove

have

theorems

number. particular

theory of Mathematics

now

to

become ing concern-

Thus

understand

it is

the

of the variable.

conceived dynamically, no doubt, the variable was as Originally, thing somethingwhich changedwith the lapseof time, or, as is said,as somewhich successively all values of a certain class. This assumed view cannot be too soon dismissed. If a theorem- is provedconcerning not be supposedthat n is a kind of arithmetical Proteus, ", it must which is 1 on Sundays and 2 on Mondays, and so on. Nor must it be its all values. If n stands for assumes supposedthat n simultaneously that is cannot it is 2, nor yet that we that n 1, nor yet any integer, say

*

Here

implies py t

Wo*

"there and

e

x

is

a

is a c'

Jtind and

wots

c," where

impliesp

is defined as equivalent to c is any class, for all values of "*"" then p is true."

gotten die Zahlen

?

1885. J AUgemeine Arithmetik,Leipzig,

Brunswick,1893.

"If

p

87-89]

The

Variable

91

it is any other particular number. In fact,n justdenotes any number, and this is somethingquitedistinct from each and all of the numbers. It is not

that 1 is any

true

number, though it is

whatever

that

true

holds of any number holds of 1. The in short,requires the variable, indefinable notion of which was explainedin Chapter v. any We 88. what may distinguish may be called the true or formal variable from

the restricted variable.

the true variable a

if

;

u

restricted variable.

defining concept of value

every

of

a

be The

Any term is a conceptdenoting all terms, any u denotes a class not containing terms included in the objectdenoted by the

variable

a

about such

called the value* of the variable

are

variable is

a

any number

as propositions they offer no implication, function x is a prepositional "

formal the

"

for all values of it is object, that

may

number"

"any concept "any number" of

This

one

number that

x

The

cannot

be

"

number

thej"e a

a

be

taken or

2

or

are

at

all.

3

or

definite

a

any number there are, so

fact is that the

The

number, but

one

be

to

all the numbers

number

denote

does

is

*

is justthe distinctive pointabout

but class,

a

for

"any

Yet

thus

certain

number."

is

plainthat it is not identical with 1

be mentioned.

that

one.

if

But

x\

:

difficulty a by Interpreted for they assert merely that difficulty, holds is a number number x implies There

constant.

not

a

particular

any, that it denotes

a

term

with

distributive manner, no preference impartial Thus and is no one a number, over althoughx it is recognized as so soon yet there is here no contradiction, in

an

another.

term

is x, is not

one

notion

definite term. be avoided,exceptin regard the introduction of a suitable hypothesis,

of the restrictedvariable

can

functions,by prepositional the the restriction itself. But in respect namely hypothesis expressing of prepositional functions this is not possible. The x in ""#,where "f*x itself is a prepositional is an unrestricted variable ; but the "f"x function, to

is restricted to the class which that the class is here vicious

could

we

call $.

may

fundamental, for

to discover any common circle, be defined,since the statement

we

(Itis to be remembered

found it

without impossible,

characteristic of any

common

by

which

the

a

class

characteristic is

function.)By making our x alwaysan unrestricted prepositional identical in we can variable, speakof the variable,which is conceptually Logic,Arithmetic, Geometry, and all other formal subjects.The terms dealt with are alwaysall terms ; only the complexconceptsthat occur the various branches of Mathematics. distinguish of any, some, We 89. to the apparentdefinability return may now Let a and b be class-concepts, and a, in terms of formal implication. and consider the propositionany a is a b.n This is to be interpreted is a 6." It is plain that,to beginwith, as x meaning a: is an a implies the two propositions the same do not mean thing:for any a is a concept need not be an a. x denotingonlytfs,whereas in the formal implication with But we altogether any a is a 6," might, in Mathematics, dispense

itselfa

"

"

"

of Mathematics Indefinables

The

92 and

the symbolically

best

this is, in- fact, implication: questionto be examined,therefore,

the formal

ourselves with

content

[CHAP,vni

The

course.

and a enter into the formal far,if at all,do any and some implication?(The fact that the indefinite article appears in "# is

is : How

for these are merelytaken as typical irrelevant, have, to begin with, a class of true functions.)We propositional *

an

and

a

"

is

x

b

a

""

is

of each asserting propositions, b.

a

that if it is an

term

constant

some

We "

class." We

of any

the truth

assert

this restricted variable.

But

included among

term

in order

obtain

to

a

it is

of proposition

then consider the restricted variable, any "

the

this

the values of

suggestedformula,

from the proposition as a whole necessary to transfer the variability " is an a implies obtain its variable term. In this way we x oc is b"

it is to

But

the

for essential,

genesisremains

here

not

are

we

"

is

expressinga

and is a b" functions If x an a; a pr-opositional the both times. should not require same x were we expressed, Only function is whole formula. the Each one involved, namely propositional relation the of one term class of the proof a proposition expresses is a b ; and we may is an a of to one function x x positional say, "

relation of two

"

this

"

"

if

"

"

choose,that the whole formula expresses

we

is

"#

a"

an

to

of "#

term

sume

the first x

again,we may say that term, namelythe firstx.

have

We

as

relation of any do

not

variable

a

so

of

term

much

have

implication.Or

term, but the second is some

is any a

a

b."" We

a

variable

containinga implication

an

is

class of

not containing implications

of this class. If any member consider any member is true, the fact is indicated by introducing a taining contypicalimplication and variables,

a

we

variable.

This

is what implication typical

is called

a

formal

of a class of material implications. it is any member Thus implication: in mathematical it would seem that any is presupposed formalism,but in legitimately replacedby their equivalents terms of formal implications. of in terms 90. Although some by its equivalent may be replaced There is, any, it is plainthat this does not give the meaning of some. in fact,a kind of duality of any and some : given a certain propositional function are asserted, to the propositional function,if all terms belonging

that

we an

and

some

be

may

any, while if

have

one

at least is asserted

The

we existence-theorem), get some.

comment, to

a

as

that

mean

in

"

(j"xis

might equallywell

is

x

true

have

a

man

implies"r taken

to

is called

out proposition "j"xasserted with-

for all values of been

(whichgiveswhat

is "r

is to be taken mortal,1' (or for any value),but it a

that

mean

fa

is true

for

In this way we x. might construct a calculus with two the and the disjunctive, in which the variable, conjunctive

some

value of

kinds

of

latter

would method 91.

occur

wherever

an

existence-theorem

was

to

be stated.

But

this

does not appear to possess any practical advantages. It is to be observed that what is fundamental is not particular

but functions, propositional

the

class-concept propositional function. A

89-93]

The

Variable

93

function is the class of all propositions which arise from prepositional the variation of a single but is this not be considered as a to term, for reasons definition, explainedin the precedingchapter. 92. From functions all other classes can be derived prepositional with by definition, the help of the notion of such that. Given a profunction ""#, the terms such that,when x is identified with positional of them, $x is true, are the class defined by ^r. This is the any one class as many, the class in extension. It is not to be assumed that every class so obtained has a defining will be discussed : this subject predicate afresh in

Chapterx.

extension

is defined

But

it must

think,that a class in function,and in particular by any prepositional form a class, since many functions (e*g* propositional

that all terms all formal

true are implications) it is necessary that implications,

of oil terms. the whole

truth defines the class should be is

for possible

functions.

For

every value if a example,

their common respectively, the producthas to be afterwards. in

be assumed, I

Here,

as

with

function propositional

kept intact,and not,

formal whose

where

even

this

of x, divided into separatepropositional and b be two classes, defined by "$"x and tyx

where partis defined by the product"f"x-$\r, .

made

If this is not

and *fyx. Thus "$"x

for every value of "r, and then x varied have the same done, we do not necessarily

but functions, multiply propositional class of function the is new products propositional and of corresponding the to functions, belonging propositions previous is by no the productof fac and tyx. It is only in virtue of means a definition that the logical productof the classes defined by $x and tyx is the class defined by far "tyx.And wherever a proposition containing

x

we

do not

: the propositions

.

an

apparent variable is

is asserted is the

truth,for all

function corresponding propositional and is never a relation of propositional proposition,

values of the variable to the whole

what asserted, of variables,

or

the

functions. 93.

It appears

from

the

above

discussion that the variable is

a

analyzecorrectly. complicated by no means logical entity, I can make. be correct to as as following nearly any analysis appears of let a be one Given any proposition function), (not a propositional the virtue of its terms, and let us call the proposition ""(a). Then in be if of can idea function, x a propositional primitive any term, we of substitution arises the which from x consider the proposition ""(#*), We thus arrive at the class of all propositions ""(#). in placeof a. If all are (x) may then be called a true, $ (x) is asserted simply: "f" (x\for every value ofx, states "f" formaltruth. In a formal implication, and the assertion of ""(x) is the assertion of a class of an implication, is sometimes If "j"(x) not of a single true, implication. implications, easy to

very The

the values of : the by "f"(x)

values of

#,

x

which make

it true form

a

which class,

class is said to exist in this the class defined

by $(x)

case.

If

is said not

is the class defined

""(#)is false for all to

exist,and

as

a

of

matter

fact,

taken

are

in

term,

would

one

the

in

speaking,

not

particular

one

different

in

terms

term

any

;

relation

This

arises,

When

for a

be

values

of

then

analogous that to

vary

order

the

A into

function,

functions,

many

is

type the

analysis

the

is

"r.

notions

reader

of

of

in

in

any

thus

in

have

had

that,

which

to

has as

more

leave

I

to

am

where

of as

of

the

as

thus

plained. ex-

entering

propositional

class

of

tions proposi-

propositional

any9

fundamental,

been

are

this of

one

able

unanswered.

be

regards

With

complete,

formally difference

term a

is

process

no

be

$x

is

a

by \|r(a).

prove

to

any

if

employed.

far it

and

assertion,

The

x.

regarded "p (#, y)

(", y\

makes

but

proposition

appears

be

the

appears

say,

denoting,

as

rendering I

simply,

symbolism

earned

made

variables

functions.

function

necessary

are

may

implication,

I, is

which

We

class, of

the

of

term

any

It

of

succeed

questions

not

term

formal

Part

;

is

of

has

individuality.

represented

values

to

term

any

must

6

be

may

it

variations

function.

is

presupposed

problems some

x

the

and

individuality

variable

that

whose

the

The

propositional

a

being

which

in

result.

the

i^("r)

integration

function

all

for

relation

consider

must

we

and

#,

it

propositional

propositional

assert

of

only

different

propositional

variables,

y"

:

kind

a

from

the

and

jc

involve

not

and

have

two

If

of

the

does

a,

has

steps. values

proposition

ternx,

some

from

denote,

not

denote

has

any

point

new

one

may

term

any

:

show,

to

function

of

double

to

say

variables

tried

all

?/,

This

may

denotes

term

any

different

a

but

by

a

does

by

different

for

elicits

term

any

terms,

Thus

Thus

have

for

of

term.

successive

by

constant.

We

I

as

asserted

all

itself.

that

denoted

denoted

object

however,

vni

class, if classes

object

the

the

This,

namely

AVe

such

no

sense,

yet

unique.

quite

is

propositional

a

is to

definite

[CHAP,

strictly maintained,

be

assemblage

an

this to

obtained

as

is

places.

and

some

a

denoting,

properly

some

proposition,

suppose, of

theory

x

is

there

vi,

is, in

hardly

can

in

occur

may

Chapter

Thus

this

yet

variables

in

saw

extension.

term;

any

we

as

Mathematics

of

Indefinable*

The

94

to

and

the

carry in

conclusion,

principal it.

answering

Maythe

CHAPTER

IX.

RELATIONS.

94.

NEXT

after

come subject-predicate propositions appear equallysimple. These are

which propositions in which

relation is asserted between

a

terms

two

said to be two.

are

The

considered

hereafter; the former

often been

held

latter class of

propositions in which

those

at

It has

once.

that

be reduced to one can every proposition subject-predicate type, but this view we shall,throughout the

work, find abundant that all

a

relation.

we

shall

be

This

be

might

present held,however,

subject-predicate' type,and

asserting

not

containingtwo propositions

to

of the

terms

and

but this too, difficultto refute,

opinionwould be more no good grounds in its favour*. relations having more than two are

complex,it

more

We terms

therefore may ; but as these

well to consider first such

will be

have

as

two

only.

terms

between

relation

A

which

propositionin and

reduced

rejecting.It

find,has

allow that there are

of the

not propositions

numbers, could

for

reason

be

will propositions

be considered

must

the

terms, and

two

types of

two

in which

the

from proposition with

"

b and

a

where

by aRb,

there

one

are

R

is

of the

concept which

a

the

two

type

a

and

b

is the relation and and

will then

always,provided a That from bRa, proposition

b

and

a

not

are

relational

a distinguish

are

b

is identical

two," which

relational proposition may

A

a

not

requiredto "

in

occurs

occurringas conceptsf, terms givesa different proposition.

terms

two

are

mark

two."

is

terms

interchangeof last

This

two

be

the terms

are

denote identical,

symbolized ; a

and

aRb

different

of two

say, it is characteristic of a relation to speak, terras that it proceeds, so from one to the other. This

is what

may

the

source

be called the of order and

impliesand *

sense

series.

impliedby

a

of the

and relation,

It must

relational

be held

as

is,as an

axiom

propositionbRa,

IV, Chap, xxv, " 200. above ("48),excludes saw as we description,

we

shall

find,

that aRb

in which

the

See inf.,Part

t This to

is

is to

predicate.

the

of subject pseudo-relation

of Mathematics Indefinable^

The

96

relation

b to

proceedsfrom

relation K

But

R.

as

even

it must

maintained be strictly

We

the distinguish

may

and referent^ of

the

relation is

a

The

a

or

difference of

which

it

b and of

converse

sense

;

and

are

the

as proceeds

relation of R

The

Schroder)by R.

these

holds between

relation which

and b will be called the

a

aRb

the

impliedby bRa, propositions. relation proceedsas the different

the relatum.

The

this must

R

is the relation

not

sense

capableof definition.

whenever

holds between

R, and will be denoted

to R

same

is

is not

a

be

not

may

that

notion, which

fundamental

or

may

impliesand

from

term

to which

term

and

a,

when

[CHAP,ix

be defined

of

(following

oppositeness,

(aswould

at

seem

mutual

in any single implication by the above sightlegitimate) all its in the given of for which cases holding case, but onlyby the fact The relation occurs. grounds for this view are derived from certain terms are related to themselves not-symmetrically, in which propositions is not identical with itself. These Le. by a relation whose converse be examined. must now propositions is a certain temptationto affirm that no There be 95. term can and there is a stillstrongertemptationto affirm that, related to itself; first

if

a

term

be

can

Le. identical with

the relation itself,

related to its

But

converse.

if resisted. In the firstplace, never

be

able

to

a

term

were

these

since self-identity, and as identity,

notion

symmetrical, be

temptationsmust

related to this

assert

But since there is such

no

both

be

must

is

should we itself,

plainlya

since it seems

relation.

undeniable

allow that a term must that every term is identical with itself, we may be related to itself. Identity, however, is stilla symmetrical relation,

and far

may

be admitted

without

any

great qualms.

The

matter

becomes

have -to admit

relations of terms not-symmetrical to themselves. undeniable ; seem following propositions is 1 has has is : being; Being is,or one, or unity; concept conceptual is a class-concept. All these are term is a term ; class-concept of one of the three equivalent at the beginning of typeswhich we distinguished Chapterv, which may be called respectively subject-predicate tions, proposithe relation of predication, and propositions asserting propositions membership of a class. What we have to consider is,then, asserting the fact that a predicate of itself. It is may be predicable necessary, for worse

when

we

Nevertheless the

our

presentpurpose,

has

since humanity),

relational. We ? unity

Now

to take

the

our

in propositions

form subject-predicate

the second form is not

(Socrates

in the above

sense

take,as the type of such propositions,unityhas may undeniable that the relation of predication it is certainly "

since is asymmetrical,

of their subjectscannot in generalbe predicated predicates.Thus "unity has unity" asserts one relation of unity to and impliesanother, namely the converse relation : unity has itself, to

itselfboth

to predicate

the relation of

subject.Now

It is plainthat the relatum

and subjectto predicate,

if the referent and

the

relation

of

the relatum are identical, has to the referent the same relation as the

Relations

94-96] referent has

to

the

relatum.

were

defined

case particular

a

if the

mutual

by

of

converse

in implication

relation in

a

particular

that

it would

case,

relation has two appear that, in the present case, our since two different relations of relatum to referent are implied

converses,

has

by "unity relation

unity." We fact that

by the

That them

any

and

by bR'a" where

R

three

Thus

is

we identical,

bRa"

whatever

term

and

be

and

sense,

so

a

be may and which the

of

converse

whatever

bRa

impliedby

if

give

we

is

implied

R. of

regardto relations that, provideda and b from bRa; (2) they all impliesand

aRb

them.

holds between

with

that

a

is

two not

are

have

a

impliedby

b may relations hold between a be; (3) some such relations are not necessarily symmetrical,

itself,and

For

noted

such

the

and variables, essentially find that aRb implies and

may

relation R

a

is

the relation R

aRb distinguish

i.e. there

96.

here

points must can

i.e.

converse,

not

define

relation other than

some

(1) they all have

terms:

impliesand or

value,we

constant

therefore

must

aRb

b may be, and whether is to say, a and b are

a

Hence

97

which different relations,

two

both

hold

between

generaltheory

of

a

and

term

other's

each

are

verses, con-

itself.

in its mathematical relations,especially

axioms

certain developments,

relatingclasses and relations are of great importance. It is to be held that to have a given relation to a that all terms so having this relation to this giventerm is a predicate, form a class. It is to be held further that to have a given relation term at all is a predicate, that all referents with respectto a given relation so form a class. It follows,by considering that all the converse relation, relata also form domain two

and

These

two

converse

of jfteld

may

of predicated These

mentioned

be stated

as

themselves. the

are

the

relation

;

respectto

a

the respectively

logicalsum

of the

the relation.

the

at

follows.

end

We

Consider

referents

given relation

and limitation,

some however, to require

contradiction

classes I shall call

of the

domain

that all referents with

axiom

class seems,

case.

class.

I shall call the The

the

a

the

(and

saw now

of

that

Chapter

that those

some

vi.

on

account

This

of which

a

of

diction contra-

can predicates

this is not

also the relata)in what

seems

be the like

with non-predicability complex relation,namely and which attaches to all of them identity.But there is no predicate will either be predicable not For this predicate to no other terms. or of of i t is those is itself. If it of one predicable itself, predicable defined,and therefore,in virtue referents by relation to which it was of itself. Conversely, if it is not it is not predicable of their definition, said of all of the of of itself, then again it is one referents, predicable and therefore againit is predicable it is predicable, which (byhypothesis) the

a

of

itself. This

considered have

predicatesare

is no

a

combination

form

contradiction,which

exclusive

common

of

shows

that

all the

referents

and therefore, if defining predicate,

do not form essential to classes,

a

class.

[CHAP,ix

Indefinable^ of Mathematics

The

98

In defining the would-be class of may be put otherwise. of themselves have been used up. all those not predicable predicates,

The

The

matter

all these

of predicate

common

there is at least

since for each of them

be

of

them, of (namelyitself) predicate

cannot predicates one

one

But again,the supposed common predicable.

predicate of be any cannot predicable were, class of of the member since would be it a Le. predicates, itself, supposed Thus no predicate defined as those of which it is predicable. these were considered. which could attach to all the predicates is left over which

it is not

other

forms

terms

class defined

a

borne in mind, and collection

above

the

It follows from

not

predicate.This

endeavour form

to

definable collection of

every

common

in order

have

must

by a

must

we

that

that asserting proposition

fact must

discover what

to

such

class.

a

established by the above contradiction may be stated containingonlyone variable may apparently to any

be

it would

for if it predicate,

the variable in

a properties

The

be

not

point

exact

follows

as

be

:

A

position pro-

equivalent

has question

a

certain

predicate.It remains an open questionwhether every class must have a defining predicate. all terms That having a givenrelation to a giventerm form a class results from the doctrine of exclusive defined by an common predicate aRb can be analyzedinto the subject Chapter vn, that the proposition a

and

the assertion

appears

that to be a

term

a

predicate.The

that

of which, for

class I shall call the domain

The

domain,

converse

of

doctrine

having

all terms

referents.

as

domain, children togetherits field. its

aR asserting

of which

Rb

it does

not

be doubted

some

its

"/,

Ry

functions prepositional

of the

well

value of

property should

of the

domain

series. regards

It may

term

a

the latter

relation R

converse

the

as

togetherwill be called the as

be

plainlya predicate.But

be

to

To

Rb.

fieldof

whether

be asserted, is

requires, however, a

well

as

relata.

the relation "

domain, and

converse

can

class. This the class of

relation will be also called the

class of

if paternity be the

Thus

asserted

follow,I think,

form

as

be

can

The a

two

notion

domains

important chiefly

fathers form relation, fathers

and

children

aRb be regardedas can proposition of 6, or whether only Ra can be asserted of b. In other relational proposition only an assertion concerningthe a

words, is a If we take the referent,or also an assertion concerningthe relatum? " shall have,connected with (say) a is latter view, we greaterthan b? " " four assertions, namely is greaterthan 6," a is greaterthan,"" is less than a" and "Z" is less than."

I

but I know

either side.

97.

We

of

no

argument form

the

on

am

inclined

myself to adopt this view,

and productof two relations or sum logical of a class of relations exactly in the case of classes, as exceptthat here have to deal we with double variability. In addition to these ways of have also the relative product, we combination, which is in general noncan

96-99]

Relations

99

commutative, and therefore requires that the number of factors shduld be finite. If j?,S be two relations, to say that their relative product RS

holds between

and

husband

#, z is to say that there is a term y to has the relation R, and which itselfhas the relation S to z. Thus which x brother-in-law is the relative productof wife and brother or of sister :

two

terms

father-in-law is the relative

productof

wife and

father,

the relative product of father and wife is mother

whereas

or step-mother. temptationto regarda relation as definable in extension as a class of couples.This has the formal advantagethat it for the primitive avoids the necessity that every proposition asserting has relation between of a other But it is no holding couple pair terms. to distinguish the referent from the necessary to givesense to the couple, becomes essentially relatum : thus a couple distinct from a class of two idea. It would seem, terms, and must itselfbe introduced as a primitive the matter that viewing philosophically,sense can onlybe derived from relational proposition, and that the assertion that a is referent and some involves a purelyrelational proposition b relatum already in which a and b are of terms, though the relation asserted is only the generalone

There

98.

is

a

referent to relatum. occur

and

doctrine of

no more

them

There

otherwise than

as

rather with

in

couples

can

to take

correct

are, in

terms

an

fact,concepts such

greater,which terms ("" 4*8,54);

as

havingtwo propositions It propositions.

evade such

intensional view of

than class-concepts

with

seems

and relations,

classes. This

fore there-

identify procedureis facts. logical to

also nearer to the convenient,and seems formallymore there is rather curious relation of the same Throughout Mathematics the symbols other than intensional and extensions! pointsof view: and relations) stand for variable terms (i.e.the variable class-concepts intensions,while the actual objectsdealt with are alwaysextensions. it is classesof couples that are relevant, Thus in the calculus of relations, of them relations. with This is deals but the symbolism by means similar to the state of thingsexplainedin relation to classes, precisely at length. and it seems unnecessary to repeat the explanations in Appearanceand Reality, 99. Mr Bradley, Chapterlu, has based of relations upon the endless regress an argument againstthe reality terms must from the fact that a relation which relates two arising be

related to

each

of them.

regress is undeniable, if to be ultimate,but it is very doubtful

The

endless

taken are propositions had occasion We have already whether it forms any logical difficulty. two kinds of regress, the one merelyto proceeding ("55) to distinguish the other in the meaning of a new impliedpropositions, perpetually itself;of these two kinds,we agreedthat the former, since proposition has ceased to be objectionable, the solution of the problemof infinity, have to inquire which kind We while the latter remains inadmissible. It may be urged that it is in the presentinstance. of regress occurs that the relation part .of the very meaning of a relationalproposition

relational

The

100

should

involved it

relates

(" 54?)

in

itself.

It

of

part

be

from which

relates this

in

and

a

it

so

be

may

far

just

the

endless

We

b"

the I

though thing

same

see

can

the

any

in

to

its

though we

work.

may

has

held of

meaning

to

the

undeniable,

relation is

further

of

the

the

solely

logically theory

a

of

or

b

quite of

J?,

to

and

harmless.

relations

to

the With

later

Parts

exceeds

a

does

without

we

not

endless these of

state states

"

Hence

aRb that

than

",

to

a

thus to

seems

But

relations.

proposition

which

the

relatum.

relation

further

of

relational of

of

2*

which

meaning,

b"

while

referent,

relation

implications a

a

mean

from

some

than The

greater

usually

have

greater

relation.

relation

greater

express

that

any

the

to

is

a

and

terms

the

greater b

conclude

leave

two

of

part

"

to

kind.

is

"a

principle,

must

be

to

impossible

people

the

word

b, except

question,

and

b"

but

differ";

and

a

it

difference

objectionable

that

On

genuine

every

the

deny

to

b

seems

exceeds

"#

propositions.

two

of in

be it

of

is

between absurd

be

a

may

assertion this

","

and

"

Thus

involved

think,

form

must

to

that be

may

have

escape,

more

a

I would

that

than

difference.

regress

these

by

than

has

similarly b"

no

and

is

contains that

it

point

and

a

difference

the

difference

of

of

no

tinguished dis-

is

Against

of

proposition

found

we

element

difference

is

relation

concept.

a

"

the

in

that

bare

distinguish,

may

than

from

assertion

implied,

relating

a

we

relation

a

the

though

indefinable

the

by

that

view,

terms,

that

concept as

specific from

that

the

in

the

and

itself

much

as

indistinguishable prove

against

and

relation

this

that

which

distinction,

the

IX

saying

in

expressed

relating

a

and

proposition

rejoined

some

as

makes

between

in

that,

b

what

proposition,

a

retorted

be

is

however,

relation

a

relation

relation

the

distinguishes

might

to

urged,

original

the

this

that

between

the

terms

unexplained,

may

relation

a

left

the

to

and

them,

formerly

of

have

[CHAP.

Mathematics

of

Indefinables

cluding inshall

include regress,

remarks, the

present

CHAPTER

THE

100.

BEFORE

examine

to

with

regardto

takingleave

not predicates

to solve this

it,and

with

be

it will puzzle,

to state

has propositions) Let w

is

be

w

a

is

w

a

w9

that

be

w

is

w

can

and

a

v

;

be asserted

can

asserted

be asserted

essential

that

there

I

can

all formal

to

of members*.

of itself.

is

a

Hence

of

v

which

by contraposition,

members

of whose

class v, is

in another

term

are

class-concepts

contained class-concept

of

of itself.

(a) if w be contained

there consequently

none class-concept

a

be

be

number greatestpossible

Then

not-man.

that class-concept

(ft)if

proof that

plausible suppositionthat

to

seen

mention

I may

Cantor's

the very

have

(which we

the necessarily

class-concepts, e.g. since

with

connected

which i.e. such that be asserted of itself, can class-concept Instances and the negations of ordinary are dass-coticept^

a

w"

a

is necessary

deductions

some

different forms. reconcile

to

number

the class of all terms

questions,it

be well to make

it in various

greatestcardinal

no

of fundamental

singularcontradiction,alreadymentioned, Before attemptpredicableof themselves. ing

led to it in the endeavour

was

"

CONTRADICTION.

in detail the

more

X.

themselves,no further,(7) if

in

u

can

ever, whatu be any class-concept of u which members not are

Hence

of those class-concept in itself, and is contained predicableof themselves, this class-concept of members hence is themselves not its of none are by (/3)u predicable ; predicableof itself. Thus u is not a ?/,and is therefore not a " ; for of themselves, the terms of u' are all predicable of u that are not terms which

the

u

is not.

u

(S) if

Thus

contained class-concept of those

in

u

that class-concepts

u

be any

which are

not

class -concept whatever, there

of w, and a of themselves. predicable

is not

member

is also

So

is

a

one

far,our

take the last question. But if we now be that cannot the class of those class-concepts of them, and admit contain a class-concept asserted of themselves,we find that this class must of itself and yet not belonging to the class in question. not a member

deductions

We class of

seem

may

scarcelyopen

observe

also

that,in virtue of what

which class-concepts *

to

See

cannot

we

have

be asserted of

Part V, Chap. XLIII,

" 344

proved in (/9),the

themselves,which

ff.

we

of Mathematics Indefinables

The

102

will call w, contains

of itselfall its sub-classes, althoughit is

members

as

[CHAP.X

sub-classes than terms. that every class has more Again, if y be any term of w, and w' be the whole of w excepty" then w',being sub-class of w, is not a w but is a w, and therefore is y. Hence each a easy to prove

which class-concept extension. is

absurdities can 101. exact

us

may

or

all other

proved. paradoxical consequences,

may

its

as

and

has been

attempt the

first the statement

have

be

givenalready.If #

of itself. Let predicable

be

not

w

teaspoon,and teaspoon of similar number any

leave these

which predicates,

of

terms

is a concept bicycle

of the contradiction itself. We

statement of

has

w

plainlyabsurd, and

is

be

Let

in terms

of

term

a

It follows that the

bicycle.This

a

x

is

that "not-

assume

us

predicate,

a

predicableof oneself is a predicate.Then to suppose either that this of itself, is self-contradictor predicateis,or that it is not, predicable The conclusion, of oneself" in this case, seems obvious : not-predicable "

is not

a

Let A

predicate. us

the

state

now

class-concept may or

conceptwhich

is not

a

be

not

may

of its

term

a

of its

conclude, againstappearances,

that

extension"" is not

own

term

of its

extension

own

concept. But if it is a term which is not a term of its own its

"

own

extension,and vice

class-concept

Thus

versa.

class-

a

is not

must

we a

term

of

class-concept.

a

even

many.

Do

"Class-

appears to be

which "class-concept

classes is a class; the class of all the terms and

class-concepts.

extension.

own

extension,it is a

In terms of classes the contradiction appears class as one may be a term of itself as

A

of

contradiction in terms

same

that

are

extraordinary.

more

Thus not

the class of all is not

men

all the classes that have this propertyform

a

man,

class?

If is it member as one of itselfas many If it is,then it is so, a not ? or of the classeswhich,as one of themselves as many, ones, are not members and vice versa. Thus we must conclude again that the classeswhich as so

ones

on.

not

are

members

of themselves

as

many

rather,that they do not form a class as one, show that theydo not form a class as many. 102.

A

do

not

form

which, however,does not lead result, may be proved concerningany relation. Let R be Then

all of them such to a we a

"

for

would x

find term

and

term, the

a

#

it is

be

no

of terms

other terms

or

to a

a

tion, contradic-

and relation,

have the relation R.

For, if there

were

function "x propositional to equivalent

"

a:

is When

a

"

do not have the relation R to selves. themthat there should be any term a to which impossible

contradiction.

to

class

which

which throughout, a

a

for the argument cannot

similar

consider the class w

a

since legitimate in placeof R

which class-concept

contradiction. The have taken it as we

reason an

axiom

does not have the relation R has the relation R to a" Substituting

that

can a

the

is formal, equivalence

we put e, the relation of be asserted of it,we get the above contradiction emerges here is that

that any

function containing propositional

The

100-103]

Contradiction

103

to asserting onlyone variable is equivalent membershipof a class defined that function. Either this axiom, or the principle by the propositional be taken as one and there is no false, term, is plainly every class can fundamental objectionto dropping either. But having dropped the arises : Which functions define classes former, the question propositional terms which are single well as many, and which do not ? And with as real difficulties this question our begin. or Any method by which we attempt to establish a one-one manyof omit at correlation all terms and all propositional functions must one

least

function. propositional

one

functions propositional is provedas

would

if all

exist

be

will all terms, the denial of "f"x then, if the correlation covers (*r) of values is all since for it function, x. a proposition propositional

with be

method

a

in the form ...e?/, since this expressed with ...ew. But the impossibility tion of any such correlafollows. Let "j"x function correlated be a propositional

correlates u

form

could

Such

x

a

;

be included in the correlation ; for if it

it cannot

But

with rt, ""a(x) would

4"x(*r)

tmt

be

th*s

its

to equivalent ff"a(a)

denial.

own

functions than propositional

althoughthe

The

103.

notion

meanings that

seen

as

result which

seems

the

which in

But

e. as

any

each

more

are

plainly impossible,

suggestsitself is to seek

Chapter vi

distinction

meaning the state

to

there

We as convincing any in Mathematics. is removed impossibility by the doctrine of

of

however, attempt

a

"

that

as

first method

far

with

terms

It follows

proofis

how

shortlysee logical types. shall

in the

of #, to the denial of value /?, since it makes

for all values equivalent, is impossible for the equivalence

*

correlated

were

seemed

we

ambiguity the various distinguished and possible, we have just an

contradiction

same

the contradiction

Let emerges. throughoutin terms

us,

of

functions. function which is not null, Every propositional propositional d efines and be defined by a we class, supposed, every class can certainly Thus that class function. to is not a a a as one propositional say of itself as many is to say that the class as one member does not satisfy is defined. Since all propositional the function by which itself as many

except such

functions

as

all classes considering the

have

above

are

null define

havingthe

property.

above

If any

all will be used up, in classes, property,exceptsuch as do not

function propositional

by every class having the above property,it would be

satisfied also

one

singleterm. of

terms

is

w

but not

by

w

the class class be

some

w

does

not

necessarily as

a

itself belongto the class ze, satisfied

function propositional

by

the

the contradiction re-emerges, and is no such entityas ze?,or that there

itself. Thus

might be thought that a solution could functions. of variable propositional legitimacy It

therefore

satisfied

of all such classes considered

w

suppose, either that there function satisfiedby its terms propositional

must no

the

therefore there must

and

we

Hence

by

were

and

by

be found If

we

no

others.

by denyingthe by k^ for

denote

Indefinables of Mathematics

The

104

the moment, the class of values is the denial of "f"(^\ where

[CHAR

function prepositional

satisfying "",our

variable.

"f"is the

x

doctrine

The

of

might make such a variable Chaptervn, that ""is not a separable entity, be overcome ing but this objection can seem by substitutillegitimate; the relation of $x to x. for "f" the class of propositions or ""#*, variable exclude functions to Moreover it is impossible propositional variable class or a variable relation occurs, a altogether.Wherever variable propositional have admitted function,which is thus a we every class

essential to assertions about of

definition of the domain

a

away

by

some

further characteristic by which

function of

found, I think,in the independent function and argument. In general, "f"xis itself either of a nd be given a these, x ; variables, "f" may

two

value,and either

Chapter,the argument have

instead of ""#,we

"". Thus

#"

an

dependentupon

the variation

"f".For this

of

when definite proposition

function,in

the

the variable

enters

when

be may in a way

type

into them

"

which, in Algebra,a variable appears

be asserted of

can "f"

being x. time

same

function of

a

"/"is asserted is

in

If here a

manner

cj" {/*("")} though

reason,

,

is not is assigned,

x

ordinarysense,

functions of this doubtful

the

at

as

which

class of terms

is varied

"j"is varied,the argument a

of

to: equivalent

is

^","this satisfying

the class of terms

it is

is defined f("f")

where "f" {/("")},

"a? is

Thus

are

of the

is itselfa function

reference to the other. in this considering function : propositional

we

"f"is varied,the argument

when

varied too.

be varied without

may

functions propositional

type of

in the

But

require

kinds of variation.

two distinguish

to

we

the

of the variability

constant

Thus

be swept

characteristic is to be

This

a

would relations,

type of variation.

refusal to allow this

the

every relation. The example,and all the general

calculus of

constitute the

which propositions

about

or

for relation,

propositional is variable. x Propositional called quadratic forms, because somewhat analogousto that in in an of the second expression a

degree. 104.

Perhapsthe

that,if

best way collection of terms

a

only be

can

positional function,then, though a

class as

be denied.

must

one

itselfmade

never

In such it a

as

there is

cases

axiomatic

class

as

many

and appears

to

;

terms

class

as

When

as

many

be

may

admitted,

that

sitional propo-

the

source

will difficulty

shall say, is one, function propositional we

; i.e. any of the terms is substituted for

x

is

function. original propositional not

is to be

one

this axiom

have been

as

it appears stated,

so

class as many,

the class

the whole it,therefore, A

a

but

class

in the subject

only

that

a

solution is to say suggested defined by a variable pro-

collection varied,providedthe resulting

be may into the

functions

the

state

to

need

not

class as

a

found be

We

one.

wherever

took

there

is

admitted, universally By denying

of the contradiction. be an

overcome.

objectof

(x) which "f"

is also

the is

same

type as

when significant when the class as significant

its one one

103-105]

The

is substituted.

But

Contradiction

the class

as

does not

one

105 the class

alwaysexist,and

is of a different type from the terms of the class, the when even many class has only one functions "f" term, I.e. there are propositional (u) in as

be the class as

which u may substitute one is not

a

class

as

of the terms

many;

if,for meaningless

are

And

"

so

x

is

one

all if the relation involved is that of

at proposition

and

which

many,

of the class.

this is the

only relation of whose

a

",

we "

among

x*$

term

to its a

presence

pro-

function alwaysassures In this view,a class as us. positional many may but in propositions be a logical of a different kind from those in subject, which its terms are subjects other than a single ; of any object term, the it whether is have different answers one or question according many will in which it occurs. to the proposition Thus have is one "Socrates we are men," in which men men are one plural species ; but among among in which men It of animals,71 is distinction of the are singular. logical types that is the key to the whole mystery*. Other ways of evadingthe contradiction, 105. which might be that the suggested, ground they destroytoo appear undesirable,on of kinds I t quite necessary propositions. might be suggested many that identity is introduced in is not an in a way which is not x x But it has been alreadyshown that relations of terms permissible. "

"

"

to

themselves

or

self-made

by

men

the

or

into

formal

be observed

it may

suicides

that

all defined

Smiles^s

of

heroes

Self-Helpare in a very enters identity generally, that it is quite impossible to so implication,

themselves.

relations to

similar way

unavoidable,and-

are

And

rejectit. natural

A to

demur

to

urgedthat our

no

such

escape from

have

the contradiction would

for escaping from suggestion

the notion

or

-total is conceivable

sum

the

of ail terms

contradiction

alreadyabundantly

that

seen

of all classes. ;

and

It

might

if all indicates

requiresus

to

if this view

were

admit

a

this.

be

whole, But

maintained

be

we

against

and Mathematics, whose be impossible,

any term, all formal truth would of truths characteristic is the statement abolished

at

the requires notion

one

stroke.

notion

of any

the

Thus term

or

concerning any term,

correct

every

statement

term, but

would

of formal not

the

be

truths

collective

of all terms.

is involved peculiar philosophy from common in the above contradiction, which springsdirectly sense, and can common-sense be solved some assumption. by abandoning only itself nourishes contradictions, which on the Only Hegelianphilosophy, because it finds similar problemseverywhere.In remain indifferent, can demands on an direct a challenge pain so answer, any other doctrine, similar other of a confession of impotence. Fortunately, no difficulty, of of the other in far I Principles so as know, occurs portion any It should be

that observed, finally,

no

Mathematics. *

On

this

subject,see

Appendix.

Indefinables of Mathematics

The

106 106. Part

We

Mathematics

Pure

L

review briefly

now

may

defined

was

formal asserting

and implications

And

logicalconstants

constants. term

to

a

class of which

of relation,and

notion

it is such

found

the

a

conclusions

as

:

constants

the Implication,

member, the

further notions

as

are

propositions except logical relation

of such

notion

in

at

of

a

that^the

involved in formal

be the

: following prepositional and any or every term. This definition brought class*,denoting, function, Mathematics into very close relation to Logic,and made it practically of SymbolicLogicjustiidentical with SymbolicLogic. An examination fied of mathematical indefinables. In Chapterin the above enumeration The former holds and formal implication. we distinguished implication between any two propositions providedthe first be false or the second

which implication,

of the variable

or

variable

value of the

what distinguished

the (including shown

was

substance

or

may

is of

that

and

to

for every value relation,but the assertion, o f function variables, a propositional which, for every

latter is not

The

true.

we

(" 93)

arrived

class of

the

containingno are

[CHAP,x

a

asserts variables,

be called

predications among

this

an

Chapteriv implication.

thingsfrom

and relations predicates relations for this purpose). It

distinction is connected

but does not attributes,

lead to

with the

the

doctrine

of

traditional results.

In the former Chaptersv and vi developedthe theoryof predicates. shown that certain concepts,derived from of these chaptersit was in propositions not about themselves,but about binations comoccur predicates, of terms, such as are indicated by att^every^ any^ a, some^ and the. Conceptsof this kind,we found, are fundamental in Mathematics, and enable us to deal with infinite classes by means of propositions of finite complexity.In Chapter vi we classdistinguished predicates,

classes as concepts,concepts of classes,

many,

and

classes as

one.

We

agreedthat singleterms, or such combinations as result from and* are classes,the latter being classes as many ; and that classes as many the objects denoted by conceptsof classes, which the plurals are are of class-concepts. But in the present chapterwe decided that it is from the class whose onlymember term a single necessary to distinguish the null-class may be admitted. it is,and that consequently In Chaptervn we resumed the studyof the verb. Subject-predicate and such fixed relation a to as a fixed term, could propositions, express be analyzed, and an assertion ; but this analysis we found,into a subject when becomes in a a given term enters into a proposition impossible than it referent of a relation. more Hence as complicatedmanner became take to notion. a as ^function propositional primitive necessary A of function is variable of a set one propositional anv proposition defined by the variation of a single term, while the other terms remain *

The

by that

notion

of class in

of the class of

we general, as an decided,could be replaced, indefinable, defined by a propositional function. propositions

The

106] But

constant.

a

certain

The

of

notion

if

not,

a

and

that

which

in

we

in

and

different,

single be

can

The

in

of

terms

one

type. relatum

type

of

be

to

as

its

it

Thus

;

The

constituents define as

and

class ;

only one,

as

but a

if

of

where

it

quadratic

class it

ever

On

this

point, however,

x

as

and is

Appendix.

the

classes

"fxc in

belong e

said,

of

of

function

contradiction

certainly

to

requires

are

propositional

exists,

see

which

exists, is, we

many,

many.

function

should

one

various

classes,

because

objects

absent. *

of

meaningless,

composed one,

classes

that

as

w

as

distinguish

propositional

a

be

then itself

of

sulting re-

which

classes

all

many,

to

characteristic

a

contradiction of

class

terms,

greater

tions, analysis. Relaof couples*. classes

member

meaning, to

a

as

as

a

is

from

the

the

necessary of

any

class

a

is

that

held

was

be

it

have

be

to

classes

on

be

themselves

not

that

to

#e#

class

of

not

examined w

is

propositions, though This

resulting

we

if

that,

and

so

is

loss

such

as

containing

"A

in

as

two

intensionally,

terms,

should

to

the

the

ultimate,

concept

proposition

such,

These

chapter,

was

and

if

of

members

not

referent.

appears that

given

a

are

the

being

constituents.

same

fact

apparent

terms,

the

proves

present

both

that

general

variable,

by

propositions

as

A"

taken

be

to

are

the

are

of

type

place they A

denoted

set

is another

concept

instance

an

general requires,

same

by

in

functions

the

function.

the

relation

there

than

greater

suggested

the

function,

propositions!

variables,

of

agreed

We

propositional

relational

i.e. the

:

same

objects, namely

couples

some

sense

precisely the

proved of

individuality;

certain

a

propositional

that

out

terms,

is

the

solution

types

the

proposition

any

the

agreed, from

as

case

and

and

Finally,

by

question

complicated.

exceedingly

certain

a

several

variable

in

term

in

is

variable.

a

indistinguishable.

in

of

two

contain

we

be

term

with

"5

relations,

of

in

have

terms

B"

than

the

multiply

all

they

same

in

pointed

ix

proposition

a

the

any

remains,

function.

Chapter that

is with

when

proposition,

a

the

by

term

would

qua

Thus

found,

distinguished

or,

term

propositional and

term

we

of

the

isolate

or

what

since

out

replaced

but

term,

left

is

define

to

function,

entity.

variable,

are

given

said, is the

but

any

occur,

a

omitted,

variables

variables

they

occupy

is

of

107

impossible

occurs,

kind

any

two

any

variable

that

it

the

simply

is not

x

for

simply

not

is

prepositional

a

wherever

discoverable

no

be

must

it

general in

term,

general

The

in

element

constant

Contradiction

sometimes

the

the in

PART

NUMBER

II.

CHAPTER

DEFINITION

with

notions

branch

of

Mathematics

this

greater

of

favour

Logic*.

its

seems

modern

outlines

of

deduction

in

though

precise sense, Given

any

when,

and

notions

of

set

sense;

into

its constituents.

*

which xxix,

t

which

numbers, both

are

xxxviu, See

Entiers,"

essential

that are

to

been usage

it

is

distinct

ordinary

attains

have

the

As

by

even

order

been

has

relations.

examine

in

and

the

most

forth

process

covertly

intruded

to

term

itself

is

by

having of

one

not,

as

restricted is

a

to

and,

necessary

entities,

to

of

Mathematics.

that

wholes

the

separate which

the On

Ordinal

of

But

an

in

idea

think, useless not"

are

study

former

these

employed

analysis I

of

notions.

rule, been

the

notions

certain

said

the

inconvenient

fact

to

Congre*,

JF.

1901,

Vol.

m,

p. p.

6

ff. and 314

ff.

Padoa,

"Theorie

a

notions'}*.

these

of

means

of

set

are

has

Mathematics,

given

some

numbers

of

are

Cardinal

simpler,

numbers,

cf.

Algebrique

des

;

as

a

and but

of

Chaps.

infra.

Peano,

if

discover,

to

particular in

which,

only

the

overlook

to

shown

a

definition has

This

seems

lias

Cantor

Ordinal

which

word

in

chapters setting

then

is definable

the

is

has, in fact,

this

it

it

number word

is relative

term

a

relation

philosophically,the

moreover

one

only when,

it

both

which

notions,

certain

a

held

years,

and

Peano,

of

special

a

recent movement

known

four

by

as

new

argument.

that

definability is

Now

and

logic

shall

I

or

Weierstrass,

imperfectly Part

form.

the

of

course

is often

It

108.

bv

unperceived assumptions

any

the

but

in

The

philosophical standpoint,

a

possible, whether

indefinable.

begin

made,

Frege,

the

of

this

non-symbolic

a

in

is

integers

has

Arithmetic.

Cantor,

It is to

indefinables

new

inaugurated

means

theory

from

themselves

theory

goal by

shall

I

of

general logical

present Part,

cardinal

subject

by Dedekind,

final

mathematicians, its

theory

deduction,

mathematical

the

In

of

of

apparatus

suffices,without

whole

the in

correctness

the

mathematical

than

brilliantlycontinued what

the

No

advances

NUMBERS.

operates.

apparatus

establish

postulates, to

CARDINAL

briefly reviewed

now

which

how

shown

be

have

WE

107.

OF

XL

Nomhres

Number

112

[CHAP,-xi

rule,determinate when their constituents entities (which may be in some new mathematical

sense,

given,but

are

defined,in simple),

sense

certain relations to their constituents.

bv

and

in future, ignorethe philosophical therefore, sense, is done

which

and

by

Professor

branches

various

and his

Peano

of Mathematics

They disciples.

various

have

it is

the

shall,

speak only

of

more

hold that the

indefinables, by

defined. are remaining ideas of the said subjects an importantpart of my purpose to prove that

the

I

I shall,however, restrict this notion definability.

mathematical than

themselves

are

"

of

means

I hold

"

all Pure

rational

Dynamics) contains namely the fundamental logicalconcepts only one set of indefinables, the various logical have been constants discussed in Part I. When enumerated, it is somewhat arbitrarywhich of them we regard as which must be indefinable some indefinable, though there are apparently (including Geometry and

Mathematics

even

of Pure contention in any theory. But is,that the indefmables my all of this kind,and that the presence of any other Mathematics are indefinables indicates that our subject belongsto AppliedMathematics.

three kinds

Moreover, of the

of

definition admitted

by

Peano

the

"

the definition by postulates, and the definition by definition, the others,it would I recognizeonlythe nominal: abstraction* seem, nominal

"

onlynecessitated by

are

Peano's

refusal to

regardrelations

undue

and by his somewhat apparatus of logic,

fundamental

part of the

as

haste in

class. These remarks will be a regardingas an individual what is really their application best explainedby considering to the definition of

cardinal numbers. It has

109. numbers and

to

as

define the so

made

a

on.

in the

common

make

definable,to

% -h 1, and

and

been

remainder This

exceptionas

an

by

method

its was

regardsthe

Thus

means.

"

1 and

We meaning of 4- was commonly not explained. this In to method. the days improve greatlyupon Cantor

has

shown

how

deal with the

to

-h

finite

other numbers

the

regarded number

1

was

to onlyapplicable

tiresome difference between

who

those

past,among

are

1,

3

1, numbers, was

; moreover

able

now-a-

first place,since

it has become infinite,

both

desirable and

of numbers to deal with the fundamental possible properties in a way which is equally In to finite and infinite numbers. applicable the second place, the logical calculus has enabled us to give an exact definition of arithmetical addition ; and in the third place, it has become as

easy to define 0 and

define any other number. In ordei4 to this is done,I shall firstset forth the definition of numbers 1

as

to

explainhow by abstraction ; I shall then pointout it by a nominal definition. and replace Numbers It is true *

C"

p. 294

are,

it will be

' c

defects in this definition,

to classes. admitted,applicable essentially

that,where the number

Burali-Forti,Sur ff.

formal

is

individuals may be enumerated finite,

les differentes definitionsdu nombre

reel,"Cortgres, HI,

109]

108,

make

to

Definition of the

up

be

must

of

virtue

But class-concept.

a

that

what

the number

where

And

and applicable,

of the class.

class.

a

the

the number

of numbers

individuals whose

in

which

it relieves

number

certain

way ; and all numbers Similarly

a

in conjoined a

number. The

the

Numbers, then, are -number?

The

their terms

either that

to corresponds

there

class and

if in

should

whose

what

one

be

and

regarded

has rendered

be

property

a

the possible

of necessity

considered.

rating enume-

view

This

all,the numerical conjunction

do two

circumstances

onlyone

domain

converse

class-concept

as

for

to one,

one

the

classes have

number

same

that any one This

so

of the other.

term

of

term

requires

is the

domain

relation whose

one-one

some

this

is,that they have

answer

be correlated

can

of

property in is class-concept

any

of the

us

is to

class.

a

joined conexample,denotes men it is as thus denoted that they have a all pointsdenotes numbers or or points n umbers thus as or conjoined pointshave of classes. to be regarded as properties

questionis : Under

next

same

when

certain way, and

a

be

of

be enumerated,

common

when

therefore

may

It is this view

some

Thus

dependsfundamentally upon the notion as we agreedto call it ("59). All men, number.

number

of individuals to which

since theoryof infinity,

whole

all finitecollections of individuals

by intension,i.e. by

they form

without

by one

is infinite, the individuals cannot

defined

which

113

one

results is after all the

there is a certain number given, is

Numbers

given number, and may be counted

of any mention form so classes, but

Cardinal

one

Thus, for example,

is the other class.

and all the women are married,and community all the men number of men the be the must polygamy and polyandryare forbidden, the number It might be thought that a one-one of women. same as 1. relation could not be defined exceptby reference to the number But a

this is not

the

A

case.

relation is

when, if

one-one

and

x

x' have

the

questionto y, then x and x' are identical ; while if x has the it is in questionto y and y\ then y and y' are identical. Thus

relation in relation

without possible, relation.

one

have

no

what

is meant

there must

is

a

in order to

terms, it is necessary

are

say

But

the notion of

by Bayingthat terms, the

no :

Two

relation whose

one-one

two

terms

classes have

unity,to define what is meant by a oneprovidefor the case of two classes which the above account of to modify slightly

the

classeshave

same

domain

number

when,

includes the

such that the class of correlates of the terms

with the other class. From terms

alwaysthe

have

same

number.

same

correlated

be

cannot

the

one

and

one.

onlywhen,

We there

and which class,

one

of the

to

one

if

For

is

class is identical

this it appears that two classes having no of terms ; for if we take any onenumber

and the class includes the null-class, relation whatever, its domain two classes of correlates of the null-class is againthe null-class. When one

have

the

Some

number, they are

same

suppose that a classes have the same

readers may

sayingthat

two

said to

be

similar.

definition of what number

is

is meant

by

whollyunnecessary.

Number

114 The

to

way

notions

as

both classes. It is such find out, theymay say, is to count this which have, until very recently, preventedthe exhibition branch

of Arithmetic

as

arises : What

is meant

a

of Pure

Logic. by counting? To

irrelevant

only some

[CHAP,xi

answer, psychological

In order to count

of attention.

successive acts

immediately question this question we usuallyget that countingconsists in as, the

For

that ten

10, I suppose

useful definition of the most : certainly a required is not 10 ! Counting has, in fact, a good meaning, which number But this meaning is highlycomplex; it is onlyapplicable psychological. of attention

acts

to

are

classes which

classes; and finite

be well-ordered,which

can

onlygivesthe

and

rare

a

"

it

number

of the We

case. exceptional

known

not

are

to

be all

this number

class when

is

not, therefore, bring in

must

is in

relation of

question. the jbhreeproperties of

is similar u classes,

is to say, if u9 v9 w be be similar to z", v is similar to u ; and

definition of numbers

the

counting where

between classes has similarity and transitive ; that symmetrical, beingreflexive, The

to itself; if

u

These properties if u be similar to v9 and v to w9 then u is similar to w. from the definition. Now of a these three properties all follow easily relation

held

are

relation holds between

property,and

vice

and

Peano

by

terms, those

two

versa.

sense

common

This

terms

have

propertywe

common

This is the definition of numbers

two

to indicate that when a

certain

the

common

call their number*.

abstraction.

by and generally the process by abstraction, fatal formal suffers from an absolutely employedin such definitions, defect: it does not show that only one objectsatisfies the definition^. Thus instead of obtaining which one common propertyof similar classes, the is {he number in of classes obtain a class of such we question, of h ow this class contains. terms with deciding properties, no means many this pointclear, In order to make what is meant, in the let us examine is meant is,that any property. What presentinstance,by a common its number, a relation which it has to nothing class has to a certain entity, similar which all classes (andno other entities) but have to the said else, 110.

Now

number.

is,there is

That and

its number

to

relation which many-one every class has to else. far the definition by Thus, so as nothing a

show, any set of entities to each of which some and to one and onlyone of which certain many-one relation,

abstraction a

this definition

class has

can

this

and relation,

given class have as

this relation to

the set of

are one

such that and

the

numbers,and any entityof

appear class. If,then, there some

*

which

are

many

class has

any given all classes similar to a

same

entityof

the set,

this set is the number

such sets of entities

"

of

and it is easy

CTf.Peano, F. 1901, " 32, '0,Note. the necessityof this condition,cf. Padoa,loc. tit., 324. Padoa appears p. not to perceive, however,that all definitions define the singleindividual of a class : when what is defined is a class, this must he the only term of some class of classes. t

On

109-111]

of Definition

prove that there are have many numbers, and to

of

class. This

a

abstraction

by

defect.

One

Numbers

infinite number

115

every class will the definition wholly failsto define the number

an

of

them

"

and shows that definition argument is perfectly general,

is

never

There

111.

Cardinal

a

valid logically

two

in which

are ways of these consists in

process.

may attempt to remedy this defining the number of a class the we

as

whole class of entities, chosen one from each of the above sets of entities, to which all classes similar to the givenclass (and no others) have some relation or many-one without all entities,

class will have

cases

method

as

a

others a

moreover

;

relation which set.

own

Thus

and applies to all practicable, employs definition by abstraction. This

Peano as

common

is

the number

given class.

is a predicate)

is practically since useless,

that every so every such class, the class of all entities of every sort and

other remedy

define

is,to

similar to the

this method

exception, belongto

which

in

But

its number

as

The description. the

other.

more

of

a

class the class of all classes

Membership of this class of classes(considered propertyof

all the similar classesand

is

being different for two

classeswhich

definition of irreproachable

an

no

every class of the set of similar classes has to the set it has to nothingelse, and which every class has to its the conditions are completelyfulfilledby this class of

and it has the merit of being determinate when classes, and of

of

the number

not

are

of

a

a

class is

similar. class in

given,

This, then,

purelylogical

terms.

regarda number as a class of classes must appear, at firstsight, Peano a whollyindefensible paradox. Thus (F. 1901, "32)remarks that with the class of classes in the number of [aclass] cannot a we identify have the class of classes similar to a],for these objects question[i.e. He does not tell us what these properties different properties." are, and it for my part I am unable to discover them. Probably appearedto him class of classes. But somenumber that is e vident not a thing a immediately this in of view. be said the to paradox mitigate appearance may does denote a such a word In the first place, as coupleor trio obviously class of classes. Thus what two have to say is,for example,that we "there and men" are means "logical productof class of men and couple," is also In the which of class a men" "there is a men two means couple." is not itself a collection, when we remember that a class-concept second place, but a property by which a collection is defined, we see that,if we number is really the class-concept, not the class, a define the number as and of defined as a common nothing propertyof a set of similar classes to of This view the a else. removes paradox great degree. appearance and this in in There is,however, a philosophical view, generally difficulty It may be that there are many the connection of classesand predicates. and to no others. In to a certain collection of objects common predicates all regarded this case, these predicates are by SymbolicLogicas equivalent, Thus if the be is to any other. of them said to and any one equal, To

"

"

xi [CHAP,

Number

116

should we objects,

defined by the collectionof

were predicate

not

obtain,

for this class of but a class of predicates; a single predicate, general, and so on. The only should require a new class-concept, we predicates of the givencollectionof would be "predicability available class-concept in

terms

and of

But in the

others."

no

where the collectionis

presentcase,

by a certain relation to one of its terms, there is some dangerof of w, we said, is the Let u be a class;then the number error. logical

defined a

class of classes similar to "similar to v" definesthe

some

the

same

of the class of similar classes, the defining as predicate require, relation to one or more of conceptwhich does not have any special number that may constituent classes. In regardto every particular we

such whether finiteor infinite, mentioned,

of

but when all we fact,discoverable; of

is the number

real

to u"

or

similarto

u

to

"similar to u?

Thus, to

sum

v.

It is such

that must

up:

number

is that it

a

same

whether

we

the

use

is similar u v? provided

to

v.

that is or defining predicate class-concept

the class itselfwhose or

a

matter

a

reference to u special This,however,is not the pointat issue.

"similar to

This shows that it is not the but defined,

told about

is defined is the

what

pointis,that

"similar predicate

are

is,as predicate

a

class u^ it is natural that

some

should appear in the definition.

as

be the actual

of to; for,if v be similar to ?/, class, althoughit is a differentconcept.

be

The

u* cannot

to

constitutes the number

conceptwhich Thus

"similar

But

u.

terms

are

the various classeswhich

and not therefore, classes,

are

such predicates

be taken to constitute numbers.

a number Mathematically,

similar classes: this definition allows the

is nothing but

deduction

a

class of

of all the

usual

and is the onlyone (so of numbers, whether finite or infinite, properties in terms of the fundamental conceptsof far as I know) which is possible we general logic.But philosophically may admit that every collection of similar classes has some entities to no common predicate applicable and if we that can find,by inspection, exceptthe classes in question, there is a certain classof such common of which one and only predicates, to each collection of similar classes, then we may, if we see one applies classof predicates callthis particular the classof numbers. For my fit, and part,I do not know whether there is any such class of predicates, I do know

that,if there be such Wherever

a

it class,

Mathematics

derives

is a

irrelevantto wholly common

thematics Ma-

propertyfrom

a

and transitive relation, all mathematical purposes reflexive, symmetrical, of the

served when it is supposedcommon propertyare completely class of the terms replaced by havingthe givenrelationto a giventerm; and this is precisely the case presented by cardinal numbers. For the I shall adhere therefore, future, once

and adequate to precise

to the above

all mathematical

since definition, uses.

it is at

CHAPTER

ADDITION

112. find

IN

the

of

applicable sufficient alone

rationals,

to

does

absolute

integers I

are,

shall

numbers now

at

are

concerned

to

but

the

have

they

which

involve There

113.

kind.

logical

All

is

by

the

same

is

sum

the

logical

of

in

is

which

both confined

classes,

whether sum

if

and

u

of

and

u

to

to

be

the

as

v

defined

class

classes,

but

infinite.

or

classes

F.

1901,

class

k

"2,

the

belonging

k

(called

1-0.

be for

a

class

short

I,

logical their

belongs

every

logical

sum

logical product to

to

of the

class

every is

definition

extended

be if

Prop.

of

be

Fart

in

their

The

r.

logical is to

classes,

which

to

terms

This

Thus

integers

are

v

to

terms

may

composing

*

in

of

contained*.

are

u

and

explained

and

the

namely

this

propositions,

if

belongs

or

u

considered.

addition,

was

;

infinite

propositions

numbers

of

and

integers

to

and

all

of

are

we

addition

finite

to

terms

as

the

z',"i.e.

the

so

addition

are

q

and

q?

or

may

two

finite the

""

v

negative

of

in

the

and

p

or

belongs

propositions,

two

logical

class

and which

the

kind defined

too

with

only applicable

infinityof

be

positive

rigidly exclude

addition,

Logical

either

classes

two

be

can

disjunction;

the

the

or

present chapter

the

proposition "p

which

term

of

sum

I shall

between

numbers

And

are

fact

real

integers

signless.

fundamental

one

means.

as

The

chapter

present,

kinds

other

stage;

in

even

what

at

preceding and

poasibly

and

are,

later

a

the ;

cannot

being equally applicable

finitude

only

In

its

the

the

is

multiplication. defined

for

either

in this

of

merit

Indeed,

integers.

at

be

integers

present, in

fractions,

unity,

we

shall

dwelling

fractions

to

fractions

positive, but

defined

be

is

excluded.

present

not

are

multiplication

explain

to

For

and

integers

rational

the

integers, given

denominator

What

endeavour

also

extension

of

between

strongly emphasized.

integers.

which

without

numbers,

admit

operations

definition

a

of

whose

fractions

once

definition

not

difference

and

of

theory

at

real

to

even

The

us.

occupy

give

to

arithmetical

of

accounts

or

the

upon

chapter, obviously the

MULTIPLICATION.

endeavouring

length

will

AND

mathematical

most

error

XII.

not a

tially essen-

of

class

classes, the sura

of

k)

is

Number

118 class of terms

the

which

is

If Jc be

addition.

short

(calledfor

terms

class of classes

a

an

or

the

metical arith-

of the various classes of k is the number

Jc

order to

infinite. In

common

any

then classes),

of Jc. This definition is

sum logical whether appliesequally

in the

terms

have

of which

two

no

contains every class underlies arithmetical

which

exclusive class of

numbers

of the

sum

class which

every

It is this notion

of k.

terra

a

belongingto

[CHAP,xn

of-

and general, absolutely

any of its constituent classes be finite number ourselves that the resulting

or

assure

onlyupon the numbers of the various classes belongingto AT,and depends class Jc that happensto be chosen,it is necessary not upon the particular exclusive class of classes,

another

done) that if Jc'be (asis easily

to prove

of k is similar to its correlate in Jc', and similar to ", and every member of Jc is the same the vice versa, then the number of terms in the sum as

number

in the

of terms

of terms

in

have

common

no

of

sum

suppose u and in the logical sum

v9 and

terms, u and number

and

u

regardto

of

the number

this definition of

number occurs emerges when one is to be observed that the numbers so

have

we

such

no

symbolism,which

in

causes

an

correlative order in what

order,if one a

number

firstand

of supposition can

above numbers whether

its

of

sum

by

certain

a

numbers, it is to be

twice

or

the

order of

no

number

essentially of

or

summation,

we

the said

the summation

:

and therefore classes, is repeated or

a classes, particular

If

number, there is of

a

:

this position, prodefective has

for there is

plainthat

no

exclude

we

class of numbers

repeated.In

the

defined

the

are

number

1, and there

as

it is not

of numbers

sum

cannot

a

in the

sense

of which

take the number

are

some

multiplication. a special considering case,

we

of

necessary to decide But in order to define, without

not.

may

It is

no

distinguish

it is necessary first to define repeated, This point be made clearer by

1-f 1.

It

the absence

cannot

of the said number.

have

some

summation,

law

commutative

symbolized.But owing to a

the

this reference to classes

in that case, no number be can definition of a sum, the numbers concerned

any

have

oftener in the summation.

concerned have

twice occurring

of certain

is

class of classes

but defined,

reference to

of the number

sum

summation

of necessity

twice in

occurrence

reference to classeswhich

be

is the

Arithmetic,results only from a the symbols which order among

is

occurs

second

a

a

obtained

as proposition

introduced

as

v

n

similar class of similar classes. The

that

and

the

be similar to ", and vf to t",and ?/,v is similar to the the sum of it and v

if

question.The number of the logical of sum

in

u

part.

common

no

only two Then

be freed from reference to classes which

observed that it cannot numbers

have

v

Jc has

example,suppose

v.

With

114

and

in v;

part, then

and

u

of Jc'. Thus, for

sum

such

1 itselftwice

instances of it.

as

over,

And

if a ddition of 1 to itself were in question, logical should find that we 1 and 1 is 1, according to the generalprinciple of SymbolicLogic. Nor

the

one

are

not two

Addition

113-116] define 1

we

can

method

This

"+" 1 as

Multiplication

the arithmetical

be

involved

member,

can

not

two,

each

certain class of numbers.

a

2, or any

in which

sum

define 1 4- 1 by its means. follows : 1 -f 1 is the number of classes u and The

term.

of numbers

is

which

v

chief

have

Whitehead*.

It is

Let

k be

term

common

observed

is,that

metical notion, while the arith-

whollysubsequent. is due multiplication

follows.

as

full

which

class w

a

no

be

point to

the

Thus

generaldefinition of

The

no

1, the only class of numbers is 1, and since this class has one

of classes is the fundamental

addition 115.

+

119

cannot

we

only one

addition logical

regards1 +

as

definition of 1 H- 1 is as of two is the logical sum and have

of

sum

employed is repeated; but as regards1 is the class whose onlymember

number

which

and

class of

a

Mr

A.

N.

no classes,

two

of

to

have

Form what is called the multiplicative any term in common. class of ", i.e.the class each of whose terms is a class formed by choosing and

one

only one

the number

from

term

each of the classes belonging to Jc. Then

in the

of terms

class of k is the productof all multiplicative of the various classes composing". the numbers This definition, like that of addition givenabove, has two merits,which make it preferable to any other hitherto suggested.In the first place, it introduces no the numbers need of the order among that there is no so multiplied, in the

case

with what

is

law, which, here

commutative

as

rather with the

symbolsthan

the place,

definition does not

above

definitions of the

givenf

requirea decision definitions

These

number allow

of finite the

these

be extended

can or

whether

to

as

productof

and

sum

the

sum

or

finite

are

they

productof

infinite.

or

productof

and

sum

do not

numbers, which

two

numbers

the

to

infinite numbers; but

definition of

the second

symbolized.In

to decide,concerning us require any has Cantor infinite. finite or are they

involved,whether

of the numbers

addition,is concerned

of

any

finite

do

not,

an

infinite number

as

they stand, of

in the above

which definitions,

pursue Arithmetic, as it ought to be the distinction of finiteand infinite until introducing

without pursued,

numbers. enable

This

grave defect is remedied

to

us

Cantor's definitions have

it.

order among case,

a

the

numbers

defect in the

mere

symbolizes.Moreover or

sum

productof

two

cumbrousness resulting

also the formal

summed

or

defect of

multipliedbut :

in the

symbolschosen,not

wish

study an introducing this

numbers,

to

avoid

this formal

to

is,in his

ideas which

in the desirable, practically

it is not

becomes

we

case

he

of the

defect,since the

intolerable.

definitions the usual the above It is easy to deduce from be thus stated. which may connection of addition and multiplication, each of which contains If k be a class of b mutuallyexclusive classes, 116.

a

terms, then the *

sum logical

of k

contains

Journal of Mathematics, Oct. 1902. Annalen, Vol. XLVI, "3.

a

x

b

termsj. It

American

t Math.

J

See

Whitehead,

toe. tit.

is also

Number

120

obtain

to

easy

and

laws, it

is

independent

as

done

is

is

by

general infinite

as

together From

ba.

to

number

simply

there

of

For

is

this

for

no

ordinal

Powers,

products

regarded

in

so

of

all

the

the are

numbers

a

defined, Moreover is

be

in

not

result

to

new

cation. multipli-

ab

since

define

therefore,

But

as

doing.

notions,

which

in

be

cannot

we

reason

exponentiations.

abbreviations

can

ab+c.

independently

be

tributive dis-

=

application

an

xn

and

abac

as

to

not

advantage

introduces

unavoidably

equal

is

associative

such

powers,

merely

is

it

the

prove

exponentiation but

Cantor*,

exponentiation

for

laws

since

that

true

to

exponentiation

that

operation, It

and

"*,

of

formal

the

observed

be

to

definition

the

[CHAP,

of

an

regarded

multiplied

equal.

are

the

hold

equally

step,

therefore,

data of

which

finite is

to

and

now

we

infinite

consider

the

all

possess,

numbers

can

distinction

infinite.

*

Loc.

rit.9

"

4.

those be between

propositions deduced. the

which The

finite

and

next

the

CHAPTER

FINITE

THE

117.

difficulties V.

Part

For

the

cardinal understood Let term

For

before

be

u

the

u.

example, if

all finite numbers

of the terms

similar

of u, and

to

some

If there

is

similar

class

u'9 it

instead

away

take

possibleto that

say

u

be

u

and

a

no

finite

an

x

also

get

From can

these

When

the

numbers

of

operation. 118. has

one

part

*

On

a

not

the

of what

while

It is easy

Among smaller the

finite classes

to

finite

present

prove

topic

of

of terms But

cf.

among

Cantor,

will be found.

is

is

than

a

proper

the

from

w,

that

that

u

the

is finite

;

definition

are

altered

by

this

of 1.

part of another, the

other.

Anna/en,

to

prove

of the addition

(A proper

infinite classes,this

Math.

it is

say

unaltered

are

holds

same

to

is

null-class is

to

null-class

classes

infinite

one

easy term

one

the

than

the

that

y

similar

u

possible,

It follows

u.

other

term

we

It is also

by adding

that the

classes,if

number

whole.)

follows

those

class

a

to

u

When

u.

is

is not

u

other

to

it follows

it.

from

class,the class formed

leave

this is not

definitions

be taken

similar

n

from

away

if any

n' is

TZ, where

except 0, then

that

and

u

to

up

ince

Thus

over.

be taken

can

class

a

from

term

one

which

easilyproved

we

twice

1 to each

of u' and

one

u'.

to

class of

u' the

by adding

with

u

taken

all these

of x

class. infinite

term

by subtracting1,

most

term is

of

term or

of

be

must

is similar

u

numbers, and

of all finite numbers

itfconsists one

that

obtained

are

u

one

converselyif this class is finite,so

that

a

of

away

finiteclass.

finite,since if

is

of

omitted

consists

u

u.

a

"

if

to

taken

is

But

all finite

terms

being

number, and

leave

u

of either

it.

of

theory

form, and

happen

not

the

in

matical mathe-

adequately explained*. formed by taking away one

be class

a

brieflythe

it appears

can

may

this correlates

finite

similar

we

class

except 0, the

term

no

versa,)

the

are

forth

set

philosophical postponed to the

discuss

to

fundamental

let ur be or

to

as

infinite

it may

be

u

infinite

ordinal

Then

merely

is its most

class,and

any

from

x

This

not

infinite,which

the

I wish

finite and

numbers.

INFINITE.

present chapter is

concerning

present

of

theory

AND

of the

purpose

XIII.

Vol.

no

XLVI,

part is

longer holds.

"" 5, 6, where

[CHAP,xm

Number

122

distinction is,in fact, essential part of the above definitions of an the finite and the infinite. Of two infinite classes, have a one may

This

greateror

a

smaller number

than

of terms

the other.

class u is said

A

greaterthan a class t", or to have a number greaterthan that of v9 the two are not similar, It but v is similar to a proper part of u.

to be

when

is known

part of

that if

(a

u

is similar

which

case

to

is similar to

u

greaterthan M."

is

"u

It is not

infinitenumbers,one

proper

onlyarise when

can

hence

u;

a

greater than

at

be

must

part u

and

v^

present known greaterand the other less.

different infinite number.

This

the

denoted,in

is made

it is known

But

any

of finite

is the number

mention

u

"r" is

is less than

which

which integers, is capableof number

presentwork, by a0*" This no

proper

different

whether, of two

number, i.e.a number

several definitions in which

a

then infinite),

are

v

to

v

is inconsistent with

that there is a least infinite

will be

of v9 and

of the finite numbers.

In

done by Cantorf)by means be defined (asis implicitly of the principle of mathematical induction. This definition is as follows:

the firstplaceit may

OG

of a one-one of any class u which is the domain is contained in but not coextensive domain

is the number

R, whose and

converse

is such

which

the

successor

not

a

successor

of

again,we

which

u

belongsto

define

may

and

of P

relation either

term

or

its

x

with

u9

has the relation R

P) have

belongto

the field of P

immediate

an or

u

which

have predecessors

then the number

Let P be

to which

successor,

i.e.a

of u ; term precedesome but has no predecessors, and

successors

of terms

is

belongsto

u

s.

transitive and

a metrical asymof the fieldof P have the

let any class u contained in the field

(Le.terms

having successors

of

term

every

different terms

Further

converse.

,9,then

follows.

as

a0

and let any two relation, relation P

to which

of -r, if $ be any class to which belongsa term of u which the successor of any other term of u, and to which belongs

of every term Or

the that,calling

relation

also have

an

of

every term whose term let there

predecessors

be

immediate

is a0. it is not equivalent

Other

of

term

one

let every term

in the field of P

has the

u

which

has

predecessor ;

definitions may

but as all are suggested, necessary to multiply them. The followingcharacteristic is important: Every class whose is a0 can number be arrangedin a series having consecutive terms, a of any beginningbut no end, and such that the number of predecessors term of the series is finite ; and any series havingthese characteristics be

has

the number

a0.

It is very easy to show number is "0. For let u Then

u

is similar to the class obtained

call the class

*

Cantor

notatiou t

that every infiniteclass contains be such a class,and let x0 be

^^l.

Thus

employs for

z^ is

an

this number

is inconvenient.

Math.

Anualen, Vol.

XLVI,

" 6.

by takingaway

infinite class. From

the Hebrew

Aleph

with

classes whose a

#0, which

this

of

term

we

the suffix

we can

u.

will take

0, but this

119]

118,

term

a

away terms

be

now

119. 1

to

may

class

in

number

number,

since,

is

similar

is

finite

to

finite

numbers

induction,

it

of be

is the

filled

and

it

is

without

mathematical

theory the

course

the

to

by

us

to

of

finite the

and work.

set

forth

infinite,

than

ae

that

the

with

define

may

mathematical

step,

or

those

as

obtained both is

a

the

leaving

by

frequently consequence the

for

hereafter;

much

controversy, of

that

are

either

a

number

prove

we

themselves

definitions

realize

to

of

such

as

by each

at

not

to

Thus

every

number

greater easy

is

the

such

no

reached 1

parts

two

occupy

new

be

can

is

definition

equivalent.

increasing

important will

the

is

a0

is

it

be

can

of

successor

number

no

We

1.

which

numbers

proved,

But

with

These

terms.

a

already

that

and

similar

not

Both

during

0

the

0

induction.

number Now

also

are

those

as

and of

definition. finite

of

number.

given

of

mathematical

0

with

that

numbers

class

successor

definitions

intended,

in

a0

the

adding

by

beginning

additions

those

belongs

Hence

is

Thus

obey

is

the

new

from

that

which

obtained

n.

as

propositions

the

two

single

and

itself.

of

are

other.

only

of

either

away

the

which

induction,

number

choose,

we

the

to

than

the

which

employed,

definition

alternative

an

the

has

which

type

successive

by

numbers

to

s

1

to

from

where

s,

part

a

different

steps,

virtue

starting

classes

taking

of

in

less

Hence

of

to

according

old.

the

mathematical

the

numbers

finite

adding

number

every

of

of

series

The

on.

so

to

of

means

number,

if

class

belonging by

of

of

every

obtained

is

such

by

0

the

is,

contained

and

numbers,

from

That

finite

any

finite

obtained

is

advance

can

by

and

"j,

and

",

we

infinite

series

a

define

in

point

finite,

form

can

we

be

n

also

is

class

123

explained.

If

n

the

Infinite

infinite

an

this

and

and

contained

From

cr0.

must

is

x^...

finite

the

leaving

#13

a?ls

number of

Finite

bare the

present outlines

details

to

XIV.

CHAPTER

THEORY

HAVING

120. we

customary, define

best

the

or

shown

follow.

to

merely

as

branch

certain

shall and

general Logic.

Peano^s the

best

has

the

exposition in the from the point of inestimable

and

Logic)

also

three

For

this

that

showing

fundamental

that, if the three

notions

regarded

renders

interpretationwhich

to

here

of

a

only remains, in order adopted, to give a definition

demonstration

has

been

finite

*

Except

t

F.

since

the

1901,

Part

be

can

of

veloped de-

general

notions.

It

determined

as

the

five

propositions,

Peano's

three

the

with

theory

fundamental

notions

propositions. When certainty that

with

once

It that

and this

everythingin

the

integersfollows.

Frege's Grundgwetze

"number

it is

know

of

out

connect

five fundamental

accomplished, we

theory of

making

of

exposition

remaining proposition false.

the

therefore

take

by the mutally independent. This

these five propositionsare propositions, shown by finding,for each set of four

an

I

know,

I

those

to

propositionsconcerning these be

here.

method,

as

This

five fundamental

five is

far

Arithmetic

(in addition

notions

question

shall

I

purpose, so

a

usually taken

are

rigour.

all

present

usual

more

is,

and

of accuracy

view

in

adopted

what

which

Fonnulaire^^

of

merit

the

dependent in-

indefinables, of

method

that

proofs of

and

indemonstrables.

and

reason,

-expositionof

an

definitions

to

this

For

results an

the

in

or

the

certain

into

is done

to

not

ordinary

Aiithmetic

axioms

new

It is

with

begin

to

as

infinite,

Arithmetic*,

all the

of analysisthan

degree

begin by

proceed

from

proves

less

a

nevertheless then

regarding it,

development, without

a

of

indefinables

as

but

makes

method

of

of

elements

the

on

This

instead

indicate

to

seems

treatises

or

study, work,

consideration

the

to

the

numbers.

finite

of

particularfinite numbers, primitive propositions,from which

number

axioms are

ourselves

in

finite from

the clearlydistinguished

now

devote

can

NUMBERS.

FINITE

OF

II and is

implied by

a

class" "0

is

a a

der

1899, "

F.

Arithmetik 20

ff.

F.

(Jena, 1893). 1901

differs

primitiveproposition.

number."

I therefore

I

from

regard

follow

the

earlier editions

this

as

unnecessary,

earlier editions.

in

Theory of Finite

120-122]

three indefinable

Peano^s

assumed, as part of think,be better to state

I

has

and

one

only one is

following.(1) 0 is

numbers

(5) If

be

a

two

a

a

(4)

(%) If 0

it

(though

is

a

have

is not

belongs0

number, the

a

the

the

same

successor

and

also the

of course, the then

successor

the

successor,

of any

demonstrated

meanings

by 0,

and

Padoa but

successor,

all the usual

Giving the

(")

of independence

Peano

and

0

to

than

other

mutual

The

above

meanings to

integer,all the

above

which exceptthe fifth,

is not

last of

five

has been propositions follows f. (1) Givingthe usual as denotingby number finite integers

propositionsexcept

less than onlyfinite integers

number

these

two

of every

successor

belongsto s. The s, then every number of mathematical is the principle these propositions induction. 121.

of

number.

belongingto

number

of. would,

separateaxiom), that every number

numbers

class to which

successor

is meant, (By successor are primitivepropositions

number.

identical.

are s

as

successor.

(3) If

number.

a

idea of succession

successor.)Peano's

immediate

a

it

125

and Q,"Jinite integer*,

are

the

It is

Numbers

0

and

10, or

successor,

the

first

but

less than any

are

true.

denotingby other

specified

secondL propositions and then becomes periodic (3) A series which beginsby an antiperiod (forexample,the digitsin a decimal which becomes recurringafter a of places) will satisfy all the above propositions certain number except the hours the series the third. (4) A as on (such clock) periodic satisfiesall exceptthe fourth of the primitive (5) propositions. Giving of 0 is ", the meaning greater by 2, so that the successor to successor satisfied and of " is 4, and so on, all the primitive are propositions finite

0. including

deduced

Thus the

from

no

one

are

satisfied if of the

five

s

true

except the

be the class of

even

numbers

can primitive propositions

be

other four.

that other classes besides that of pointsout (loc.cit.) the above five propositions. he says What the finite integers satisfy is the all of is as follows : There an infinity systemssatisfying primitive number 0 by all and verified, They are propositions. e.g., by replacing the primitive number other than 0 and 1. All the systemswhich satisfy Number have a one-one with the numbers. correspondence propositions all these systems by abstraction ; in other is what is obtained from enunciated words, number is the system which has all the properties and those only.*" This observation appears in the primitive propositions, the question In the first place, correctness. to me lackingin logical fying arises : How which agree in satisthe various systems distinguished, are the primitive propositions?How, for example,is the system from that beginningwith 0 ? To this 1 distinguished beginningwith 122.

Peano

"

*

Throughout

finite integer. t F. 181)0,p.

the rest of this chapter,I sh^ll

30.

use

number

as

synonymous

with

Number

126 different

questiontwo 1

0 and

1

be

can

We

given.

say that

may

at

which, by the way, will have and succession ideas,number to the other primitive if

But be extended

take this view

we

and

"

we

to

method, 0, number,

In this

in the definition of number.

speaksof

ideas which must as appear, like other indefinable*, what mathematicians Their yields recognized. recognition

be

succession

it

existence-theorem,i.e. or

to

"

I call constants, what say that these three notions are that there is no need of any such process of abstraction as Peano

shall have

But

0 and

least that 0 is so, and that therefore tinguish as intrinsically yellowand blue are disdistinguished,

or ideas, primitive

both

are

be

may

answers

[CHAP,xiv

numbers this process leaves it doubtful whether not, and therefore makes Arithmetic, accordingto the

I, Chapteri, primti faciea branch of

Part

simply

call

definition in

AppliedMathematics.

which

the

reallyare numbers. constants are logical

there

that

us

assures

and

over More-

in mind.

has

Peano

The not evidently process other answer to the questionconsists in regarding0, number, and succession as a class of three ideas belongingto a certain class of trios It is very easy so to state defined by the five primitive propositions. the matter that the five primitive become transformed into propositions the

it is

the nominal

longerany become

definition of indefinables

pure pieceof

a

since they are variables, the

moreover

certain

a

indemonstrables

or

be

there

are

any

constant, and

a

such

trios at

thus

becomes

now

at

least

all.

What 0, number, and succession. such trio,there are an infinite number

out

the

first term

from

any

class

number, we concerning conditions in question.But number

is still in

is to l"e avoided.

abstraction from

even

question,must Moreover

be

For

onlya

properties by which in in

by striking

is therefore

of

variable member

meaning of if circularity

:

it, to

of

process of

a

no

Thus

Peano

as

class is the term

false when

the

of the

the class is defined.

Is any

five axioms, such

term

merely the properties definingthe class and of Peano's process abstraction really to is amounts members.

For

again satisfies the

has

variable members

that, if there

conditions laid down

ask ourselves

logically contemplates, possible Every term satisfies and which becomes some is, proposition

the class and

is

worded differently

the satisfying

There

class,

givingconstant

show

of them.

?

of the class is substituted.

cannot

we

this statement, since the

must

we

all systems

of

class which

a

:

trio,however, would

can

we

the satisfying

alwaysobtain

class of trios

actual trio of this

method

requiresome

we

no

has

succession become

of the

actual

values to is one

and

doubtful,since

one

One

then

are

theory,which

our

0, number Logic. onlydetennined as one

of know5 except by the discovery that

in

But

existencertheorem

There

class of trios.

no

of

another a

it

term

class which

others.

What

the consideration of

exclusion

of

class will have Peano

does not

constant

only the succeed

indicating meaning for 0, number, and succession,nor any constant since the existenceshowing that any constant meaning is possible,

123]

122,

Theory of Finite

Numbers

127

is not

is to say that at proved. His only method, therefore, such constant least one be but is meaning can immediatelyperceived, is not logically not definable. This method unsound,but it is wholly different from the impossible abstraction which he suggests. And the of his five primitive is proof of the mutual independence propositions in show that the order to definition of the class of trios onlynecessary is determined them not redundant. by Redundancy is not a logical in error, but merelya defect of what may be called style.My object, the above account of cardinal numbers,has been to prove, from general

theorem

Logic,that

there

is

meaning which satisfiesthe above five meaning should be called number,

constant

one

and that this constant propositions, or

rather finite cardinal number.

and

indemonstrables

are

the

class of trios in

questionhas

of trios has

by

whollyavoided

been used

has

member

to

define

infinite number

an

And

at

in this way, ;

for when

least

number,

means

For

the

show easily

of members, and

of the five properties enumerated

indefinables

new

have shown

member, and

one

we

we

we

in Peano's

when

that this

that the class

define the

class

tions. proposiprimitive

of the connection between comprehension

Mathematics

and similar pointswill Logic,this pointis of very great importance, occur throughoutthe presentwork. constantly In order to bring out more the difference between 123. clearly and mine, I shall here repeat the definition of the Peano^s procedure the definition of jimie class satisfying his five primitive propositions, in the case of finitenumbers, of his five primitive number) and the proof, propositions. his axioms is the same The class of classes satisfying as the class of class whose of classes, cardinal number is cc*, i.e.the classes to according is defined which It is follows most the is as : a0 simply o^t my theory,

and

class of classesu each of which is the domain

(the relation least has

of

a u

one

of

term

a

term

and

its

u

relation R

one-one

successor)which is such that there is

at

no

and also predecessors,

having no

some

other term, every term which succeeds is contained in any class $ which contains a term

succeeds

which

successor,

to

of

contains the

of every definition includes Pea-no's five successor

belongsto s. This Thus of every such class all the and no more. primitive propositions be proved: usual propositions in the arithmetic of finite numbers can the be whole of etc. can defined,and addition,multiplication, fractions, in so far as complexnumbers are not involved. be developed, can analysis But in this whole development, the meaning of the entities and relations since the entities and is to a certain degreeindeterminate, which occur

term

of

u

which

the relation with which

Moreover, in this whole classes

as

start

are

variable members

nothingshows development,

of

a

certain class.

that there

are

such

the definition speaksof.

In the We

we

we logical theoryof cardinals,

first define

a

start

from

the

and then show certain class of entities,

oppositeend. that this class

Number

128

of

belongs

entities

(1)

0

number the the

of

belong

to

if

classes

two

sums

numbers.

We

assumes

if

n

a

n

class,

belongs

(4)

to

a

0

it,

Hence is

mathematical indemonstrables

one

have,

be

n

to

all

-f

this

finite

class

definite

of

Meaning number.

the

a

n

class,

no

whole

and

by classes

a0

of need of

class has

a0.

whatever Arithmetic

it.

to

of

and

is,

and

0.

of

If

it,

to

the

the

as

class from

indefinables

Analysis.

?z,

then

1,

4-

m

all

therefore, new

finite Peano

(5)

.numbers

members,

of

=

Thus

finite

There

of

belongs

n

which

to

successor

1

4-

from

when

belong the

n

the

number

which

the

y

that

numbers

and

0,

by If

different if

the

definition

1

-f

n

is

1

+

is

propositions

(3)

numbers

member

standpoint, in

(2) is

satisfied

are

the

1

five

the

regards

as

the

and

x

Finite

belongs

to

each,

to

(5)

completes

This

if shown

1

-f

is

1

belongs

x

that,

n

terms.

n

which

to

if

added

number,

A

(3)

Having

be

term

a

of

s

number,

any

belongs then

n

a

(4)

one

be

n

class

class

number. then

properties

defined. number

If

and

essential

is

number,

a

77i.

=

then 0

to

every

of

if

that,

belongs.

n

are

that, such

or

;

(")

themselves.

such

are

identical.

class

a

unit

a

and

xiv

follows.

as

null-class.

of

one

any

done

is

the

is

null-class

the

y

and

to

if

(1)

:

be

x

define

belonging

belongs

and

to

null

not

is

x

similar, we

similar

are

then

adding

from

those +

class, be

1

n

which

member

only

classes

This

defined.

above

c^

whose

without

similar,

resulting

all

of

class

the

are

classes

classes

all

the

class,

are

class

is the class

of

class

the

is

class

the

to

[CHAP,

be

s

n

-f- 1

five

above

finite the or

CHAPTER

ADDITION

OF

TERMS

XV.

AND

ADDITION

CLASSES.

OF

.

124.

HAVING

cardinal

it is time

numbers,

questions raised remarks

by

the

to

as

this

forth

set

to

turn

I

attention

shall

between

mathematical

the

our

theory.

distinction

function

the

briefly

now

begin by

philosophical preliminary

the

to

few

a

philosophy and

of

theory

and

mathematics,

philosophy in such a subject as the foundations mathematics. The not necessarily to are following observations of philosophy, since they regarded as applicable to other branches derived specially from the consideration of the problems of logic. as

to

distinction

The

point

of

view

and critical,

:

of

in

is constructive

certain

a

mathematics

and

philosophy

mathematics

deductive

have

of

impersonal

reasoning, we

controversial.

sense

between

such

entities,are

fact, mainly

perceived by

the

some

not

so-called

senses,

philosophy,except as regards the it seems whether highly doubtful In with not

know

however,

case,

any

sensible

objects,we

regarded

as

anything

existingin about

them,

space

and

must

be

be

mathematical

reasoning

philosopher.

When

its

deduction

results

may

inquiriesthat disproof will

;

can

proceed. consist

may in

also

in

fonn

it is this

Now be

it follows

out

;

but

maintained.

entities

which

entities,if sense

of indefinable

has in

been

entities

can

contradictions

the

plished, perfectlyaccom-

nature

very be

never

and

any

concerns

which

premisses from

the

be

must

starting-point for

the

to

are

we

also

relations

body

are

perceived,and

starting-point that

disproved,but

pointing

to

some

from

of

scope

essentiallyunconcerned

their

certain

for

axe,

relations be

can

Such

time.

philosopher'swork be wholly embodied

the

results

and

is

remarks

our

another;

from

distinguishedone in part immediately apprehended. A and indemonstrable propositions must must

exclusion

such

the

their

of

abstract

sounds,

within

coming

as

more

confine

may

distinguishing

the

and

colours

as

present work

the

since

such

any

we

principlesof

philosophy. Philosophy is,in are perception. Entities which

regarded

commonly

of

one

of

business

insight and

of

question

a

reason,

the

;

entities, and

deduction, the recognitionof indefinable

are

Wherever

the

but

be

philosophy is

deductive,

and

mathematics

have

broadly

is

of

of

proved.

such The

inconsistencies

;

Number

130 absence

the

but

of these

can

[CHAP,xv to

amount

never

proof. "\l depends,in philosophical argument,

perception;and the reader to consists mainlyof an endeavour to cause speaking, strictly The the author. argument, in perceivewhat has been perceivedby of exhortation. but Thus the the of nature of proof, short, is not indefinable set of entities : Is there any questionof the presentchapter immediate

end, upon

the

to be settled by inspection question, essentially philosophical of chains rather than by accurate reasoning. the question examine shall whether In the presentchapter, we

defined?

is

125.

an

in any There are

definition of cardinal numbers

the above

fundamental

more

the set of entities above

different from

called numbers, and

commonly

of number.

sense

way presupposes some several ways in which

In the first place, the individuals this may be supposed to be the case. which to be each in some sense one, and it might compose classes seem

be

thought that

the number whether questioned from

that

one

relation could not

one-one

a

a

In

1.

class which

And

term.

second

the has

in the

be defined without

it may place,

only it place, one

term

third

can

well

very

be

be

distinguished

be held that

may

troducin in-

the

different from that above of class presupposes number in a sense defined: it may be maintained that classes arise from the addition of notion

addition and, and that the logical

indicated by the word individuals, as of classes is

demand I

subsequentto

this is of one

course

"nd

which

follows y, and

that seems

defined

they are :

x

fact that any individual or term is in some sense one, the that of it notion But does not follow

of

one

no

reason

by

means

individuals of term

or

is derived. to

of

it may be, on the individual is the fundamental one,

are

This

spoken of: view

rejectit. And without identity,

was as

adoptedin for

any xf when have and relation a one-one if, x has the relation R to y and y',then x and x R

is

one-one

mention

Part

are

are

I,

relations, of one,

the relation R

as

to

and identical,

y and y '. It is true that here x, y, #',y' are each one is this not (it would seem) in any way presupposedin the

so

questions

and

one

undeniable.

presupposedwhen

there

of

inquiryinto the

that the notion "contrary, from

These

and here, of class, meaning shall find ourselves aided by the theories set forth in Part I.

new

a

hope,we As regardsthe is

of individuals.

this addition

term, but definition.

of the (pendinga new disposes classes) firstof the above objections. is as to the distinction between a class containing The next question onlyone member, and the one member which it contains. If we could class with its defining a or no identify predicate difficulty class-concept, would arise on this point. When certain attaches to one a predicate and only one term, it is plainthat that term is not identical with the in question.But if two the attach to precisely predicate predicates should same terms, we are different, say that, althoughthe predicates the classes which they define are i.e. there is only one class identical,

This

inquiryinto the nature

of

124-126] Addition all

define.

both

which

of

Classes

and

131 and

men,

are featherless bipeds

differsfrom

featherle"s biped. This shows that a be identified with its class-concept cannot or defining predicate. be to might seem nothingleft except the actual terms, so that there is onlyone term, that term would have to be identical with

class There when

for many

class. Yet

meaning of

the

formal

symbolswhich

that

is

"

member

a

point is

The

have

terms

class of a

the relation of

class contained

the

But

to

we

can

classes

a

not

another to

number;

a

the

one

whereas

" is

To

is to cause havoc distinguishes of the formal destroy precision many

indubitable It seems, in fact, that some be found of way must

a

number, but

arguments and

term a discriminating

third

only. this point,it

and difficulty,

the notion

Peano's

distinction is from

is necessary to pass to our notion

of class itself. This

of

in Part I, denoting, explained of which one Chapterv. We there pointedout five ways of denoting, This was the kind indicated by aU. called the numerical conjunction* we is relevant in the This kind of conjunction appears to be that which aU men the will of classes. For example, man case being class-concept,

appeal's

to

be connected

be the class.

class,but what

particular way since any

man

But

with

is

the two identify in the theoryof

just,and class containing that a

reconsider the notion

nected con-

it is contained.

that

In order to decide

is

S, give 5, is contained in the

in the class of numbers.

term

term.

individual to its

an

in which

definitions.

126.

say

peculiar they belong to.

SymbolicLogic, and

Peano

relations which

This

3, give 5.

to

%.

the relation of

class to

which,added

numbers, but is

and infinity,

to

in Peano^s

prominentone

a

the class of numbers

Thus

which,added

except the number

with his distinction between class and

symboliclogic. For

indicate that the class is different from

to

seems

numbers

give the

cannot

of this class,i.e. it has to the class that

indefinable relation which This

this view

reasons

stand for classes in

example,consider the class of terms is a class containing no

not

Addition

If,for example,all featherless bipedsare

identical, though man

the

and

featherless bipeds, the classes men

are

men

Terms

of

qua conceptwhich will be the denotes,i.e.certain terms combined in the

it will not

be aU

this concept indicated by aU. or

some

man

men

is essential, way of combination is plainly not the class, though either denotes The

might seem as though,if of its terms, we must a class with the numerical conjunction we identify deny the distinction of a term from a class whose onlymember is that But we found in Chapterx that a class must be always an object term. of a different logical type from its members, and that,in order to avoid be extended even to classes the proposition #"#, this doctrine must far this forbids us to identify How member. which have only one to decide ; in any I do not profess classes with numerical conjunctions, and the class whose only member term a case, the distinction between be taken extensionally to the it is must be made, and yet classes must combinations

of

the precisely

same

terms.

It

Number

132

[CHAR xv

are degree involved in their being determinate when their members and cardinal given. Such classes are called by Frege Werthverlaufe; this in sense. numbers to be regardedas classes are which is this: a certain is There still,however, a 127. difficulty, class seems to be not many terms, but to be itself a single term, even of the class. This difficulty would seem when many terms are members

be identified with all its

members, but In order, the whole which they compose.

to indicate that the class cannot

regardedas in an unobjectionable must we however, to state the difficulty manner, of it,since these notions from the statement exclude unityand plurality of the notion of class. And here it may be to be defined by means were Is the to the reader. to occur well to clear up a point which is likely of one notion presupposedevery time we speak of a term ? A term, be

is rather to

said,means

be

it may

a concerning seems proposition

term

without

is,is one, yet truth of

by all men, or the class has only one

absent, and

is in any this view there is not

a

classes as many, but the in other assertions. as suitors

is

this view,to one.

assertion

an

we

certain

a

yet have

may

one-ness

is

and

"a

term"

128.

It

of

use

beingand

:

It is difficultto be

whatever

are,

objectwhich by any concept

one,

as

whole

a

identical with

But the many. i.e.the kind class,

are a

of

class,is

a

as contradiction,

;

not

But in many. in the theorythat verbs and as

be made subjects ; for assertions can of such assertions is subject many, not one "Brown

about

seems

as

Jones

are

class "Brown

be

about them.

and

Jones," but

classeswhich

in which

every

A

class

a u "

implies x

class

has

one

be may member

said to

is identical with

*

t

when

y?

have u

one

is not

about

distinction

as

objectis one,

Here

the

member

null,and one-ness

is "x

is a

Ed. Gerbardt, ir, p. 300. Grandlagen der Arithmetik,Breslau,1884, p.

40.

not

are

a

term,

regards is

which

involved in speaking of an object, apparently is,as Fregeurges f, a to everything alike. But the shadowysense, since it is applicable in which

not

conclude that

we

in statements presupposed, regardedas an indefinable.

sense

Smith's

in belongs,

one-ness

Hence

a

only

of Miss

not

however,to make

about

two

Thus

a

necessary, The

one.

and the

assertions made

is to

one

but this is sometimes

the class

singleterm. of but logical subjects, type

but implied,

as

how

sure

althoughwhatever

is

is the class

this class considered

about

sense

of one, this

term, and must not be made a single said in Part I,Chaptervi, in simplecases an

not

case

be made

can

be made

cannot adjectives

the

that

true equally

There is, as subject. logical term which associated single

"

some one

merely grammatical.For

are

objectdenoted

except where

In

be that the kind of

to

seems

statement

no

is,is

Whatever

convertible terms*.

statements

it is

thus

one. presupposing

indubitable.

Leibniz remarks, are far such

term, and

one

very sense

quite precise. and y

are

propertyof

wV the

126-130] Addition class,which the

be itselfa

may

of

sense

and

therefore be called

may

onlymember

Terms

of

involved

one

Addition

unit-class. The

a

class of many

in

term

one

of

*r

terms, and

or

which

133 is its that

this shows

is not

term

a

Classes

relevant

to

of a class Arithmetic, for many terms as such may be a single member is One, therefore, not to be asserted of terms, but of classes

of classes.

member

having one is

"w

and y

a

or

if

many

of

member

the

is not

"u

means

identical V"

are

one

or

unit"

a

in the above-defined

The is

u

a

null,and

i.e."

;

and y

*x

is

n

one," or better wV

are

*

impliesx

member

of w, in this case, will itselfbe none class of classes; but if u is a class of terms,

will be neither

u

sense

none

nor

one

nor

many,

but

simply

term.

The

129.

commonly received view, as regardsfinite numbers, is that would preferto they result from counting,or, as some philosophers those who hold this view have Unfortunately, say, from synthesizing. the notion of counting:if they had done so, they would not analyzed that it is very complex,and have seen presupposes the very numbers it is supposedto generate. which The process of countinghas, of course, a psychological aspect,but this is quiteirrelevant to the theoryof Arithmetic. I wish now What is to pointout the logical which process involved in the act of counting, is as follows. When we are necessarily say one, two, three,etc.,we relation which the numbers used holds between some one-one considering in countingand the objects What counted. is meant by the one, two, is that the objects three their correlates indicated by these numbers are in to the relation which we with respect have mind. (This relation,by the way, is usually is involve and to a reference extremelycomplex, apt correlate a class of objects to our state of mind at the moment.) Thus we a

"

with

class of numbers

a

numbers drawn

from

from that

of

this number

fact,when

is

n

or

indication to how

as n

are

we

onlyimmediate

Let

numbers

us

the

to what

a

certain

Moreover

numbers).

numbers

other

plurality.

than

we

objectsis the

as

same

to

as

why theyform

where

cases

a

series,

it is true)that

countingis

there

irrelevant in the

conclusion,it may

be dismissed

ordinal numbers.

to the notion

objectsas are

one

I, in relation

Now

inference to be

is ctg (the wider sense, when n the process of countinggivesus

are,

proved (inthe

it is of such

Part

be identical.

return

of

A further process is required to 1 up to n. is n, which is onlytrue, as a matter of numbers

to order and

come

in objects,

The

Hence 1 up to n. with this of Arithmetic ; and

plainthat

that

n.

from

numbers

130. is

as

consists of all the

class of numbers

from

finite, or, in

it is to be

foundations until

the

number

some

of numbers

smallest of infinite no

and

this correlation is,that the number

the number show

1 up to

;

must

to

to

of the numerical "A

and

B?

be asserted.

with which classes,

their investigate

"A We we

conjunction.It and

B

and

examined

C? such

found them

relation to numbers

to

and

Number

134 The

notion

to

be

notion

of

is not

to

This

numerical

a

be

identified,

,

of

the notion

treatment. independent

By

but is class"

a

collection I

a

mean

receive

to

what

is

a

and

new

conveyedby

of definite and B and C," or any other enumeration of mention the actual the terms, collection is defined by

and B"

A

"

or

The

terms.

is the

examined

now

collection. a shortly,

conjunction or, more to begin with, with "A

xv [CHAP,

by and. It would seem that and represents fundamental of combining terms, and it might be urged that a way justthis way of combination is essential if anythingis to result of which and the terms

Collections do not presuppose the terms togetherwith and:

other than 1 is to be asserted.

number

a

connected

are

numbers, since they result simplyfrom

case particular numbers. presupposed

of the collection themselves

terms

which, since grammaticaldifficulty be

must

whereas

one,

strict

word

and J?, or

A

meaning of is needed

no

for.

collection is the whole

to denote

collection in this sense,

so

method

exists of

There

the is

a

avoidingit,

is collection, grammatically, The A and B and C, are essentially many.

allowed

and

pointedout

where

in the

numbers

they could only premppose

and not one. many regardswhat is meant

a

but since

composed of

many, I choose to themselves,

the many that

A

collection, accordingto

a

the word

use

here

the usage

is adopted, As

giveswhat

we

B

is denoted

is what

the

only terms, and by all. We

a

class of which

by the concept of is precisely A and B

conceptwhich A and B B

B

are

if

u

by and" is A

conjunction.That class of which

a

denoted

be the

the

A

and

in the way

B

it

and

B

are

which

is

class-concept corresponding

only terms, combined

that

"

all ?^s

"

certain way, J, Thus A in precisely that way. combined

denotes the terms

in

a

is

a

and and

from the class, though distinguishable the if the concept of class. Hence be a n

from indistinguishable

from

and class-concept

class of

say,

and

those terms

are

appears

the

may

A

indicated

the combination

called before the numerical

indicated to

by

than

more

but many, from the whole

since

one,

term, it

one n

is

seems

necessary

both distinguished

to

hold that

from the

u

is not

and class-concept are broughtback

composedof the terms of u*. Thus we to the dependenceof numbers upon classes;and where it is not said that the classesin question it is practically are finite, necessary to begin of the with class-concepts and not with the theoryof theory denoting, and which has justbeen given. The theoryof and applies practically finite which and is to to finite numbers numbers, a only gives position from that of infinitenumbers. at least psychologically, There different,

*

A

conclusive

is,that

terms

class/'or is of this

an case.

a class with the whole againstidentifying composed of its be class "class is a the in the case as itself, may

of these terms

one

rather

reason

*c

classes are

classes." The logical type of the class claw among " infinite order^and therefore the usual objection xex" does not apply in to one

130-132] Addition short,two

in

are,

of of

ways

Terms

and

Addition

of

Classes

but finite classes, defining particular

135

there is

infinite classes, only one practicable particular namely way of defining classes primarily by intension. It is largelythe habit of considering side of extension

the

from correct

method

add B to A^

do not obtain the number

we

is a collection of two

follows

is

u

:

of

is a term

u

differs from from

x

notion of

a

coupleif u

a

just two

terms

when, if

-f 1 terms

differ from

which

is obtained

of

one

1 and

is

x

from

term.

mean

couple. And

a

It

8

collections. If

of ?/, there

term

a

be

x

And

n.

that of the

the

in

#, y

above

be defined

can

For, if

has

n

means

"

term

one

in terms proposition

of

and

by

induction

been

as

of

a

class of

class

defined,a

take it

as

a

one

term

"if variables,

u

that possible take 1

of

has

as

an

+

n

u u

1

class

a

should

we

individual,

the rule of Symbolic class, in the

Thus

1

can

couple. That

have

one

corresponding of which

side terms

two

are

and

terms

n

a

without

of w, the number of terms the notion of the arithmetical sum

sum logical

we

definition : but

term

a

we say 1 + 1 = 2, it is not 1 : if we 1, since there is onlyone

while if

togetherwith the be objected that we

doubt

might no

to which 1 and 1 is 1. Logicapplies, according have on the left-hand we logical proposition, be asserted,and on the right-handside we =

a primarily

not

and B9 coupleis defined

a

if,if x be

When

1 and 1 is nonsense,

1 4- 1

a

obtain A

we

definition, only diversity occurs,

of fact any finite number than one term. more introducing n

2, but

has terms, and

class havingterms.

take

to

terms, or

matter

has

of

way

different from a?, but if #, y be different terms of u, and z and from T/, then every class to which z belongsdiffers this

In

u.

have

in the

infinity. be carefully observed,is of but formingnumbers, formingclasses or

of

which as

has hitherto stood

Addition,it should

131.

we

which

logical theoryof

a is, t he terms,"" or, stating

and

term

v

has

one

It is to be has two terms." sum term, and u differs from z", their logical of observed that on the left-hand side we have a numerical conjunction have a proposition side we cerning conright-hand in of But numerical the true terms. a premiss, the conjunction of the three propositions, but is not the conjunction proposition, logical product. This point,however, has littleimportancein the

while propositions, above their

on

the

presentconnection. 132.

Thus

the

onlypoint which

remains

is this : Does

the

notion

that all have seen presuppose the notion of 1 ? For we numbers except0 involve in their definitions the notion of a term, and and 1 will if this in turn involves 1, the definition of 1 becomes circular, of

a

have

term

to be allowed

is answered

which

by

the

to

be indefinable.

is relevant to

Arithmetic, or

to objection

This

doctrine of " 128, that

a

term

is not

our

one

which

procedure in the

sense

is

opposed to in indefinable, presupposed logical

in the

sense

The notion of any term is a many. formal truth and in the whole theoryof the variable of terms, which that of the variable conjunction

in

; no

but this notion way

is

involves the

Number

136

number

by

There

1. of

means

To

sum

similar

up

to

but

terms

a

;

from

that

classes

true, the

term,

least

at

to

seems

may that

so

the

in

that

fact

are

contrary

presupposes

first

propositions,

addition that

a

not

derivative.

class

of

in

Chapters

to

xiv.

and

as

the

of

of

subject

in

terms

can

it

mathematical

be

appeared theory

of

as

one,

no

many

Only

One

latter

object" Finite the

on

addition,

arithmetical

depends

upon

the

without

philosophical

cardinal

not

from

which

logical

subject

is

it

appears "

the

distinct

terms.

object,

counting,

which

logical

classes

proposition.11

numbers

that

its

one

primarily

from

a

1

in

is

term

one

terms,

by

all

having

of

is

some

is

of

assertion

classes

sum

it

generated

classes,

number

understood

of

many

addition

and

be

called that

say

class

of

class

a

commonly to

namely

of

;

then

Thus the

xi

regarded

many

a

logical

The

one.

overthrow

"a

be

be

may

them

is

arithmetically could

to

is

the

to

case

simply

not

required,

sense

merely

mean

is

what

of

have

the

terms,

no

class

a

object

the

numbers

of

have

numbers;

have

in

defining

classes,

of

classes

Here

in

xv

term.

any

classes

conjunctions

class

of

or

are

class.

given

a

term

a

Numbers

:

numerical

of

sense

of

notion

the

circular

nothing

therefore

is

[CHAP,

numbers

fact

being argument set

forth

CHAPTER

XVI.

WHOLE

the

FOR

133. the

of

notion

AND

PART.

comprehension of analysis,it

whole

and

part,

is necessary to investigate which has been wrapped in

notion

a

without certain less valid more or obscurity though not logical who the writers be roughly called Hegelian. reasons In the by may present chapter I shall do my best to set forth a straightforwardand non-mysticaltheory of the subject,leavingcontroversy as far as possible It may side. be well to point out, to begin with, that I shall on one the word whole as strictly correlative to part^ so that nothing will use be called a whole unless it has parts. Simple terms, such as points, instants, colours,or the fundamental concepts of logic,will not be called "

"

wholes. which

Terms

chapter,

of

not

are

two

kinds. not

assertingthe being of

such

of terms

in their

case,

Whatever

that

their

is not

are

be, may kind first

The

characterized, though kind

classes

not

as are

defined, by the have

terms

classes,on

in

saw

we

that

preceding

these

simple: fact

the

be

may

the

propositions presuppositions.The second other hand, are complex, and

no

the

other the being of certain being presupposes either class is called a unit, and thus units are

terms.

a simple complex. A complex unit is a whole ; its parts are other units, whether simple or complex, which are presupposedin it. This suggests the possibility of logicalpriority, of definingwhole and part by means it be must a ultimatelyrejected,it will be suggestion which, though to examine at length. necessary one-sided formal have 134. Wherever a we implication,it may be from functions involved are obtainable one urged,if the two prepositional is the other by the variation of a singleconstituent, then what implied "Socrates man" what is is simpler than it. Thus a implies implies Socrates is a mortal," but the latter propositiondoes not imply the the former, since former: also the latter proposition is simpler than

or

"

man

is

a

concept

of

which

mortal

relation

proposition asserting a propositionimpliesthe being of

a

A

and

forms of

part.

two

the

Again,

entities

A

being of J?, and

if and the

we

take

I?, this

being

of

Number

138 the

another

one

the

"

the "

in

and "5

than B"

definition : following

the A

and each of which is the proposition, implies proposition.There will onlybe equal complexity as theorythat intension and extension vary inversely is greater such as "A of mutual cases implication, be is less than A* Thus we might tempted to set up of which

none relation,

simplerthan accordingto

but A

w,

and

derivative from

is said to be

A

implyB

is does not

whole

[CHAP,xvi

part would

is.

be

not

part of

B

when

B

is

implies

tained, If this definition could be main-

but would be indefinable, however, reasons why such

new

a

logical priority.There

are,

opinionis untenable. is not a simplerelation : The first objection is,that logical priority of A to B requires is simple, not but logical priority implication only does not imply B" but also "A UJB impliesA? (For convenience, A is impliesB is.) This state of I shall say that A implies B when is it is true, is realized when A things, part of B ; but it seems necessary which must to regardthe relation of whole to part as somethingsimple, whole to another which is be different from any possible relation of one an

part of it. This would not result from the above definition. For example,"A is greater and better than B" implies"B is less than A? but the converse is does not hold : yet the latter proposition implication

not

part of the former*.

not

Another

is objection

derived from

These two

concepts appear

other and

simplerthan

to be

man.

there

is here

(when taken

althoughit in all close

cases.

the way red is is no

For

these

connection,the

not, I

redness and colour. is no

specification,

be added

can

to

colour to

will turn mortal into specifications A Is coloured, more although complex than in one-sided a Redness, fact,appears to be implication. to mean one shade) a simpleconcept, which, particular impliescolour,does not contain colour as a constituent. A

implication. 135. 'Having

in which

does not and intension, therefore,

attempt

must

we

reasons,

to

in reject,

define whole

failed to define wholes

think,find

whole, and or

which itself,

inverse relation of extension

The

as

cases

simple:there equally

redness

produceredness,in Hence

such

it

define

to possible

and

spiteof

their very of by means

part

we by logical priority,

them

at

hold

all. The

shall

relation

of

is,it would

indefinable and ultimate relation, an part seem, rather,it is several relations, often confounded,of which one at least

is indefinable.

relation of

The

a

part

to

a

whole

must

be

differently

discussed

the

Let

those that

are

us more

accordingto the nature both of the whole and of to beginwith the simplest case, and proceedgradually

parts.

elaborate. Whenever

have any collection of many terms, in the sense there the terms, providedthere is explainedin the precedingchapter,

(1)

we

*

See

Part IV,

Chap. xxvu.

Whole

135]

134,

and

Part

139

function which theyall satisfy, non-quadratic propositional together class t he form a whole. In the preceding formed w e as chapter regarded by to show no reason all the terms, but usage seems why the class should not be regardedas the whole composed of all the terms in those cases equally some

there is such

where

the class

The

first is the class

of the terms

Each

one.

as

whole.

a

which is one indefinablerelation*, whole

The

part.

I shall call

is,in this

(") to

of

called

of the Greek

was

of

value

;

sense

(3)

is

""r

that

means

If #,

is

be

can

defined,

aggregates,and for every

two

z*,"then, providedthe

a

converse

*

B

of

senses

which

the

A

parts are

by

relevant in

symboliclogicand mathematics.

that the

three

of B

one

in

"

sense,

are

while B

inferred

(ingeneral)that

we

make

fourth

Which

if we may, for " is that not

required. The whole

first two

difference ; but

that

sense

senses

to

A

is

part of

in which

choose, l"e taken every

as

function prepositional the class

as

many

1901, " 1, Prop. 4. 4,

note

(p. 12).

mast

c.

The

defines

of hy being-

not

is

part in

objection to a

whole the

same

be But

senses.

anythingwhich

Peano's

be held

if A is part I.e.,

another, it

in

part is

indefinable,

be

part of C in any of the three

generalsense,

differs from

C

of

sense

It must

no

is

chiefly usually

are

those

are

This third

terms.

t Of. e.g. F,

and

B

it. way of defining be distinct : kept always to

?.#.,I know

senses

a

"-dTs difference from

analysis: it appears

to corresponds

like the first sense

and

parts. It is parts in this third

while the philosophers,

which

or

asserted

an

since senses, part is different from the previous and the has all in is not an aggregate, no parts at

considered

sense

be

need not

and

of whole

sense

meaning

relation

hold, u is "a proper part (in the second sense)of v. of whole and part,therefore, is derivative and definable, But there is another kind of whole, which may be called a unity.

first two

*

of the

The

Peanof.

to

the

primesto

of " to the whole

though it proposition, example,"A differs from B"

may

relation

of Socrates

that

and the first kind of implication

be

v

implies""r

u*

a

example,the of the

be

would

which

one

in which it is contained of

all the terms

not

differsfrom

the

but not

is different from

race

one

I, by

whole.

proposition.For is a complex of B* "A

are

whole is alwaysa

a

by the

known.

For

vital distinction is due

to

part

x

and

onlybetween

is different from

aggregateto

does implication

this

of other kinds

the relation of the whole

in Part explained

relation of

Such

and

numbers

subordinate

This

wholes

part to whole.

to the human

This most

numbers.

as

a

race

of the

whole

relation of

nation

to the human

certain

a

kind, which particular

a

aggregate,is a different relation, though also

our

a

of

some aggregateof aggregatescontaining

our

commonly

of

as

whole

the relation of whole

its constituents

relation holds

the

to

the second

many,

the aggregate and of the collection composingthe aggregate: the relation

terms single

the

whole

a

case,

soon

the above

But

has

meaning of

it differs from

aggregate:

an

fact that it is definite as

then

as

of the

this kind

tyjteas its

Number

14:0 any sense, or part in one This sense, however,has 136. and

part in another,is to be called a part.

of

sense

in actual discussion. seldom,if ever, any utility

it in other words.

collection whatever,if defined

Any

is

important,

I shall therefore

pointin Logic.

fundamental

illustrates a

of wholes

kinds

the

difference between

The

[CHAP,xvi

repeat

by a non-quadratic

function,though as such it is many, yet composes a whole, propositional whose partsare the terms of the collection or any whole composedof some of the terms tetween

of the collection. It is whole

a

The

is a minimum.

to the whole than

Such

whole is

a

all its

no

occur,

being parts of contain

which

not occurring simplyas predicates,

one

in

aggregate.

inter se, but and the same

are

only the whole.

may be called but as relating collection,

relations

terms

an

simpleconstituents

direct connection

involved in

the collection.

as

of the collection I call

when specified completely

other wholes

strictly has led expression

parts; but convenience of and speakof all the terms

its parts have specified; But

the difference

more collection, applies beingsingular,

word

of the terms

indirect connection

realizethe difference

where

case

to all the

formed

whole

The

in this

and all its parts,even

neglectgrammar,

to

me

highlyimportantto

a

or

what

These Such wholes are not are alwayspropositions. qualifying. all their when known. Take, as a simple specified completely parts are where A A differs from and B are simple the B? instance, proposition The simplepartsof this whole are A and B and difference ; but terms. the whole, since there of these three does not specify the enumeration two other wholes composed of the same are parts,namelythe aggregate "5 of A and B and difference, and the proposition differs formed from A? different from In the former case, althoughthe whole was its parts; but all its parts,yet it was specified by specifying completely in the presentcase, not onlyis the whole different, but it is not even its parts. We cannot explainthis fact by saying by specifying specified or

"

that

the parts stand in certain relations which of " A differs from case analysis ; for in the above included in the

thingwhen in

term

it

however

Similar remarks

Being,but Being. A

raises the

the relation

it is

as

the whole

is

a

our

whole

formed

relation

is

merelyenumerated

irrelevant to

is,which

a

in

was one

as

a

difficulties in this

presentpurpose*. A

and

of the collection A

and

composed of

point,and so does A and B are two. and we raise this point, them propositions may distinguish that tenns fact the raise it. complex by they

Indeed

is

one

same

all

among

Thus which

is different from

B*

certain fundamental

are

I leave aside

applyto A

omitted

be that

to

seems

and another when relates,

collection. There

a

view,which

fact

analysis.The

the

are

that

very different classes of wholes, of the firstwill be called aggregates,while the second will be called

unities.

we

see

there

are

two

since what(Unit is a word havinga quitedifferent application,

*

See Part I, Chap, iv, esp.

" 54.

Whole

135-139] class which

and

Part

141

is not

null,and is such that,if # and y be members is a unit.) Each class of wholes consists of of it,x and y are identical, to all their parts; but in the case of unities, terms not simplyequivalent is not even the whole specified by its parts. For example,the partsA9 greater ihan" 5, may compose simplyan aggregate, or either of the is greaterthan B? "A is greater than A? "B Unities propositions thus involve problemsfrom which aggregatesare free. As aggregates ever

is

are

more

a

relevant specially

to

than

mathematics

confine myselfto the former. generally is It 137. importantto realize that a whole

unities,I shall in

future

is

a

singleterm,

new

distinct from each of its partsand from allof them: and is related to the

parts,but has

it is one, not many*, beingdistinct from theirs. The

a

reader may perhapsbe inclined to doubt whether there is any need of wholes other than unities; but the following to make seem reasons unavoidable. aggregateslogically and it would seem manifold,etc.,

speak of

We

(1)

that in all these

one collection,

one

there

cases

is really

term. shall as we somethingthat is a single (2) The theoryof fractions, shortly see, appears to dependpartly upon aggregates. (3) We shall find it necessary, in the theoryof extensive quantity, that aggregates, to assume when have what even they are infinite, may be called magnitudeof number and divisibility,that two infiniteaggregatesmay have the same this theory, of terms without havingthe same magnitudeof divisibility: we

it

shall

in metrical geometry. find,is indispensable would seem, the aggregate must be admitted as

all its constituents, and and indefinable relation.

from

havingto

For an

each of them

these

reasons,

entitydistinct certain ultimate

a

doctrine, importantlogical the I mean prominence" be analyzedis a is falsification. Whatever doctrine that analysis can is in some of wholes that analysis whole, and we have alreadyseen the is it falsification. But measure importantto realize very narrow I have

138.

which

the

alreadytouched

theory of whole and

limits of this doctrine. are

not

We

on

very

part bringsinto

conclude that the parts of

cannot

that the

its parts, nor really

a

whole

in presupposed

not

partsare

a

the

the whole is not

in the parts,nor presupposed prioris not usuallysimplerthan the logically yet that the logically givesus the truth,and nothing subsequent.In short,though analysis whole in

but

the

a

in which

truth, yet it

onlysense it becomes

dislike the 139.

sense

can

never

the whole

give us

truth.

the doctrine is to be

accepted.In by givingan excuse merelya cloak for laziness, labour of analysis.

in which

It is to

be observed that what

always,exceptwhere

they contain

one

or

none,

or

is the

any wider sense, to those who

called classes

we

term

This

are

one may defined by

as

as functions,be interpreted aggregates. The quadraticpropositional

*

I.e. it is of the

same

logical type

as

its

simpleparts.

Number

142

logical

product

second

of

be

the

sense)

but

allowing forms

which

previous wholes

the

of

predicates this

reason

it

whole

is to

and

and

part

class-concepts

or

that

*

Cf.

+

See

the

e.g.

begin obtain

or

less

his

F.

and

1001,

Algebra

with

a

in

class-concept

it

" 2, Prop. Logik,

Vol.

been

by

writers infinite

than

that it to

is

so

is

or

the

Thus

and

1890).

it

predicate

postponed

(Leipzig,

fact

peculiarity

this.

(p. 19). i

the

cases

or

is

and

wherever

functions;

has

1-0

this

logically

prepositional

der

from

aggregate

is

other

many

fundamental

of

But

but

relation

by

developed

as

Schr""der)t.

the

consideration

Peano,

Calculus

necessary,

is

It

second

transitive

relations

such

other

multiplication.

(including

Frege

function,

prepositional

from

the

The is

senses,

will

parts,

are

the

sum

in

(again

aggregate*.

three

our

Logical

the

of

unavoidable,

practically

theory

basis

concerned

are

of

and

addition

and

Peano

to

second

the

their

aggregates

other

any

distinguished

is

logical

of

part

of

part

or

given

two

and

xvi

(in

part

common

aggregates,

two

with

the

which

nor

in

part,

asymmetrical, of

of

with

and

whole

the

the

be

will

one

is identical

which

identical

as

of

senses)

aggregate

any

classes

two

three

our

aggregate

neither of

of

"CHAP.

of for

late

CHAPTER

XVIL

INFINITE

Ix

140. not

(") If so, must three senses our to

called

be a

the

whole*

part

a

Thus

whole.

and

part

a

;

in the

and

terms

The belong to unities. is must concerned, infinity

where

parts in

third

in

parts

the

first sense

the

second order

In

?

of to

I propose henceforward senses, is to part in the first sense

A

in the

second will

sense

parts belong

to

consideration

of

be

infinite wholes?

any

third

following phraseology:

of the

part simply ; and

of the

of

contains

the first, second

to

there

Are

which

aggregate

an

specialdifficulties of infinityare postponed to Part V. My object

questions: (1)

two

be

term

are

infinite whole

an

use

a

all these

:

the reference

avoid

the

present chapter

consider

is to

now

the

considered

be

to

WHOLES.

is to be

sense

be

called

a

called

constituent while

aggregates, aggregates and

separatelyconducted.

stituents con-

unities,

I shall

begin

with

aggregates. infinite aggregate is an An class, i.e. an aggregate which defined by the are aggregates as

terms

many

that

Our

first

they

is

:

of

concerned,

parts which there any Are

it is

make

possibleto

about

not

all the

valid statements

about

about

the

this

condition

believe

not

who

deny

they

are

*

A

t

Of.

have such

that

that

part

view is

space

in this

Phil.

ed.

can

be

a

finite space

not

may

will also be sometimes

Gerhardt,

n,

is

true

no

defended. successfully

given whole,

a

call

sense

Werke,

fulfilled there

is not

this

pleasedto

terms,

nor

p. 315

a

be

would

few a

which

called

a

the

(as he

maintain and But

Among admit

as are

criticised

whole.

given whole,

; also i, p.

of

term

any

whole

yet would say) they do not compose"}".Kant, again, has been much infinite given whole. for maintaining that space is an Many finite number of terms, have that a aggregate must every where

Such

terms.

Even Leibniz, favourable that, where infinite classes

the

to

class,but

number contain

question

often denied. aggregates are actual infinite,maintained

Infinite was

themselves.

infinite

an

fact

infinite

an

?

aggregates he

as

corresponding to

aggregate has

that

for

that I do

those what

instance,

simple or indivisible part, 338, v, pp.

Number

144

[CHAP,xvn

the space in a room, a box, a bag,or a book-case. onlyfinite in a psychological sense, i.e.in the sense in at

glance:

a

it is not finite in the

But that

that it is

sense

such

an

we

a

is

space

can

take it

aggregate of

a

finitenumber

of terms, nor yet a unityof a finitenumber of constituents. be a whole is to admit that there to admit that such a space can wholes which are not finite. (This does not follow,it should be

Thus are

objectsapparently occupying finite spaces, for it is alwayspossible to hold that such objects, though apparentlycontinuous, consist reallyof a largebut finite number of material points.)With argument holds : to respectto time, the same time certain of between sunrise that for a elapses length example, say,

observed,from

the

admission

and sunset, is to admit

an

of material

at least

infinitewhole, or

a

whole

which

is not

to deny the reality of space customary with philosophers and time, and to deny also that, if they were real,they would be aggregates. I shall endeavour to show, in Part VI, that these denials

finite.

It is

resolved difficulties and by the now of faultylogic, the in science S ince and common sense view, it join infinity. opposite will therefore be accepted thus, since no argument a priorican ; and be adduced againstinfinite aggregates, derive from space and now we

by supported

are

time

a

argument in their favour.

an

Again,the

of all

sum-total

numbers, or the fractions between 0 and 1, or the and seem to be true aggregates colours,are infinite, :

natural

about any be made can that, althoughtrue propositions position about all numbers, could be number, yet there are no true propositions Leibniz supportedit,by the supposedcontradictions as supportedformerly, of infinity, but has become, since Cantoris solution of these contradictions, a whollyunnecessary paradox. And where a collection be defined by a non-quadratic can function,this must be propositional held,I think, to imply that there is a genuine aggregate composed the

of the terms were

collection. It may be observed infinitewholes,the word Universe would be

no

also

of the

that,if

there

whollydestitute

of

meaning. 141. a

must, then, admit infiniteaggregates. It remains

We

difficult question, namely: Are

more

we

to

admit

to

ask

infinite unities ?

This

questionmay also be stated in the form : Are there any ? This question is one of great logical infinitely complex propositions and shall much both in we care require importance, statingand in it. discussing The first pointis to be clear as to the meaning of an infinite unity. A unity will be infinite when the aggregate of all its constituents is but this scarcely constitutes the meaning of an infinite unity. infinite, In

order to

simpk of

a

obtain

constituent.

constituent is

relation of

part

to'

the We a

meaning, we may

must

observe,to

constituent

beginwith, that

of the

of

introduce the notion a

a

constituent

unity,i.e. this form

of

the

whole, like the second,but unlike the firstform, is

141]

140,

Wholes Infinite

transitive. A

simpleconstituent

which itselfhas the to

no

constituents.

may We

then

aggregate,or, if there be

an

this constituent is rendered

is to be taken

may

this the definition of We

define

now

may

a as

assume,

a

an

as

constituent

a

in order to eliminate

constituent of

no

constituent which

is

simple. (Thisview

by the fact that legitimate

and does not have that kind of With

be defined

now

questionconcerning aggregates,that be

145

aggregate,

an

of

an

gate aggre-

aggregate is a

an

which complexity

simpleconstituent infinite unity as

unityis

our

term, single propositions.) is completed, to belongs

follows : A

unity is finite

when, and onlywhen, the aggregateof its simpleconstituents is finite. In all other cases a unityis said to be infinite. We have to inquire whether If

there

are

any

such unities*.

it unityis infinite,

a

againcontains

a

constituent

is possible to find and unity,

any unities of this nature, two cases may be simpleconstituents of our

on

constituent

without

end.

unity,which If there

prima faciepossible.(1)

unity,but these must

be

There

be infinite in

simple constituents at all,but all without exception, constituents, may be complex; or, to take a slightly more complicated happen that, althoughthere are some case, it may simpleconstituents, yet these and the unities composed of them do not constitute all the constituents of the original unity. A unity of either two of these two kinds will be called infinite. The kinds,though be considered distinct, together. may it will infinite unitywill be an infinitely An : complexproposition It be of constituents. not in any way into a finite number analyzable thus differs radically from assertions about infinite aggregates. For is composed of example,the propositionany number has a successor into it of conceptsentering of constituents : the number a finitenumber

number.

(#) There

may

be

are

so

a

no

"

u

can

be

enumerated, and in addition

of terms

denoted

constituent.

to these there is

an

infiniteaggregate

indicated by any" which counts one as way is which the be that it may said logical purpose

in the

Indeed

of finite propositions to deal with infiniteclasses of terms : this objectis effected complexity by o#, any, and every, and if it were not effected, every generalproposition about infinite class would have to be infinitely an complex. whether propositions Now, for my part,I see no possible way of deciding that of infinitecomplexity not ; but this at least is clear, are or possible all the propositions all propositions known to us (and,it would seem, that we can know) are of finite complexity.It is only by obtaining enabled to deal with about infiniteclassesthat we such propositions are and fortunate fact that this method is infinity ; and it is a remarkable whether or not there are infinite unities successful. Thus the question is must be left unresolved ; the onlything we can say, on this subject,

served

by

*

the

theoryof denotingis,to

In Leibniz's

enable

all contingent thingsare philosophy,

infiniteunities.

Nmiber

140 that

such unities

no

therefore

such

none

occur are

[CHAP,xvn

in any departmentof human and knowledge, relevant to the foundations of mathematics.

second

question: Must an infinite whole which contains parts be an aggregate of terms ? It is often held,for example,that spaces have parts,and can be divided ad lib.,but that theyhave no simpleparts,i.e. they are not aggregatesof points. The I

142.

to

now

come

our

regardsperiodsof time. Now it is plain of terms of the second (i.e. that, if our definition of a part by means of the first) was of part by meann correct,the presentproblemcan sense to never arise,since partsonlybelong aggregates. But it may be urged is

view

same

put forward

therefore it

pari ought

of

the notion

that

apply

mav

as

be

to

to

other

add

to

taken

wholes

as

and that indefinable,

an

than

Aggregates. This

will

kind of aggregates and unities a new of part. This will be a whole to the second sense whole, corresponding

requirethat which has Such

partsin the second

whole

a

should

we

sense, but is not

to be what

seems

a

fond of

philosophers

are

many

continuum, and space and time

aggregateor

an

unity. a calling a

often held to afford instances of such

are

whole.

infinite wholes, we it may be admitted that, among is in distinction which relevant,but which, I believe, seenix

find

Now

In some merelypsychological. but

great doubt

obvious,but 0 and

the

as

terms

cases,

whole, while

the

to

or

0 and

as

others, the

in

reality

the terms,

whole

seems

1

be

to

seems

of the nature

of

a

hand, sensible spaces and times but the inference to indivisible pointsand

On

inference.

be obvious wholes

to

to

a

aggregate of ratios between seem

doubt

no

inference. The ratios between precarious indivisible entities; but the whole certainly

seem

1, for instance,are

construction

feel

we

a

;

the other

be

This regardedas illegitimate. distinction seems, however, to have no logical basis,but to be wholly A slight with codependenton the nature of our senses. familiarity oi'dinate geometry suffices to make a finite lengthseem strictly analogous

instants

to

is

the

so

obscure

often

stretch of fractions between

that nevertheless,

parts

to

as

are

evident

does

not

in on

where, as

cases

0

and

with

It must

1.

the

the indivisible fractions,

the problem with inspection,

arise.

But

to

infer that

indivisible parts merelybecause this is known

be admitted,

which

we

are

cerned con-

all infinite wholes

have

be the

to

case

with

some

The be rash. them, would certainly generalproblem remains, therefore,namely: Given an infinite whole, is there a universal reason

of

for

supposingthat 143.

it contains

be held to deny that it has

do not has

reconstitute it. For

three

indivisibleparts?

the definition of In the first place, an

an

infinitewhole

number assignable

of

must

not

simplepartswhich

example,the stretch of fractions from 0 to 1

simpleparts,J, i-,".

But

these do

not

reconstitute the

whole, that is,the whole has other parts which are not parts of the assignedparts or of the sum of the assigned parts. Again, if we form a

^Vhoks Infinite

141-143] whole has

of the number

out

1 and

a

simplepart,namely 1.

one

asking whether every part simpleparts. In this case, infinite

line an

Such

of if

our

whole, the

a

inch

case

whole

our

147

whole

long,this

whole

certainly

this may be excluded by as either is simple or contains

be formed

by adding n simple

simpleterms can be taken awav, and questioncan concerningthe infinitewhole which is left. But again,the meaning of our questionseems hardly to be: Is our infinite whole an actual aggregate of innumerable simpleparts? This is doubtless an importantquestion, but it is subsequent to the questionwe is Are which there : are asking, alwayssimpleparts at all ? We may of simpleparts be found, and taken observe that, if a finite number from the whole, the remainder is alwaysinfinite. For if not, it away terms

to

an

n

be asked

the

would

have

a

finitenumber

originalwhole

the finite,

and since the term

;

would

then

finite.

be

that every infinite whole contains one shown infinite number of them. it contains an

simplepart,the remainder

of two

Hence

if it

therefore has

is

be

can

simplepart, it follows For, taking away the

infinitewhole, and

is an

finite numbers

that

a

one new

It follows that every part of the whole either is simplepart,and so on. contains simpleparts,providedthat every infinite whole has or simple, least hard to prove this as to prove at one as simplepart. But it seems that every infinite whole is

If

parts must and be infinite,

will reduce all the

so

of

no

finite number

Successive divisions

endless series there is

in such manner

a

Thus

on.

partsto finitude.

series of parts, and Parts IV and V) no

finitenumber

parts,one be infinite. If this be again divided,one of

least of these

partsmust

aggregate.

be divided into

infinitewhole

an

an

of contradiction.

(as

Thus

a

its

of divisions

give an endless shall

we

there

is no

actual division that every infinite whole aggregate. So far as this method can show, there is no more simpleconstituents of infinitewholes than for a first moment of

at

provingby

see

in

method be

must reason

in time

an

for or

last finite number. But

contradiction may emerge in the presentcase from the of whole and part with logical seems a priority.It certainly

perhapsa

connection

greaterparadoxto maintain that infinite wholes do to maintain

partsthan

that there is

no

not

first moment

have

in time

indivisible or

furthest

by the fact that we know many might be explained infinite wholes undoubtedlycomposed of simple simpleterms, and some a beginningof time or terms, whereas we know of nothingsuggesting have .solidbasis in logicalpriority. more a perhaps space. But it may For the simpleris alwaysimpliedin the more complex,and therefore truth about the more be no there can complex unless there is truth in the analysis infinite whole, \ve are of our about the simpler.Thus always dealingwith entities which would not be at all unless their limit to space.

constituents

example :

a

This

were.

moment

This makes does not

a

the

real difference from

logically presuppose

a

for time-series,

previousmoment,

Number

148

and

if

did

it

moment,

deny

a

have

Being

presupposed is

presupposition special

a

of

does

and

whole

The

are

definition

aggregate),

is

will

complexity of

a

defining

could It

side

(as and

doctrine

logically assume,

wholes

be

in

In

case

of

a

finite

method

in

a

not

way

in

all

we

VI

those

simpler, of

the

remainder shall

have

of to

all

the

VII) in

this deal

work, are

the

that

aggregates

are

and

demands

imperatively

view.

on

and

Geometry

results

I

of and

paradoxical,

opposite

issue.

at

arguments

of

short,

less

than

point

infinity,

of

procedure

and

applications, far

that

it

the

that

Parts

the

difficulties

supposed

are

the

of

conclusive

of

importance

great

also

than

throughout which

an

of

known

less

is

admitted,

support

the

shown

advocated

be

the

upon

satisfactory,

with

of

istic character-

whole

a

any from

is

some

other

no

is

case

the

which

give

such

obtaining

must

fallacious;

instants.

here

it

urged

depend

will

of

or

be

to

the

(in

will

infinitely

there

unity.

be

wholly

Dynamics

whole,

it

i.e.

our

obtained

(in

definition-

any

seems

if

it.

how

be

not

be

whole

almost to

asking

Now

numerous

as

But

There

terms.

by

cannot

the

intensional,

considering

however,

therefore

be

argument,

wished,

other

points

of

infinite

any

may,

the

an

complex

as

infinite

this

relation

regard

must

unity.

infinitely

aggregate).

an

necessarily

above be

not

collection

involving The

themselves

or

"which

whole*

either

are

infinite

an

relations

that

in

stated

complexity,

finite

these

since

parts,

of

is

which

require

infinite

the

of

fundamental

definition

The

seems

of

case

transitive

adopt

we

otherwise

be

may

the

the

there

peculiarity

and

theory

defined.

would

the

of

the

upon

be

to

this

since

complex,

of

depends

regress

important

so

in

whose

where

Thus

asymmetrical

instance

relation

a

:

argument

same

wholes

infinite

another

is

philosophy

our

the

part

other

where

exist

not

This

concerned.

are

all

reason

infinite

the

to not

Beings For

also.

false

is

first

a

would

simple wholes.

infinite

the

consequence

completing

for

which

wholes,

the

false,

of

that

wholes

innumerable

were

deny

to

infinite

that

xvn

self-contradictory

reason)

same

follow

to

there

in

the

(for

seems

all, unless

at

is

Being

It

Cause.

First

self-contradictory

be

perhaps held

been

has

it

as

would

it

[CHAP,

shall all

more

therefore

the of

the

infinite

terms.

CHAPTER

RATIOS

THE

144.

AND

no

But

I

FRACTIONS.

present chapter,in

confined integers,is essentially have

XVIIL

relations

far

so

finiteintegers :

to

strictly analogousto

distinguishratios,as

shall

fractions,which

relations

it deals

as

what

that

those

between

of

infinite

are

usually called

are

relations

relations

with

ratios.

integers,from

aggregates, or rather between shall find, may and magnitudes of divisibility; fractions,we

their

are

between

hold where both aggregates are infinite. It will express relations which be necessary the mathematical to begin with definition of ratio,before

proceedingto more general considerations. Ratio is commonly associated with multiplication and division,and in this way becomes indistinguishable from fractions. But multiplication and division are equallyapplicable to finite and infinite numbers, though in the case of infinite numbers which they do not have the properties them

connect

ratio

developa theoryof

to

and

in the

shall be

ratio which

finite numbers

having one class has

of

the

said to

are

numbers, and

the

other

and

one-one

the

number.

rath

of this

by

desirable

it becomes

Hence

independentof multiplication

between 0 -f

and

a

If

mn.

n

is

given.

equation expresses when also aV of B

and

from

b.

the

b is

=".

the This

three

have

we

between

Then converse

a

has of

relation

advantage that

to

a

define it

c

a

=

power V. Also a,

i,

c

B' is derived the

appliesnot

ratio

only

to

+

a

of

n

=

relations

such

that

from of to

06

which

Suppose

is the

relative

b' as

a' to

a.

B

This

b.

minate is deter-

define

may

we

b

number

a

relation

this

of

relation

one-one

relation which

as

resulting

relation which

call this relation B.

us

d

mn

numbers

H, where

we

+

and

a

Let

given.

a!

the

class

a

relation which

a

have

be

u

(powers

we

one-one

mth

the

now

shall have

6',we

now

If

J, a

t^,

has

a

of consecutiveness

and

a

to

is thus

number

a

now

then multiplication),

equation expresses, between when

be added

term

if

when,

consecutive

be consecutive

If

relation

relative

be

one

To

asymmetrical.

power being defined

has

finite case.

division. Two

is

with

was

This

holds as

mn

this

",

=

is determinate we

have

product derived

theory

finite integers,but

to

[CHAP,xvni

Number

150 all other

series of the

type,i.e.all

same

series of

type the

which

I call

progressions, 145.

observe

to

it is

only pointwhich

The

regardsthe

as

above

for important,

definition of

our

presentpurpose,

ratios

is,that they are

with one which are exception integers, between holds and such that which one are only one any asymmetrical, of which in finite definable terms of consecuare integers, specified pair themselves form a series having no first or last term a nd which tiveness, of terms, between infinite number and having a term, and therefore an relations

one-one

results that

ratios

no

speak of

the

be contained

to

relations it

are

: the integers

to be identified with

are

ratios

ratio of 2 to

When, therefore, whollydifferent entityfrom " the integers said series of ratios as containing integers,

1, for example,is we

the fact that

From

terms. specified

two

any

finite

between

a

cardinal

not

are

relations which

numbers, but

have

a

with cardinal numbers. same correspondence numbers. wth of The the relation and to positive negative applies power number is the positive of consecutiveness + "5 which is plainly a wholly certain

The

one-one

different concept from with

others

which

to

error

to

which

the cardinal number

they have

are

is well to

realize,as

distinctions is sure, consequences. There is the

no

of entities

relation is

liable,and

which

one

has

mathematics.

philosophyof

other instances of the

an

We

error, and

same

it

that any failure in subtlety of earlyas possible, in this subject the most disastrous at least, to cause

the in connecting difficulty

theoryderived

usual

very

in the

shall find hereafter innumerable

confusion

important one-one

some

mathematicians

greatesthavoc

producedthe

The

n.

remark

from

above

theory of

and multiplication

ratio with

division.

But

the

usual

theorydoes not show, as the present theorydoes,why the infinite integersdo not have ratios strictly analogousto those of finiteintegers. fact

The

is,that ratio dependsupon

above

consecutiveness.

defined does not exist among as unchangedby the addition of 1. It should a

set

new

of

be observed relations

that what

and consecutiveness

infinite integers, since these

is called addition of ratios demands

ratios,relations which

among

are

may

be

called

and negative ratios,justas certain relations among are positive integers This however, need not be positiveand negative integers. subject, further developed. The above theoryof ratio has,it must 146. l"e confessed, a highly artificialappearance, and one which makes it seem extraordinary that in daily life. The ratios should occur fact is, it is not ratios,but and that occur, fractions are not purelyarithmetical, but are fractions, concerned with relations of whole and part. really fractions show an Propositions asserting importantdifference from those asserting We A is can say integers. two, and one, A and 11 are so

on

;

but

we

cannot

say A

is

or one-third,

A

and

B

are

two-thirds.

Ratios

144-147] is

There

alwaysneed

of

fractional relation. two-thirds of C, and a

simple part

is not the

of

to

fraction

the

number

wholes will show

is

There

147. is

to entity,

is one-third

in Fractions,

on.

In

the

which

our

short,are

wholes the

to

first has

and B

of C, A

one

some

togetherare

either relations of another.

wholes, the

it

But

simplepart, should

of finite

case

151

be part

matter

seems

of parts in the expresses the ratio of the number in the other. But the consideration of infinite

that this

us

second

say A

so

the

Fractions

whole, or of two the one whole,or

a

necessary that other whole.

simple: one

to

some

We

and

to the facts. simpletheoryis inadequate that the notion of half a league, half or

doubt

no

notion. It is therefore necessary to find some legitimate for fractions in which theydo not essentially sense depend upon number. hours is to be divided into two For, if a givenperiodof twenty-four each of which is to be half of the whole period, continuous portions, there is onlyone that every way of doingthis : but Cantor has shown of t he into two continuous dividing period possible portionsdivides way the it into two of number There terms. must same be, having portions other respectin which two of twelve hours therefore,some are periods equal,while a periodof one hour and another of twenty-threehours to say upon this subject in Part III ; are unequal. I shall have more for the present I will pointout that what we want is of the nature of a wholes. a property of ordered magnitude,and that it must be essentially that A is To say now I shall call this property magnitudeof' divisibility. a

day,

a

one-half of B

means

have

parts which

:

is

B

both the

a

whole,and

be divided into two

if B

same

similar

each other, divisibility as

magnitudeof

each of these parts. as magnitude of divisibility the fraction ^ somewhat We more simply,by regarding may interpret to ratio so long as finite wholes are concerned) it as a relation (analogous finite integral Thus fractions between two magnitudesof divisibility. of the of the relation will measure divisibilityan aggregate (suchas n/1) of a single relation will term ; the converse of n terms to the divisibility then

the

has

A

same

I/?*. Thus here again we dangerof being confused with be

Fractions, as

quitedistinct.

number

now

infinite

smaller among We shall see more

of terms.

and

of the usual accounts

essential the notion

of

magnitude of

then, in the

in

which

"

and

reality advantage(upon a

tion discrimina-

aggregateshavingthe

same

inadequacy logical how light, absolutely is. divisibility Fractions, really more,

as

the

they

may

express

relations

infinite

of

have in dailylife theyusually of the nature magnitudesof divisibility ; really of measured number of are parts by only magnitudes divisibility

aggregates are

the

is broughtto

of measurement

sense

is in

though in integers,

have interpreted,

greaterand

of

class of entities which

new

a

geometry depends)that they introduce

all metrical

which

have

finite cardinal

where

and

this is the

sense

which

"

of relations between

the

aggregates concerned

(though this

remark

is

are

finite.

that, anticipatory)

It may whereas

also

be

observed

ratios, as

above

Number9

152

defined,are them, this

are

rational,fractions,in essentially also capableof irrational values.

148.

the

sense,

But

the

here

given

to

developmentof

be left for Part V.

topicmust

up the results obtained in Part II. In the the modern mathematical theoryof cardinal integers, chapters, We

first four

[CHAP,xvm

may

sum

now

jointlabours of arithmeticians and symbolic set forth. was Chapter xi explainedthe notion of logicians, briefly of integers and showed that the usual formal properties similar classes, result from defining them as classes of similar classes. In Chapter xn, arithmetical addition and multiplication both depezid showed how we and how in be defined both may a addition, logical way which upon applies equallyto finite and infinite numbers, and to finite and infinite introduces nowhere and products, and which moreover sums any idea of In Chapterxni, we gave the strict definition of an infinite class, order. from takingaway one of its which is similar to a class resulting as one as

it results from

terms

;

and

we

the

showed

in outline how

definition of finite numbers

by

to connect

mathematical

this definition with

induction.

The

the

special

Chapter xiv, and it was shown in Peano proves to be sufficient

theoryof finite

discussed in

was integers which primitive propositions, all be deduced from this subject, definition of finite cardinal can our integers.This confirmed us in the opinionthat Arithmetic contains no indefinables or indemonstrables beyond those of general logic. We then advanced,in Chapterxv, to the considei-ation of philosophical with the mathematical view above of questions, a critically testing deductions. We decided to regardboth term and a term as indefinable, and to define the number 1, as well as all other numbers,by means of these

how

the

indefinables

(togetherwith

certain

others). We

it necessary class may have

also found

since one class from its class-concept, a distinguish We decided that a class consists of all class-concepts. the terms denoted by the class-concept, denoted in a certain indefinable of manner common ; but it appearedthat both usage and the majority mathematical a class with the whole purposes would allow us to identify formed of the terms denoted by the class-concept. The only reasons this view of distinguishing against a class containing were, the necessity term from fact that classes ai-e that one some only one term, and the to

several different

members

of themselves.

infinite classes, that

We

also

distinction between

finite and

former can, while the latter cannot, be defined then We actual enumeration of their terms.

involved in

"

may be called the addition of individuals, less A and B" ; and we found that a more or

independent theoryof finiteintegerscan But

a

the

i.e. by extensionally, proceededto discuss what i.e.the notion

found

it

be based

upon

this notion.

in virtue of our analysis of the notion of claas^ appearedfinally, that this theorywas from the theorypreviously really indistinguishable expounded, the only difference being that it adopted an extensional

definitionof classes.

Ratios

148]

147,

Chapter there

that

which

could

we

Logic

as

in

parts

an

the

definable

and be

must

should

terms

of

infinite

Chapter

In

were

the

unities,

found latter

to

were

divisibilities in

which

all

(They

assumed

somewhat

we

between

nature

be

is

no

however,

relations

is

to

if

even

be

must

mitted ad-

unities

not

the

that

necessary

owing

the

accessible

are

means

and

ratios

exclusion

the

to

be

fractions of of

discussion

further

quantity

found,

and

xvn,

unities,

which

divisibilities

the

Symbolic

wholes

complexity.)

oijimte

considered

their of

by

of

anything

in

null-

to

Chapter

in

that,

the

of

infinite

wholes

complicated

magnitudes,

general

infinite it

relations

we

also

saw

calculus

appear

aggregates,

must,

to

that

never

though

terms,

relations

being the

that

terms

wholes,

of

sorts

and

next,

appeared rate

any

the

to

definable

one

We

traditional

considered

infinite

finally,

xvin,

be

It

But

be

the

single

applicable We

at

simple.

be

aggregates of

and

different

found

We

part.

relation,

to

whole.

of

aggregates

this

and

respectively.

sense.

seemed

it

;

whole

aggregates

specially

knowledge.

human

153

correspondingly

whole

possible,

logically

be

they

of the

infinite

an

of of

senses

and

regard

Fractions

relation

two

notion

algebra

of

notion

the

are

unities the

extending

class,

there

called

we

by

to

that

with indefinable

two

are

and

sense,

dealt

xvi

and

finite

:

former

the

integers,

These

aggregates.

belongs

considered.

while

to

Part

III,

PART

III.

QUANTITY.

CHAPTER

THE

149. few

are

as

from

is evident

his

MAGNITUDE.

problems

than

this relation

to

OF

traditional

important

more

Opinion as

MEANING

the

AMONG

XIX.

the

has

of mathematical

relation

undergone

definitions of ratio

of

many and

philosophy,

quantity

number.

to

revolutions.

Euclid,

proportion,and

indeed

whole

his

of not procedure, was persuaded of the applicability Des and the Cartes spatialmagnitudes. When Vieta, by of co-ordinate introduction mental fundaa Geometry, made this applicability method postulate of their systems, a new was founded, which, from

numbers

to

fruitful of results,involved,like most

however

the

seventeenth

century,

subtletyof distinction. all spatialmagnitudes for

questions

whose

instrument done

before

a

diminution

What

was

of

by measurement,

meant

decision, until was

lacking;

advances

logicalprecisionand

susceptibleof

were

mathematical

very and even

a

numerical

lately,the BOW

much

and

of

loss in

a

whether

measure,

necessary remains

were

matical matheto

be

given. The view prevailedthat investigation, objects of mathematical that and the two similar as not to require careful so were separation. Thus and number to without was hesitation, applied quantities any found were ment, conversely,where existing numbers inadequate to measurecreated the sole ground that every new ones were on quantity

number

have

must

a

and

complete answer quantity were

numerical

a

can

be

the

measure.

different lines of argument, happily changed. Two both conducted in the main by different men, have laid the foundations On for largegeneralizations, in detail. the and for thorough accuracy All

this is

now

their followers, have Cantor, and to be significantly that, if irrational numbers are employed as reference of quantitative fractions,they must be defined without

hand, Weierstrass, Dedekind,

one

pointed out measures

to

quantity;

definition

and

same

men

who

showed

the

necessityof

such

a

In this way,

supplied the want which they had created. subject,which has added during thirty or forty years, a new quite immeasurably to theoretical correctness, has been created, which for, startingwith integers,it legitimatelybe called Arithmetic ; may the

have

the

last

Quantity

158

rational limits,irrequiresrationals, It results that, for all Algebra and on. assume any material beyond the integers, themselves be defined in logical terms.

succeeds,in definingwhatever and continuity, it is unnecessary Analysis,

which,

have

we

as

It is this

seen,

science,far

more

so

to can

[CHAP,xix

than

else it

"

non-Euclidean

Geometry, that

is really

theory of a prioriintuitions as the basis of the strongholds mathematics. and irrationals were formerly Continuity of the school who may be called intuitionists, but these strongholds are theirs no longer. Arithmetic has grown to include all that can so as in traditional be mathematics. the strictly called pure advance with this purist^ 150. reform,an opposite But, concurrently fatal to

the

has been

effected.

with

Kantian

number

nor

New with

branches

of mathematics, which

quantity,have

been

Geometry,and LogicalCalculus,Projective of Groups. Moreover it has appearedthat the

with correlation,

aggregates "

is not

a

"

invented; such

in its

essence

measurement

numbers, of entities which of prerogative

deal neither

quantities :

are

some

"

"

not

are

the

the

Theory

if this

means

numbers

or

cannot quantities

thingswhich are not quantities(forexample anharmonic ratios projectively defined)can be measured. Measurement, series certain is all in fact,as we shall see, of a kind to a kind applicable includes and which excludes some which some are things quantities The number and is b etween thus not quantities. quantity separation each o f the is other. : independent wholly Quantity, complete moreover, has lost the mathematical importancewhich it used to possess, owing to it can be generalized the fact that most theorems concerning to so as order. be become would It therefore natural theorems concerning to discuss order before quantity. As all propositions order concerning instances of for particular can, however, be established independently less order,and as quantitywill afford an illustration, slightly requiring effort of abstraction, to be applied to series in general of the principles ; of w hich forms of of the the a as, further, theory theory distance, part controversial opinionsas to the nature order, presupposes somewhat I shall follow the more of quantity, traditional course, and consider a theory quantityfirst. My aim will be to give,in the presentchapter, of quantity which does not depend upon number, and then to show the relation which is possessed to number classes of peculiar by two special wherever of quantities quantities, upon which dependsthe measurement this is possible.The whole of this Part,however and it is important to realizethis is a concession to tradition ; for quantity, shall find, we is not definable in terms of logical constants, and is not properlya notion belonging to pure mathematics at all. I shall discuss quantity because it is traditionally to supposed occur in mathematics,and because this supposition a thoroughdiscassion is requiredfor disproving ; but did not exist, of any such if the supposition I should avoid all mention notion as quantity. be measured, and

some

"

"

"

The

149-152] In

151.

meaning of Magnitude

the meaning of fixing

one

is faced with the

one

must

such

a

term

that,however difficulty

as

one

159

quantityor magnitude, may

define the word,

This difficulty arises wherever appear to departfrom usage. characteristics have been commonly supposedinseparable which,

two

closer examination,are

discovered

to be

capableof existing apart. case meaning appears to imply (1) a for the relations of greaterand te, (2) divisibility. Of these capacity the first is supposedto imply the second. But I characteristics, as I must either admit that some things propose to deny the implication, which indivisible are are magnitudes,or that some thingswhich are upon

In

of

the

magnitude,the usual

less than

others

partures magnitudes. As one of these deshall the which choose unavoidable,I former, usage I believe to be the lessserious. A magnitude, be defined as is to then, anythingwhich is greateror less than somethingelse.

greateror

from

are

not

is

be

should be mentioned, along with thoughtthat equality the definition of magnitude. We shall see reason greater and less,in such to think,however view that what as a paradoxical may appear be greateror less than some be equalto any term can never term, can This will require whose necessity a distinction, whatever,and vice versa. will become more and more evident as we proceed, between the kind of and the kind that can be greater or less. The that can be equal, terms actual footshall call the latter magnitude*. An former I quantities* its lengthis a magnitude. Magnitudes are rule is a quantity more : abstract than quantities two are : when they have the quantities equal, of this abstraction is the first pointto same magnitude. The necessity It

might

"

"

be established.

Settingaside magnitudesfor the moment, let us consider A quantities. quantityis anythingwhich is capableof quantitative is to be distinguished to somethingelse. Quantitativeequality equality from other kinds,such as arithmetical or logical equality.All kinds of being reflexive, of equalityhave in common the three properties which has this relation at all and transitive, i.e. a term symmetrical, 152.

has this relation to itself;if A has the relation to B, B has it to A ; tinguish it is that disWhat if A has it to B, and B to C, A has it to C*.

equalityfrom other kinds, and whether this quantitative difficult question, is a further and more is analyzable, kind of equality to which we must now proceed. of quantitative views far I know, three main There as are, so equality.There is (1) the traditional view, which denies quantityas *

On the

independenceof

these three

Peano, see properties,

Revue

de

Mathtmatique,

; what is properlynecessary necessary vii, p. 22. The reflexive propertyis not strictly of is,that there true and what is alone (atfirstsightat any rate) equality, quantitative exists at least the

other

question.

two

the relation in question. It follows then from pairof terms havingthat each of these terms has to itself the relation in properties

one

160

idea,and independent

an

[CHAP,xix

Quantity

onlywhen, they have

the

asserts

that two

number

same

of

terms

equalwhen,

are

is what

parts. (")There

and

may

be called the relative view of

to which equal, greater according quantity, this view we In and less are all direct relations between quantities. have is of magnitude need of magnitude,since sameness no replaced the of There relation is and transitive by symmetrical equality.(3) the absolute theoryof quantity, is not a direct relation, in which equality but is to be analyzedinto possession of a common magnitude,i.e.into

of relation to

sameness

kind

of relation of

of the

term

less will

equal,greater and relation

magnitudes.

to

to its

other series, and 153.

number indeed no a

(1) of the

between

The

difference which

regard to

space

of very considerable which kind of equality

special

a

magnitudes

two

greater and

less; while

virtue

the second

of their

and

third

arises in the

case

of many

and

The

decision

time.

importance. consists in havingthe

same

alreadydiscussed in Part II. If this be then quantityintroduces quantitative equality,

parts has been

meaning But

idea.

new

magnitude;

difference between

a

notablyin

matter

there will be

case

apply to quantitiesonly in

The

of theories is exactly typical

is,therefore, a

In this

there will be the relation of

kind

same

a

third term.

a

wider field than

arguments may

of

it may

whole

be

shown, I think,that greaterand less have

be

and

part,and

enumerated

independentmeaning.

an

follows

as

(a) We

:

admit

must

The visible indi-

(y3)where the number of simpleparts is infinite, quantities; there is no will give the recognized of number which generalization results as to inequality; relations must be allowed to be (7) some and relations divisible not are even conceivably quantitative, ; (8) even the axiom where there is divisibility, that the whole is greaterthan the and not a result of definition. part must be allowed to be significant, indivisible. For it is generally admitted are (a) Some quantities that some such and as existents, psychical pleasure pain,are quantitative. in the If now of indivisible parts,we number means sameness equality shall have to regard a pleasure of a collection or a pain as consisting of units,all perfectly simple,and not, in any significant sense, equal of compound pleasures results on this hypothesis, inter se ; for the equality of into their composition, the from number simpleones entering solely is formally to indivisible pleasures. that equality so If,on inapplicable allow pleasures to be infinitely that no the other hand, we so divisible, unit

we

can

then the number indivisible, and if there is wholly arbitrary, take is

pleasureis shall have to admit that we pleasures, called equal or unequal*. Hence we meaning * mean

other

I shall

never

than

use

sameness

the word

as

to

any two

shall

the

unequal to

units may

requirefor

number

mean

greater or /ess,i.e. not equal^though of the

same

of units in any given to be any equalityof

be

significantly equalitysome

of parts. This

latter

merely not equal,but always to kind of quantities.

The

153]

152,

meaning of Magnitude

unavoidable. theory,however, seems to regard pleasures of as consisting but further

as

"

candid

a

can pleasures

two

However

small two

say that

they are

in

and who

I cannot

(#)

we

in

that,where reduced that

to

may of

our

be

to

power

rightand an

reduced

number

of

simpleparts; for

be taken

to

make

parts. The

than equality

these,whatever

in

that

other

the

number

to

the

up

of the

whole

cardinal number as

number

this it follows

parts,if equality

at

all,it

must

be

of

complex parts wholly arbitrary.But than

narrower

partsin any from Cantor ;

in the

sameness

of

know

we

From of

parts

is

of the quantities wrong, though it is

two even

continuous the ordinal

for any two lengthswhatever. if Hence of the kind to which Geometry and inequality spatial

accustomed

us,

obtained from

I shall be told that

we

must

seek

the number

some

of

other

parts.

At

meaning this

for

point

meaning is very obvious : it is obtained from trenchingtoo far on discussions which belong

the

Without superposition. not

it follows.

infinitenumber

be

to

there is to be any have common-sense

a

both

able, whollyunwarrantheld by those* consciously

decide the alternative.

consist of quantities

of space is the same, portions number or type is the same

to

"

seems

concerningtwo

for example in Geometry,is far equality,

number

anyone

when significant

and divisible, infinitely

definite:

must

answer

is inequality

or

view

a

premissesfrom which

alwaysbe

kind, one

same

the

are quantities

must inequality

often not

reason

take of infinite number, equality is not coextensive with the number of parts. In the first place, equalityor

in

sameness

Such

be

to

believe that it has been

have advocated

definition

only no

of indivisible units,

sums

I think, convince will,

suddenlycease

indivisible units.

Some

definite

always be significantly judged equal or unequal. to pleasures may be, it must always be significant But the the on equal. theoryI am combating, judgment

question would

were pleasures

there is not

For

consideration

161

later part,I may observe (a) that to space, (")that as a criterion of

appliesto matter, superposition it equality, presupposes that the means constancy as regards rigidity

(r)that superposedis rigid, This shows that we cannot, without a vicious properties. define spatial Spatialmagnitude is,in circle, by superposition. equality matter

metrical

fact,as indefinable

of parts,in this case every other kind ; and number is whollyinadequate the number is infinite, even

as

as

in all others where

as

a

criterion.

This is suggested by the above quantities. where it is very natural to base equality discussion of spatial magnitudes, shall see hereafter,is not this view, as we upon distances. Although whollyadequate,it is yet partlytrue. There appear to be in certain series (forinstance that of the in some there certainly are spaces, and

(7)

*

Mind,

"g. N.

relations

Some

Mr

"

Bradley,

S. Vol.

iv

;

see

are

What

do

esp. p. 5.

we

mean

by

the

Intensityof PsychicalStates

?"

[CHAP,xix

Quantity

162 rational

relations of distance numbers),quantitative

terms.

Also

for

and similarity

shades of colour.

exampletwo

of red

are

is

Red

mere

a

shades,and the only reason lies in the close resemblance resemble

a

undeniable that two

seems

propertyin the

common

the other also.

regardedas

It

Consider quantities.

similar to each other than either is to

more

yet there is no

difference appear to be

other.

And

(8) Finally,it

case

for

since

virtue

of which

relations

are

relations cannot

is well to consider

not

;

found in

is not

certain series of

a

Hence

its terms.

shades

shade of blue

a

which

givinga collective name

between

divisible, greaterand less among of parts.

one

collective name

propertyin

common

each

for

the various

among

to

this series

red must two

not

shades

even

be

of red

conceivably

depend upon

number

directlythe meanings of greater

hand, and of whole and part on the other. Euclid's axiom, that the whole is greater than the part,seems ficant; undeniablysigniand less on

the

one

would this axiom be quantity, mere a tautology.This point is again connected with the question whether superposition is to be taken as the meaning of equality, as or a criterion. On the latter view, the axiom must be significant, and mere cannot we identify magnitudewith number of parts*. There is therefore in quantity 154. (") somethingover and above

but

on

the ideas which

the

we

the relative and

traditional view

have

absolute

of

hitherto discussed.

It remains to decide between

theories of

magnitude. of any not possessed as theoryregardsequalquantities but as common property over and above that of unequalquantities, relation of the mutual merelyby equality.There is no distinguished such thingas a magnitude,shared by equalquantities. We must not say : This and that are both a yardlong\ we must say : This and that are equal,or are both equal to the standard yard in the Exchequer. is also a direct relation between between not Inequality quantities, is are magnitudes.There nothingby which a set of equalquantities from is which to the not relation of distinguished one equal them, except itself. The of definition is,therefore, follows: We course as equality of which there are various have firsta quality or relation, say pleasure, in the case of a quality, instances,specialized, by temporalor spatioin the and of a relation, case temporalposition, by the terms between it holds. Let us, to fix ideas,consider quantities which of pleasure. consist merelyof the complexes at such a Quantitiesof pleasure pleasure at such another time (to which place time?and pleasure may be added, if that have it be thought in space). In the analysis of pleasures position there to a is,according the relational theory, no pleasure, particular other element to be found. But on comparingthese particular pleasures, The

*

when

relative

Compare, with the above discussion, Meinong, Ceher

die

Bedeutung des

1890 ; especially Gesetz"t,Hamburg and Leipzig, Chap, i, " "

153,

The

154]

find that

we

greater,and to question answer or

less. which

Why it is

for there

;

have

two

any

meaning of Magnitude and

one

have

some

This

not

agree that some merelyhappens

stillmore

when

so

of one being all quantities A

is

irrelevant. Equal quanobviously tities respectin which unequal ones

have be

examine

we

relational theoryobliges to us A

give an

in any

it must things,

of

state

relation,some

one

relations, equal, another, is a

is,ex hypotkesi, no pointof difference excepttemporal

do pleasure

differ: it

of three

and strictly to theoretically impossible

which position, spatio-temporal of

only one

163

the

relation and

one

some

admitted,is curious,and indemonstrable

assume.

They

are

the

axioms

another. it becomes

which

the

(A^ B, following

C

kind):

is greaterthan

B, 1?,or A is less than B. (a) (b) A beinggiven,there is alwaysa 5, which may be identical with A, such that A=B. (c) If ,4 =5, then 5=^. C,then^=a (d) If ^=JBand5 If A is than 5, then B is lessthan A. (e) greater or

"

=

If A

(f)

is

greater than JB,and

B

is greaterthan

C, then A is

greaterthan C.

(g) (A) From

than is greater

If A

B9 and B=

C, then A is greaterthan C.

5, and B is greaterthan C, then A is greaterthan C. A *. From (e)and (/) it (b\ (c),and (d) it follows that A If A

=

=

j",and B is lessthan C, then A is lessthan C ; from (c),(e\ and (h)it follows that,if A is less than By and B C, then A is less than C ; from (c), (e\and (g) it follows that,if A J5,and the C. of A is lessthan B is less than C, then (In place (b)we may put If then These A be axiom the a : axioms, it will be A~A.) quantity, observed, lead to the conclusion that, in any proposition asserting be substituted an defect, or equalquantity equality, excess, may where any-

follows that,ifA

is less than

=

=

falsehood of the proposition. the truth or affecting is essential A A an part of the theory. Now proposition the first of these facts strongly suggeststhat what is relevant in quantitative but the is actual not some quantity, propositions propertywhich this suggestionis almost And it shares with other equalquantities. For it may be laid down that demonstrated by the second fact,A "A. and transitive relation which a term the onlyunanalysable symmetrical the Hence if this be indeed a relation. have to itself is identity, can Now to say that a relation is should be analyzable. relation of equality relations is to say either that it consists of two or more analyzable not the case here,or that,when it is between its terms, which is plainly without

Further, the

=

said to hold between related in ways

are

* ever

two

terms, there is

which, when

This does not follow from

equalto

B,

See

Peano,

some

third term

both

relation. compounded,givethe original

(c)and (d)alone,since they do

he, tit*

to which

not assert

that A is

Quantity

164 to assert that A

Thus

third person C, who be Hence if equality

is j?s

[CHAP

is grandparent

to

assert that there is

xix

some

daughterand J?'s father or mother. both be related to two equalterms must analyzable, arid third since term ; a term some may be equalto itself, any two equal have the same relation to the third term in question.But must terms to admit this is to admit the absolute theoryof magnitude. when of what we A direct inspection terms mean we say that two relational to the are equal or unequal will reinforce the objections maintain that equalquantities have theory. It seems preposterousto in what is shared common beyond by unequal absolutelynothing different Moreover not are : they unequalquantities merely quantities. different in the specific manner are expressedby sayingthat one is quiteunintelligible greater,the other less. Such a difference seems where unequalquantities unless there is some are point of difference, is where Thus the which absent tional relaare concerned, equal. quantities is not absolutely theory,though apparently self-contradictory, and paradoxical.Both the complication and the paradox, complicated shall find,are entirely absent in the absolute theory. we 155. (3) In the absolute theory,there is,belongingto a set of definite concept, namely a certain magnitude. one equal quantities, Magnitudes are distinguished concepts by the fact that they among is A\

son

or

have the relations of greater and less (or at least one terms, which are therefore also magnitudes. Two

of them) to other

magnitudescannot defined as possession

and is belongsto quantities, equal,for equality of the same magnitude. Every magnitude is a simpleand indefinable concept. Not any two magnitudesare one greater and the other less ; the contrary, given any magnitude,those which are on greater or less within which than that magnitudeform a certain definite class, any two be

greater and

one

are

magnitude.

A

kind

which

the of

magnitude may,

class is called

a

way,

to be

connected

however, with

be

also

a

kind

of

defined in

by an axiom. distance, Every magnitudeis a magnitudeof something pleasure, area, relation to the somethingof which and has thus a certain specific etc. and appears to be it is a magnitude. This relation is very peculiar, further of definition. All magnitudeswhich have this relation incapable and the same to one are something (e.g.pleasure) magnitudesof one

another

has

other less. Such

the above

"

"

kind

;

and with this

of magnitudes

the

it becomes axiom to say that,of an definition, same kind,one is greaterand the other less.

two

objectionto the above theory may be based on the a magnitude to that whose magnitude it is. To fix our ideas,let us consider pleasure.A magnitude of pleasureis so much such and such an intensity of pleasure. difficultto It seems pleasure, regardthis,as the absolute theorydemands, as a simpleidea: there to be two constituents, need not and intensity. seem pleasure Intensity of and be intensity is distinct from of pleasure pleasure, intensity 156.

An

relation of

154-156]

The

meaning of Magnitiide

165

pleasure.But what we requirefor the constitution of a certain in general, is,not intensity but a certain specific magnitudeof pleasure intensity of pleasure cannot be indifferently ; and a specific or intensity of somethingelse. We first settle how much cannot will have, and we then decide whether it is to be pleasure A specific or mass. intensity be of a specific must kind. Thus dependent inand pleasure not are intensity

abstract

and coordinate elements in the definition of a given amount of pleasure.There are different kinds of intensity, and different magnitudes in each kind ; but magnitudesin different kinds must be different. Thus

it

that the

seems

is not any magnitude^ of a single term, but is

or

element,indicated by the term intensity that can be discovered by analysis thingintrinsic,

common

merelythe fact of beingone term in a relation of inequality.Magnitudes are defined by the fact that they have this and theydo not, so far as the definition shows, relation, thing agree in anyelse. The class to which theyall belong, like the married portion of a community,is defined by mutual its terms, not by relations among relation to some a common outside term unless,indeed, inequality itselfwere taken as such a term, which would be merelyan unnecessary complication.It is necessary to consider what may be called the "

extension

or

field of

a

relation, as

well

as

that of

and class-concept:

a

Thus magnitude is the class which forms the extension of inequality. magnitudeof pleasureis complex,because it combines magnitude and but a particular pleasure; magnitude of pleasureis not complex,for magnitudedoes not enter into its conceptat all. It is onlya magnitude because it is greateror less than certain other terms; it is onlya magnitude of pleasure because of a certain relation which it has to pleasure. This is more understood where the particular easily magnitude has a A yard,for instance, because it is greater is a magnitude, name. special than a foot ; it is a magnitude of length, is called because it is what all magnitudesare simpleconcepts, and are classified a length. Thus into kinds by their relation to some relation. The quantities or quality which are instances of a magnitudeare particularized by spatio-temporal the of relations which case o r are (in quantities) by the terms position between which the relation holds. Quantitiesare not properly greater or

less,for the relations of greater and

magnitudes,which this

When

axioms,

we

are

distinct from

theory is

find

a

very

the

appliedin the notable

less

hold

No

(b) (c) (d)

L is greater than

M

or

of the

The simplification.

magnitudeis greateror

their

quantities. enumeration

equality appears have all become demonstrable,and following(L, Jf, A7 beingmagnitudesof one kind): (a)

between

less than

L is less than

we

axioms

necessary in which

requireonlythe

itself. M.

greaterthan M^ then M is less than L. and M is greaterthan N9 then L is If L is greater than M

If L is

greaterthan N,

[CHAP,xix

Quantity

166 The

formerlycalled (b)is avoided,as are the concerningequality;and those that remain are simpler

difficultaxiom

other axioms than

our

157.

former The

which

we

set.

relative theories

the absolute and

decision between

can

of very be made at once to a certain generalprinciple, by appealing of Abstraction. wide application, which I propose to call the principle This principle asserts relation,of which there are a that, whenever has the instances,

then

the

two

relation in

and transitive, of being symmetrical properties but is analyzableinto questionis not primitive,

of relation to

sameness

is such

that there is

other term

some

onlyone

to which

most

at

term

and that this

;

giventerm

a

related,though many terms may be so related (That is,the relation is like that of son to father : have only one father,) sons, but can many

so

This

which principle,

cardinals, may

have

we

with

to

a

a

man

be

can

given term. may

in connection

It is,however,

elaborate.

somewhat

seem

alreadymet

relation

common

have

with

capableof assumption. or identity

of a very common proof,and is merelya careful statement into It is generally held that all relations are analyzable this view, I retain,so far of content. diversity reject Though I entirely transitive relations are a what someconcerned,what is really as symmeti'ical of the traditional doctrine.

modified statement usual

adopt more a

are phraseology,

property. But

common

common

a

and will not, in conception,

Such

to relations, alwaysconstituted by possessionof property is not a very precise

ordinarysignifications, formally A question. common of of those terms. two is terms regardedas a predicate quality usually the only form the whole doctrine of subjectand predicate, But of as which propositions are capable,and the whole denial of the ultimate of are reality relations, rejected by the logicadvocated in the present work. Abandoning the word predicate,we may say that the most generalsense which can be given to a common property is this: A

fulfilthe

property

common

and

one a

function of

the

terms

is any In this

third term

to

which

both have

generalsense, the possessionof but not necessarily transitive. In symmetrical,

relation.

property is

common

relations in

analyzingthe

of two

same

of its

most

order that must

it may be transitive, the relation to the common property be such that only one be the property of any term at most can

giventerm*. an

event

Such

to the time

is the relation of at which

namely the referent,the one

*

of

The

no

means

proof of

it

a

common

these assertions

occurs

is

determinate.

by of that the possession is

other

a

quantityto :

given one

term

of the

of

relation,

determinate,but given the other,the Thus

it is

capableof demonstration the type in questionalways

propertyof is mathematical,

Relations ; it will be found in my article M. vn, No. 2, " 1, Props. 0. 1, and 0.

If. d.

its magnitude,or

"

'2.

Sur

and la

depends upon the Logic Logique des Relations,"

The

156-158] leads to

of Magnitude

vieawng

167

symmetrical transitive relation. What

a

principleof

the

abstraction asserts

is the converse, that such relations onlyspringfrom of the above type*. It should be observed that the properties

common

relation of the terms be that

never

to what

which

or predicate,

of

is

I have called their

usuallyindicated by the relation of subjectto

the individual to

received view) can

propertycan

common

its class. For

have

subject(in the can belong

no

individual

and no onlyone predicate, onlyone class. The relation of the terms to their common property different in different cases. In the present case, the is, in general, quantityis a complex of which the magnitude forms an element : the relation of the quantityto the magnitude is further defined by the fact that the magnitudehas to belongto a certain class, namely that of It must then Ixi taken axiom magnitudes. an (as in the case of as coexist in one kind cannot colours)that two magnitudesof the same spatio-temporal place,or subsist as relations between the same pair of this the required of the magnitude. It terms; and supplies uniqueness is such synthetic judgments of incompatibility that load to negative judgments; but this is a purelylogicaltopic,upon which it is not to

in enlarge

necessary to 158. of

We

results.

may There

now

this connection. up the above

sum

are

greater and le$#; these relations inconsistent the

are

other,in the other

sense

that,whenever

holds between

relations

some

concept.

Two

said to be of the

are

the

The

one

is the

Each

terms

which

are

same

and sufficientcondition

of the

converse

A

holds between

magnitude*. Every magnitude lias a by sayingthat it is a concept,expressed magnitudeswhich have this relation to

are

relation to

and A.

B

and and transitive, asymmetrical

are

the other.

with

one

brief statement

a

pair of indefinable relations,called

certain

a

discussion in

and

J?,the

capableof these certain peculiar magnitudeofthat the

same

concept

kind is*the necessary a for the relations of greaterand less. When ; to be of the

kind

same

or spatio-temporal by temporal, spatial, magnitudecan be particularized be particularized it can by taking or when, being a relation, position, it into a consideration a pairof terms which holds,then the between is called a quantity. Two magnitudesof magnitudeso particularized cations. specifithe same kind can be particularized never by exactlythe same

Two

magnitude are Thus

The

a

the relation R

are

Our

eqwiL (1)greaterand less,(2) every particular

are

indemonstrable

are: propositions

principleis proved by showing that, if

and relation,

hy

indefinable*

our

magnitude.

*

said to be

its converse,

a

term

taken is

of the field of as

a

equalto

X,

a

has, to the

whole, a many-one K.

the same particularizing

result from

quantitieswhich

a

be

a

symmetricaltransitive

class of terms

relation which,

a

to which

it has

multiplied relationally

far

as magnitude may, so class of equalquantities.

Thus

concerned, he identified with

R

formal

arguments

[CHAP,xix

Quantity

168

(1) Every magnitude has it of

certain

a

(2) Any

to

the relation which

term

some

makes

kind.

two

the

magnitudesof

kind

same

one

are

greaterand

the

other less.

(3) or

magnitudesof

Two

both have the

time,cannot

can

kind,if capableof occupyingspace if relations, position; spatio-temporal

the

same

same

the

be both relations between

never

pairof

same

terms.

(4") No magnitudeis greaterthan itself. (5) If A is greaterthan #, B is less than J, and lice versa. (6) If A is greater than B and B is greaterthan C, then greater than C*. Further

above

axioms

alone

seem

depend in any undismayedin measured,

of

way

upon

but the of magnitudes, species magnitude in general.None of them

to

number

the

presence of which, in the next

is

various

characterize necessary

A

hence

measurement;

or

magnitudeswhich cannot shall find an we chapter,

we

be

may

be divided abundance

or

of

instances. Note

to

work

The

CJwpterXIX.

of Herr

Meinong

on

Weber's

Law,

alreadyalluded to, is one from which I have learnt so much, and with which I so largely desirable to justify myself on agree, that it seems the pointsin which This work it. I depart from begins("1) by a is limited towards zero. characterization of magnitude as that which Zero

is understood

as

the

and after a discussion, negationof magnitude, is adopted(p.8):

the

statement following of terms which allows the interpolation That is or has magnitude, itself and between its contradictory opposite." "

Whether

doubtful a

this

constitutes

but in either (*".),

fundamental

definition, or

a

case, it appears

of

characterization

a

to

me

magnitude.

mere

to

is left criterion, be undesirable

It derives

as

support,as

Kant's to Meinong points out (p. 6 n.\ from its similarity But I of it t." is,if am not mistaken, liable "Anticipations Perception to several grave objections. In the first place, the whole theory of zero is most rather than prior, and seems to the theory difficult, subsequent, of other magnitudes. And to regardzero as the contradictory opposite of other magnitudes seems The phraseshould denote the erroneous. class obtained by negationof the class "magnitudesof such and such of that kind of would not yieldthe zero a kind"; but this obviously we magnitude. Whatever interpretation give to the phrase,it would to imply that we must seem regardzero as not a magnitudeof the kind Herr

whose

it is.

zero

the kind in *

above would

But

in that

and question,

case

there

It is not

it is not

seems

necessary in (5)and (6)to add relations of greater and less are what therefore be tautological.

f Item

Vmnnift,

ed. Hartensteiii

the

of magnitudes

meaning particular

no

"

less than

in

saying

A, B, O beingmagnitudes/*for the define

(18(57), p.

158.

magnitudes,and

the

addition

The

158]

that in

magnitude

lesser

a

the

case,

any

which

I

what

for

as,

that the

first

that

that

it

is

of

distances,

among same

relation such

inversion

involved

relations

have

be

resumed

of

where

the to

zero

pleasure.

as

in

been in

the

introduction

specified Chapter

from

xxn.

wise,

whether

magnitude, zero

is

the

or

simply

which

of is

'between it

could

be

urged, have and

unimportant

in

be

must

any

arise.

;

is

there

as

before

to

magnitude

identity,

reason

appear introduce

might

non-existence

main of

kinds

zero

then

therefore,

give

to

stage

will

objections all

and

magnitudes,

it

;

Other

negation But

be

and

greater

later

a

and

zero

hardly

can

doubtful

kinds

discrete

in

of

at

it

relations,

than

of

endeavour

magnitude.

;

qualities

will

of

account

It

other

none

demands

IV,

Part These

betweenness

theory

true

is.

the

shall

in

see

And

magnitude,

concerned.

terms

to

I

the

subject

instance,

zero

be

to

this

the

in

zero

a

conceive

shall

we

169

greater

a

magnitude,

prior

definition.

to

difficult

how

of

and

as

the

case

Magnitude

zero

fat-ween,

among

therefore

of

between

of

the

in

are

suitable

more

notion

are,

seem,

kss9

is

relations

asymmetrical would

meaning

the the

hardly case

of

logical

asymmetrical This

subject

XX.

CHAPTER

RANGE

THE

THE

159.

kinds

What number

of

there

are

all such

which

will

that

ensure

of terms

sorts

present chapterare

class of in

Is there

? to

a

set

these

relation

common

quantitiesof

common

related

is thus

term

a

in the

which, by their

magnitudes, constitute a terms anything else

Have

What

be discussed

questionsto of terms

QUANTITY.

OF

of

one

a

kind

?

mark

any

magnitudes ?

or capableof degree,or intensity, greater

are

:

to

and

less? mark of all as a common regards divisibility We have there is no that seen already having magnitude. the question We to examine now are a prioriground for this view. undoubted find of instances to as as possible, inductively, quantities many other and to inquire whether or common they all have divisibility any The

traditional

view

terms

mark.

Any it in

a

term

collection

of which

contains greater or less degree is possible

magnitudes of one primd facie evidence

of

is

grammar

a

conclusive,we

should

have

admit

to

susceptibleof magnitude.

kind. of

the

Hence

quantity.

that

under

comparative-form

If this evidence

all,or almost

were

are all,qualities

by praises and reproaches addressed mistresses would afford their and to superlatives comparatives poets known of most adjectives.But some circumspectionis required in There is always, I think, using evidence of this grammatical nature. a some qitantitative comparison wherever comparative or superlative is but often it not a occurs, comparison as regardsthe qualityindicated by grammar. "O

ruddier

than,

the

cherry,

O

sweeter

than

the

berry,

O

nymph

Thau

are

lines

The

more

moonshine

bright light," and

containing three

have, we brightness,

I

comparatives. As regards sweetness think, a genuine quantitative comparison ;

regards ruddiness, this may generallywhere

colours

are

be

doubted.

concerned

"

The

comparativehere

indicates,I think, not

more

but "

as

and of

a

159,

The

160]

of Quantity

range

171

likeness to a standard colour. Various shades of givencolour,but more colour are supposedto be arrangedin a series, such that the difference is greateror less according of quality the distance in the series is as

greateror less. One called

of these shades is the ideal

"

ruddiness,*and others

less

ruddyaccordingas theyare nearer to or further I think, same explanation applies, to such terms as whiter.Hacker, redder. The true quantityinvolved to be, in all these cases, a relation, seems namelythe relation of similarity. The difference between two shades of colour is certainly difference of a of not quality, merely magnitude;and when we say that one thing is redder than another,we do not implythat the two are of the same shade. difference of shade,we If there were should probablysay one no was than the other,which is quite a differentkind of comparison. brighter But though the difference of two shades is a difference of quality, yet,as of serial arrangement shows,this difference of quality the possibility is itself susceptible of degrees. Each shade of colour seems be to simple but in colours and unanalyzable the neighbouring ; spectrumare certainly

are

similar than

more

more

B

or

A does.

this

and

B,

givescontinuityto we

that C

means

should say, there resembles A or B

for such relations of immediate

But

resemblance,

be able to arrange colours in series. The resemblance be immediate, since all shades of colour are unanalyzable, as appears

should

we

It is this that

shades of colour,A

two

third colour C; and

alwaysa than

colours.

remote

Between

colours. is

or

more

this shade in the series. The

from

must

from

not

at

attempt

any

indubitable

case

resemblance

colours

of two

is

I have dwelt upon

160.

Thus

have

have

we

this

case

or

of

an

difference or

The

magnitude. and is a magnitude; for relation,

a

other differences

less than

greater or

definition*.

or description

of relations which

it is

resemblances. since colours,

it is

one

instance

be arranged of terms can of a very importantclass. When any number in a series, it frequently happens that any two of them have a relation relation which sense, be called a distance. This may, in a generalized

suffices to

generate

magnitude. if these

the

suppose

have in

is

a alwaysnecessarily

of the series have

and

names,

the comparativesindicate,not comparatives,

but question,

distance from the is not itselfa

arrangement,and

more

of the quality

in which

one

than

recent

presentwas

likeness to that

more

time-series to be

is said to be

event

serial

In all such cases, if the terms

names

of the term

a

term.

there is distance,when

another,what

is meant

less than that of the other.

time

or

Thus,

more

if

What

of the event.

Thus are

we an

is that its recentness

quantitatively

und of colours,see Meinong, "Abstrahiren ff. Vol. 72 d. Sinmsorgane, xxiv, J%". p. Ze*t*chrifi f. P*gch" Vergleichen/' but his of whole with the I general Meinong's that I am not sure argument, agree *

On

the subjectof the resemblances u.

conclusion,"dass an

to

me

to

be

AUgememheiten darstellen, Antheil hat" (p. 78), important logicalprinciple.

des Umfangscollective

die Abstraction

denen

appears

die

a

Aehnlichen

wenigstens unmittelbar

correct

and

keinen

172

[CHAP,xx

Quantity

compared in

not qualities. The case of colours relations, because colours have and the for illustration, names,

such

is convenient

cases

are

colours is

difference of two

admitted generally

be

to

But qualitative.

The importance of this class is of very wide application. principle of clear notions as to their and the absolute necessity of magnitudes,

the

nature, will appear of space

and

and

more

more

as

proceed. The

we

whole

the doctrine of so-called extensive

time, and

philosophy magnitudes,

of series and distance. depend throughoutupon a clear understanding be distinguished from mere difference or unlikeness. Distance must connected with It holds onlybetween terms in a series. It is intimately that the terms between which it holds have an ultimate order,and implies into constituents. not and simple difference, one capableof analysis also that there is a more It implies less continuous passage, through or other terms the from one of the distant terms same series, belongingto

the other.

to a

difference per

Mere

relation,being in fact

alwaysabsolute,and terms

members

of certain

the

that

It is

degrees. Moreover it holds between is hardlyto be distinguished from the the But distance holds only between

of is incapable

whatever, and

any two assertion

series.

a

of appears to be the bare minimum of almost all relations. It is precondition se

they are

two.

series,and

and relation, specific

a

distance of A

B

from

from

the

is then

its existence it has

from

that of B

last mark

This

A.

the

distinguish

can

we

sense;

of

source

alone sufficesto

distance from bare difference. distinguish there is It might perhaps be supposed that, in a series in which than the than or less distance AB must be greater distance, AC, although For than the AC. distance be less BD need either not yet greateror

example, there derivable from from

and

"5

is

obviouslymore and

"5

that

that from

derivable from

"20.

But

the

difference between

need

there

be

pleasure

between

than

"100

either

that

equalityor

between the difference for "1 and "20 and that for "5 inequality For This questionmust AC be answered affirmatively.

and

"100?

greater or BC

less than

BC, and BC

and also BC, BD

are

is greateror

magnitudesof

the

same

less than BD

kind.

;

hence

Hence

must one kind,and if not identical, greaterand the other the less. Hence, when there is distance in

are

magnitudes of distances

the

is

AC,

AC, BD be the

same

a

series,

comparable. quantitatively kind form that all the magnitudesof one and that their distances, a series, therefore,if they have distances,are again magnitudes. But it must not be supposedthat these can, in kind as the be obtained by subtraction,or are of the same general, as a magnitudes whose differences they express. Subtraction depends, to is in and therefore rule, upon divisibility, generalinapplicable indivisible quantities. The is important,and will be treated point in detail in the following chapter. Thus have and distance are relations which nearness magnitude. any

two

It should

be

are

observed

160-162]

The

of Quantity

range

173

any other relations havingmagnitude? This may, I think, be doubted*. At least I am of unaware though any other such relation, of I know their no existence. disproving way there

Are

161.

There

is

difficultclass of terms,

usuallyregarded as magnitudes, apparentlyimplying relations, not though certainly always relational. These differential coefficients, are such as velocityand acceleration. about

They

a

be borne in mind

must

This

will be

given in

differentialcoefficientsare

segments in 162.

All

? magnitudes

Fart

V

and

;

shall then

we

find that

but onlyreal numbers, or magnitudes,

never

series.

some

the

magnitudesdealt

indivisible. speaking, is

attempts to generalize their complexity they requirea special

magnitude,but owing to

discussion.

in all

Here

essentially one,

Thus

the

I think

with

hitherto have

questionarises :

distinction must

a

Thus

been, strictly

there any divisible be made. A magnitude Are

magnitudeis correctly expressed which has as a quantity magnitude may of parts, be a sum and the magnitudea magnitudeof divisibility ? If so, of parts will be a single of the proterm possessed every whole consisting perty of divisibility. The more parts it consists of,the greater is its On this supposition, is a magnitude, of which we divisibility. divisibility and the degree of divisibility corresponds may have a greateror less degree; of parts. But though the whole in finite wholes,to the number exactly, which has divisibility which alone is of course divisible, yet its divisibility, The is strictly is d ivisible. not properly a magnitude, divisibility speaking does not itselfconsist of parts,but onlyof the property of having parts. It is necessary, in order to obtain divisibility, to take the whole strictly Thus its adjective. as as although,in one, and to regard divisibility many. of terms. But

number

this case,

not

have

we

no

not

numerical

the

measurement,

and

mathematical

all the

magnitudeis

our speaking, yet,philosophically consequences of division, still indivisible.

as difficulties, however, in the way of admittingdivisibility but of the kind of magnitude. It seems whole, to be not a property

There

a

are

merelya relation to the parts. It is difficultto decide this point,but a as a simple good deal may be said,I think, in support of divisibility convenience w hich for The certain we whole has relation, a may quality. call that of

to all its parts. inclusion,

there be many

parts or few

;

what

This relation is the

a whole distinguishes

that it has many such relations of inclusion. But it seems from a whole suppose that a whple of many partsdiffers

intrinsic respect. In

some

Cf. Meinong,

Leipzig,1896,

Ueber

p. 23.

die

of many

whether

parts is

reasonable to of few

parts in

fact, wholes may be arrangedin a series fewer parts,and the serial arrangement or

as according theyhave more series have already we as seen, some implies, when less from each other,and agreeing or *

same

Bedeutung

des

of

more differing properties

two

Welter schen

wholes

have

the

Genetee*, Hamburg

same

and

Quantity

174 finite number wholes.

These

of

parts,but distinct from

can properties

be

none

[CHAP,xx number

of

parts in finite other than greateror less degrees

would appear to be a Thus magnitude of divisibility divisibility. number the distinct from of parts included simpleproperty of a whole, be finite. If in the whole, but correlated with it,providedthis number be allowed to remain as a this view can be maintained,divisibility may class of magnitudes. In this measurable,but not divisible, numerically and should have to placelengths, class we areas volumes, but not later that find distances. At a the divisibility stage,however* we shall in which this is not measured by cardinal of infinite wholes,in the sense numbers, must be derived through relations in a way analogousto that in which distance is derived,and must be really a propertyof relations*. it would Thus divisibl inappeal4, in any case, that all magnitudesare mark is which all This one common they possess, and so far of

as

I

know, it is the only

one

to

be

added

to

those

enumerated

in

there seems to be no Concerningthe range of quantity, Chapterxix. further generalproposition. Very many simple non-relational terms the principal have magnitude, instants being colours, exceptions points,

and numbers. it is important to remember that, on the theory Finally, adoptedin Chapterxix, a givenmagnitudeof a givenkind is a simple concept,havingto the kind a relation analogousto that of inclusion in class. When such as pleasure, what the kind is a kind of existents, a exists is never the kind,but various particular magnitudesof actually the kind. Pleasure, taken,does not exist,but various amounts abstractly of it exist. This degree of abstraction is essential to the theoryof be entities which differfrom each other in nothing : there must quantity except magnitude. The grounds for the theory adopted may perhaps from a further examination of this case. clearly appear more famous Let Bentham's with start us proposition: "Quantity of Here the qualitabeingequal,pushpinis as good as poetry."" tive pleasure difference of the pleasures is the very pointof the judgment ; but in order to be able to say that the quantities of pleasure are equal,we be able must to abstract the qualitative and leave a certain differences, the qualitative magnitudeof pleasure.If this abstraction is legitimate, difference must be not trulya difference of quality, but only a difference 163.

of relation to

other terms, as, in the present case, a difference in the causal relation. For it is not the whole pleasurable states that are

compared,but only as the form of the judgment aptlyillustrates of pleasure.If we to their quality suppose the magnitudeof pleasure be not a separateentity, will arise. For the mere element of a difficulty be identical in the two cases, whereas we require must a possible pleasure difference of magnitude. Hence neither hold that only the we can "

"

whole concrete

state

and any part of it is an exists, *

See

Chap.

nor abstraction,

that

exists

what

say

The

163]

162,

and

pleasure.

the

pleasures.

and

The

would

it

must

they

abstract

bare

different

magnitudes.

pleasure

as

but

pleasure, pleasure where

only

and

has

abstract

quality

Having consider and

the

is

nature

extent

and

to

of

the

whole.

a

Thus that

numbers measurement.

is

can

be

and of

for

it

is

a

is

that

this

of

in

cular parti-

unanalyzable, class

to

have

next

that

magnitude.

a

indivisible,

are

of

magnitude

magnitude

of

magnitude

states,

discussion

inclusion

that

seen

magnitude

the

of

one

the

every

it

which

most

the

magnitude

pleasurable

at

which

magnitudes

And

whole

a

;

not

possibility

is into

analyzed

is

from

have

any

abstract

to

be

have

to

are

have

to

of

of

we

analogous

all

limits

if

which

we

in

pleasure

of

general

in

And

of

and

pleasures,

whole,

a

magnitude

save

theory

relation

that

seen

the

the

or

as

possible.

relation

as

we

magnitude

magnitude

a

states

not

difference

no

confirms the

only

we

must

part

a

comparison fully

pleasure, what

us

can

comparison

being

in

magnitude.

no

abstracted

as

is

there

quantitative case

be

exist

must

have

This

must

the

to

Nor

elements

quantitative

a

give

not

pleasure.

two

agree

abstract

Thus

whole.

a

would

cannot

we

wholes

as

not

this

get

would

magnitude

a

Hence since

states,

not

175

of the

states,

should states

But

Quantity

magnitude

not

whole

we

two

give

admissible.

the

then

For

magnitudes.

being

we

pleasure, from

abstract,

We

:

abstract

is

of

range

used

to

we

express

magnitudes,

to

CHAPTER

NUMBERS

XXL

EXPRESSING

AS

MAGNITUDES:

MEASUREMENT. 164.

IT is

of the

one

assumptions

of educated

that

common-sense

kind be numerically comparable. must magnitudes of the same to that People are apt they are thirtyper cent, healthier or happier say than they were, without suspicionthat such phrasesare destitute of any The of the present chapter is to explain what is meaning. purpose of which it meant what the classes to are by measurement, magnitudes applies,and how It is appliedto those classes. of magnitudes is,in its most Measurement generalsense, any method by which a unique and reciprocalcorrespondenceis established between all or some of the numbers, of the magnitudes of a kind and all or some integral,rational,or real,as the case may be. (It might be thought that ought to be included ; but what can complex numbers only be measured of numbers in is fact an tudes magniby complex always aggregate of different kinds, not a singlemagnitude.) In this generalsense, two

demands

measurement

some

relation

one-one

the

between

numbers

and

question a relation which may be direct or indirect, in this Measurement trivial,according to circumstances. be applied to very many classes of magnitudes ; to two great sense can distances shall see, in a more and divisibilities, it applies, we classes, as and intimate sense. important in the most Concerning measurement general sense, there is very

magnitudes important or

in

"

little to be said. of the

the numbers

magnitude also forms magnitudes measured all relations

that

measures.

measured measure

by

the

should

165.

There

there number

fulfil if than are

shows accepted,-

measurement

a

in the

series,it

should

of between

Wherever

rather practical if

Since

a

will

be

series,and desirable

a

zero.

be the zero,

These

same

it

is

and

for well other

every kind the order of

since that

of the

correspondto that

should is

form

numbers,

magnitudes that

this

and

i.e.

their be

should

conditions, which

a

possible,may be laid down; but they are of importance. generalmetaphysicalopinions,either of which, all magnitudes are capable of theoretically

theoretical two

that above

sense.

The

first of these

is the

theory

that

164-166]

Measurement

either axe,

all events

causal series.

In

or

correlated with, events in the dynamical so-called secondary this view qualities,

are

regardto

far acted upon so-called intensive

has been

177

the

science that by physical

it has

providedmost of the that appear in space with spatial, quantities thence and And with regardto mental numerical,measures. quantities the theory in questionis that of psychophysical the Here parallelism. which is correlated with motion cally theoretiquantityalways any psychical of measuringthat quantity.The other metaphysical affords a means is one opinion,which leads to universal measurability, suggestedby of Kant's "Anticipations Perception*," namely that,among intensive increase is alwaysaccompaniedby an increase of reality. an magnitudes, with existence; hence Reality,in this connection, seems synonymous so

doctrine

the

be

may

stated thus:

of which, where magnitude, than

more

doctrine

where

a

a

case, since two

(That

but it is at least

instances of the

is

a

kind

of intensive

greater magnitude exists,there is always

less magnitude exists.

improbable;

seems

Existence

a

this is

tenable

exactlyKant's

view.)

In

this

two equalquantities) magnitude(i.e. have more existence than one, it follows that,if a single must magnitude kind can be found havingthe same of the same amount of existence as then that magnitude may be called the two equalquantities together, In this way all intensive double that of each of the equalquantities. That this capableof measurement. magnitudesbecome theoretically has any practical method importance it would be absurd to maintain ; of meaning belongingto twice but it may contribute to the appearance for It as example,in which we may say that a givesa sense, happy. child derives as much pleasurefrom one chocolate as from two acid drops;and on the basis of such judgments the hedonistic Calculus be built. could theoretically other generalobservation of some There is one importance. If it be maintained that all series of magnitudesare either continuous in Cantor's or

sense,

are

similar to

same

series which

be chosen

can

of

out

continuous

then it is theoretically to correlate any kind of magnitudes series, possible of the real numbers, so that the zeros and with all or some correspond, if any But to the greater numbers. greatermagnitudes correspond series of magnitudes,without being continuous, contains continuous and theoretically then such a series of magnitudeswill be strictly series, the

of incapable

Leaving now

166. the

sense

*

Reine

in which

t See

may

Vernunft, ed. Hart

illustrates better than Erdmaun's

we

numbersf.

these somewhat

usual and concrete

more

some

by the real

measurement

that

edition,p. 161. Part V, Chap, xxxra

sense

vague

of measurement.

say that

one

(1867),p.

of the

ff.

let us generalities,

second

we

magnitudeis double

160.

the

What

The

wording

doctrine to which

of the

examine

requireis of another.

first edition

I allude.

See e.g*

Quantity

178 In the above

[CHAP,xxi

derived

by correlationwith spatiowith existence. This presupposed that in these or temporalmagnitudes, found for the had been Hence cases a measurement phrase. meaning there should be demands intrinsic in an that, some cases, meaning to the is of that." double "this what the sense (In magnitude proposition is will Now intrinsic as we so proceed.) long as meaning appear there is a perfectly divisible, quantitiesare regardedas inherently obvious meaning to such a proposition : a magnitudeA is double of B each of these it is the magnitude of two when quantitiestogether, having the magnitude B. (It should be observed that to divide a since there magnitudeinto two equalparts must alwaysbe impossible, such thingsas equalmagnitudes,)Such an will no are interpretation of have admitted still apply to magnitudes divisibility ; but since we different be found for other magnitudes, must a (ifany) interpretation these.

Let

instances this

us

to the other

where

was

the

first examine

cases

case

measurement

of a divisibility

The

167.

sense

correlated with the number

of

of is

and then proceed divisibilities,

intrinsically possible. is immediately and inherently

finite whole

simplepartsin

the whole.

In this case,

of addition of the sort incapable

magnitudesare even now be added in the manner in Part II. can explained quantities The addition of two magnitudesof divisibility yields merelytwo magnitudes, not a new magnitude. But the addition of two quantitiesof i.e.two wholes,does yielda new singlewhole, providedthe divisibility, which results from addition by regarding addition is of the kind logical their formed classes as the wholes terms. Thus there is a good by is double of magnitude of divisibility meaning in saying that one it applies to a whole another, when containingtwice as many parts. is by no means But in the case of infinitewholes,the matter so simple. the of infinite number of simpleparts (in the number Here only senses in the magnitude of hitherto discovered) may be equalwithout equality which does not go back to We requirehere a method divisibility. simpleparts. In actual space, we have immediate judgmentsof equality as regardstwo infinitewholes. When we have such judgments,we can regardthe sum of n equalwholes as n times each of them ; for addition althoughthe the required,

of wholes of known

does not some

demand

their finitude.

pairsof wholes

methods, by continual

becomes

In this way

possible.By

subdivision and the method

numerical

parison com-

the usual wellof

limits,this

parisons pairsof wholes which are such that immediate comWithout these which immediate are possible. comparisons, and be both are logically psychologically*, nothing can necessary in the last resort to the immediate are alwaysreduced : we accomplished judgment that our foot-rule has not greatlychanged its size during measurement, and this judgment is priorto the results of physical

is extended

to

all

*

Cf.

Meinong, op. tit.,pp.

G3-4.

166,

167]

science

the extent

to

as

immediate

where

But

Measurement to which

179

bodies do

comparisonis

actually change their sizes. we psychologically impossible, may

substitute a logical variety of measurement, which,however, theoretically relation or class givesa propertynot of the divisible whole,but of some of relations more less anak)gous to those that hold between or pointsin space. That a

in the divisibility,

propertyof

Part

a

requiredfor

sense

whole,results from

the fact

VI) that between the pointsof

and

areas

space there

a

different space. Thus two sets of form set of relations, equalareas, form

one

respectto another a

volume.

of

set,or

form

even

one

an

in the relevant divisibility

If

pointswhich, with

were

with

unequalareas

and

area

sense

alwaysrelations

are

which generate a

regardto

volumes, is not

(which will be established in

the other

a

intrinsic

an

line

or

property

wholes,this would be impossible.But this subjectcannot

be

fully Geometry. Where not onlydo numbers oxir measure magnitudesare divisibilities, but of with tions, limitathe difference certain two them, measuringnumbers, of disthe magnitude of the difference (in the sense similarit measures of the magnitudes be the divisibilities. If one between discussed until

we

to

come

difference in the number

of

that,if A, B, C, D generally A"B~C"Dy

two

for

inches

the difference of the

I do not

the

dependsupon

think it

be shown

can

measuringfour magnitudes, of the magnitudesare equal.

be the numbers differences

difference between

the

that instance,

that between

is greater than

as

difference

parts. But

the

then

seem,

increases

increases; for this

measuring numbers

It would

other

the

fixed,its difference from

and

Metrical

inches

1001

inch

one

and

and

inches.

100"

importancein the presentcase, since differences of never are divisibility required;but in the case of distances it has a curious connection with non-Euclidean Geometry. But it is theoretically be indeed a magnitude as the importantto observe that,if divisibility of areas to require then there is strictly no and volumes seems equality twice of is two units of for as that the sum a ground saying divisibility be strictly cannot great as that of one unit. Indeed this proposition taken,for no magnitudeis a sum of parts,and no magnitudetherefore is of two units conthat the sum double of another. We tains can only mean not a quantitative, twice as many parts, which is an arithmetical, judgment,and is adequateonlyin the case where the number of parts is is in general since in other cases the double of a number equalto finite, contains numbers the measurement of divisibility it. Thus by even This remark

has

no

"

"

an

of convention

element

the

prominentin 168.

the

In

fundamental

case

;

and

this

in other

The

sum

cases

of two

shall find,is stillmore

of distances.

above

case

we

still had

senses, Le. the combination

But

element,we

magnitude we is not a pleasures of

addition

of wholes do not new

to

have

in

form

any but pleasure,

of its two

one

new

a

such is

whole.

addition.

merely two

pleasures.The But

in this

such

b, c

the

kind

and

we

we

are

now

for

it holds.

Supposing

some supposed,

case

uniquelydetermines of the

quantityb

relation to

one-one

exampletwo

kind

same

that

as

ratios have

position proand is to

relation,

a

is itselfwhollydetermined

by

arithmetical sense, of the in a series in which there is

in the difference,

the

ratio,namely

which

relation

a

which

the

which difference,

call their

be terms

If a, /8,7

given ratios.

two

is

by some belong. Thus

c

may

another

those between

as

we

B

itself a

has

quantities, have, in

be such

a

which

which relation,

a

same

aBc, where uniquelydetermined which

which

sense

be

addition in abstract explainthis generalized to distances. illustrate its application which are not capableof happensthat two quantities,

addition proper, have #,

restricted

Some

is to

measurement

shall first

It sometimes

to

where possible

natural and

more

then

of quantity

an

distance.

one properly

idea of addition.

of the

extension

always be

must

I discussing.

terms, and

have

we

case

extension

distances is also not

of two

sum

effected in the

a

[CHAP,xxi

Quantity

180

a/3,ay have a relation which is measured by (thoughnot identical with) the distance /$y. In all such cases, by an of addition,we may put a 4- b extension Wherever c in placeof aBc. if further of aBc this have relations kind, bAc, implies a set of quantities distance,the

distances

=

that

so

4- 6

a

b + #,

==

shall be able to

we

able

shall be

addition,and

in

introduce

to

consequence

had

if we

proceedas

ordinary numerical

measurement.

distances

how a

far

come

kind of distance

by a

which

for the

present I

to be measurable.

concerned

am

The

Part

fullyin

word

IV,

show

only to

will be used

to

in

cover

that of distance in space. I shall set of quantitative relations of asymmetrical

than conception general

more

mean

order:

with

connection

discussed

will be

distance

conceptionof

The

and

one

class; which

holds between

only one

that,if there

such

are

a

is

a

of a given any pairof terms relation of the kind between a

and by and also between b and c, then there is one of the kind between the and between the relation and c being relative productof a a c, those between

a

and

of independent

i. e.

b,

b and

greaterthan the distance db is greater the class, than dc. therefore indivisible and natural convention

simpleand

ac,

be commutative, if the distance and finally,

productis

the order of its factors ;

ab be

and

this

c;

to

then,d beingany

Although distances

member

other are

thus

of

relations,

addition proper, there is a cally such distances become numeri-

of incapable

by

which

measurable. The

OftO}, #1^2

said to be

by a

a

is this.

convention a"n-ian

...

n

number

times n

distances

are

each

times

convention,but

are

an

as

all

Let it be

agreedthat,when

equaland

in the

same

sense,

the

distances

then

a0#n

is

of the distances aQaly etc.,i.e. is to be measured been regardedas not great. This has generally obvious

truth

distance no indivisible,

;

owing,however,to

is

a really

sum

the

fact that

of other distances,

167-170]

Measurement

numerical

and

measurement

convention, the

numbers, become

such

upon

members

the

to

choice of

the

to corresponding

definite, except as unit.

a

Numbers

constant, dependingupon stillfurther

capableof

In order

IV.

of

the terms

to

show

common

assigned by

the

are

dependent

this method

distances hold; these

additive factor, an arbitrary arbitrary

choice of

that

have

axiom

further axioms, the

which

there

factor

origin.This method, which is will be more generalization, fullyexplainedin

set,can

our

the

a

also

this

With

where distances,

to

are

of the class between

have, in addition to the

numbers

Part

be in part conventional.

must

numbers

181

aU

the distances of

numbers

of

kind,and

our

to them, assigned

Archimedes, and

what

all

two require

we

be called the

may

of

linearity*. of the numerical measurement The importance of distance, at least as appliedto space and time, depends partlyupon a further fact, of by which it is broughtinto relation with the numerical measurement series In there all intermediate terms between are divisibility. any two

axiom

169.

distance is not the minimum.

whose the

distant terms

two

called the stretch from and has quantity, providedtheir number is

a

of consecutive terms "o

and

OH,

the

a0 to

"z"f. The

measured divisibility

a

is finite.

If the series is such

of the distance is

stretch

stretches

to equaldistances. correspond measures

becomes

It then

whole

of terms,

infinitenumber

and

we

stretch. estimate

axiom, which

an

the

Thus

the

may

the

number

not

as

we

other, the

equal

of terms

and

in

the amount

contains

stretch

equalstretches or

be

between

and proportional,

are

When

may

may

Thus, if

n.

not

both the distance of the end terms

of the divisibility

of

to proportional

the distance

of the

when

that the distances ?i~l terms

are

terms, but

end

of the

one

terms

composed of these terms by the number of terms,

all equal, then,if there

stretch

determinate

are

intermediate

whole

measures

the stretch

terms

The specified.

are

are

measure

include in the

These

an

above. explained

hold in

a

given case,

equal distances.

In this case, coordinates equal stretches correspondto distinct two measure magnitudes,which, owing to entirely

that

their

common

measure,

The

are

confounded. perpetually

explainsa curious problem which must analysis about have troubled most peoplewho have endeavoured to philosophize Geometry. Startingfrom one-dimensional magnitudesconnected with tho"e the straight line,most theories may be divided into two classes, and volumes, and those appropriate to angles appropriateto areas 170.

*

into

See n

above

Part IV,

Chap. xxxi.

equalparts, itnd forms

This axiom

part

of Du

asserts

Bois

Reymond's

Bettazzi,Teoria delle that,givenany

exceeds

the

greater.

t Called Strecke

a

magnitude can definition

be divided

of linear

tude. magni-

(Tubingen,1882),Chap. iy " 10 ; also of Archimedes Grmidezze (Pisa,1B90),p. 44. The axiom two finite multipleof the lesser magnitudesof a kind, some

See his AHgeineiue Fimctionetitheorie asserts

that

by Mainong, op. ciL3 e.$*

p. 22.

[CHAP,xxi

Quantity

182

different planes. Areas and volumes are radically -which hold and are from angles, generallyneglectedin philosophies to relational views of space or start from Geometry. The protective of this is plainenough. On the straight line,if,as is usually reason have two such there is relation as distance,we a cally philosophisupposed, the distinct but practically distance, magnitudes, namely conjoined of the stretch. The former is similar to angles and the divisibility ; the and volumes. to areas latter, Angles may also be regardedas distances in a series, between terms namely between lines through a point or planesthrough a line. Areas and volumes, on the contrary,are sums, or Owing to the confusion of the two kinds magnitudesof divisibility. and of magnitude connected with the line,either angles, else areas or with the philosophyinvented to volumes, are usuallyincompatible is at once this incompatibility suit the line. By the above analysis, between

lines

or

and explained

thus

We

171.

distances

and

overcome*.

what

cover practically

convenient

will be extend

to

name

amenable

to

continue

to

allow

extensive the

all distances and

cover

These

measure.

classes

two

magnitudes,and

I shall

them.

to

name

it

whether divisibilities,

they

But the word extensive any relation to space and time or not. not be supposedto indicate, as it usually does,that the magnitudes

have must so

this

"

usuallycalled

are

to

great classes of magnitudes divisibilities

two

rendered

are

"

how

see

divisible. We

are designated

divisible. Quantities are of wholes which

case

have

that

no

magnitudeis

in the onlydivisible into other quantities of divisibility. Quantitieswhich quantities

are

distances,though

I shall call them

smaller distances

but

;

alreadyseen

extensive,are

they allow the

not

importantkind

one are

divisible into of addition

plained ex-

above, which I shall call in future relationaladdition f. All other

called intensive* magnitudesand quantities may be properly unless causal of some or relation, Concerningthese, by means by some less roundabout

relation such

those

at the beginning explained of the present chapter, is impossible. numerical Those measurement mathematicians who are accustomed to an exclusive emphasis numbers, on think that will not much be said with definitenessconcerning can magnitudes of the is This, however, incapable measurement. by no means The immediate case. saw) judgments of equality, upon which (as we all measurements still w here measurement a re as fails, possible depend, also the immediate are judgments of greater and less. Doubt only or

more

arises where

*

In

the

Part VI,

difference is small

we

shall find

is stilla distinction between such

as quantities

form

a

areas

one-dimensional

t Not

to

and

;

and, all that measurement

does,

deny distance in most spaces. But there and of the terms of some stretches, series, consistingvolumes,where the terms do not, in any simple sense, reason

to

series.

be confounded

It is connected

as

with

the relative addition of the

rather with relative multiplication.

Algebra of

Relatives.

171]

170,

in

this

Measurement

which

Quantities

We

a

scale

third

magni

magnitudes

of

is

pair

another

the

whole

which

will

of

is

in

merged us

occupy

the

subject

for

which

the

might

at

of

wider

length

What

question

is been

of

The derived far

chief from

more

the

is

correlation

its

of

why

traditional

summarily

treated.

I

and

ment, measure-

added,

fact,

of the

to

all

have

the

from

one

theoretically

reason

one

the The

numbers.

is, in

quantities

importance.

hereafter.

have

of

assignment

is

correlated

at

precise

numerical

nothing

"

as

magnitudes

of

of

pair

though

just

are

that

difference

the

as

Without

capable

measurement

much

thus it

the

of

propositions,

relations

are

by

same

approximate, Arithmetic.

of

they

the

such

and

(theoretically,

difference

the

or

And

appear

theoretical

than

than,

the

differences

the

always

the

ranged ar-

is

this

another,

since

is

whether

quantitative

standpoint, subject

it

may

propositions

of

practical in

same

the

theoretical

kind.

also,

be

measurement.

than

greater

there

less

than,

and

numerical

even

them;

question

greater

therefore, definiteness

is

importance. thus

can

magnitudes, of

achievement

an

philosophical

no

smaller

magnitudes, the

they as

and

between

the

of

mathematician definite

smaller

measurement

magnitude

one

to

of

achievement

always

are

answer

an

doubt

numerical

greater

intermediate

is

least)

of

that

tildes

and

of

quantitative

know

can

of

margin

psychological, susceptible

a

strictly

the

"

not

in

make

to

purely

is

only

Is

respect,

183

of

more

important series, have

importance,

which treated but

CHAPTER

XXII.

ZERO.

THE

172. numerical of

yet

nor

zero,

magnitude.

points

out

that

immortalityof

that, though can

zero

while zero

the

soul*. of

remaining is

exist.

a

definite

This

Kant

the

same

magnitude,

kind

of zero, we of the points

quantitativenotion, and is one quantity presents features peculiarto

fundamental

itself. The theory of both connection with the number certain 0 and has a quantitativezero with the null-class in Logic, but it is not (I think) definable in terms of realized is its complete independence either. What is less universally

in

which

a

the

magnitude,

and

;

of

magnitude is

quantity whose

shall find, is

zero

proof

zero

concerned, not with any form of the but with the pure infinitesimal, zero Kant which has in mind, in his refutation

is

the

intensive

an

become

kind, can

with

is the

This

Mendelssohn's

of

no

present chapter

of the

the

infinitesimal.

latter

The

notion

will not

be

discussed

until

the

followingchapter. The meaning of zero, in any kind of quantity,is a questionof much be bestowed, if contrawhich the greatest care must dictions difficulty, upon Zero seems to be avoided. to be definable are by some general of the kind of characteristic,without reference to any specialpeculiarity quantityto which it belongs. To find such a definition,however, is far from easy. distinct conceptionaccordingas Zero seems to be a radically discrete or that the magnitudes concerned continuous. To are prove various suggesteddefinitions. this is not the case, let us examine 173. Herr the con(1) tradictory as Meinong (pp. dt^ p. 8) regardszero kind. of each its of The opposite magnitude phrase is not which free from ambiguity. contradictory opposite11is one The opposite of a class,in symbolic logic,is the class containing all "

individuals of

an

not

individual

belongingto should

be

the

first class; and

all other

individuals.

hence But

the

this

opposite

meaning

is

evidentlyinappropriate: zero is not everything except one magnitude of its kind, nor yet everything except the class of magnitudes of its kind. It can hardly be regarded as true to say that a pain is a zero *

Kritik

der

Reinen

Vernunft, ed. Hartensteia, p.

281

ff.

172-174]

Zero

pleasure.On and

this is

the other

shall find this view

as

It

friends

our

to

seems

this

be

to

is said to be no pleasure, pleasure But althoughwe Meinong means. correct, the meaning of the phraseis very not mean somethingother than pleasure, zero

Herr

that it is no

us

is neither a

be

way

of

saying nothing,and

magnitude,and

meaning of zero, then zero is not one nor yet a term in the series formed

faults.

our

yet anythingelse. But

wholly dropped. This

for all kinds of

same

pleasureto tell us

nor pleasure,

cumbrous

merely pleasuremight

is the

which

assure

what

mean

would

reference

be

to

seize. It does

difficult to when

hand, a

what evidently

185

the

among

the

gives a

if this be

zero

the

magnitudesof

a

true

kind,

by magnitudes of a kind. For though nothing smaller than aU the magnitudesof a kind, yet it is always false that nothing itself is smaller than all of them. This zero, therefore, reference has no special of kind to any particular the of fulfilling magnitude,and is incapable functions which Herr Meinong demands of it *. The however, phrase, shall see, is capable of an interpretation which avoids this difficulty. as we But let us firstexamine other suggested some meaningsof the word. 174. (") Zero may be defined as the least magnitudeof its kind. kind of magnitudeis discrete, Where and generally a when it has what Professor Bettazzi calls a limiting magnitude of the kindf, such a definition is insufficient. For in such a case, the limitingmagnitude the least of its kind. to be really And in any case, the definition seems be sought which must givesrather a characteristic than a true definition, in some in some be for fail to cannot more zero notion, purelylogical denial of all other magnitudesof the kind. The sense a phrasethat is the smallest of magnitudesis like the phrasewhich De Morgan zero it is often

commends Thus

that

true

there is

for its rhetoric: "Achilles

it would

or integers,

be

was

the

strongestof all his enemies."

say that 0 is the least of the positive between A and A is the least interval

false to obviously the

that

interval

alphabet. On the other hand, where a kind of magnitudeis continuous, and has no limiting although magnitude, have apparently and unlimited approachto zero, yet now a we a gradual such as arises. Magnitudesof this kind are new essentially objection

between

any

have

minimum.

zero

no as

letters of the

Hence

their minimum.

sayingthat zero,

two

there

is

unless that

We

may,

always

other be

express contradiction take however, avoid this contradiction by

cannot

we

a

without

magnitudeless than

zero.

This

emendation

any

other, but

avoids any

not

formal

of because a onlyinadequate else is a magnitude of the kind than its true meaning. Whatever zero in question might have been diminished ; and we wish to know what it This of any further diminution. is that makes zero incapable obviously definition does not tell us, and therefore, the suggested though it givesa

and contradiction,

it givesrather

is

*

See note

to

Chap, xix,

supra.

t Teoria delle Grandezze, Pisa,1890, p. 24.

mark

characteristicwhich often be

cannot

than

other

no

zero

(3)

so

Where

these regarding

from

there less

as

our

to have

definedseems

rather than another a

where

sufficient. Moreover, philosophically

considered

kind,it

zero magnitudesare differencesor distances, But here again, firstsight, obvious meaning, an namelyidentity.

has,at

for

the

of magnitude

zero.

175.

as

to belongs

it precludes us negativemagnitudes,

are

the

[CHAP,xxn

Quantity

186

distance in space. This can, however, be member with some identity simply,identity zero

kind of distances

one

distance in time would

zero

a

:

relation to

no

to be

seem

the

same

avoided, bysubstituting, class of terms

of the

hold. By this device,the zero question of any class of relations which are magnitudes definite is made perfectly and have both zero quantities and free from contradiction ; moreover we for ifA and B be terms of the class which has distances, zero magnitudes, This with B are distinct zero quantities*. with A and identity identity be clear. And yet the definition must is thoroughly case, therefore, that zero has some : for it is plain meaning,if logical rejected general for all classes of only this could be clearly stated,which is the same the same distance is not actually quantities conceptas ; and that a zero identity. 176. (4) In any class of magnitudeswhich is continuous,in the between which

of

sense

the distances in

havinga

between

term

introduce

any two, and in the manner

we magnitude,

can

obtained from

rational*?.Any collectionof

zero

less than allof them. This magnitudes and can actually be small as we please, contain

no

also has

limiting

no

in which real numbers

are

defines a class of magnitudes be made as classof magnitudes can made

to

i.e.to be the null-class,

if our collection for instance, (Thisis effected, magnitudesof the kind.) The classesso defined form a at all.

members

consists of all

which

and in this magnitudes, original the first term. Thus takingthe series the null-classis definitely new the null-classis a zero classesas quantities, quantity.There is no class finite number of members, so that there is not, as in a containing

related series, closely

the series of

to

to Arithmetic,a discrete approach

the null-class;

on

approachis (inseveral senses of the word) continuous. defining zero, which is identicalwith that by which the is

and introduced,is important,

presentwe of

may

observe

that it

makes

and magnitude,

the

This

it not

one

among

the

zero

the

same

of

method

real number

will be discussed in Part V.

againmakes

contrary,the

But

zero

for the

for all kinds

whose magnitudes

zero

it is. in this question, to face the problem compelled, of negation. No pleasureis obviously to the nature as a different when these terms are taken strictly as even concept from "no pain," that and painrespectively. It would seem denials of pleasure no mere

177.

(5)

We

are

"

"

"

*

On

this point, however, see

" 55

above.

174-178]

Zero

187

*

the various magnitudesof pleasurehas the same relation to pleasure as pleasurehave, though it has also,of course, the specialrelation of negation. If this be allowed,we see that, if a kind of magnitudesbe defined by that of which theyare magnitudes, is one then no pleasure the various of magnitudes pleasure.If,then, we are to hold to among all that our axiom, pairsof magnitudesof one kind have relations of shall be compelled to admit that zero is less than all other we inequality, magnitudesof its kind. It seems, indeed,to be rendered evident that this must be admitted,by the fact that zero is obviously not greater of has a than all other magnitudes its kind. This shows that zero with

connection

less which

adopt this

it does

shall

we no theory, but above, given and strictlyand merely7/0 distance,

of

it would

Thus

that

seem

with

greater.

longeraccept the clear and

distances

zero

have

not

shall hold that

we

is

Herr

a

if

And

we

simpleaccount distance is

zero

onlycorrelated with identity. Meinonglstheory,with which we

began,is

the above correct; it requiresemendation,on substantially that a zero view,onlyin this, magnitudeis the denial of the defining not the denial of any one concept of a kind of magnitudes, particular shall have to hold that We magnitude,or of all of them. any concept which defines a kind of magnitudesdefines also,by its negation,a particular magnitudeof the kind,which is called the zero of that kind,

and is less than benefit

a

kind of

relation which

of which

would

and magnitude,

we

we

made

the various

allowed between

a

And

magnitudewas not generis;there

be sul

be in most

we

between

reap the

now

the

defining

magnitudesof the kind.

magnitudeand particular

identified with

it is a

held to

was

of the kind.

members

of the absolute distinction which

concept of The

all other

is thus

no

that

but the class-relation,

contradiction,as

in supposingthis relation theories,

to hold

there

between

no

and pleasure, between no distance and distance. or pleasure the zero 178. it must be observed that no pleasure, But finally, denial of pleasure, and is not magnitude,is not obtained by the logical the of n otion On the not the same as logical contrary, no pleasure. is a pleasure essentially quantitative concept, having a curious and intimate relation to logical denial, justas 0 has a very intimate relation relation is this,that there is no quantity whose to the null-class. The is the null-class*. magnitudeis zero, so that the class of zero quantities of The zero of any kind of magnitude is incapable that relation to existence or to particulars, of which the other magnitudesare capable. But this is a synthetic to be accepted onlyon account of its proposition, self-evidence. The zero magnitudeof any kind,like the other magnitudes, is properly but is capableof specification by means indefinable, speaking of its peculiar relation to the logical zero.

*

This

distances.

must

be

appliedin

correction

of what

was

formerly said

about

zero

CHAPTER

179.

ideas

all mathematical

ALMOST

of difficulty

AND

INFINITESIMAL,

THE

INFINITY,

XXIII

CONTINUITY.

present

one

great difficulty:

as usually regardedby philosophers an are antinomy,and as showing that the propositionsof mathematics this received From I not metaphysically true. compelled to opinion am dissent. Although all apparent antinomies, except such as are quite of logic,are, easilydisposedof, and such as belong to the fundamentals of infinite in my opinion,reducible to the one number, yet this difficulty itself appears to be soluble by a correct difficulty philosophyof any, and been generatedvery largely to have by confusions due to the ambiguity in the meaning of finite integers. The problem in general will be of the present chapter is merely to discussed in Part V ; the purpose show that quantity,which has been regarded as the true home of infinity, and this the infinitesimal, in must continuity, give place, respect, to of the difficulties which arise in regard to order; while the statement ordinal and arithmetical, quantitycan be made in a form which is at once reference to the specialpeculiarities but involves no of quantity. three 180. The the infinitesimal, and problems of infinity, tinuity, conin connection with related. as they occur quantity,are closely

the

None

of

them

infinity.This

can

be

is

fullydiscussed

at

this

stage, since all depend

order,while the infinitesimal depends also upon number. essentially upon The considered more question of infinite quantity,though traditionally than that of zero, is in realityfar less so, and formidable might be brieflydisposedof, but for the great devotion by commonly shown which call I shall the axiom of finitude. to a proposition philosophers kinds of magnitude (for example ratios,or distances in space Of some and time), it appears to be true that there is a magnitude greater than is,any magnitude being mentioned, another any given magnitude. That found which from be is can greater than it. The deduction of infinity this fact is,when .fiction to facilitate compression correctly performed,a mere in the statement of results obtained of limits. method by the

the Infinitesimal, and Continuity 179-181] Infinity, class

Any

arise :

of

u

magnitudesof be

(1)There

and this be such

may class of terms

new

but class,

a

a

kind

our

class of terms have

may

it may

have

being defined,three

cases

may

greaterthan any of our class u, smallest member; (%) there may

a

smallest member

no

189

(3) there may

;

be

magnitudeswhich are greaterthan any term of our class u. Supposing kind of magnitudesto be one in which there is no our greatest will arise the class where contains a finite u always magnitude,case (2) no

number

of terms.

condensed has

and

no

greatestterm this is

be

to

admit

are

of the

device,and

such.

there is

that

such.

many

capableof

is called

infinite

an

condensed

series is not

in

class, itself,

kind

a

an

of

magnitude

measurement,

Thus,

of which so

(3).

case

To

avoid

cumlocuti cir-

limit is infinite.

the

ticians admitted generally by mathemaspecialcircumstances,there is no to magnitudeshas no maximum,

magnitudesof

numerical

in

it is

infinite

Archimedes, in virtue

connected

is

u

that in which

as

from

Apart

When

kind is finite.

our

limits,except

(3) is defined

merely because

reason,

series be what

our

arise when

never

if

if

any two, another which has this property from it*. Thus all infinite series which have

will have

mere

a

hand,

between

term

a

alwaysbe obtained

But

and

greatestterm;

no

case

are

(") will

case itself^

in

but does have can

the other

On

a

kind,

of the

kind

they very

having often

the ratio of any

far,there might appear

two to

or

that there maximum

no

obey the axiom magnitudesof be no problem

with

infinity. is apt to step in,and to declare pointthe philosopher that,by all true philosophic principles, every well-defined series of terms have a last term. If he insists upon creating this last term, and must from which deduces intolerable contradictions, it infinity, he easily calling he infers the inadequacy of mathematics For to obtain absolute truth. the To I axiom. for show, philosophers my part,however, see no reason if possible, let take underthat it is not a necessary philosophic principle, us its analysis, involves. and see what it really it has now The as emerged,is not properlya problem of infinity, quantitativeproblem,but rather one concerningorder. It is only because our magnitudesform a series having no last term that the problem arises : the fact that the series is composed of magnitudesis whollyirrelevant. With this remark I might leave the subjectto a But

at

this

later stage. But the

it will be worth

axiom philosopher's

while

now

to

if not elicit,

to

examine,

of finitude.

place,to show how the problem and the that concerning is the same as continuity concerninginfinity infinitesimal. For this purpose, we shall find it convenient to ignorethe when we absolute zero, and to mean, speak of any kind of magnitudes, of the kind except zero. This is a mere all the magnitudes change of 181.

It will be

*

This

well,in

the

will be further

first

explainedin

Part

V, Chap, xxxvi.

Quantity

190

diction,without there

which

are certainly

hold

axioms

intolerable kinds of

some

[CHAP,xxm

would repetitions magnitude where

be necessary. the three

Now

following

:

(1) If A and B be any two magnitudesof the kind, and A is greater than J5; there is always a third magnitude C such that A is greaterthan C and C greater than B. (This I shall call,for the of continuity. the axiom ) present, (2) There is alwaysa magnitudeless than any givenmagnitudeB. (3) There is alwaysa magnitude greater than any given magnitude A. these it follows

From

:

"

magnitudesof the kind are consecutive. That there is no least magnitude. (3) That there is no greatestmagnitude. kinds of magniThe true of some above propositions are certainly tude all of remains kinds whether be true examined. The to are they ; which directly contradict the previous three propositions, three, following of finitude is to be axiom be always true, if the philosopher's must (1) (2)

That

two

no

accepted :

(a) no

There

magnitude of

other

greaterof the

the

than

(6)

consecutive

are

There

(c) There

is

a

is

the

magnitudes,i.e. magnitudes such kind is

same

greaterthan

the less and

that less

given magnitudes. magnitudesmaller than any other of the same kind. a magnitude greater than any other of the same two

kind* As

would

these three seem

that

contradict propositions directly both

be

sets cannot

We

true.

the have

previousthree,it to

examine

the

groundsfor both, and let one set of alternatives fall. Let us begin with the propositions 182. (a),(6),(c),and examine of their nature the grounds. definite A (a) magnitudeA beinggiven,all the magnitudesgreater whose differences from A are magnitudesof a new than A form a series, If there be a magnitudeB consecutive to A^ its difference from A kind. will be the least magnitude of its kind, providedequalstretches correspond to equal distances in the series. And if there be conversely, between 'difference two smallest these two then a magnitudes,A, 5, magnitudesmust alwaysbe consecutive; for if not, any intermediate *

Those

Hegelians who

search

for

chance

of

antinomy may proceed to (2) propositions.When the and (b)both hold, they may magnitude satisfyingsay, (b)is called zero ; when (3)and (c)both hold, the magnitude satisfying (c)is called infinity.We have seen, that is be to otherwise defined,and has to be excluded before (2) ssero however, becomes is not a magnitude of the kind in questionat all,but true ; while infinity in general,that is,but merely a piece of mathematical shorthand. (Not infinity infinite magnitude in the cases we are discussing.) the definition of

zero

and

by infinity

a

means

of the

an

above

the Infinitesimal, and Continuity 182] Infinity,

181,

magnitudewould

have

smaller difference from

a

A

if (b)is universally true, (a) must also be true ; and true, and if the series of magnitudesbe such that to

We

(a) to (b\ and proceedto to offer

direct

a

proof,such

has.

Thus

if (a)is conversely, respond equalstretches cor-

of the distances between

then (b)is true equaldistances,

magnitudesconsidered.

B

than

191

might rest the proof of (b); as presumablythe

but

it

the

the reduction

with

content

worth

seems

of

while

has in finitist philosopher

his mind. and B

unless A magnitudes, all have and order,so are magnitudes that in passingfrom A to B all the intennediate magnitudeswould In such an be met with. enumeration, there must be some magnitude wise, othernext after any magnitudeC ; or, to put the matter which comes it must begin somewhere, and since the enumeration has to begin, with which it beginsmust be the magnitude next to A. the term If A

Between

this

have

In

the

number.

an

of many

terms

must

number and

of

be consecutive.

namely: self-contradictory,

contain

such

and

B

magnitudes,A

and

B

of

If there

them, say

to assigning

al*e {;/z+l)th

If the axiom

then

such

are

Since

n.

them

the

of

jirntenumber

some

A

is decided.

way

be

to

magnitudesbetween no

definite series ; for if all the

no

argument, what is importantis its dependenceupon The whole argument turns upon the principle by which infinite

is shown

are

of

above

number

the

certain number

a

intermediate

The

the case, there would be of them must order,some

not

were

terms

there is

consecutive.

B

form are

a

A

given collection We

terms.

say:

given collection.

consecutive,and

the

ordinal numbers

from

1 to

is n.

If there

question

be

magnitudes,there must they form a series,there

All

a

a

finite

definite The

mth

consecutive.

in italics be

denied,the whole argument collapses ; and shall find,is also the case as regards (b)and (c).

this,we (b) The proofhere

is

similar precisely

to the

proof of (a). If there least of its kind,and the no are A, then A is there decided. If form definite collection, are a question any, they finite and therefore (by our have Since they a axiom) number, say n. ordinal numbers form a series, may be assignedto them growinghigher the magnitudesbecome more distant from A. Thus the nth magnias tude

magnitudesless than

is the

is the smallest of its kind.

of

The

proofhere is obtained magnitudesgreater than A. (c)

as

in

the collection (6),by considering

Thus

everythingdepends upon our out againstcontinuity, or absence of the and least a greatest magnitude against the axiom itself, As regards that it has no particular it will be seen reference to quantity, and at first sightit might seem to have no axiom, without

which

reference to order.

no

But

case

can

the

word

be made

jimte^ which

in

it, requires in the form suited to the presentdiscussion, definition;and this definition, has, we shall find,an essential reference to order. occurs

Quantity

192 183.

"11 the

Of

number, I doubt between Finite

whether

finite and numbers

numbers

do

to

class

obey

is to

which

after any

inveighedagainstinfinite has

who

one

of

law

the

That to

s

is

there

have

infinite numbers.

not.

every number next

who philosophers

[CHAP,xxm

The

mathematical

which

is

if

n,

which

,?, then

an

.simplythis.

induction; infinite

number

given any belongs,and to

number

the difference

difference is

say,

0

known

belongs

n

belongsalso the is finite ; if not,

n

It is in this alone',and in its consequences, that finiteand

not.

numbers

infinite

differ*. be otherwise stated thus : If every proposition principle may holds concerning 0, and also holds concerningthe immediate of every number holds concerning of which it holds, the number

The

which successor

n, then

This

is finite ; if not, not.

n

is the

of what

sense precise

may be be reached

by sayingthat every finite number can popularlyexpressed from 0 by successive steps, or by successive additions of 1. This is the must be held to laydown as obviously which the philosopher principle to all numbers, though he will have to admit that the more applicable is stated,the less obvious it becomes. his principle precisely It may be worth while to show 184. exactlyhow mathematical Let the take the proofof (a\ above proofs. induction enters into us Then to begin and suppose there are n magnitudesbetween A and B. with, we supposedthese magnitudescapableof enumeration,Le. of an there

order in which

consecutive

are

immediatelyprecedingany dispute.Hence

we

be

which would

must

series of

if it

was

presuppose

first. This

numbers

term, and

belongsto

property presupposes

to

the

must

be

definite way

a

havingm + 1 terms, Hence, by mathematical

havingm terms. havinga finitenumber of terms

kernel of the

series

every

belongsto all series

if it be allowed that the number

term

This property belongsto

terms.

belongsto

series

every

the

to

a

in fact the very property in the possibility of enumeration,

But to come prlncipii. petitio there supposedthat,in any series,

one

it induction,

not

firstterm, and

a

a

argument : we of assigningordinal a

except the

term

induction,and

mathematical

and

terms

may

not

be

But

of terms.

the whole finite,

argument collapses. As a

regards(b)and (c)9the argument of terms

finite number

have

a

firstand last term

be shown

can

;

but

no

series,or of provingthat induction,in short,like the axiom other

in its proper

*

place;

finite and

definition. This has been be further

of

discussed in Part V.

that

infinite

by

Every series having

mathematical

way exists of all series are

but' to suppose

It must, however, be mentioned

difference between

is similar.

induction

to

provingthis concerning finite. Mathematical

is useful and convenient parallels, it alwaystrue is to yield to the

one

of these

numbers,

which

consequences may

be

taken

givesa logical as

alreadyexplainedin Part II,Chap, xm,

an

pendent inde-

and will

the Injimtedmal, and Continuity 183-185] Infinity,

tyranny of rest to

of principle

a

and affirm,

antinomies 186.

The philosophers finitistarguments,therefore, prejudice.

mere

on

193

every

which he is ignorant, which to

reason

deny.

With

this

there is no

the conclusion,

reason

apparent

be considered solved.

may It remains

consider what

to

kinds of

the magnitude satisfy from which (1),(2), (3). There is no generalprinciple propositions be but there are certainly these can where cases provedor disproved, and others where It is generally held by theyare true, theyare false. that numbers while are discrete, philosophers essentially magnitudesare continuous. This shall the find to be Real not we case. essentially numbers possess the most complete known, while many kinds continuity of magnitude possess no has at all. The word continuity continuity in but mathematics it has onlytwo one old,the other many meanings, For old new. presentpurposes the meaning will suffice. I therefore the following definition: set up, for the present, whenever these are Continuityappliesto series (and onlyto series) such that there is a term between any two giventerms*. is Whatever whatever is a series not fulfilling not a series, a compound of series, or or "

the above

is discontinuous. condition, the series of rational numbers

Thus

of two

mean

of them

letters of the

The

have

We

distances

A

and

or

the

between

continuous,for the arithmetic

is alwaysa third rational number

alphabetare that

any

not

two

between

the two.

continuous. in

terms

a

series have

a

distance, or

a

discrete series magnitude. Since there are certainly discrete magnitudes, there are certainly alphabet), namely,the

stretch which the {e.g.

seen

is

J?,but

has

the

stretches

letters A there is

and no

of terms C

is

in discrete series.

greaterthan

magnitudewhich

The

is greaterthan

distance the letters

that between one

of these

and less than the other. and

a

In this case, there is also a greatestpassible least possible that all three propositions so distance, (1),("),(3)

fail. It must

be

supposed,however,

that the three

propositions for example, of the integers, have any necessary connection. case is and there least consecutive distances, there are a distance, possible is consecutive but there that between no integers, greatest namely, not

In the

(3) is true, while (1) and (2) are false. In of the series of notes, or of colours of the rainbow,the series the case and end,so that there is a greatestdistance ; but there has a beginning Thus (1) least distance, and there is a term between any two. is no series is false. take Or if the and (2) are true, while (3) again, we

distance. possible

composedof

zero

Thus

and the fractions havingone

for numerator, there is a

shall see in Part V) is,that it does not to this definition (aswe objection of the existence of limits to convergent series which are give the usual properties commonly associated with continuity.Series of the above kind will be called *

The

compact, except in the present discussion.

[CHAP,xxm

Quantity

194

least distance, but no though the series is discrete. greatestdistance, Thus (2) is true,while (1)and (3) are false. And other combinations

might be

obtained from other series.

(1),(2),(8),have propositions

Thus the three and

connection,

necessary

no

to any given them, or any selection, may be false as applied to prove their truth magnitude. We cannot hope,therefore,

all of

kind of

magnitude. If theyare ever to be true,this must be in each particular discovered by mere or inspection provedindependently, That they are sometimes case. true, appears from a consideration of from the nature

of

the distances between numbers. has or

no

Either firstor

of the number-continuum

terms

of these series is continuous

last term

(when

is

zero

the wholes which

parts.

the number

where

But

do not divisibility

divisible consist of

are

of

sense, and

its distances

might be inferred what is to be said anticipate

The

time, but I do not wish to

space and of of these. Quantities

in the above

excluded), Hence

stretches fulfilall three conditions.

from

of the rational

or

a

same

fulfilthese conditions when of indivisible

finite number

parts is infinite in

whole

a

class of

all three conditions are satisfied, as magnitudes, differing appears from of the number-continuum. the properties and continuity We thus see that the problemsof infinity have no but are essential connection with quantity, due, where magnitudes

presentthem order.

at

Hence

characteristics depending upon

all,to

the discussion of these

can problems

We

Chapterxix

may we

now

sum

determined

and

onlybe undertaken

after the pure theoryof order has been set forth*. Part. be the aim of the following 186.

number

To

do

this will

up the results obtained in Part III. In to define a magnitudeas whatever is either

less than

somethingelse. We found that magnitudehas no and that greater and less are indefinable. necessary connection with divisibility, relation has certain we a to, analogous Every magnitude, saw, but not identical with,that of inclusion in a class to a certain quality relation ; and this fact is expressed or by sayingthat the magnitude is a magnitudeof that quality defined a relation. We in question or contained under a magnitude, I.e.as the complex as a particular quantity of a magnitudewith a certain spatio-temporal or with consisting position, of between it of is which relation. We terms a a pair decided, by means a generalprinciple concerningtransitive symmetrical relations,that it is impossible to content ourselves with quantities, and deny the further abstraction involved in magnitudes; is not a direct that equality relation between quantities, but consists in beingparticularizations of the same instances of the same are magnitude. Thus equalquantities greateror

"

"

*

Cf.

Couturat,"Sur

Morale, 1000.

la

du Cantmn" Definition

Revue

de

et Metapfiysique

de

185,

the Infinitesimal, and Continuity 186] Infinity,

195

magnitude. Similarly greater and less are not direct relations between but between quantities, magnitudes: quantitiesare only greater and less in virtue of being instances of greater and less magnitudes. Any two magnitudeswhich are of the same relation or are one quality greater,the

less ; and

other

greaterand less are

asymmetricaltransitive

relations. the terms

Among also

but

which

magnitudeare

asymmetricalrelations

constituted. These

may

series, any

terms

in

have

two

which

by

not

certain

be called distances. of the series have

onlymany kinds

When

qualities,

of series

there

are

are

distances

which distance,

is the same less than, the distance of any two other terms in the as, greaterthan,or class of magnitudesdiscussed in Chapterxx is series. Another peculiar a

a

of different wholes. by the degreesof divisibility This, we while there is no found, is the onlycase in which quantities are divisible, instance of divisible magnitudes. Numerical measurement, which was discussed in Chapterxxi, required, and all magnitudesare indivisibl owing to the decision that most quantities somewhat unusual The treatment. a problem lies,we found, relation between numbers in establishing and the magnitudes a one-one On certain metaphysical of the kind to be measured. (which hypotheses this was found to be alwaystheoneither accepted retically nor were rejected), actual as or regardsexistents possible possible, though often feasible In classes of to not two or important. regard practically measurement found was namely divisibilitiesand distances, magnitudes, defines what is to proceed from a very natural convention, which meant by saying(what can never have the simplesense which it has in with finite wholes and connection parts)that one such magnitude is

constituted

double

of, or

n

times, another.

and discussed, there was that effect,

was

it

a

priori

reason

to equalstretches. corresponding discussed In Chapter xxn we

of

was

zero

found

being as

to

the

nature

in

to

fact

of

have

for

the definition of

related closely We

to

with

the

decided

stretch

axiom special regarding equaldistances

connection

no

negation.

relation of distance to

that, apart from

found

was no

The

a

zero.

that

The

to as

problem

of the

tesimal, infini-

purelylogicalproblem

that,justas there

are

the

third fundamental and arithmetical negations, so a logical that is this but negation of kind, the quantitative negation; relation of which the magnitudes that quality or are, not of magnitude relation. Hence able to regardzero as of that quality we were one or and to distinguish the magnitudescontained in a kind of magnitude, among that of different kinds. We showed also the zeroes quantitative that there fact with the cannot negationby logical negationis connected whose magnitudeis zero. be any quantities the infinite, and In the presentChapterthe problemsof continuity, to the theoryof the infinitesimal, shown to belong,not specially were distinct

there is

quantity,

but

there

kinds

are

magnitudes

;

magnitudes

The

in

those

to

of

to

which,

supposed

and

does

form

a

not

is

series

in there

consequently, contradiction

was

induction

mathematical

a

"

the

philosophy

of

us

to

no

which

there

is

term

no

shown

to

the

shown

was

infinite

admit

in is

a

consecutive result

full

and

from discussion

an

least

kind

a

between to

though no

infinitesimal

or

supposing

term

xxm

that,

greatest

no

contradiction

principle, order.

is

there

require

there

that

which

in

It

order.

and

number

of

magnitude

fact

this

magnitude,

and

[CHAP,

Quantity

196

two,

any a

given

undue of

which

of

term. use

of

supposes pre-

PART

OEDEE.

IY.

CHAPTER

THE

THE

187.

XXIV.

GENESIS

of

notion

SERIES.

OF

order

series is

or

with

one

in connection

which,

"

had

deal.

to

prepared us time

for

examine

to

order, from increased

a

by

which

"

shall

properly importance of its

very

the

upon is

view

obtained.

are

and

the

understanding of 188.

The

order

the

while

which as

fundamental discussion to

more

immeasurably and

Cantor,

Peano

series of

a

order to

and

of

descriptiveGeometry

Thus

is

to

almost

of

than

order, and

an

this

that

order, which to

any

I

this

such ^topics,

as

three

the

logical analysis approach circumstances

second

chapter

analysiswill raise

the

cannot

therefore

chapter,the

reserving for This

shall

hitherto

any

even

complexity, the

reallyis. will general logic, which nature. purely philosophical

mathematical

proves

essential

become

complex

difficulties.

arises, and in

discussion

a

more

have

Owing

order

lateral quadri-

s

of mathematics.

order

cannot

what

Geometry,

new

important

Staudf

von

a

of serial only the possibility and time depends philosophyof space

foundations

notion

an

been

Analysis upon

philosophies,has

current

the

points of

of

importance

demands

order.

of

considerable "presents

discussion

high

now

on

problem gradually,considering,in

under

pass

In

work

the whole

terms analyzed. Two order. have a cyclic

of

and

ProtectiveGeometry have shown planes an order independent of metrical

PierTs

take

we

in

lacking

It is

see) entirelyby the help of order; is introduced, by which the most

Moreover

arrangement.

and

of

chapter notion,

The

Dedekind,

all Arithmetic

considerations a

order.

account.

own

last

ordinal

standpoint,has

give points, lines,and and of quantity ; large part of Geometry

to

the

in an

already

have

we

ordinals

construction how

continuity

is

developments.

base

to

magnitude,

of

those i.e. upon in virtue propertiesof finite numbers Irrationals form what I shall call a jrogrcsstan.. are

interestingresults

and

of

this

fundamental

modem

transfinite

class of

order

this concept on purely mathematical

they (as we

defined

that

the

how

kind

certain

the

discussion us

many

shown

have

with

The

showed

III

Part

of

and

distance,

with

considerable

demand

this I shall

From

types

the

several

of

series

and

Order

200

numbers, thus gradually the way for preparing

the ordinal definition of

There

and continuity in infinity

of

the discussion

[CHAP,xxiv the

Part; following order may arise,though \ve

different ways in which end that the second way is reducible to the first. In may be called the ordinal element consists of three terms

two

are

shall find in the what first,

the

", c,

#,

there is

whenever

(b say)is

of which

one

of

relation

a

of

b is between

b and

followingseems

to

is

there

when

b arid c,

and

c

relation

and

or

a,

a

between

and

a

c

a

and

are

either between

d

similar

a,

between

and

a

c

arc

c,

the or

and

c

and

a

6, or

6,

between

first case, the

else between

same

both

a

assumptionsfor the

assumptionis special b and

and

where

have four terms

we

holds between

have

we

proposition where,at

but the complicated, from b and d, separated

more

rf,d and

if

;

a

definition, or

say that

can

we

is not

of order

where

cases

relation is

while

and b

d; with

between

or

c

d,

and

and c9 and a (No further

This

is the

cases

asymmetricalrelation which

hold

must

are

happens

c, which

and satisfied,

not

element, of which

d.

b to

This

a.

other

are

are

characterize it :

an

d^ d and

and

a

conditions

ordinal

the

c*,J, as

by separated

c

there

But

to

c

of

This

sufficientcondition, of the

applicable.These obviously

is not

^

c?

above

sight,the

first

#,

and

a

the other two.

b and

to

a

relation of b to #, of c to b, or the necessary and better perhaps, "

between

requiredas

to

other two cases*. the relation between

J; it is the absence of such

assumption

an

reducingthis case to the former in a simplemanner.) prevents There are cases notablywhere our series is closed in which it seem* to the first, to reduce this second case though this formallyimpossible in is shall have to show, part deceptive.We see, appearance, as we in series the principal which arise from in the present chapter, ways which

our "

"

collections of such

Although that

assume

terms

order all

In

terms.

two

two

ordinal elements.

is

alone

cannot

have

an

possible except where

there

shall

are

we series,

find,there

oixier,we are

must

not

relations between

asymmetricalrelations

asymmetricalrelation of which there is only one instance does not constitute order. We requireat least two of pairs. Thus instances for between,and at least three for separation although order is a relation between three or four terms, it is only where there are other relations which hold between pairsof possible be of various kinds,givingdifferent ways These relations may terms. with series. I shall now the principal enumerate of generating ways I am which acquainted. method of generating series is as follows. 189. a (1) The simplest

between

Let term

*

of

two

terms.

there be

a

This

gives

an

collection of terms, finite or

exceptionof possible

(withthe

couples.

But

a

sufficient but

not

a

a

such infinite,

singleone) has to

necessary

condition

for

one

the

that

and

every

only

separation

189]

The

other term

of the collection

188, one

of

must

be

course

also

to

the relation which

must

-not

and

one

of Series

certain

a

and intransitive),

which exception, possible

excepted)has

Genesis

201

relation (which asymmetrical

that every term (with again one be the same the term formerly as

only one

other term

of the collection

is the

of the former one*. converse Further, let has the first relation to ", and b to e, then c a the have first relation not to a. does Then every term of the collection the two has t erms relation to a second term, and one peculiar except relation to a third,while these terms the converse themselves do not

that, if

it be assumed

have

each other

to

The

third terms. the

which

term

given

has

Two

terms

between

the series, or

one

natural numbers

exist

one

;

second

and

of the

two

the term

to

relation is called next

Ixrfvre question (when they exist)

which

the relations in

The

terms exceptional terms; they are called

and beginning

the

two

ends

the other the end.

that of the other

of

The

for example imply but end and need neither no beginning and positive together have negativeintegers

have

example,the

for

"

has

term

our

afterthe given term

converse

does not

one

question. Consequently,

is between

given

a

between

any pair of is called the

of the

existence the

the

called consecutive.

not

are

which

to

term

given term. are

first term

questionis called next

relations in

hold

relations in

definition of between^our

by the

the

either of the

a

"

"

neither tabove

The

be

Let R

if

Then

method of

one

clear

and let its relations,

our

of

be any term

e

perhapsbecome

may

set, there

our

formal

by a

be denoted

converse

two

are

exhibition.

by J?J.

d,J\ such

terms

that

Since each term d, e Rf, i.e. such that d R e" e Rf. onlyhas the have d JRf; and it was relation R to one of other, we cannot one e

R

that initial assumptions

the

If

andf ". a obviously is not d

between notion

of

between

between

and

c

this way,

In

be

a

between

we a

not

were

which has

term

pair of

any

that, if by defining

e, then

unless

we

c

or

have

to

c

f R onlythe We

terms.

be

between

d.

Hence

an

end

come

or

is

relation Ry then the may extend b and d, and d

d will be said to be also between

either reach

e

back

b and

e,

to the term

started,we can find any number of terms between which we will lie. But if the total number of terms be not b the term c

with which and

less than must

one

*

when

The the

seven,

be between

converse

of

of any three terms the other two, since the collection may consist

cannot

we

a

is the notation

adoptedby

x

of

The

most

hold

between

y and

j?

and y.

generatingseries given by Bolzano,

tc

Para-

Professor Schroder.

denial ofdRfis onlynecessary to this of between. essential definition d to the is fR

"

that

relation which

relation is the

givenrelation holds between

f The above is the only method doxien des Unendliehen/* " 7.

J This

in this way

show

method, special

but the denial ot

Order

202

[CHAP,xxrv

of which, if the collection is finite, distinct series, one than two ends. be closed, in order to avoid more

of two must

remark

This

series,to which

any

We

be connected.

condition

ourselves

content

by

is to

collection is to

our

number, but

reference to

without

method

sayingthat

for the

of its terms, there is a any unique)of steps from one term

certain finite number

from

the

of

the

to

to

terms

two

our

next,

must

by

we

mav

are

when, given (not necessarily

which

When

other.

assured that, of any three terms be between the other two.

fulfilled, we

present

collection is connected

our

two

one

least

give a single belong,we need a by sayingthat the collection may be expressed shall find means hereafter of expressing this of

term

which condition,

further must

that, if the above

shows

at

we

can

pass

this condition is

of

collection, one

our

connected,and therefore forms series may have two ends, a may arise : (a)our (b)it may have one end, (c)it may have no end and be open, (d)it may 'have no end and be closed. Concerning("),it is to be observed that series must be finite. For, takingthe two ends,since the collection our finite number of stepswhich will take is connected,there is some n

Assuming now that singleseries,four cases

from

us

end

one

of the

series.

to

the

other,and hence

the other

our

hand, were

not

connected.

For

have the relation R" but not R. has both relations, and can never is asymmetrical.Hence

or

a

does not

have be

Then

to which

the term

the relation R reached

to

this term

contrary to

the

or

which which

successor

a

fc; and not.

hypothesisthat

term

any

In

hence

cannot

which

(say)e

hold

exists to

of the collection

term, since R

same

has the relation R is

case

be

can

R

or

series is open

"

a

a

have

one-one

it be any term without passing

predecessors,

two

relation.

successive

by any

a

of the

from i.e.^ starting

of steps in either direction be

would

is

a

can

steps from

reached

is not

Nor

Hence, if

steps from

terms

between

", a

the collection is infinite, whether it be connected (c\ the collection must again be infinite. For here,

the by hypothesis,

number

to the

(6),on

case

this would

the end

any term.

for if it were,

and

In

other term

both

have

through ",

has

them, and

every

successive

can

k

ends is between

two

suppose

by

which

be

of terms

the relation R, but is either some new term, it cannot of "?'$predecessors. Now be the end-term a, since

one

k

+ 1 is the number

it had

that to which

not

n

any other pair of terms. and collection must be infinite,

is between

if it

except the

Every term

neither of them

even

collection is

our

finitelimit to the number

were, the series would have an suppose the series connected.

end.

any

term

",

no

there

bringsus back to e. And of possible steps,since,if there Here again, it is not necessary to the

contrary,we must that connection. assume mean we By sayingthat the series is closed, certain exists number there of stepsby which,starting from a some n In

case

(d),on

The

190]

189, term

shall be

we

#,

brought back

it makes

of terms, and

(2)

190.

above

The

closed series, but or method, which is now

is the number

n

of

In this

start.

are consecutive, Otherwise,we need the

separation.

method, as

only such

have seen, will giveeither open The second

we

have consecutive terms.

as

be

will give series in which there discussed, consecutive terms, but will not give closed series*. In this

no

are

In this case,

a.

than three terms.

more

relation complicated

more

to

203

no

the series contains

and

of Series

difference with which term we definite except where three terms

between is not

case,

Genem

method

have

to

transitive

relation P, and a collection asymmetrical of which are such that either xPy or yPx. When of terms any two satisfied our conditions terms series. these form a single are necessarily Since the relation is asymmetrical, from we can distinguish xPy yPx, Since P is transitive, both subsist"}** and and the two cannot yPz xPy we

It follows that

involve xPz. Thus

with

a

respectto

term

any

Callingthese

zPx.

of

x

asymmetricaland transitive^. all other terms of collection,

our

those for which classes,

the collection fall into two which

is also

P

classes wvr

two

and

xPy^ and

for

those

see we respectively,

irx

the transitiveness of P, if y belongsto the class mr, is contained in TTX ; and if z belongsto the class TT^T, vz is contained *iry in TTX. Taking now two terms x9 y? for which xPyy all other terms fall

that, owing

to

belongingto

Those

into three classes:

(1)

(") those

iry, and therefore to THT If z be of the first class, we

belongingto

but not to

TTX

TH/.

and

of the second,xPv and

yPu

case

yPv

;

if

is excluded

uPx

Thus

with uPx.

for

:

the

two, and P

If for *

no can

whole

terms,

x

be

such that

are

a

a

are

there

and before^

manifold/' Mind, N.S.

in which

sense

none

our

and*

Vol.

other methods

and

are

;

i. are

We

The

is inconsiste

y.

Hence

the other class

the

(")

but many tions relain the class (8). in

moments

Vivanti in the Formulaire "On

be

V

is between

If the

alwaysterms

if

which

(1)x

collection be the

our

;

wPy.

is between

said to be consecutive

followingmethod is the only one given by Mathematiqmit,(18Q5),vi, " 2, No. 7 ; also by Oilman, of

one

TT^;

belongingto

zPx, zPy

singleseries.

The

dimensional

have

we have, in the three cases, and v ; (3) w is between x

for which be assigned,

example P

those

xPy, yPu implya'Pw,

collection forms and y

(3)

;

therefore to

and

be of the third,xPw

a?

x z and y ; (")y is between collection of three terms our any

contains

TTX,

of properties

shall find that it is

a

a

de one-

general in

so.

the contrary, rather than the contradictory, as asynmietricttl have always yPxi if If xPy^ and the relation is symmetrical, of symmetriml. we relations" e.g. lexicalimplication" are have yPx. Some asymmetrical,we never t I

use

the

term

P to be asymmetrical, symmetricalnor asymmetrical. Instead of assumingthat Professor what Peirce calls an is the it make equivalentassumption may has itself. to which term (This relation i.e. a no assumption is not afiorelative, with but combined when in to asymmetry transitiveness) only general, equivalent

neither we

J

P may

spatialideas

be read are

prtt"de*,and

allowed

to

P

intrude

may

be

read

themselves.

follows,providedno

temporal or

Order

204

time,-there is

in the Similarly

collection.

our

in all

or interval,

certain

last

chapterof Part

the

presentmethod,

of the

case

between

moment

a

any two

of

magnitudeswhich, There is nothing in in the

called continuous.

III, we there

as

[CHAP,xxiv

that there

to show first,

in the

was

must

of terms in our collection be consecutive terms, unless the total number will allow not closed be finite. On the other hand, the presentmethod series

for

;

and closed,

were

because impossible

is

generatingrelation

two

generatedby Part

y^z....

are

remainingterms are

and #, if

xz

"

is distance,

is less than

xz

than

number

xiv,

we

must

of other terms

be

be

wz

presentcase

the

if these axioms

independentof

the

the distance of

xz

the

A

sequence, con-

had only and asymmetricalrelations,

be considered "

order

can

similar relations between

was

x

have

to

are

have yz less than yw.

must

involve

the

fully

more

term

less than

is less than is

ww?

that

For

zero.

is less

i.e. wz

reduced practically

the

to

;

in the the

second

But

xPy.

case

(4?) Cases

192.

requireone

further

order.

Let

and so (y,"),between u and (#,TC?), this and relation, might therefore seem of o generatingrder. We should say, way

such

a

natural

z

is between z.

We

be

be the

between

there

a

relation R

which

holds between

and

x

and ", when should need

that,if y is between *

For

in order that

axiom

unambiguously. If xz=yw^ and same point. With this further axiom, (")becomes complete. of triangular relations are capableof givingrise to

w' must

the reduction to

we

be thus effected

arrangement may

zoand szE/="ry,

#,

we

#,

Denotingby

follow when

must

xw

a

see

was

for every pairof terms "r,y will be such that xy is less than 0 else xy is greater than 0 ; and we put in the first case yPx^ may

second or

"

shall

we

greater or less,we

order may

start.

we

sense

one

this way

In

0.

the

the distances must

that

have

which

that

not

in the

as distances,

requirenothing further. But kind, certain magnitudesof the same

which

did

as

and

w

no

are

of

means

we

is less than

which

those

are

from

term particular

...,

insure

to

necessary

a

2,

y,

relations which

have

x

If there

terms. corresponding

only,

and infinite,

certain term

a

between magnitudes, relations are Accordingas these

which relations, the

III, and

with starting

In this case,

In the first case

be

it may

case

method,

so.

be

in explained alreadypartially

hereafter.

in the former

As

none.

or

one,

in this

series may

A

(3)

191.

ends,or be

it must

cases

be transitive*.

never

can

the series may have two it may be finite; but even other

of its terms, we should have xPx^ which Thus in a closed series,the asymmetrical. one

any P is

were

x

relation P, if the series

of the

the transitiveness

owing to

x

more

y

and

Between

on.

the most in such

a

(x^ z\ is itself

direct and case, that y

the relation R holds between

y and the assumptionsconcerningjR which should and

2, and

z

between

precisestatements,see

y and

o", then y and

Chap. xxvm.

couple show z

are

The

190-194] each between

and

x

Genesis

of

is,if we

have

That

w.

Series

205

we yR(x, z\ zR(y, a?),

must

yR (x,w) and zR (#,w). This is a kind of three-term transitiveness.

have

Also if y be between x and a?, and z between y and tv, then z must be between x and "?, and y between x and 2: that is,if yR(x" w) and zR(y, w)) then ##(#, a?)and yR(x, z). Also yR(x" z) must be equivalent

yR(z" #)*. With generatedamong any

these

to

will be

Whether

the relation R.

an assumptions, unambiguousonier

number

such

state

a

is a questionwhich analysis,

further 193.

We

(5)

series.

have

found

There

of terms of

such that any triad has of thingscan ever be incapable

I leave for the next

hitherto

no

way

of

chapter. generatingclosed

however, instances of such series,e.g. the elliptic angles, straightline,the complex numbers with a given continuous

are,

It is therefore necessary to have

modulus.

theorywhich allows of In the case where our their possibility. terms are asymmetrical relations, lines are, or are correlated uniquelyand reciprocally with as straight t he such relations, following theory will effect this object. In other cases, the sixth

Let

method

sr...

x" y,

be

(below) seems set

a

of

relation which asymmetrical

term

relations

holds between y and

the

let all the terms collection,

R, R

be

of the

two

in view.

let R

be

an

except #, y any y^ Afso let R be such that,if it holds

x.

of the

terms

the end

adequateto

and asymmetricalrelations,

relation to when y is the converse between x and "/, it holds between any

some

or

of

converse

to which

collection. All

x

x

x

and if

;

be

x

has either of the

conditions

these

are

series is and whenever the resulting angles, theyare satisfied, and thence hence For xRy implies xRy, and yRx; so yRx"

satisfiedby closed. that

by

of relations R

means

it is

the definition to show

Also there is

nothingin

continuous.

Since it is closed, we the notion

between; but

why

reason

to possible

of

travel from that

our

x

back

to

x.

be

series cannot

the notion of applyuniversally be always applied.The can separation cannot

it is necessary to suppose

that

our

terms

either are,

or

are

correlated with, asymmetrical relations, is,that such series often have terms as they may be called ; and that the notion of antipodes, apposite bound to be essentially seems opposite relation. asymmetrical 194. construct

In

the

same

way

of

converse

which, in (4),we showed construct we can series by relations of betioeen,

(6) a

up with that "ofthe

in

how a

an

to

series

separation.For this purpose, as have five axioms before,certain axioms are necessary. The following ordered and by Padoa to possess been shown by Vailatif to be sufficient, be deduced from its prebe such that none can decessors i.e. to independence, must we d" from b by ab\\cd, J. Denoting "a and separatec

by directly

four-term

relations

of

have:

*

See Peano, I Principiidi Geometria,Turin, 1889, Axioms

+ Rivista di

Matematica,

v,

pp. 7$, 183.

}

vm,

Ibid. p. 185.

ix, x,

xi.

Order

206

ac

db

||cd

is

equivalent

to

cd

(")

06

[Io"

is

equivalent

to

ab\dc\

(*y)

o"

||cd

excludes

(B)

For

jjbe

If

By

of

above

The with

I

to

be the

with,

logically can

are

been of

analyzed) this

and

these

on

a

ab

jjcd,

or

and

and the

meaning

with

the

to

lines of

order

of

meaning

the

methods

six

we

points

a

on

a

discuss

and

the This

chapter.

Chap,

xxvizi.

be

But

fore there-

not

in

be

to

before and

constituents done

of are

correlation

fourth has

ones

know,

nor

appears

line.

(what

logical will

the

point

the

as

method

should

where

through

I

are,

method

Geometry,

must

order,

See

last

This

case,

as

a

neither

(especially

compounded.

*

gives

terms

*.

elliptic the

independent,

is

whose

the

principal far

so

alone

this

stage.

the

are

two

which

to

later

a

acquire

e.."

between

of

methods, last

d,

c,

extent

to

series

The

series

the

postpone

other

relations

line

these

all

six.

protective

whether

irreducible

which

in

I

generating

continuous

subsequent

decide

of

a

to

b,

a,

relation

consideration

further

separation,

asymmetrical

applied points

of

"r6m

except

The

it.

of

one

closed

generating

undefined

is

terms

our

start

we

acquainted,

am

reducible

which

methods

six

which

have

must

we

||^e.

oc

assumptions,

in

relation

the

above

correlated

five

define

assumptions of

collection,

our

then

j|be,

oc

which

terms,

generally

of

and

order,

unambiguous

pairs

of

terms

these

of

means

;

;

1|cd,

a5

xxiv

ac\\bd;

four

any

ad

or

(e)

are

\\ab

(a)

\\bd,

an

[CHAP,

we

sixth) hitherto

(if any) the

following

CHAPTER

THE

set

a among inductive

MEANING

now

of terms, and

upon which, so far the authors with whom

genesisof order; enumerated

means

have

we

of order. This

is

of them

there is an

acquireda

But

have

we

order certain

not

yet

and difficultquestion,

a

all has been

acquaintedare

am

since most

in

circumstances

know, nothingat

I

and

this

is order? I

as

what

the nature

question: What

one

methods

by

ORDER.

OF

under

seen

with familiarity

the

faced

have

WE

195.

XXV.

content

written. to

give only

All

exhibit the of the

one

six

the it is easy for them to confound This confusion is rendered evident to

Chapterxxiv,

genesisof order with its nature. of the above methods ; for it is evident that we us by the multiplicity definite,which, being generated mean by order something perfectly distinct from each and all of the equallyin all our six cases, is clearly ways

in which

out to be

it

be

may

of these ways should turn elicit others to be reducible to it. To

generated,unless

fundamental,and

the

one

discussions logical connected with it,is the purpose of the presentchapter. This discussion is of purelyphilosophical interest,and might be wholly omitted in a of the subject. mathematical treatment let us separate the In order to approach the subjectgradually, have of couples. When discussion of between from that of separation we decided upon the nature of each of these separately, it will be time to this

element

common

combine I shall

them, and

begin

with

in all

and series,

examine

what

between,as

broach

to

it is that

being

the

the

both

have

simplerof the

in

common.

two.

relation may be characterized (as in Chapterxxiv)as a of one term y to two others x and z, which holds whenever x has to t/, and relation which does not have to x" nor z to#, nor z to x*. y has to ", some y 196.

*

The

Between

condition

that

does

x

not

inessential, being only requiredin have

j?

such

cases, for

between

y and

zy

or

example, as

we may drop the the contrary, seem

condition more

z

have

order

between

the in

to

x

anglesof

the relation in

x

that, if and a

y.

y be between

If

we

are

each triangle,

questionaltogether. The

essential.

questionis comparatively #

and

s,

we

willingto allow is between other

four

not may that in

the other

two,

conditions,on

Order

208

[CHAP,xxy

for betweenness, but it may undoubtedlysufficient Several possible whether they are be questioned opinions necessary. that hold the in this respect. (1) We above must be distinguished may conditions give the very 'meaningof between, that they constitute an its actual analysis of it,and not merelya set of conditions insuring conditions

These

are

is not a relation of the terras may hold that between to that of y to z" a relation of the relation of y to x X) y, z at hold that (3) We namely the relation of difference of sense. may between is an indefinable notion,like greater and less; that the above

presence.

(2)We but all,

infer that y is between x and 2, but that there that it be other circumstances under which this occurs, and even the relation without involving occur except diversity among any

conditions allow may may

to

us

("/, 2),(x,z). In order to pairs(x,"/), each in turn. will be well to develop define 197. (1) In this theory, we There

"

remains to

such that

is a relation R a

whether question

these

y is between

and z" to

xRy""yRz

are

we

"

decide between

to add

"

but not

admitted generally

will be and

and z, and w let us express "y is between

x

are propositions

We must wyz. the extremes are

from directly

concerned

is the see

whether

means

Thus

yztc

axioms

write R

us

means

yzw

ie. xyz

between,on that

J?,and

these

purpose, let xyz

:

implies zyx. regardto the

are

for not

are

consequences

and z,

w

and

z.

For

brevity,

are

This condition follows it (a)and (/9),

axioms

of

our

the

to

to hold when

premisses.Let

definition.

For

it us

this

-R.

#Ry, yRz" yRx, zRy. yRz, zRw, zRy, wRz. onlyadds to xyz the two

seiies

some

x

($) if y be between

;

only assumed

in both question

these conditions insure xyw transitive, that

following propositions

present view, is alwaysrelative

our

the axioms

relation R that is in

same

and it

;

will suppose

Then

definition. With

is to be observed that some

We

:

and z" by the symbolxyz. two our x and and : (a) xyz xwy imply yzw implyxyw ; (") xyz far as add that the relation of between is symmetrical so

our

relation

zRy

x,

mean

"

(a)If y be between

y and w^ then y is between x and w between and "/, then y is between x

between

z

to be self-evident:

yR

not

zRx?

The

beginwith that this addition is not made.

x

it theories,

generatedby

;

one-one

conditions zRw, wRz. If R is Now have seen if not, not. we relations #, which

are

not

In these cases, however, denotingby R2 the relation between impliedby xRy, yRz, and so on for higherpowers, we can

transitive. x

and

z

substitute

transitive relation R' for R, where Rf means of R" In this way, if xyz holds for a relation a

"

some

which power definitepower of #, then xyz holds for R', providedonlythat no

of R

is

positive is

positive

to R. equivalent For, in this latter event, we have yRfx whenever xR'y,and R' could not be substituted for R Now of xyz. this condition, explanation that the converse of R

power

some

should in the is not

be

to

The

197]

196,

of

positive power

a

meaning of

n

Now

-f 1 terms.

of

we

regardto ("),we

xjjz =

xRy yRz yRx

a

.

"

xRw

=

icRy

.

is here

immediate

an

condition,that

properly is not

R

axiom

our

to

(a) as

ivRx .

.

yRw. this axiom

have

we

is

xRjj and of the

consequence

if R onlypossible The

xRw.

be not

deduction

tcyz the need of

without definition,

further conditions.

any

It remains

examine

to

and zRx relation,

to be

xRy^ and we

have

we

further

have

we

we similarly

of

new

it is

condition

from

y

involve and

by implied

and

that

If

three

the

that impossible

adhere

we

should

term

a

and

x

z

would

xRy to preferable

condition must

the condition zRx is

with compatible

anglesof

to the

"

an

our

far at least

as

#,

#,

found

:

z

for if we are

have

we

necessary

set

and

to

may

(It

that

to willing

such

a:

(a);

In axiom

premisses. admit

cases

insert,in

before to

y

is not

axiom

our

identical.

which

the

would

yRz

impliedin the are

"r.

as

that the

place of

the universal

power of R is to be equivalent both xyz and yzx^ we shall have (so

(a),namely that

axiom of R

converse

we

we

admission

render possible.Or triangle

a

zRx, the condition which of validity

yzx

and

necessary, and

are

be

is different from

yRx,

then be added

necessary, if

is not

must

x

insert the condition

xyz and yzw are to imply jcyss^ unless x and w this addition is not necessary, since it is (j8), Thus

x

case

(a),

of the

up to be different

are

involve

our

relation should

we

that definition implies

our

axiom

be between

zUr,or

condition

our

to

certainly part

the

in

terms

that definition,

This

zRa\

since R

now

be different : for this is in any

to

are

z

one-one

.

.

impossible ; for it is

is

for if not, It would seem

zRy.)

a

yzx-,

in the

z\

be

have we Hence, in virtue of the definition,

should be observed that and

to

condition

and

either insert

must

we

,

R

the

is one-one,

which

xzx"

and different, Thus

If we suppose shall have

xRy yRz zRy

=

shall obtain zxy.

between

dispensewith

can

zRx" by hypothesis

xRz,

shall have

meaning

we

we satisfied,

xyz and

whether

definition of between.

in the

zRx

xyz

between is not

zRy.

.

since relation,

one-one

closed series of

a

restrictions upon

ix-f-1

have

contemplatedby

case

but since R

;

identity.Thus

series is

our

this

R*+1

=

our

to. subject

With

xrcy The

Hence

7",imposes onlysuch

power should expect it to be

a

relation of

and

*r,

RR

that agreedalready

closed series.

to applicable

be

to

x

have

we

Jp",then

"

impliesthe

from

back

steps bring us

if R

For

RR relation,

one-one

a

209

Ry is equivalent to the condition that

seriesis not to be closed. is

Order

no

concerned)JS3

=

Le. if xRy -ff,

and

yRz"

then

[CHAP,xxv

Order

210 last

This

zRx.

course

is defined by

first instance of between

our

be the best.

to

seems

substitute the relation #', which means and The relation Rf is then transitive,

condition that

shall of

positive power

"some

the

where

cases

relation J2,we

one-one

a

in all

Hence

R.""

positive

no

to the condition that to R is equivalent equivalent is simplified the whole matter Rf is to be asymmetrical.Hence, finally, into the following : to sayingthat there To say that y is between x and z is equivalent relation which relates botli x and ?/, and transitive asymmetrical is some

is to be

of R

power

y and

z.

This

shows,

and

short

contains

neither

with the

emendations

more

the

as

less than

nor

which

above

lengthyargument together definition, original found to be necessary. gradually

simplestatement, we

our

remains,however : Is this the meaning of between ? question established if we allow the instance can be at once A negative 198. the reader will as phrase: R is a relation between x and y. The phrase, excluded from the definitions of have observed,has been with difficulty between^ which its introduction would have rendered at least verbally but a linguistic circular. The or importance, phrase may have none above Let definition. real the in insufficiency again it may pointto a

The

examine

us

relation of

the

a

relation R

is such certainly

there first place,

the relation R

a

to

relation which

we

may

is

If

express this relation between

shall have *r"jR,yER.

by /, we El is by

shall have RIR no

the

means

between,if for this is transitive. such

a

Now

case.

at all the

has

which itself, to regardthe that

the

terms term to

in

which to

use

two

a

"

x

and y

admit

A

be

of between in this

case

of JB."

father

and relates, To

same

a

sense

the

J?,between term to itself,

a

be

tempted accident linguistic the subjectand the we

may

due to the

as

between On

the

other hand, it may

relation to the very peculiar

such

and

term

a

a

relation

relations objection concerning that

in

as

between

that between should denote

be answered

case,

do not

we Certainly

are

El

nor

between,in this

relations of

relations

usuallymentioned

is the

others.

it may itself,

admit

Now

to whollyinapplicable

whether cases.

to R

definition of

also neither E

apply;

impossible.Hence

relation does have it

we

JK,by ",

xER, yEIR.

thus the above

not, in the

are

that such

agreedto

we

J",and

in other

Moreover,if

relation is

object, urged that as

to

have

we

well be doubted

meaning as

this way R and other terms.

between y and

definition of between is

our

has

that of R,

to

express the relation of R

we

Thus

of

In the

which

term

a

JZ,and y

of

J?,or

only,does not

it may

same

shall have

and RIR. converse

obtain series :

we

If further

reason

Thus

and

x

be

To

and y.

x

to have a relation to J?, certainly belongingto the domain of R?

will belong to the domain x 4"jR?/j

if

we

*c

as

express

Thus we

relation.

a

other term

some

its terms

to

in relations,

any

of

be

pair of of a

system,

one

term con-

The

197-199]

meaning of

Order

211

be denied logical difficulty ; that theywould, if possible, where the relation asserted is validity;and that even philosophic stitute

grave

a

there must be two identical terms, which therefore not identity, are this raises identical. As fundamental which we cannot a quite difficulty, discuss here, it will be prudent to allow the answer it to pass*. And further

urged that use of analogy,the pointsalways to some indicated by those who deny that be

may

cases

and

;

that the

order of words

in

analogyhere

the

word

in two

the

connections

should be

of which

extent

carefully in both

is the

same meaning than profounder certainly

is

the

is,in any

sentence, which

a

same,

mere

variable

case, far more relation is between

its terms. respect than the phrasethat a these remarks, however, it may be retorted that the objector has

in this To

himself indicated the its terms

relation to

is,and

between

its terms,

to

the

not

is

relation of

a

this is what

is,I think,valid,and

retort

same

makes we

one

the

most

relation of between

the

the above definition Nevertheless, ultimatelyforced to accept it,seems,

from

a

is vague,

in which

such

no

betweenness.

the

term two

to

relation

the

of

a

others,just as

two

similar.

cases

of

last

This

relation

a

problem,is importantlogical stituted. by which order is to be conbetween* though

shall be

we

first

sight,scarcelyadequate reference to some asymmetrical and seems to require to be replaced by some phrase undefined relation appears, but only the terms and This bringsus to the second of the above opinions

pointof view. philosophical

relation

analogy:

allow that the relation of

may

a though involving as

of the

preciseextent

at

The

between. concerning

(") Between^ it may be said,is not a relation of three terms but a relation of two relations, Now at all, namely difference of sense. take this view, the first pointto be observed is,that we if we require b ut in the two not as relations, general, merely particularized opposite 199.

by belongingto familiar from

the

of

case

is both

term

then

between

a

what

in the difficulty

a

This

term.

distinction is

after that

and

it is before and reduction a

between

what

of between

arises:

it is after.

to difference of

this term

Hence sense.

which puzzlingentity, logically

a

relation of the

term

time, great advantages are the

we

questionto

secured

by

two

others.

this reduction.

a

difference of sense, Le. the and its converse.

difference between

*

Cf.

" 95.

an

("55) distinguish term,

same

the

At We

ticularize par-

same

get rid of

philosophers may namely relation asymmetrical

to which many triangularrelation, element to all cases assigna common

for necessity

and object,

in

Is

I

in Part

it necessary to deny ; and it is not quite easy to relation of two relations, as belongingto the particularized

is

there The

found

from

already

Before and afta magnitudesand quantities. and the between : it is onlywhen one

before

relation is we

same

constitute

in the abstract do not same

the

and

one

of between

9

Order

212

question whether

The

200.

relation is

there

be

can

is both

actual solution

whose

one

[CHAP,xxv ultimate

an

difficultand

triangular

unimportant,

is of very great importance. Philosophers precisestatement though not, so far as I know, explicitlythat usuallyto assume

but

whose

seem

"

"

relations

have

never

than

more

terms

two

and

;

relations

such

even

they

to Mathematicians, on the other reduce,by force or guile, predications. relations of of many We terms. hand, almost invariably cannot, speak mathematical to the settle a instances, questionby simpleappeal however,

for it remains

questionwhether

a

these

analysis.Suppose,for example,that defined

as

a

it should

that

difference.

Let

what

see

us

is the

of

point

a

makes

little

and

or

meaning of precise

different radically

There

not,

are

relation

a

as

change which

a

"

the the

points:

defined

been

have

lines intersecting

of two

or

relation of three

of susceptible ve project! plane has been philosopher may alwayssay or

are,

the

which

terms

are

can

question.

kinds, whose

terms

except as

occur

never

difference attribute.

such

;

line,

matical mathe-

no

terms two are among the doctrine of substance and constitutes the truth underlying

There

a

points,

are

of the kind terms colours,sounds,bits of matter, and generally instants, which other There the of which existents consist. terms hand, are, on otherwise

can

occur

and

relations.

of

than

Such

terms

as

terms

concepts;

than

there

subject-predicate theory "

and term, the subject, This view, for many

departurefrom

the

terms

traditional

The

terms.

are

call concepts*.It is the presence from which distinguishes propositions

there proposition

in every

holds

view

traditional

be

abandoned

which

which there and

are

onlytwo in

occurs

no

seems

there

are

place,when

of

an

smallest

lies in

holding that, where form, there are subject-predicate

opinion

is not

(The

term.

a

terms

;

for

a

be

relation may

defined

as

than

any

propositioncontaining it priori reason for limitingrelations to instances which lead to" an oppositeview.

the

collection has

;i)which

The

f-

each contain concepts may, of course, be complex,and may not tenns.) This givesthe opinion that relations are always

terms

between

be called the

which

"

proportionsare not reducible to the always two terms only,and one concept which two

concept

more

one

may in every propositionthere is one which is not a term, the predicate.

must

reasons,

is at least

that

concept

one

being,adjectives generally,

are

agreedto

we

as conceptsnot occurring

mere

such

;

a

more

concept

of

terms, there

n

is not

a

term.

existent to the

a

number are

n

In the

is asserted

terms, and second

placeand time

of

only

a

one

of its existence

But

two

In

terms, the

first

if the collection,

concept(namely

one

place,such

concept

term.

relations are

as

those

onlyreducible

method to relations of two by termsj. If, however, very cumbrous the reduction be held essential, it seems to be alwaysformally possible, a

*

See

Part

t See

The

I, Chap, iv.

Philosophy of Leibniz,by the present author, Cambridge,

ChapterH, "

10.

Part

VI

%

See

I,Chap. rjv.

1900

;

The

200-202]

by compounding part be

can

is not

as

into proposition

assertinga relation between

then

a

of the

reduced to similarly but possible, I do not reduction

formal

able to

been

I have

of them.

discover, one

be

may The

of any

remainder, which

the

this

where

cases

such

questionwhether

alwaysundertaken

and

complex term,

one

There

term.

one

213

this part and

know

be

is to

Order

of

meaning

is not,

far

however, so

theoretical or great practical

importance. There

201.

into

between

is

in favour of analyzing priorireason if relation seems relations, a triangular other reason in favour of the analysisof

considerable.

more

reference to

a

relation of two

a

So

be taken

terms, it must

of the

valid

no

The preferable.

otherwise between

is thus

transitive

some

longas either

between

is a

relation triangular

or as indefinable, relation. But asymmetrical

involvinga

as

if

make

we

in the opposition of two relations belonging essentially to be no to one term, there seems longerany undue indeterminateness. this view we now Against may urge, however, that no reason appears consist

between

relations

the

why

is

what

and that questionshould have to be transitive, important the very meaning of between involves the is they,and not their relations, And that have order.

more

as

in

it

they hold.

Thus

relation triangular

a

202.

We

(3)

come

our

This

ways

to show

seems

the

to

the

The

relation.

that the

could

that

and

x

z?

of

applya

the

terms

such

not

and

which

urged that,

that

merelyconditions

were

we

of this

definitions

true

can

insure answer,

that

of between

be

nature unanalysable may w ith symmetrical respectto

the relation is the

not

with the relations of

case

supportedby which

And

fact that

extremes, which

the two

pairsfrom

the

between

y

without

definition. to any previous having to appeal(atleast consciously) the

of

cases

definitions. suggested

conditions

such

is alwaysone

see

ultimate

an

might be

could

test to

were

is

between it

suggesteddefinitions

Do

: question

shall be between

view

generatingopen series,we we

be necessary,

not

mention

whole, the opinionthat between

of this view

imply relations of between^and

which

by

the

be abandoned.

In favour

and that arise,

did

between

of

on

must

now

and indefinable relation. in all

it would relevant,

were

them particularize

is,to

which

between is not

relations that

onlythe

fact

"

"

terms, for it if it were

in

was

inferred.

was

in the way of such a view, and is,however, a very grave difficulty of have we terms that is,that sets many different orders,so that in one There

y between This seems

have

may

z*.

y and

to the relations from to

series *

of

that

admit

these

x

and

to show

which

2, while

that between

it is inferred.

relations

are

that requireimperatively

This

case

denumerable.

The

in

relevant there

we

have

x

involves essentially

If not, to

should

the be

we

reference

shall at least have

genesisof at

between

most

one

series ;

for

relevant

by the rational numbers., which may be taken in order orders (e.g.the logical order) in which they are logicalorder is the order 1, 2, 1/2,3, 1/3,2/3,4,

is illustrated

magnitude^ or

in another

one

of the

214

Order

-

relation of

between

allow that between

with

which

respectto

of

of

apparently, alwaysbe relation asymmetrical must,

we

but series,

transitive

some

betweenness

the

arises. The

must

that

most

relation asymmetrical

this transitive

said is, that

Hence

three terms. among is not the sole source

the mention

supplementedby

[CHAP.XXY

of two

be

can

terms

may derived and relation of some from, to, sulxsequent logically three terms, such as those considered in Chapterxxiv, in the fourth way such relations fulfilthe axioms which were of generating series. When

itself be

mentioned, they lead of themselves

then

For

terms.

may

we

b follows

that

say abd

when

c

such relations

Though

that between

that

are

in such

occurs

leave the reference to

when

two

terms

and

x

have

203.

to

next

consider

the

acd

in

with

z

relation R when xRy and yRz. asymmetrical be said properly to be between y and jsr; and of but the very meaning merelya criterion, We

pairsof impliesbed, and

precedesc impliescbd,where a and d are fixed terms. merely derivative,it is in virtue of them Hence we seem cases. finally compelledto

relation asymmetrical

an

shall therefore say : A term y is between

relations between

to

b

our

definition.

reference to In

other

no

a

transitive

case

this definition

We

can

y

givesnot

betweenness.

meaning

of

separationof

complicatedrelation than between,and was but little considered until elliptic Geometry broughtit into prominence. It has been shown like between,always by Vailati* that this relation, tion relation of two terms ; but this relainvolves a transitive asymmetrical

couples.This

of

is

pairof

a

set,as, in the further

a

more

is itselfrelative to three other fixed terms

terms

of

case

between,it

evident sufficiently

which relation,

that

fixed terms.

relative to two

was

there

wherever

relates every

pair of

is

terms

of the It is

transitive asymmetrical

a

in

a

collection of not

less than four terms, there there are pairsof couples having the relation of separation. Thus we shall find it possible to express separation, as well

between,by

as

means

terms.

But

let

We

may

denote the fact that

the

us

symbol abed.

the require

relations and their asymmetrical the meaning of separation. directly a and c are separated by b and d by

of transitive

first examine

If,then, a, b,c, d, c be any five

terms

to hold of the relation of following properties

of the set

we

(of separation

which, it will be observed,only the last involves five

terms):

*

"ii

I.

abed

=

bade.

%.

abed

=

adcb.

3.

abed

4.

We

5.

abed

excludes acbd. must

have

and

acde

abed

or

aedb

Mtttematiat, v, pp. 75"78. Poxixiotie, Turin, 181)8," 7. five properties are

adht\

togetherimply abde"\.

/ttrota "ti

t These

or

taken from

See also

I Prinripii delta Fieri,

lot\ cit. and Vailati,

ib. p. Ifr).

Grometria

The

202-204]

meaning of be

properties may

These

as circle,

in the

illustrated

Order

by

215

the

consideration

of five

points of two pairsof terms between of separation

shall call a relation we possesses these properties the pairs. It will be seen that the relation is

but symmetrical,

in

on

a

not

Wherever

204. any

of

a

a

a

and

as relation, asymmetrical

of

instances

consecutive

is separation

a

For

we

in any

it, gives rise

extension

mere

of such

existence

a

R

tween be-

four terms, the relation of if four terms have the series,

to

series.

a

of between:

and aRb" bRc, cRd, then transitive, The

asymmetricalrelation

less than

separated by b and d; and have seen, providedthere

are

c

transitive

set of not

arises. necessarily separation

order abcd^ then

relation

generaltransitive. have

we

terms

two

accompanyingfigure.Whatever

a

and

if R c

are

relation is therefore

a

every transitive at least two are

Thus

in this case,

be

asymmetricaland separatedby b and d. sufficientcondition

of

separation. It is also

necessary condition. For, suppose a relation of separation and let fl, 6, c, "/, of the set to which the relation to exist, e be five terms a

applies.Then, considering a, ft,c

as

fixed,and

and

d

e

as

variable,

we properties, may arise. In virtue of the five fundamental introduce the symbol dbcde to denote that, striking out any one

twelve may

cases

the of these five letters,

remaining four have the relation of separation

is indicated

by the resulting symbol. Thus by the fifth property, Thus the twelve cases arise from permuting abed and acde implyabcde*. d and ^, while keeping#, fi,c fixed. (It should be observed that it makes no difference whether a letter appears at the end or the beginning: which

abcde is the

i.e.

either d

or

before

six will have

and

respectto e

e

same

the

sense

d. precedes

In

e

case

as

eabcd.

a.)

Of

these

before d.

We

may

therefore decide not

put

twelve cases, six will have d before e, In the firstsix cases, we say that, with

abc^d precedes e ; in the other six order to deal with

limiting cases,

say that shall say further

cases,

we

a precedes cf every other term, and that b precedes is asymmetrical and find that the relation of preceding

that

We

we

shall then

.

of that every pairof terms other follows. In this way our at

to

to least,

the

combination

and transitive,

precedesand the is reduced,formally relation of separation of "a precedes "b b^ precedesc? and "c

our

set

is such that

one

precedesd? The

above

reduction

is for many reasons firstplace, it shows the distinction between somewhat sort which

transitive

*

open and

closed series to

the be

be of the For althoughour series may initially superficial. is called closed,it becomes, by the introduction of the above but having an relation, havinga for its beginning, open series,

argument is somewhat in Vailati, loc. tit. t Fieri,op. ciL p. 32, The

In highlyinteresting.

tedious,and

I therefore omit

it.

It will be found

[CHAP,xxv

Order

216 last term, and

no possibly

not

in any

to returning

sense

Again

a.

it is

highestimportancein Geometry, since it shows how order may line,by purelyprojective arise on the elliptic considerations, straight than that obtained from which is far more in a manner satisfactory it is of And construction*. Staudt's finally, great importanceas von and since it between of the order, two separation; sources unifying that transitive asymmetrical relations are shows always present where other. the either For, by the relation of either occurs, and that implies? is between two others,althoughwe term can we preceding, say that one of pairs. started solely from separation of the

205.

At

same

time, the above reduction (and also,it would in the

allowed to be

more

relation to which

our

of

between)cannot be That is,the three terms than formal. a, 6, c by relation transitive asymmetrical was defined,are

reduction corresponding

the

seem,

the

case

and cannot be omitted. essentialto the definition, for supposing that there is any transitive no reason of independent what

other

relation asymmetrical related, though it is arbitrary

all other terms

than

those

choose.

And

the fact that the

terms

we

reduction shows

The

term

a, which

is

illustrates beginningof the series, relations independent this fact. Where there are transitive asymmetrical have an arbitrary of all outside reference, series cannot our beginning, all. the of sepait four-term relation Thus at have none though may ration and two-term to the remains logically relation, resulting prior

essentially peculiar, appears

not

be

cannot

206.

into analyzed

But when

said that

not

it is

latter.

have said that the reduction is

formal,we

have

of order. On the contrary, genesis that the four-term relation a reduction is possible transitive relation is in asymmetrical resulting

order.

such The

reality relation of five terms a

it becomes

the

it is irrelevant to the

justbecause

leads to

we

the

as

and asymmetrical

althoughbetween appliesto

;

but when

transitive such

as

three of these

are

keptfixed,

regardsthe other two.

series,and

althoughthe

essence

Thus of

order consists, in the fact that one here as elsewhere, term has, to two relations which are and others,converse transitive, asymmetrical yet such

order can onlyarise in a collection containing an at least five terms, because five terms are needed for the characteristic relation. it And should be observed that all series, when thus explained, are open series, in the sense that there is some relation between pah'sof terms, no power of which is equalto its converse, or to identity. 207. The

Thus

six methods

to sum finally,

of

distinct; genuinely

*

where from

The

up this

discussion : longand complicated series enumerated in generating Chapterxxiv are all but the second is the onlyone which is fundamental,

advantagesof this method are evident from Pieri's work quoted ahove, deduced many thingswjhich seemed incapahleof projective proofare rigidly projective premisses. See Part VI, Chap. XLV.

The

204-207]

rise

give

alwavs

between as

v

m

-7-

from

y

the

exceptional

and

2";

beginning; cases

that

is

this

the

it

all

are

was

reducible

reductibility

order

at

proposition

conclusion

21

ordinal

an

which

simple

very but

they their

relation

This

z."

that

minimum

there

Order

of

of

The

transitive and

this, virtue

wherever

and

metrical

between

in

order.

to

made

l"e

in

agree

solely

is

it

Moreover,

they

five

other

the

and

meaning

all,

could

is

the

of

means;

holds

by

the

to

second

might

discussing be

solidly

form

a:

all

the

can

and

is

^y

:

is

been

have

that

which

"There

between

second.

the

proposition,

conclusion

only

to

some

y

and

guessed

apparently

established.

7

XXVI.

CHAPTER

RELATIONS.

ASYMMETRICAL

have

WE

208.

is

main

the

found

make

the

philosophers Relations not

not

i.e.

for

that

which

xRz

I

and

symmetrical,

but

is

not

shall

a

but

if third is

*

see

This Camb.

term

Phil.

am

which

stage (in

general objections

only

with

her

the

property The

woman

own

Spouse

marriages

appears Trail*,

be

to

ix,

sister.

symmetrical

transitive.

asymmetrical would

were

and

not

p.

104

been ; x,

cases

a

man

but

is not

be

may

;

term

is

present is

be

second

Brother-in-law

the

;

his

is not

descendant

symmetrical

transitive

p. 34(5.

is

sister

or

brother

intransitive

if

in

trated illus-

htdf-xixteris symmetrical

or

it would

The

not

always be

may

brother

forbidden,

first used

call

shall

I

relation

The

Half-brother

intransitive.

have

that

shall

I

xRy., yRz

relation

allow

Half-brother is

that

these

;

called

are

property

property

All

symmetrical

opposite property, call asymmetrical*

shall

I

yRx,

we

first the

symmetry.

"rRz

imply

possess

second

if

transitive

called

are

do

they

as

and

veness*

the

the

Intransitive.

is transitive.

Son-in-law

forbidden,

do

relationships.

transitive;

it

a

classes, according

possess

which

possess call

is

transitive.

asymmetrical

not

possess

which

and

brother,

four

excludes

always not

human

symmetrical,

but

of

one

grounds later

concerned

together always

do

relations

do

those

from

own

I

attributes, transiti

two

which

jcRy

which

transitive;

answer

present,

into

tfRy""yRz

symmetrical;

exclude

of

Relations

Relations

;

the

the

scRy always implies yRjc

that

such

traiwtive. call

either

such

relations

for

divided

be

may

possess

Relations

relations

to

to

the

At

necessary.

endeavour

is

relations.

asymmetrical do

relations

such

shall

I

them

Philosophy has proceeding further, to

forth

set

to

traditional

Critical

the

which

which

admit

to

transitive

upon

kind

a

refusal

logic,and

pure

of

admission

VI, Chap. LI),

Part

or

into

of

desirable, before

be

it will

mathematics,

depends

are

the

as

contradictions

the

excursion

an

admit, and

to

of

sources

in

make

of

unwilling

relations

such

As

order

all

that

seen

relations.

asymmetrical logic

now

now

and

intransitive.

marriages not

sense

in

not

were

symmetrical

by

De

general

Morgan use.

;

208-210]

Asymmetrical Relations

and not transitive.

219

Finally, fatheris both asymmetricaland

intransitive.

Of not-transitive but not

intransitive relations there is,so far as I know, but not onlyone importardinstance, namelydiversity ; of not-symraetrical relations there seems to be similarly asymmetrical only one important

In other instance,namelyimplication. relations

occur,

Relations which

other term. and

a

is

and symmetrical

there relation, be identical, then a a.

is

it follows that

a

it holds between has

shown, property cannot

since it is The

a.

=

property

and

term

a

neither of these

a

have

we

relation which

a

If

6.

=

relation

5, 6

=

insures

a, that =

itselfis called what

contrary to be

a

if not, then, since the

and transitive, of

that

6 such

term

some

But

=

5 = a; symmetrical,

he

both

transitiveare formally equality.Any term of the field of such a relation has the questionto itself, though it may not have the relation to any For denoting the relation by the signof equality, if a be are

field of the 6

either symmetrical

of

of the nature relation in

and intransitive,

or

usually

kind that

asymmetrical.

or

209.

of the

either transitive

are

cases, of the

was

inferred from

and by Peano reflexweness^ believed,that this previously and

symmetry

transitiveness.

For

6, but only what follows in case there is such a b ; and if there is no such ", then the proofof a a fails*. This however, propertyof reflexiveness, asserts properties

that there is

" such that

a

a

"

=

introduces some without

There difficulty.

limitation,and

that

is

is

onlyone

relation of which it is true

identity.In

all other

it holds

cases,

for example, onlyof the terms of a certain class. Quantitativeequality, is onlyreflexive as appliedto quantities ; of other terms, it is absurd to assert that theyhave quantitative with themselves. Logical equality relations. or or again,is onlyreflexive for classes, equality, propositions, and is reflexive for on. Thus, with any so events, Simultaneity only transitive relation, other than identity, can we only given symmetrical assert

reflexiveness within

the

of principle and shortlyto

a

certain class: and

(alreadymentioned there discussed at length),

of this class, apartfrom

abstraction

in Part

be

need

III,Chap, xix,

be

no

definition

except as the extension of the transitive symmetricalrelation in question. the class is so defined,reflexiveness within that class, And when as we have

seen, follows from

better account

somewhat

defined* he

a

process which

shows, frequentuse *

See

e.g. Revue

de

I have called the

is made

in

Mathematiques,T.

Mathematics.

vn,

matique, Turin, 1894, p. 45, F. 1901, p. 193. identical with this t An axiom virtually and precision, necessary Trans. Vol. x, p. 345.

J

Notations

de

not

symmetry.

of abstraction f principle has Peano of reflexivenessbecomes possible. of which, as he calls definition by abstraction,

what By introducing

210. a

transitiveness and

p. 22 ; Notation* but principle,

demonstrated,will be found

LogiqueMathematiqtteyp.

45.

This

in De

de

not

process is

,

as

Logigue M"thfstated

Morgan,

with

Camb.

the PhiL

Order

220

[CHAP,xxvi

there is any relation which is transitive, and symmetrical this if relation holds between its and r eflexive, then, u field) (within r,

follows

when

:

define

we

is to be identical with

(w),which entity"f"

new

a

(v). Thus "f"

of relation to the new term analyzedinto sameness this of the legitimacy (u) or "f" (v). Now process, as set forth by "f" axiom the that, if there is any axiom, namely Peano, requiresan there is such an entityas instance of the relation in question,then relation is

our

stated,is precisely

is my

axiom

(v). This "f"

$ (it)or

Every transitive symmetricalrelation, into jointpossession instance,is analyzable

follows :

as

of which there is at least one of

a

relation to

new can

does

not

have

language,to from

this

one

being such

that

no

term, but that its converse

have

amounts, in common principle transitive symmetricalrelations arise

that

property,with

the

the terms those

It

tenns.

it,in

addition

that

this

propertystands,

relation in which

nothingelse stands often to principle, that symmetricaltransitive relations always appliedby philosophers, of content. Identityof content is,however, an springfrom identity gives,in the extremelyvague phrase,to which the above proposition which in but one no signification, way answers presentcase, a precise the reduction of relations which is,apparently, the purpose of the phrase, of the related terms, to adjectives of the reflexive property. It is now to givea clearer account passible Let 72 be our symmetricalrelation,and let S be the asymmetrical relation which two terms having the relation R must have to some There the proposition Then third term. to this : xRy is equivalent is some Hence it follows that, if X term such that xSa and ySa? a belongsto what we have called the domain of S, i.e. if there is any

to

which

than

more

relation

new

property."This

assertion

the

common

a

term, the

new

this relation to

have

term

a

of abstraction, which, principle

"

givesthe

a

of the

statement precise

"

term

a

such

that

of

course

does not and But

thus Peano^s

then xRx .r*Sa,

;

for xRx

is

and

merely xSa

ocSa.

It

follow that there is any other term y such that xRy, to the usual proofof reflexiveness are valid. objections

of symmetrical transitive relations, we analysis obtain the proof of the reflexive property,togetherwith the exact limitation to which it is subject. We 211. the reason for excluding from our accounts can now see of the methods of generatingseries a seventh which some method, have find. is the which readers may This method in to expected is position merely relative a method which, in Chap, xix, " 154, we rejected as regardsquantity. As the whole philosophyof space and time is bound of this up with the questionas to the legitimacy

by

means

of the

"

method, which it may position,

how

is in fact the

question as

be well to

to

absolute

and

of it here, and give an the principle of abstraction leads to the absolute theory of

If

we

consider such

a

series

account

as

that

of events,and

if

we

relative to

.show

position. refuse to

AsymmetricalRelations

210-212] allow absolute

time, we

shall have to admit

221

three fundamental

relations

and posteriority. Such simultaneity, events, namely, priority,

among

theorymay that

such

formallystated

be

two,

any

the

relation P,

or

relation K.

Also let

all the

Then

be

may

follows

#, have

place,accordingto

which

have

the

relational

the

an

or

symmetricaltransitive

a

place in

same

the

series.

number

a

of other

But

terms.

of abstraction that there is some principle

the

This

theoryof position,is nothing but

relation R to symmetrical

it follows from

;

either

a class of terms, asymmetricaltransitive

JcRy^yPz implyxPz, and let JcPyyyRz imply xPz. be arrangedin a series, in which, however, there can

terms

the transitive

a

Let there be

relation P,

converse

terms

many

and

x

as

relation Sy

such that, if ocRy^ there is some one entityt for which xSt^ ySt. We shall then find that the different entities 2, corresponding to different of

groups any

in which originalterms, also form a series,but one terms have an asymmetricalrelation (formally, the

our

different

two

These

product SRS). and

a^

our

is reduced series

y\ and to

having

the

t will then

terms

be the absolute

supposedseventh

our

fundamental

only relative

method

second method.

of

Thus

of positions generatingseries there

will be

but in all series it is the position,

no

positions

themselves that constitute the series*. We

212.

relations. The

argument

now

are

whole

in

a

to position

account

of order

"

that

relations

no

is not

my

the

dislike of philosophic

givenabove, and the present necessarily objectedto by

concerningabstraction,will be

philosophersand they are,

those

meet

I

hold

fear,the major part who "

validity.It possess absolute and metaphysical intention here to enter upon the general question,but merely can

of asymmetrical relations. to any analysis objections and employed a common opinion often held unconsciously, it that who advocate in argument, even those do not by explicitly and a predicate.When consist of a subject all propositions, ultimately, it has two ways this opinionis confronted by a relational proposition, be called inonadistic, of dealing with it, of which the one may the other monistic. aJ?6,where R is some Given, say, the proposition relation,the inonadistic view will analysethis into two propositions, call ar^ and 6r2, which give to a and b respectively which we may The to R. monistic to be togetherequivalent -supposed adjectives view, on the contrary,regardsthe relation as a property of the whole to a proposition, which composedof a and 6. and as thus equivalent first is the represented by, we may denote by (o")r. Of these views, second the and and the Leibniz by Spinoza whole) by Lotze, (on as these views successively, appliedto Mr Bradley. Let us examine to exhibit the

It is

"

"

*

A formal treatment

d* fidee

is given by of relative position

tfordre,Gougr**,Vol.

in, p. 235.

Schroder,Sur

une

extewwn

[CHAP,xxvi

Order

222

relations; and asymmetrical

for the

the relations of greaterand

less.

Leibniz

ratio

conceived lesser M

three ; as

a

proportionbetween

or

several ways; as a ratio of the lesser M

consideringwhich

which the

which

and subject,

M

greater

may

be

to

the

L

to

which

In the object.... M

the consequent ; of

first way

considering subjectof that

is the the lesser,

relation.

But

which

of them

will be

them ? It cannot be said considering the subjectof such an are together, with have an accident in two subjects,

the third way of of them, L and M

both

accident; for if

leg in

the

and

one,

of

should

we

so,

notion of accidents.

other in the other; which

Therefore

The

greaterand

we

must

it,is indeed considering

third way an neither a substance nor the consideration of which 214.

the

the

in subject,

one

of

antecedent,or

call philosophers

accident which that

lucidity by

and

lines L

two

ratio

is the

in the second them, L the greater, the

admirable

with

take

us

the greater L; and lastly, as both, that is,as the ratio between L and My

something abstracted from without

let definiteness,

of

followingpassage*:

in the

"The

is stated

monadistic view

The

213.

sake

is

contrary to the

in this say that this -relation, out of the subjects ; but being

accident,it

must

be

a

mere

ideal

thing,

is nevertheless useful."

third of the above

less

the relation of ways of considering that the monists which advocate, is,roughlyspeaking,

as they do, that the whole composedof L and M is one subject, holding, ratio does not compel us, as Leibniz that their way of considering so is only to placeit among bipeds.For the presentour concern supposed, the In firstway of considering the matter, we with the first two ways. the than in "L is words brackets have M)? (greater being considered But when of L. examine this it is at once we an as adjective adjective at least, of the parts greater evident that it is complex:it consists,

and

Jf, and

does not also

both

these

essential.

To say that L is greater it is meaning,and highlyprobablethat M is

parts are

all convey our of L involves greater. The supposedadjective

but what

at

some

be meant

reference to M

;

by a reference the theoryleaves unintelligible. reference to M is plainly An adjective a involving an which is adjective this is relative to Jf,and merelya cumbrous way of describing a relation. the i f L has to matter Or, otherwise, an put adjective corresponding to the fact that it is greater than M, this adjective is logically sequent subto, and is merelyderived from, the direct relation of L to M, of L to differentiate it Apart from My nothingappears in the analysis from M ; and yet,on the theoryof relations in question, L should differ M. from Thus should be we in aU cases intrinsically of asymforced, metrical to admit relations, difference the a between related specific terms, althoughno analysisof either singlywill reveal any relevant can

*

PfoV.

Werke, Gerhardt's etL,Vol.

vii, p. 401.

AsymmetricalRelations

212-214]

22S

property which it possesses and the other lacks, For the monadistic this constitutes a contradiction ; and it is a contratheory of relations, diction which

Let

condemns

the

further

examine

theoryfrom

which it springs*.

the

of the raonadistic theoryto application relations. propositionA is greaterthan B is to be quantitative into two one to A^ the propositions, analyzable giving an adjective to B. The advocate of the opinionin questionwill other givingone hold that A and B are quantities, not magnitudes, and will probably requiredare the magnitudesof A and B. But say that the adjectives he will have to admit then relation between the magnitudes, a which will be as asymmetrical the relation which the magnitudeswere as to the Hence will need and new magnitudes explain. adjectives, so on and infinite ad infinitum the will have be completed to before ; process be assignedto our original proposition.This kind any meaning can of infinite process is undoubtedly objectionable, since its sole object is to explainthe meaning of a certain proposition, of its and yet none it that to nearer Thus take cannot meaningf. we steps bring any the magnitudesof A and B as the requiredadjectives. But further, if we take any adjectives whatever exceptsuch as have each a reference shall not be able,even to the other term, we to give any formally, the of without assuming justsuch a relation between account relation, For the mere the adjectives. fact that the adjectives different will are relation. Thus if our two terms have different yieldonlya symmetrical colours we find that A has to B the relation of differing in colour, of careful handling amount will render asymmetrical. no a relation which could to magnitudes, Or if we to recur we were merelysay that A and B differ in magnitude,which givesus no indication as to which is of A and B must the adjectives the greater. Thus be,as in Leibniz's The adjectives having a reference each to the other term. analysis, be of A must adjective greater than B* and that of B must be less A? B since they have different adjectives than Thus A and differ, but the adjectives B is not greater than B9 and A is not less than A has reference to J5,and that A^s adjective are extrinsic,in the sense of the relation fails, ITs to A. Hence the attemptedanalysis and we us

The

"

*

"

"

"

"

*

This

See paper

a

paper was

ou

"The

written

the contradiction

in

Quantity,"Mind, N.S. No. 23. to the monadistic theory of relations : regardedas inevitable. The following

Relations of Number

while I stilladhered

therefore,was question,

and

the same point: "c Die reehte Hand ist der liiikeu aluilieh passage from Kant raises bios auf eine derselben allein sieht,anf die Proportion und gleich,und wenn man eine der Lage der Theile nnter einander und auf die Groese des Ganzen, so muss

vollstandige Beschreibungder ehien in alien Stueken auch vou der aiideru gelten." (V"n dem ersten tfrunde de* Uriervchiede* der Gegnidrn im Itaut"e,ed. Hart Vol. n, p. 380.) dealing infinite process of this kind is requiredwe are necessarily an t Where of Fart II,Chap. xvu. with a propositionwhich is an infinite unity,in the sense

Order

224 the

relation,i.e.one

external"

"

what

forced to admit

are

[CHAP,xxvi

theorywas designedto avoid,a so-called implyingno complexityin, either of the

related terms. The

dependssolely upon

it

since

be

result may

same

a

and

Let

the

symmetrical. Let

are

that

aRb

seen,

must

and

and

bRa. have

each

y9 to

is asymmetrical.But

and

reference

to

a,

a

b

;

the relation R, and priorto themselves have difference must

6, but

it;

the difference between

difference between be

relation must

a

relation

they have, the pointsof analogous^to#, so that

and

ft cannot

b.

Thus

reference to

far from a.

and

have

#, presuppose R9 to

intrinsic

supplyan

to

a

term

a

being prior

/3 both

used

be

we

difference between

a

involves

0

or

since

difference without

againa

tions asymmetricalrelaultimate,and that at least one such ultimate asymmetrical relation that may be be a component in any asymmetrical

difference. This

priorpointof must

involves

j3 expresses is,so

and

a

a

if

or,

itself. And a

ba.

by ft

since the /? differ,

relation

a

have

we

term) be denoted a"$ and

so

intrinsic differences corresponding

it adjective

whose

that

is in fact the relation R

a

a

which, since either

one

than

other

and

no

or

a

other

become

terms

and

to

and

the

to

b have

nothing is gained. Either

asymmetrical relation #,

an

(which,as supposedadjectives

our

and

a

b have

reference

a

Thus respectively.

a

provedof asymmetricalrelations generally, and diversity the fact that both identity

that

shows

some

suggested. criticizethe

It is easy to

by of the

point, theoryfrom a generalstandrelations springfrom the

monadistic

the contradictions which developing

terms

the

to

into adjectives

which

first relation has

our

been

connection which have no special with analysed.These considerations, and have been urged by asymmetry, belong to general philosophy, advocates of the monistic theory. Thus Mr Bradleysays of the monadistic in of fission which brief,are led by a principle theory*:"We,

conducts a

us

to

end.

no

its

within diversity

be asserted of the for

an

in Everyquality

nature, and

own

quality.Hence

internal relation.

each something in

relation,and

so

on

the

But, thus

must relation,

is fatal to the diversity

internal

without

this

has,in consequence,

cannot diversity

immediately exchangeits unity

qualitymust

set

each

relation

because free,the diverse aspects,

be

somethingalso beyond.

unityof each

limit."" It remains

;

and to

it demands

be

seen

a

whether

This new

the

in avoiding this difficulty, does not become theory, subjectto others quiteas serious. monistic theoryholds that The 215. every relational proposition aRb is to be resolved into a proposition concerningthe whole which and b compose a which a we proposition may denote by (ab)r. This

monistic

"

view, like the other,may *

be examined

with

reference special

Appearance and JKeaKty,1st edition,p.

31.

to

asym-

215]

214,

Asymmetrical Relations from

metrical relations, or

told,by those

who

the

225

of generalphilosophy. We standpoint

advocate

this

that the whole contains opinion, within itself, that it synthesizes and that it performs differences, diversity other similar feats. For unable to attach any precise part, I am my these to But let us do our best. phrases. significance is " /z The proposition greater than b? we are told,does not really say anythingabout either a or 6,but about the two together. Denoting the whole whidi they compose by (ab\ it says, we will suppose, "(ab) of magnitude.1"Now to this statement contains diversity neglecting all for the present generalarguments there is a special objectionin of asymmetry, the case (ab)is symmetricalwith regardto a and i, and thus the propertyof the whole will be exactly the same in the case are

"

"

where

is

a

greater than did

Leibniz, who

do

we

not

it is

and

differ in "

and b.

A

whole

(bo),as

For

is

the precisely

greater than b

greateris,in

same

whole

first case,

the

must

we

from

the

(ba) consist of

respectwhatever

no

a

(ab) and

the containingprecisely to

greater than

theory,and

consequent.

nor

shall be forced back

their relation.

a

is

had

a.

therefore

antecedent,which the consequent; evident that,in the whole (ab) as such, indeed sufficiently

asymmetry, we and

b

is the

consider which

(ab) from

whole

where

it

there is neither antecedent a

case

this fact,as appears clearly plausible, perceived in his third quotation. For, way of regardingratio,

the above

from

in the

as

accept the monistic

not

render

to

reason

no

b

"

same

the

save

and

u

In

order

do

if

whole

are

we

the

to

the precisely

sense

distinguish to explain

to

parts and

same

parts,

of the relation between

b is greaterthan

"

a

are

tions proposi-

and givingrise therefore constituents,

;

their difference lies solely in the fact that

a

relation of

a

to

6a in the second,a relation

the distinction of sense, Le. the distinction between an which is the its monistic and relation one converse, theory asymmetrical of b to

a.

Thus

whollyunable

explain. almost a more generalnature might be multiplied Arguments relevant. The but the following argument seems peculiarly indefinitely, and the relation of whole and part is itself an asymmetrical relation, is distinct from all fond of telling whole monists are peculiarly us as Hence and collectively. when we is its parts,both severally say "a the monistic be of if theory correct, to assert 6," we really mean, part of and of the whole a ", which is not to be composed something confounded with 6. If the proposition concerningthis new whole be not of whole and part there will be no true judgments of whole and one of relations is

to

of

"

"

part, and

false to say that a of adjective the whole. If the

it will therefore be

an parts is really of whole and part,it will require a

If,as of

a

a

and

is the

sum

desperate measure, b is not

sense

the

is one proposition for its meaning,and so on. one asserts that the whole composed new

to admit that a whole ", he is compelled of SymbolicLogic)of its parts,which, besides

distinct from

(in the

new

the monist

relation between

Order

226

beingan have

renders it inevitablethat position, its parts"a view which we as regards symmetrical

abandonment

the whole should be seen already

the view that the

[CHAP,

of his whole

to be fatal. And

hence

find monists driven to

we

has onlytrue whole,the Absolute,

partsat all,

else are quite anything true view which,in the mere contradicts a statement,unavoidably which holds all propositions to be in the itself. And surely an opinion condemned by the fact that,if it end self-contradictory is sufficiently be accepted, it also must be self-contradictory. that asymmetrical relationsare uninhave now telligible 216, We seen

and that

in regard to propositions

no

it

-no

or

"

both the usual theoriesof relation*. Hence,since such

on

involved in Number, Quantity, Order,Space, Time, and

relationsare

of Mathematics philosophy hardly hopefor a satisfactory so longas we adhere to the view that no relation can be "purely the logical external" As soon, however, as we adopta differenttheory, which have hitherto obstructedphilosophers, to be are seen puzzles, artificial.Among the terms commonlyregarded those as relational, and transitive"such as equality and simultaneity that are symmetrical called identity of reduction to what has been vaguely of are capable into sameness of relation content,but this in turn must be analyzed of a term are, in fact, other term. For the so-calledproperties to some onlyother terms to which it stands in some relation; and a common of two terms is a term to which both stand in the same property

Motion,we

can

"

relation. The

into the realm of logic is necessitated presentlongdigression of order, fundamental importance and by the total impossibilit

by the order without abandoning the most cherished and wideof explaining spread of philosophic where order is dogmas. Everything depends,

of sense, but these two concepts concerned, upon asymmetryand difference to the traditional unintelligible logic.In

are

the next

have to examine the connection of difference of in Mathematics

as

sense

we chapter

shall

with what appears

difference of sign.In thisexamination, thoughsome

will.still be requisite, we shall approach againto mathematical pure logic and thesewilloccupy us wholly the succeeding topics; throughout chapters of this Part.

*

The

groundsof these theorieswill be examined view in Part VI,Chap.LI.

from

a more

general pointof

CHAPTER

DIFFERENCE

OF

have

WE

217.

greater and connected

sign.

(though is

far

as

that

DIFFERENCE

order

always have

and

two

The

west, etc.

identical)with

not

notion

AND

seen

these

less,east

It

SENSE

now

that

relations,and

XXVII.

the

OF

depends senses,

SIGN.

asymmetrical upon before and after*

as

difference

of

sense

mathematical

is closely

difference

of

fundamental

of

importance in Mathematics, and of any other notions. explicablein terms realized its importance would The first philosopherwho to be Kant, seem Versuch den Begnff der neg"tiven Gro"c in die In the Wdtwdsheif of the difference between dn^ifuhren (1768),we find him aware logical and the of and In discussion the opposition positive negative. opposition Grunde des Unterschwdes der Gegenden im Rawm Von dem ersten (1768), full find realization of the of in a we importance spatial asymmetry be wholly relations,and a proof,based on this fact, that space cannot is, so

relational*. this

a

I

can

But

see,

it

not

seems

with

doubtful

difference

whether

he realized the

connection

of

of

not sign. In 1768 he certainlywas of aware regarded pain as a negative amount pleasure,and supposed that a great pleasureand a small pain can be view added which both and to give a less pleasure")a seems logically psychologicallyfalse. In the Prolegomena (| IS), as is well known, relations a ground for regardingspace of spatial he made the asymmetry of intuition, perceiving, from form the discussion as a mere as appears Leibniz could not of 1768, that space consist, as supposed, of mere and to his adherence relations among unable, to being objects, owing in the preceding chapter, the logicalobjection to relations discussed

asymmetry

of the connection, since he

"

to

free

from

contradiction

the

notion

of

absolute

space

with

metrical asym-

points. Although I cannot regard this advance Kantian later and as an that more theory distinctively upon for having first called to Kant of 1768, yet credit is undoubtedly due of the attention to asymmetrical relations. logicalimportance relations

between

its

*

Sec

ed. Hart, especially

t

Ed.

Hart,

Vol.

II, p.

83.

Vol. II, pp.

386,

391.

Order

228

By

218.

difference of

the difference between

[CHAP,xxvn in the

I mean,

sense

presentdiscussion

asymmetricalrelation and

an

at

its converse.

least,

It is

a

fact that, givenany relation J?, and any two terms logical to be formed of these elements, the one two propositions a, i, there are b the other to I call These a. (bRa) relating aJRb), a to b (which relating are though sometimes (as in the case two alwaysdifferent, propositions the other. In other cases, such as logical plication imeither implies of diversity) does not imply either the other or its negation;while the one in a third set of cases, the one impliesthe negationof the other. It is onlyin cases of the third kind that I shall speakof difference of sense. But here another fundamental In these cases, aRb excludes bRa. logical where aRb does not imply bRa there In all cases fact becomes relevant.

fundamental

related relation,

is another

is,there is

that

relation of

The

impliesaRb. relation is

such

relation R

a

J?,which

to

R

to

of series,of the distinction of part of mathematics.

source

219. to the

and

and

That

further,bRa

is difference of

R

signs,and indeed

This

sense.

of the

greater

and especially importanceto logic,

be raised with

theory of inference, may

aRb

Are

considerable

questionof

A

;

a.

intransitive. Its existence is the

and symmetrical,

one-one,

impliesbRa

aRb

b and

hold between

must

regardto

difference of

sense.

do they only differ or reallydifferent propositions,

bRa

relation R9 and may be held that there is only one that all necessary distinctions can be obtained from that between aRb of speechand It may be said that,owing to the exigencies and bRa.

? It linguistically

and that this gives are we compelledto mention either a or b first, writing, is greater than b and b is less than a a seeming difference between *

"

a^; but that, in

identical. But if are propositions we to explainthe indubitable we distinction between greater and less. These two words have certainly each a meaning,even when terms mentioned as related by them. no are And theycertainlyhave different meanings,and are certainly relations. take

Hence

if we

the

are

and

this

else

to hold that

into each we

two

shall find it hard

"

we proposition,

same

or

view

are

less enter

false;

these reality,

"

of

shall have

a

is greaterthan b

shall have these to

"

and

maintain

to

which propositions,

hold

"

b is lessthan

that both seems

be

one

greater

obviously

that what

is neither occurs really mentioned by Leibniz in the

of the two, but that third abstract relation In this case the difference between passage quoted above. less would

"

a

greaterand

reference to the terms a and b. a essentially involving be maintained without circularity ; for neither the less is inherently the antecedent,and we can only say

this view cannot

But

greater nor the that,when the greateris the antecedent, the relation is greater; when the R the

the relation is ks$. less,

and

R

are

Hence, it would

distinct relations.

into analysis

We

cannot

seem,

we

must

admit

that

escape this conclusion by chapter.We there

adjectives attemptedin the last

218-221] Difference of Seme analyzedaRb

into

a/9and ba.

and

Difference of Sign

229

But, corresponding to every 6,there will

" and j39and corresponding to adjectives,

be two

every a there will also Thus if R be greater,a. will be "greater than A be two, a and fc and a " less than A? or vice versa. But the difference between a and a *

presupposes that between

explainit.

cannot "

bRa

be

must

I

a

Hence

and R

R

must

be

and distinct,

"aRb

implies

genuineinference. to

now

come

between R and R, and therefore greaterand less,

the

connection

between

difference of

sign. We shall find that the former, beinga difference which onlyexists

difference of

sense

latter is derivative from

and the

between terms which either correlated with, asymmetricalrelations. But in certain cases are, or are find of detail which will demand shall some we discussion. complications The difference of signsbelongs, numbers to and traditionally, only and magnitudes,

is

intimatelyassociated with addition. It may be allowed that the notation cannot be usefully employed where there and is no even addition, that, where distinction of sign is possible, is in generalalso possible. addition in some But we sense shall find that the difference of

make

To

subtraction.

and

signhas

with addition very intimate connection this clear,we must, in the first place,

no

and magnitudeswhich have no realize that numbers clearly sign are different from such Confusion this as are on positive. radically pointis quitefatal to any justtheoryof signs. and negative numbers 220. Taking firstfinitenumbers, the positive R the between arise as follows*. relation two Denoting by integersin mRn the proposition virtue of which the second is next after the first, is equivalent to what is usually expressed by m + 1 n* But the present and does not depend upon generally theorywill apply to progressions Part in of cardinals II. In the proposition the logical developed theory and n are considered, when theyresult from the as m m fin,the integers mRn and nRpy to be whollydestitute of sign. If now definition, logical we Every power of " is an put mIPp\ and so on for higherpowers. shown is easily to be the same and its converse asymmetrical relation, =

Thus

to qR*m. mR*q is equivalent written which the two propositions are are commonly m-i-a^q and Thus the relations Ra, Ra are the true positive and q m. a and these,though associated with a, are both wholly negativeintegers;

of R

power These

"

it is itself of R.

as

=

distinct from

it.

is obvious

sense

Thus

and

in

this

case

the

connection

with

difference of

straightforward.

regardsmagnitudes,several cases must be distinguished. We have (1) magnitudeswhich are not either relations or stretches, (2) stretches,(3) magnitudeswhich are relations. 221.

*

in the

I

As

here, as givethe theory briefly " 233. chapteron Progressions,

it will be dealt with

more

fullyaud generally

[CHAP,xxvn

Order

230

(1) Magnitudes of this class are themselves neither positivenor in Part III,determine as explained negative. But two such magnitudes, and these are alwayspositive or either a distance or a stretch, negative. These

alwayscapableof

moreover

are

neither relations

magnitudesare obtained

of

are

the stretches,

nor

different kind from

a

But

addition.

since

original magnitudesthus

new

Thus

set. original

the

our

the difference

intermediate between the collection of pleasures or pleasures, in the but in the one case is not a pleasure, a relation, pleasures, of two

two

other

class.

a

in generalhave (2) Magnitudesof divisibility

sign,but when they acquiresign by correlation. other collections by the fact that it

magnitudesof from distinguished

they are

stretch is

A

no

stretches

consists of all the terms

By

terms.

series intermediate

a

combiningthe stretch with

relation which

and

from

a

to

terms

of the

two

given

asymmetrical

end-terms, the stretch itself That is,we can distinguish asymmetrical.

becomes

(1)the collectionof

sense

one

between

its

between

exist

must

acquires sense, the,terms

of

between

i, (3)the

a

and 6 without from b to

terms

complex,being compoundedof (1) and

a.

regardto order,(") Here

of the

sense

one

(") and (3) are constitutive

the other negative. be called positive, must relation. Of these two, one Where our series consists of magnitudes, usage and the connection with

addition have decided that,if

negative. But

where, as

and (3)is Z", (")is positive

is less than

a

Geometry, our series is which is to whollyarbitrary in

composed of and positive

not

be it becomes magnitudes, relation to addition, which negative.In either case, we have the same which is as follows. Any pairof collections can be added to form a new but not any pair of stretches can be added to form a new collection, For this to be possible be constretch. the end of one stretch must secutive to the beginning of the other. In this way, the stretches ab, be can

be added

greater than of them. as

to form

the stretch

either;if they have

In this second

the

case

If

ac+

ab, be have the

same

is

ac

sense,

different senses, ac is less than one addition of ab and be is regarded

the subtraction of ab and

If respectively.

our

subtraction of their

adding or addition

eb, be and cb beingnegativeand positive stretches are measurable, addition or numerically

the subtracting or

as negative,

subtraction. is

will

measures

give the

where stretches, But

the

whole

evident,depends upon

measure

these

are

of the

result of

such

to

as

allow

and oppositionof positive

the fundamental

fact that

our

series is

generatedby an asymmetricalrelation. (3) Magnitudeswhich are relations may be either symmetrical or relations. In the the field former of asymmetrical case, if a be a term of

one

of

them, the other

of the various

terms

are

in fulfilled*, may be arranged

are

greateror smaller. This arrangement may *

series

Cf.

if certain conditions fields,

accordingas

" 245.

their relations to

be differentwhen

we

a

choose

221,

222] Difference of Sense

some

term

be chosen

When

some

or

231

shall suppose a to therefore, we present, have been arrangedin a series,

the terms

all places in the series are

by more occupied

one

some

a

once

for the

Difference of Sign

of terms between term; but in any case the assemblage other term m is definite, and leads to a stretch with two

than

or

a;

for all

happen that

it may

We

other than

and

a

aad

senses.

may then combine the magnitudeof the relation of a to m with one other of these two senses, and so obtain an asymmetrical relation of the Hke to my which, will have magnitude. Thus the relation, original of

relations may be reduced to that of asymmetrical symmetrical relations. These latter lead to signs, and to addition and subtraction, in exactly the safbe way as stretcheswith sense ; the onlydifference being case

that the addition and subtraction are we

called relational. Thus

difference between source

The

which

case

in all cases

the two

of the difference of

now

senses

of

of the kind

which,in Part III, of magnitudes having sign,the

an

relation asymmetrical

sign.

discussed in connection

we

is the

with

stretches is of

in Geometry. We have here a magnitudewithout importance and some relationwithout magnitude, intimate an sign, asymmetrical

fundamental

connection between

magnitudewhich

the

two.

The

combination

of both

then

givesa

sign. All

geometrical magnitudeshavingsign curious in the case of a complication volumes. Volumes are, in the first instance, signless quantities ; but in or negative.Here analytical Geometrytheyalwaysappear as positive the asymmetrical relations (forthere are two) appear as terms, between but one which yet has an opposite which there is a "symmetrical relation, relation. of an asymmetrical of a kind very similar to the converse discussed, be here briefly This relation, an as exceptional case, must The descriptive line is a serial relation in virtue of 222. straight which the points of the lineform a series*. Either sense of the descriptive line may be called a ray, the sense being indicated by an straight has

But

arise in this way.

there is

relations, or other of two non-coplanar rays have one f. This which be called rightand left-handedness respectively may of the usual and is the essence relation is symmetrical but not transitive, distinction of rightand left. Thus the relation of the upward vertical and to a line from south to to a line from north to east is right-handed,

Any

arrow.

*

two

See Part VI.

t The

between

two

cases

the two

are

sorts of

illustrated in the

coordinate

axes.

figure. The

difference is the

same

as

that

Order

232

[CHAP,xxvii

though the relation is symmetrical,it is by changingeither of the terms of the relation changedinto its opposite That is,denoting by #, left-handedinto its converse. right-handedness But

is left-handed.

east

if by L (which is not j?), shall have we right-handed,

ness

A

ALB, ALB, ARB, BRA,

ARE,

rays which

be two

and B

are

mutually

BLA, BLA, BRA.

lines gives rise to eightsuch .straight is,every pairof non-coplanar

That

and four left-handed. The right-handed, difference between L and R, though not, as it stands,a difference of and negative, and is the the difference of positive sense, is,nevertheless, always reason why the volumes of tetrahedra,as givenby determinants, in the have signs.But there is no difficulty following plainman's relations. The plainman reduction of rightand left to asymmetrical of which relations,

takes

four

of the rays

one

are

when

(sayA ) as fixed "

he is

sober,he takes

be the upward vertical and then regards rightand left as

A

to

of properties

"

relations of any as thing, In this way, rightand left become two of transiliveness, of and even have a limited degree relations, asymmetrical series (inChapterxxiv). in the fifth way of generating the kind explained It is to be observed that what is fixed mast be a ray, not a mere straight line. For example,two planeswhich are not mutuallyperpendicular section, not one are rightand the other left with regardto their line of interto this but only with regardto either of the rays belonging

the

single ray B, or, what comes pointswhich determine B.

line.*

But

when

this is borne

the

to

same

mind, and

in

when

consider,not

we

but complete rightand planes, throughthe ray in question, semi-planes, left become asymmetrical Thus the signs and each other's converses. associated with rightand left,like all other signs, depend upon the

therefore, asymmetry of relations. This conclusion, may to be general. Difference of

223.

sign,since is unable

it exists in to

deal.

relationswhich

are

sense

cases

And not

is,of

than general

course, more

mathematics

with which

difference of or transitive,

signseems

are

not

now

be allowed difference of

(at least at present) to scarcely applicable

connected intimately

with,

transitiverelation. It would be

absurd,for example,to regardthe relation of an event to the time of its occurrence, or of a quantityto its' of sign. These relations are what as conferring a difference magnitude,

some

Professor Schroder calls mchvpft^, i.e. if

theycan

never

hold

their square is null. of

they hold between a and b, between b and some third term. Mathematically, These relations, then,do not giverise to difference

sign.

^* This

that the passage from requires anglesmade by their t Algebrader Logik,Vol. Ill,p. 328. in Schroder, (reference repeating ft.). ta"

one

of the acute

the

one

planeto

the other should be made

intersection. Professor Peirce calls such

relations

""w-

Difference

223]

222,

relations

either

are

what

But

pair

I

and

are

should

emit

from

kind

seem

that

of

the

what

by

the

two

while

incompatible the

such

of

rise

See

*

it

f and

The

Thus

not

without

examine)

obtain

pleasure.

money

paid sense

be

with

to

"f;

is

latter

difference

the

from

is,

sign

so

transitive

extensions

present

pleasure

and

only we

of the

patibility incom-

tions, proposinotion

in

difference a

that

special gives

conclude

may

tions, rela-

variously

terms

always

of

the

by

of

fact,

to

are

difference

from

same

asymmetrical

by correlation

such

of

the

in

case

"

in

of

one

class

with

special of

the

important

identified

All

is

coexist of

kind

some

be

it

Leibniz,

to

consists

propositions

This

most

a

patible incom-

two

contains

one

stituted con-

are

oppositions

cannot

true

other.

This

but

extended

but

that

respect

series)

derived

derived

of

mathematical

in

negative need

sign.

may

Philosophy

the

relations.

primarily

relations

fact

the

predicates into

and

different

two

which

only

are

incompatible

enter

the

means

no

very

of

or

to

different

incompatibility

be

them,

regard

subsequent

to

sense.

author

(Cambridge

1900),

20.

19,

pp.

the

of

original opposition

the

\ye

such

to

by

converse

is

"

which

from

in

given

a

The

both

contains

incompatibility;

argument

related

is

difference

the

to

our

but

mutually

between

only

there

cannot

or

cannot

other

of

terms

logic,

general

the

blue,

that

thus

are

usually belongs,

(which to

differ

which

form,

fact series.

place,

generally,

more

or,

certain

case

which

terms

and

The

relations,

these

the

whole

a

spatio-temporal

same

existent, a

that

fact

the

of

a

oppositions, synthetic incompatibility*,

in

only

instead

terms,

From

called

be

differ

mentioned

above

kind.

With of

and

?

of

analysis

asymmetrical red

eater.

good

:

aversion

an

opposition

an

of

and

opinions.

converse

opposition

same

may

have

desire

believe,

to

us

relations

opposites

attempt

condemned

mutually

the

to

were

233

led

which of

ugliness,

if I

to

me

two

to

of

and

to

into

instances

and

universally

they

magnitudes

in

complex,

rather

analogous

usual

of

has

account

concepts

the

of

Dfference

above

compound

say

some

and

the

so

pain, beauty

very

others,

the

to

Seme

sign,

or

we

are

evil, pleasure last

with

magnitudes

All

of

of

and

Economics,

logical that

a

opposition

money

received, Arithmetic.

the

of

theory be

must

man

The

elementary

by

error,

pleasure

which

is

an

paid and

pain

pain

endure is thus

opposition

be

taken

psychological

(whose to

may

of

pain,

and

correlated

positive

and

as

positive

correctness mast

with

pay that

negative

to

of in

XXVIII.

CHAPTER

BETWEEN

DIFFERENCE

THE

ON

SERIES.

CLOSED

WE

224.

the

have

can

of the

discussions purelylogical with

attention

our

free mind

a

solution

the

As

in the

respectablecontradictions

and

ancient

most

turn

aspects of the subject.

mathematical

more

the end

to

come

now

order, and

with

concerned

AND

OPEN

of

notion

to

of the

infinity

philosophy of order,it has been necessary depends mainly upon to go into philosophical length not so much because questionsat some But we they are relevant,as because most philosophersthink them so. work. of this remainder shall reap our reward throughout the mately ultidiscussed in this chapter is this: Can be The to we question in does if what and so, distinguishopen from closed series, We the distinction consist? have seen that, mathematically,all series tive that all are are generated by an asymmetrical transiopen, in the sense relation. But must we distinguishthe different philosophically, not in which must this relation may arise, and especiallywe ways correct

a

"

confound

the

terms

with

that

plain

that

there

between,

for

where is

difference 225.

society. precisely.

Where is

xxiv,

intransitive

the

relation powers,

method

relation

of any term the relation

identity. (It

R"\

with or

in

Rn

is assumed

the

other

of

our

series,two next

but

have

that

R

hand, it may

series

the

other

series

closed

"

pedigree and to

is

is

a

the

express

finite, and

explained in ways of the relation out

different according radically

generatingrelation,and n be arise. cases Denoting the may for higher and one on so by R\

be the

the can

is

start

we

If R

the

transitive

obtaining a

which

closed.

to

first

the

in

of

a

quite easy

in

terms

and

circle,or

a

to

it practically,

And

open

it is not

of

the first term, if there be one, Rn term, and gives no new On

between

reference

no

essential.

are

But

number

generated

the

the series is open as the number of terms

thus

terms

difference

some

involves

relation

instance,a straightline and

series

Chapter

with

such

admiration

mutual

the

this

where

case

only is

a

Rn^ there

and values, zero F or relation.) starting

of

one

one-one

bringsus is

no

two

to the

instance

last term of the

happen that, startingwith

;

and

relation

any

term,

225] Differencebetween Open

224,

bringsus back

B*

to that

and

again. These

terra

Closed Series

two

the

are

235

onlypossible

In the first case, we call the series open ; In the secood, we call it closed. In the first case, the series has a definite beginningand

alternatives.

end

in the second

;

the

In

terms.

case, like the

first case,

anglesof

transitive

our

relation "a power of R disjunctive this relation,which By substituting of the second

becomes

simplereduction of any two

terms

second

and

a

of

m

types. is

type

call JT, for j?,our series in the second case such no

may But

possible.For now, be justas well

series may

our

power of J?,and the questionwhich the other two becomes whollyarbitrary. We

of R

power

a

of the six

to the

is between

a

as

peculiar

no

asymmetrical relation is the greater than the (n"iyOo^

not we

polygon,it has

a

the

relation

taken to be

of any three terms

duce, might now introof four terms, and then the resulting separation in We should then regard explained Chapter xxv.

firstthe relation of

relation

five-term

of the

three

relation resulting before

case

not

This

method.

the

and

;

of two.

one

terms

and

a

before

a

to

a

and

a

before

then

order

from

different,as fixed,and c and

the

m

relation between

c

case

and

an

open series may

the other

same

and

c

In

get a

comes

m

g that the that of the order from

as

be.

series, any two be described,

in which

other terms

may

g variable,we from g" obtained

In

:

the

OT, and

say of any two c to g is the

or

m,

in which

senses

comes

can

of the

the

is transitive and

two

be illustrated as follows

may

We

a.

find that

series is

define two

m

fixed,and

as

asymmetrical.But which not the our was whollyarbitrary, the generatingrelation is,in reality, of five one terms, There is,however,in the case contemplated, a simpler of

in which

one

sense

of the other

first term

the

here

five-term relation

in the

terms

this way, considering transitive asymmetrical

a

transitive symmetricalrelation

paira, m (or m, a, as the case may be). But this pairc9 g be of abstraction, transitive symmetricalrelation can, by the principle analyzedinto possessionof a common property,which is,in this case, and the relation with the same the fact that #, m generating c, g have to the

of the

in this case, not essential. the four-term relation is, of the series, closed series, do not define a sense even m a and Thus

sense.

told that

are

is to

a

But

either direction. from

to start

are

a

series is defined. the other.

not

by

the east

must

by

go

will have

some

The

by

m

may

the west If

east.

definite

; now

a

m

in this

case

when

get

to

a

we m

d, and decide that

in we

on

England to are

to

positionin

the

sufficient to

Vailati'sfive-term relation will then

Zealand

New

take India

consider any

by

seems

and

a

in

of the the way, then a sense but includes one portionof the series,

go from but if we we

from

third term

takingd

stretch adm we

start

can

we

take

we

and reach

Thus

the

if now

:

other

series which

k will come In this series, way of d. Thus the between d and wz, or after m.

reaches

c?,or a, d" m

or

precedem

But

on

the

either way,

we

term, say fc,this starts

with

either between three-term

a a

and and

relation of

definite series. generatea perfectly consist in this,that with regard to

Order

236 the order adm, k

(orafter)any other

before

comes

[CHAP,xxvm I of the collection.

term

But it is not necessary to call in this relation in the presentcase, since the relation may be formally three-term relation suffices. This three-term of our collection a defined as follows. There is between any two terms relation which is a power of R less than the nth. Let the relation between Then if x is less than y, we Ry. a and m a and d be J?% that between

assignone

will be also between

There 772

if

adm;

to

sense

If

the relation Rn~v.

The

y^ then

n

x

"

other.

between

a

is greaterthan

n

and z/;

"

to that of R and JR. corresponds are simply ordered by correlation with their with smaller numbers precedingthose with

series

of the

terms

numbers

is less than

x

assignthe

z/, we

and d the relation Rn~x^ and

a

asymmetry of the

hence the

greaterthan

is

oc

two

cases

y" those Thus there is here

and

x

need of the five-term relation, no ones. larger thing everyis which itselfreduced the three-term effected to relation, by being But the closed an asymmetricaltransitive relation of two numbers. from the series is stilldistinguished by the fact that its first open one is arbitrary. term where A very similar discussion will apply to the case 226. our three To relations of is terms. series generatedby keep the analogy relation of the above case, we will make the following with the one-one assumptions.Let there be a relation B of one term to two others,and let the

others the extremes. the two called the mean, Let the extremes when determined be uniquely are given,and let

the

mean

one

extreme

Further

as

uniquelydetermined by

be

let each

each term also

be

term

one

that

occurs

are

as

extremes, let there be

be

Then

extremes.

b and

fact constitutes

are

two

onlyone,

c

this

togetherin

occur

terms, exceptional

d is the

b and

or

c.

if,our

extreme, and

as

occur exceptions)

two

b

c or

is mean, d is one

b is the

relation in which

relation to

new

most

relation in which

in which

relation between

a

besides b will have there

will

also

occur

mean

always(exceptwhen

of the extremes, and another

one

a

and the other extreme.

mean

(with at

extreme

as

possible terms) a exceptional

two

is

occurs

if there Finally,

mean.

and d

that

term

the

and

c one

other

only one

c

'of the

relations.

onlytwo

c, and

of the

mean

and

mean

and b

This term

relation,if there collection beinginfinite,

By

means

of this

relation be open series. If our two-term be evident ; but the same this is sufficiently result can asymmetrical, we

proved if our

construct

can

relation is

symmetrical.For there will be at asymmetricalrelation of a to the onlyterm which is other term. This relation multiplied and some by

two-term

either end, say a, the mean between

an a

the nth than

of our two-term power the number of terms in

holds between

a

and

of which collection,

than

an

"

givesa

a

where n 4- 1 is any integerless relation, will give a relation which our collection, number of our (not exceeding n-f 1) of terms

terms

relation of

one a

and

onlyone

to this term.

is such that Thus

we

number

no

obtain

a

less

correlation

between Open 225-227] Difference of

with

terms

our

series with

has

for

a

natural

numbers, which

of its ends.

one

Let

other

each

the

hold between

least :

this may

number

to

of terms

of

c

will have

all the relations of the form

Of

given terms, there will relation of principal

two

relation

every term

others,which

two

be called the

collection will have

of the collection be

be our

Then

n.

Pm

in which

one

two

Let

terms.

of

term

every

is

m

our

relation P*, where x is every other a principal integernot greaterthan n/2. Given any two terms c and g of the

some

to

do not have

collection,providedwe if n

P*.

three-term original

extremes.) Then

the relation P

relation

which

the

to the z*egard

will have

collection

to

generates an open collechand, our tion

If, on is finite,then we shall obtain relation be P, and first suppose it

two-term

our

symmetricalwith

was

237

the other

(Itwill be symmetricalif our symmetrical. our

Series

exceptionalterms, but

no

closed series.

a

the

Closed

and

be odd),let us have

defines

a

of the

sense

where cPxg-,

which series,

We

a\

have

may

cases

gPy~*k,or, if

be shown may x

+y

which

case

is less than

a

may

where y is also less than n/2, three than

cP^g (a

will not

arise

ra/2.This assumption as

follows.

If cP*fc,

arise,assumingy is greater is less than n/2,we may

gP**yk" or, if x+ y is greater than w/2,we may havegp* *~*k. These three cases trated illusare relation.) (We choose alwaysthe principal shall say, in these three cases, in the accompanyingfigure.We

have

that,with regardto the k

before

comes

c

is such

eg*,(1) k

comes

c

after

But

if

n

is even,

that cPn'*c'. This term

we

have

c

and g,

(") and (8)

shall say all is odd, this covers which to consider the term c\

If y is less than #, and and g in the sense eg. If n

and g.

that A* is between cases. passible

sense

"P*~y^,

we

to c ; certain sense, antipodal the above method in the series when

is,in

a

may define it as the firstterm definition is adopted. If n is odd, the first term

will be that term

we

of

of

definite order, a (3) for which cP**-1"1*. Thus the series acquires is the first term but one in which,as in all closed series, arbitrary, class

onlyremainingcase is that where we start from four-term five terras. relation has,strictly and the generating speaking, relations, This is the case of projective Geometry. Here the series is necessarily closed; that is, in choosingour three fixed terms for the five-term 227.

The

there relation, these three may

any restriction upon be defined to be the first.

is

never

our

choice

;

and any

one

of

Order

238

Thus,

228,

when

it

be

the

but

R

only

fixed)

when

besides

the

an

the

addition

relation

the

them "

the

and

open, nature

a

mathematical

all

of

the

distinction,

be

variable

every

or

of

exists,

it

to

which

closed the

open

however,

regard

with

every

closed,

is

a

of

yet

genuine

be

or

to

and

there the

there

can

be

transitive

it

series

is,

terms

generates

in

distinction

philosophical

metrical asym-

may

fixed

which

a

definite.

beginning

more

closed

if R

considered

transitive

discrete,

not

genuinely

perfectly

the

is

having

term

is

series, though

one

is

Now

a

may

that

involving

relation, which

is

it

is

series

be

(which

series

terms

R

series

a

which

is

xxnn

transitive

a

series,

series

must

variable),

as

one

the

Whenever

JR.

term

if

the

beginning

a

mathematically,

generating

importance.

of

other

of

cases

relation

although,

Thus

if

regard

must two

relation

regarded

one-one

to

series.

be in

Hence

asymmetrical

rendered to

are

arbitrary.

asymmetrical in

to

terms

by

arbitrary beginning.

an

some

with

two

generated

beginning

beginning,

involves

R

(which be

the

relation,

is

has

the

the

not

being

beginning, it

relation,

constitutive

two-term

It

two

any

no

when

series

Every

;

relation

the

either

closed

is

:

between

has

it

arbitrary

up

sum

relation

asymmetrical open

to

[CHAP,

be

can

regard between

rather

than

CHAPTER

PROGRESSIONS

is

IT

229.

that

namely

postpone

infinity of

the

time

which

the

to

series

for

term

order

the

in

with

form

supposed myself

infinite

series, I shall

belong.

here

arising out only to give

not

presupposing numbers*.

are

those

But

since

series, and

since

terms.

of

themselves

numbers,

natural

such

of

a

simplest type

difficulties

concern

considered

the

the

of

in

be

to

the

NUMBERS.

numbers

all the

series, and

of them

now

term,

natural

Part

such

ORDINAL

consider

to

the

next

elementary theory The

AND

now

to

XXIX.

which

without the the

be

can

correlated,

requiring any

change

numbers

natural

of the

are

a

whole

of Arithmetic and particular case be of out such without one series, developed Analysis can any any it better definition is to of to which number, a give appeal progressions involves no appeal to number. A progression is a discrete series having consecutive a terms, and The end, and no being also connected. beginning but meaning of connection of number, but this by means was explained in Chapter xxiv be

series Speaking popularly, when a each or more being a series parts, and Thus numbers instants series which for itself. a together form is not do Whenever two connected, and so parallelstraight lines. of a transitive tion, series is originallygiven by means a asymmetrical relaconnection that terms two can we by the condition express any have the of our series are But to generating relation. progressions six of the the of our that in kind first series be generated are may relation. In order to namely, by an asymmetrical one-one pass ways, from this to a transitive relation, we before employed numbers, defining

explanation cannot is

connected

not

definition is

not

the

triumphs

principleto

ancient

*

will

of

one

The

present

Mathematique*, in

RdM,

(leorg

given

Vols. Cantor,

now.

falls into

relation

transitive

the

it

as

serve

the

any

II, "2.

VII

and

needs

I have The

the

numbers

modern

of

VIII.

of

power since

now,

chapter closely

Vol.

two

are

mathematics

of this

case.

follows

Peano's

given subject

a

due,

to to

in

have

See

treatment

the

main,

This

excluded.

be

Arithmetic.

mathematical is

relation.

one-one

to

It

adapted

Fbrmnlaire of the Dedekind

an

de

subject and

Order

240 definition which

The

used to be

which principle,

This

induction.

obtained

is to be

want

we

[CHAP,xxix

been

It is now investigated. closely

more

which and

as

extension possible

are

to be the

seen

have

principle upon law

which principle, gives is the distinguishing mark finite,

the

to

of mathematics

concerned, the commutative This

distributive law*.

of the

form

one

the widest of

far

depend,so

ordinals

mathematical

fuge regardedas a more .subterhas proofwas forthcoming,

results of which no other eliciting foundations gradually grown in importanceas the for

from

may be stated as follows : class of terms #" to which belongsthe first term

It progressions.

Given

any

of any

next after belongsthe term of the progression of the progression belongingto ", then every term of the any term belongsto $. progression in another form. Let ""(.r) be We principle may state the same as soon function,which is a determinate proposition a propositional will is function of in and is given. Then ""(#) a as x generalbe x, of a be the of If member value to false x. a ~x true or according after .r. Let "f" next let seq x denote the term (x) be true progression, to which

and progression,

when be

x

true

It then

and let ""(seqx) progression, whenever (x)is true, where x is any term of the progression. "j" the of mathematical induction,that "j" follows,by (x) principle of

first term

is the

certain

a

in question. if x be any term of the progression is as follows. Let completedefinition of a progression

is alwaystrue The

and relation,

one-one asymmetrical

any of u

has the relation of R

to

there be

at

least

the relation R

to

any term

Let

at least one term

relation R as

of the terms to which

of u, and

to

be

Then

to

term

some

whollycontained

of

of

u

Let

u.

s

class

u

belongingto in any

does not

which

be any class to which

do not

which

that

every term also belongingto the class u. a

have

the relation R

belongsalso every term

ordered

can

be

one

term

We

then define the term x

of the

class such

be

both

u

and

of s ;

u

which

have

belongs to any

has

the

and let u be such

class s

the above conditions. satisfying as R, is a progression!. by r elevant to finiteArithmetic progressions, everything proved. In the first place,we show that there can only be

w, considered Of such 230.

of

term

some

term

one

u

R

of

which

u

(x being

a

does not

M),which

the relation

have

to which

may

x

the relation R

of any term has the relation R as the successor

be written seq

x.

to

u.

The

definitions and

of addition, and division, subtraction, properties multiplication, positive *

The other form, a(" + y) Namely (a -f /% a" -t-ay. a" -f ay, holds also numbers,and is thus independentof mathematical induction. t It should be observed that a discrete series generatedby a transitive open relation can alwaysbe reduced, in the preceding as we saw to one generated chapter, by an asymmetrical one-one relation., providedonly that the series is finite or a =

for infiniteordinal

progression.

=

Progressionsand

230]

229,

and

rational

shown

that between

two

From

this

terms, negative

fractions rational

any it is easy

point

Ordinal

to

are

Numbers

given; easily

fractions there is

advance

241 and

it is

alwaysa

easily third.

to

irrationals and

the

mathematical

what induction,

is chiefly

real

numbers*.

Apart

from

the

of principle

about interesting

this process is,that it shows that onlythe serial or of finite numbers ordinal properties used by ordinary mathematics, are be called the

logical properties beingwhollyirrelevant. By of numbers, I mean the logical of their definition by means properties ideas. This process, which has been explained in Part II, purelylogical here be We briefly recapitulated. show, to beginwith,that a onemay what

may

one

correlation

any

two

"r,

then

classes w, cannot

x

of such

be effected between

can v

be

which a

w, with

correlation

one-one

such

are

we

a

between or any two null classes, that,if x is a ", and J differs from

like condition call

for

v.

of the similarity

The two

possibility classes

w,

z".

be analyzable must beingsymmetricaland transitive, Similarity, (by the of into of possession a common principle abstraction) property. This define

we

u,

as

of either of the classes. When

the above-defined

have

v

the number

the two

classes

is otie ; and say their number generaldefinition of finite numbers

property,we

highernumbers ; the of whole demanding mathematical induction,or the non -similarity terms. part,but being alwaysgiven in purelylogical defined that arc used in dailylife, and that It is numbers so so

on

for

essential to

any

assertion of numbers.

It is the fact that numbers

and

are

have

that makes them logicalproperties important. But it is not mathematics that ordinary these properties employs,arid numbers might without to the truth of Arithmetic be bereft of them and any injury is solelythe fact that is relevant to mathematics Analysis. What form finite numbers maticians a progression.This is the reason why mathe-

these

and Kronecker have maintained e.g. Helmholtx,Dedekind, the ordinal that ordinal numbers are priorto caruinals ; for it is solely "

"

the conclusion that orBut relevant. that are of number dinals properties have resulted from confusion. to cardinals to seems a are prior and have exactlythe Ordinals and cardinals alike form a progression, be proved ordinal properties.Of either,all Arithmetic can same the without the other, beingsymbolically propositions any appealto to In order in different but identical, meaning. prove that ordinals it would be necessary to show that the cardinals are priorto cardinals, can onlybe defined in terms of the ordinals. But this is false,for the of the ordinalsf definition of the cardinals is whollyindependent logical There seems, in fact,to be nothingto choose,as regards logical priority, .

between

ordinals and cardinals, exceptthat the existence of the ordinals

See my article on the Logicof Relations,RdM, t Professor Peano, who has a rare immunity from *

See

Formulaire,1898, 210, note

(p.39).

VII. error,

lias

recognizedthis

fact.

Order

242

the series of cardinals.

is inferred from

the

in

cardinals; but the Similarly,

defined,they

when

cardinals

without

without

shall

we

appeal

any

the

imply

to

seen

are

defined

be

can

as ordinals,

The

defined

be

can paragraph,

next

[CHAP, xxix to

see

the

cardinals.

appeal to the and all progressions, progression, any

form a they essentially I shall now show, necessarily as implythe ordinals. of ordinals has been prevented hitherto by 231. The correct analysis the prevailing againstrelations. Peoplespeak of a series as prejudice taken in a certain order, and in this idea of certain terms consisting All sets of terms element. there is commonly a psychological have, a ll of which orders considerations, they are apart from psychological fields there w hose serial that a are relations, are is, givenset of capable; order. In some terms, which arrange those terms in any possible cases, either account of serial relationsare specially on more one or prominent, of their importance. Thus the order of magnitude their simplicity, or after seems numbers, or of before and instants, cally emphatiamong among the natural order,and any other seems to be artificially introduced choice. But this is a sheer error. Omnipotence itself by our arbitrary cannot : all that giveterms an order which they do not possess already order. Thus is the consideration of such and such an is psychological that in it is said when can we arrange a set of terms any order we please, the serial relations that consider of what is really meant we can is, any whose field is the given set, and that these serial relations will give of before and after that are compatible between them any combinations ordinals

;

but

with transitiveness and

connection.

fieldis the

relation whose with

this it results that

From

is not, properly a propertyof speaking,

a

Given

given set.

order

an

given set of terms, but of

a

serial

its fieldis given the relation,

given the field,the relation is by no means given. The notion of a set of terms in a given order is the notion of a set of terms considered as the fieldof a given serial relation ; but the consideration it ; but

the

of

is

terms

and superfluous,

that

of

the

alone

relation

is

quite

sufficient. We sets

then,regardan

may,

ordinal number

of serial relations which

relations have

what

their fieldscan

be

I shall call so

correlated term

of which

terms

vice z*er"d, virtue a

As

of the

in the

for term

that

class of such in order

of

that two

terms

of which

alwaysbe correlated with

second

the relation Q, and

numbers*, so here,we may, define the ordinal number abstraction,

as

in of

the class of like relations.

relations of generating

are progressions

relations will be the ordinal number

magnitude.

propertyof

common

of cardinal

case

of principle

the

will

the first has to the

given finite serialrelation

show

a

similar series. Such generate ordinally i.e.if P, Q be two such relations, likeness,

the firsthas to the second the relation P two

as

When

a *

Of.

of the

It is easy to all alike ; the finite

class is finite, all series that

"

111.

integers can

be

and Progressions

230-232]

of its terms

formed series

V, there is no have

unless 1 is to be

terms, where

n

of terms.

n

n

243

differentfrom ordinally Hence

there k

for which, as cardinals,

analogyin respectof

therefore define the ordinal number domains

Numbers are

of finite ordinals and

correlation

shall see in Part may whose

and similar, ordinally

are

different cardinal number

having a

one-one

Ordinal

as

infinitenumbers.

a we

We

the class of serialrelations

is a finitecardinal.

It is necessary, instead of' fields here,for no

excluded,to take domains

relation which

have one in its field, term can diversity implies though it This has a practical inconvenience, owing to the fact may have none. obtained be that n 4- 1 must by adding one term to the field; but the for is conventions involved to notation,and is quite one as point destitute of philosophical importance, 232. The above definition of ordinal numbers is direct and simple, but does not yieldthe notion of be regarded 7*th,"which would usually number. This far notion is ifieordinal more as complex: a term is not and does become not the so intrinsically wth, by the mere specification "

of

n

A

1 other terms.

"

is the nth

term

in

respectof

a

certain serial

has the term in question when, in respectof that relation, relation,

n

"

1

This is the definition of nth" showingthat this notion predecessors. serial but also to a specified is relative,not merely to predecessors, the various relation. By induction, finite ordinals can be defined cardinals. the A finiteserial relation without mentioning is one which relation is not like (inthe above sense) but it not implying equivalent any of finite serial If to it ; and a finite ordinal is one relations. consisting "

n a

ordinal such that,if the last term* of finiteordinal, n -f 1 is an series of the type n + 1 be cut off, the remainder,in the same order,is be

3,

of the Ti

In

type w.

+ 1 is

which,

one

becomes

of the

technical

more

This

n.

finite ordinal,in particular

serialrelation of the instead of its

its domain

confined to

when

type

a language,

givesby

induction

which cardinals

are

never

a

type field,

definition of every mentioned. ITius

though they are more say that ordinals presuppose cardinals, whereas relations, complex,since they presuppose both serialand one-one

we

cannot

cardinals

onlypresuppose

relations,

one-one

of the finite ordinals in order of

Of the ordinal number

magnitude,

equivalentdefinitions may be given. One of the simplestis, is such that any which that this number belongs to any serial relation, has first class contained in its field and not null term, while every a

several

of the serieshas

term

firsthas in

no

an

way

immediate

not transitive, The

successor, and

above discussions

one-one.

last term

but

but

before other terms.

not

The

of a-series

to the domain

domain not

every term exceptthe Here, again,cardinal numbers are predecessor. immediate

presupposed.

Throughout the *

an

one-one

our

serial relations

relations

are

are

is the term belongingto (ifit exists) * .". the term relation, generating

of the

taken to be

derived from easily the which

is after

Order

244

the

transitive the

Moreover and

they

unless 233. here

be

are

any

the

to

convenient Oth

ordinal,

need

we

the

from

*r,

shall x

assign or

the

positive other

as

the

two

they

a

and have

characterizing

+n

all

the

terms

"

to

the

w

have

signs.

they x.

chosen

we

recognize

negative This

start

by

R.

The

the

to

positive

progression.

origin

To

shall

we

belong

in

is a

terms

converse

is

the

as

or

other

other

mutually

express

it

progressions

two

double

a

assigned

purpose

positive

the

as

arbitrarily

which

properties

it,

from

starting such

be

progression

jff, the

according

form

and

and

progression.

a

progression.

relation

0,

progressions

to

of

taken away

this

any

to

seem

be

may

smaller

series

ones.

form

for

series,

infinite

ordinals

ordinals

double out

serial

ordinals

which

still

the

a

#

ordinal

relation

a

finite

of

progression

giving

called

a

themselves

essentially

progressions, Thus

by

negative

or

of

ordinals

negative express

assign

then

we

of

term

any

the

negative

but

series be

generated

one

and

negative

a

may

choosing

that,

such

series

what

transitive

abstracted;

have

to

the

remainder

the

beginning

the

order

In

term.

n

been

have

regard

to

of

terms

progression,

new

that

terms

first

the

number),

the

to

positive

study

xxix

complicated. define

to

the

to

from

concerning

If

finite

regard

With

extended

derivative

as

words

few

place.

in

being

(n

taken

somewhat

is

adequate

only

are

be

cannot

use

A

derivation

converse

relations

one-one

their

thus

the

while

ones,

[CHAP,

one

and

They of

the

two

relations.

Chapter

xxvn

CHAPTER

DEDEKIND^S

we

234.

THE

have

been

occupied and

"

concerned

with

Dedekind^s

theory

to

He

subject

which

now

as

work,

relevant

was

und

been,

and

of

time

that

is

Dedekind's

writing,

invented

acquainted un-

of

much

as

adopted phrases

convenient

so

and

die Zahlen?'"*

sotten

he

two

theory

theory,

naturally

he

always

not

were

It is his

strictlyto

although

to

present;

the

"

the

at

main

at

was

adhere

his purpose,

to

usual, and

account

which

being specially

postponed.

an

not

have

to

be

to

give

shall

I

considered

Wassind

"

symbolic logic ;

not

were

in his

appears

with

is also to

the

in

contributions, be

with

numbers,

due

chapter, is

not

irrationals

NUMBER.

of ordinal

Cantor's

I wish

this

phraseology. this

of

is contained

reviewing

In

Cantor.

which

which

say,

OF

last

the

in

infinity,need

integers of

of

THEORY

theory of progressions and

Dedekind

men

XXX.

their

as

ventional con-

equivalents. fundamental

The the

(})

chain of

ideas

of

representation (Atbtidung) (37)

(3)

;

the

mathematical

ordinary

of

induction

an

these

Arithmetic.

Let

five

(5)

;

(4)

;

Dedekind

of

a

generalized form

of

a

singly

deduces

numbers

definition

the

notions

and

explain the notions,

first

us

(44)

the

f:

these

are

(SI); (") the notion

system

a

element

(59)

From

(71).

system

chain

of

question

in

pamphlet

the

then

infinite

and

examine

deduction.

the

235. of

term

w,

class u,

or

is

different

are

A

terms

2nd

t*,

by

ed.

which

The the

which

by

numbers work

the

in is

$(x)

a

class

terms,

1893

to

as

be

may

u

which

(1st

the

is

a

of

brackets

refer,

not

as

same

or

to

will

be

pages,

when

to

this

also

not

in

found

but

to

the

my

and

y

:

domain to

article

w,

of

contents

small

No

the

to

"r

belong

principal

every

""(#).

relation, whose

The

1887).

$(y\

amounts

may

term

one

to

""(*r)belongs

whether

many-one

may

ed.

only

thus

Relations,

Algebra

divided*

and

by which,

law

is any

u

one

some

definition

The

u.

of

Brunswick,

book, expressed VII, 2, 3. t

of

class

a

begin with,

to

whether

representation

contains

*

made,

to

as

corresponds

x^

say

assumption

representation of

A

(1)

in

sections

are

this

RdM,

into

Order

246 correlated

of the

each

with

one

[CHAP,xxx of u*.

terms

The

representationis

fax) differsfrom

similar when, if x differsfrom "/, both being u\ then is one-one. that is,when the relation in question ""(y); classes is

between similarity

(34)that classescan

remarks

of suggestion

a

which correlates with

a

class u

with

respectto class

say, any

contained

u

one-one

or

of

many-one,

then belongingto that class, of ?/ in itself (36), representation

a

is called

u

chain

a

(37).

the

That

relation, a many -one relation,and the correlate of

is to

chain,if u is

respectto any

domain

in the

work.

onlyterms

this relation

is,with

in Cantor's

relation,whether

a

this relation is said to constitute and

that

"

is fundamental

idea which

an

shows

and reflexive, symmetricaland transitive, to a given class be classifiedby similarity

(2) If there exists

236.

He

a

is

u

The collection of correlates of a class is called the always itself a u. image (Blld)of the class. Thus a chain is a class whose image is part or the whole of itself. For the benefit of the non-mathematical that a chain with regardto to remark reader,it may be not superfluous relation, a one-one providedit has any term not belongingto the image contain the same of the chain,cannot be finite, for such a chain must of terms

number

(3) If

237.

with respectto

number

term

common

calls the chain

of

than

",

a

by

n

is the

will be

1"

is denoted

example,if

For

(44).

of which

be,

may which in chains

relation,many of these chains,which all part

set of numbers

or

terms, there

collection of

or

any regardto the relation "less

with

is

a

by a^ be the

a

the chain of least,

a

less

all numbers

not

theorem

which

n.

238. a

part of itselff.

proper

givenmany-one

Dedekind

is what

a

be any

a a

The

contained.

as

(4) Dedekind

form generalized

now

proceeds(59)

of mathematical

to

a

is

theorem

This

induction.

is as

be any term or set of terms contained in a class *?,and let the image of the common part of $ and the chain of a be also contained

follows : Let in

#

a

then it follows that

;

the chain

of

is contained in

a

clearer become may relation by which the

rather the

so succession, relation)

image

successor, to one

converse a or

term a

of this

will be its

successor.

collection of such terms.

Let A

a

This

that

be

chain in

a

the

term

correlate which

A

has

or a

(withregard general

will be any set of terms such that the successor succession) of them also belongs The chain of a will be the to the set.

*

what some-

by being put in other chain is generated (or

theorem complicated language. Let us call the of

"v.

of any common

relation is one in which, as in the relation of a quantity to its many-one magnitude, the right-handterm, to which the relation is,is uniquelydetermined: when the left-hand term is given. Whether the converse holds is left undecided. Thus a one-one relation is a particular of a many-one case relation. f A projier part (Echter Theil)is a phraseanalogousto "proper fraction"; it means

a

part not the whole.

Dedekind's

235-240] the

part of all inform an

",

that

us

of

chain

containinga.

is contained

a

is its

so

chains

This

Number

Then

247

the data

of

and, if any term of the chain of a be the conclusion is,that every term in the

theorem, as

is

evident,is mathematical induction,from which it differs, firstby

need

a

be

not

an

s.

singleterm, secondlyby

a

relation need

be

not

the theorem

in $,

and

successor;

is

Theory of

but

one-one,

remarkable fact that Dedekind^s

the

fact that

very similar to the fact that a the

be many-one.

may

constitutive

It is

most

a

sufficeto demonstrate previousassumptions

this theorem.

(5) I come (71). This

239. class

or

itselfby

of

means

next

is defined

class not contained in the the

relation R, (1) The

one-one

to which

N

an

there

it, Le. it is

to

relation R

of relation,

one-one

are,

image

the class.

the

not

in other

is one-one,

be

of the

class Ny and

remarks, four points in in JV; that is,every

is contained is

N.

an

is such

term

one

a

to

as

singleterm

Callingthe

Dedekind

as

of N

has the relation R

(3) This

of its terms.

one

a

image of

this definition. term

as

singlyinfinitesystem be represented in can

a

class which

and which is farther such relation,

one-one

a

chain,with regardto this

the

j?

the definition of

to

(2) N

that

is the

has the relation

N

no

chain of

image of any other term of N. (4) The is similar. The words,the representation

these properties, is defined system,defined simplyas possessing by Dedekind as the ordinal numbers (73). It is evident that his singly

abstract

system is the

infinite

deduce

proceedsto form. of

n

the

induction

mathematical

number

One

and he progression, various properties of progressions, in particular (80),which follows from the above generalized

same

is contained

(88,90) that of

what

as

is said to

m

in the

be

image of

less than

different numbers, one

two

infinite

systems

are

(133)any system which conversely is singly infinite. When

ZH, where vice

Zn

in

regardto

to

this last

any

is

onlyone

given finite

of elements we

if I may order of

reach the

a

of which

from number

system,and cardinal the

1

said

cardinal numbers.

to n

n

all that do not

that

contained

not

ordinals may

be

similar

in the to

follows:

and

the

be

Here

ordinals,

owing

to

class of ordinals

a

They

image of

the chain

class,which

;

property

(161).

dependenceon

as

succeed it.

another

this

is said to

consists

system is

has

system

some

inclusive

considered in relation

when

number, and

to

to

both

which

Their

singly

and that ordinals,

singlyinfinite system

a

interpretDedekind, the ordinals, every ordinal n defines venture

of consisting are

is similar to

present

(132) that all the

to

this

From

importantfor our

it is similar system is finite,

propertyit is called

number at

means

it is shown

be the less.

It is shown

other and

all the numbers

(160). There

verm

a

the chain

when

w,

(89); and

m

must

seems

is the definition of cardinals. similar to each

a

another

the chain of

simply. pointeverything proceeds The onlyfurther pointthat 240. purpose

called

we

may of

be defined n.

as

the

,Zm, all

This class of

is then said to have

the

Order

248 cardinal number that each

n.

of them

[CHAP,xxx

it is only because of the order of the ordinals and thus this order is presupposedin defines a class, But

obtainingcardinals. Of

241.

In

call for discussion.

induction,while

Peano

it is not

necessary for

universallyacknowledged. But

speak,for they are

to

me

deduction

of the above

the merits

first case, Dedekind regardsit as an axiom. the

which apparent superiority,

must

points

some

proves mathematical This givesDedekind

be examined.

In the second

place, which Dedekind the numbers because obtains merely reason, have an order, to hold that they arc ordinal num)"erh ; in the third and the complicated, place,his definition of cardinals is unnecessarily dependenceof cardinals upon order is onlyapparent. I shall take these pointsin turn. the proof of mathematical As regards induction,it is to be observed that it makes the practically equivalent assumptionthat numbers form

an

there is

no

of

the chain the

is most that as

theorems

rule have

a

be deduced

can

from

the

other,and

an axiom, which a theorem, is mainly as the whole, though the consideration of chains On of taste. and has the disadvantage it is somewhat difficult, ingenious,

to which

choice

matter

a

Either

of them.

one

is to be

concerningthe finite class of numbers to be

deduced

from

greater than

the infinite class of numbers

not

greaterthan

theorems corresponding For

n.

these

n

concerning reasons,

and

it seems of any logical simplerto begin with .superiority, And induction. it should be observed that, in Peano's mathematical

because

not

method, it is only when

be

proved concerningany that induction number required. The elementary Arithmetic of our numbers, childhood,which discusses only particular of mathematical induction ; though to prove that is whollyindependent mathematical number this is so for every particular would itselfrequire In Dedekind^s induction. method, on the other hand, propositions the demand numbers, like generalpropositions, concerningparticular theorems

to

are

mathematical

consideration of chains.

is

Thus

there

is,in Peano's

method,

a

distinct

and a clearer separation between the particular advantageof simplicity, and the generalpropositions of Arithmetic. But from a purelylogical of the methods two seem point view, equallysound ; and it is to be remembered both Peano's and that,with the logical theoryof cardinals, Dedekind's axioms 242.

what of

a

On

become

the second

point,there

are

is

of clearness deficiency

some

Dedekind

in

His words are (73): If in the contemplation says. infinite singly system N9 ordered by a representation "",we disregard u

the j"eculiar nature entirely of

demonstrable*.

of the

elements, retaining onlythe possibility

them, and considering distinguishing onlythe relations in which they then these elements are placedby the ordering representation*^,

called natural

numbers

or

ordinal numbers *

Cf.

Chap. xin.

or

simply numbers"

Now

Dcdekind's

240-243] it is

that impossible

that

the

and

that

Theory of Nwnber should be

this account

quitecorrect.

For it

implies

of all

other than the ordinals are complex, progressions ordinals are elements in all such terms, obtainable by

terms

the

But

abstraction.

this is

not plainly

the

A

case.

be

can progression

of transfinite ordinak,or

of

formed

249

of cardinal*, pointsor instants,or shall the ordinals which, as we not elements. over More.shortly are see, that the ordinals should be,as Dedekmd it is impossible suggests, such of relations but the terms constitute as nothing a progitission. If theyare to be anythingat all,they must be intrinsically son*ething ; theymust differ from other entities as pointsfrom instants,or colours from What Dedekind sounds. intended to indicate was probably a of the principle of abstraction, definition by means such as we attempted to give in the precedingchapter. But definition so made a always class of entities having (or being)a genuiiie indicates some iiature of their own, and not logically in which they dependentupon the manner in

been

have the

defined.

mind's

The

what

eye;

entities defined should

the

asserts is principle

be

at least visible,

that, under

certain

to

ditions, con-

such entities, if onlywe knew where to look for them. have found whether, when we them, they will be ordinals or there

But

are

cardinals,or

something quite different,is

even

not

be

to

decided

And

off-hand.

what it is in any case, Dedekind docs not show us in have for nor giveany rea-son progressions common, supposing it to be the ordinal numbers, except that all progressions obey the sauje

that all

laws

ordinals

as

would

do, which

prove

equallythat

any

assigned

have in common. progressions Tliis bringsus to the third point,namely the definition of 243. remarks in his preface of ordinals. Dedekind cardinals by means (p.ix) their old friends the natural niiinl"ersin will not recognize that many he introduces to them. In this, it seems the shadowy shapeswhich in the right in other words, I am one to me, the supposedpersons are is progression

what

all

"

What them. Dedekind presentsto us is not the numbers, among he says is true of all progressions what but any progression: alike, where he comes not even to cardinals and his demonstrations nowhere "

"

involve No

mnntars property distinguishing

any

evidence

brought We are progressions.

other

have

in common;

have progressions in the

The

forward

is

fact is that

all

show

that

other

numbers

but

not

no

reason

progressions. priorto

are

all gressions prois given for thinkingthat

told,indeed, that they are

anythingin

which do definition,

to

from

what

assigned beyond the properties themselves constitute a new progression.

common

dependsupon

one-one

which relations,

Dedekind

that theyalone suffice perceiving between relation of similarity for the definition of cardinals. The combined with the principle of he employs consciously, which classes,

has been

usingthroughoutwithout

suffice for the definition of abstraction,which he implicitly assumes, cardinals; for the definition of ordinals these do not suffice;we

as require,

we

cardinals:

if

(or the

notion

corresponding

n

In order definition).

serial relation, a finite series generating

one-one

domain, but not

the

the

to

If

but not the relation R.

number, the nth

of

term

again,it

relation 12 H"1,or,

has

is the

r

domain,

converse r

or

a

define

to

term

first

belonging is a finite

n

the first term

relation

having the

is the term

tion rela-

a

?.".,havingthe relation

which

to

the

n,

the progression,

terms, where

more

or

n

is

if It be any

will call r) is the term

(whichfieldwe

of the field of R

Thus

of times.

itselfa finitenumber

71

their domain

in

terms

of

into multiplied term

particular

"/2th," we need, besides the ordinal number Le. of the relative productof of powers of a relation,

notion

R

have

preferthis

we

the

of

terms

number, the ordinal number

finite cardinal

a

field,if

their

in

in explicitly

serial relations which

class of

the

be

n

definition of

The

serial relations.

is effected

finite ordinals

relation of likeness

chapter,the preceding

in the

saw

well-ordered

between

to

[CHAP,xxx

Order

250

has the

J2"-1but

not

powers of a relation,the introduction of cardinals is here unavoidable ; and as powers are defined by mathematical induction,the notion of "th, accordingto the above relation

the

R*.

of

notion

beyond finite numbers. followingdefinition: If

be extended

cannot definition,

extend

the

Through

the notion

the

by

We P

however

can

be

transitive

a

nth term of p is the aliorelative generating a well-ordered series p, the such that, if P' be the relation P limited to x and its preterm x decessors then

upon

has the ordinal number

P

cardinals results from

onlybe

defined

by means importantto

It is

order rather than in relation to the

of

a

the

the fact that

of the cardinal

observe that

another,and

that

the

Here

n.

ordinal

n

dependence in general,

can,

n.

set of terms

no

is the nth of

term

no

inherentlyone

has

a

set

except

field is the

relation whose particular generating

set

or

example,since in any progression, any finite number terms includingthe first may be taken away, and the remainder will stillform a progression, the ordinal number of a term in a progression may be diminished to any smaller number part

For

set.

of consecutive

we

choose.

to

which

term

be It in

Thus it

a

is relative to the series

term

may be reduced to a lest a vicious circleshould

relation to the

first

be

it may suspected, thej?r*"term can alwaysbe defined non-numerically. in DedekirKf infinite system, the onlyterm not contained s is, singly the image of the system ; and generally, in any series, it is the only which

w, but

*

last

of

belongs. This

of the series; and

has

the other*.

as

number

explainedthat

term

to

the ordinal

Thus

also to

when

Though

to

which

the constitutive

we

the relation the

first term

the series has two

will call

illustratesthat of its

which first,

relation with

expressed by

one

"th

sense,

is not

of the series ; and ends, we last. The

correlative, firxt.

have to make

but

onlya

not

with

relation

Jir#titself depends an

selection arbitrary

obviouslynon-numerical

nature

of

Dedekind's

243] the

upon

they

ordered, first

not

was

assigned,

the

term

and

relation, either

as

more

than

and

progression;

dwelt

with

does, For

the

general,

in

general

defined,

theorems

concerning

question,

whether

the

one

of

numbers as

to

series

largely

must

begin and seem

which

with

be

be

order

or

simplicity

have

;

to

occupied

a

applied

precede in

the

the

work

this

as

order.

with

order,

number.

without

of But

appeal

numbers

them.

resolves

point

present

be

before

Hence

very

to

can

progression

this

from

at

properties

numbers,

with

us

to

is

view

with

the

of

proof

a

either

whatever.

begin

cardinal

and

form

Dedekind's

of

form

to

seen

can

naturally

of

since

is

requiring

than

independent

are

opinion

my

begin,

most

capable

"wth," cardinals

essentially

to

error

even

progressions,

and

progressions to

and

be

must

of

convenience

cardinal

be

to

number

properties

an

ordinals,

of

If to

serial

cardinals

rather

absolute

progressions,

seem

of

properties

independently

into

of

it

between

that like

and

error

numbers

chain

the

progression

authorities.

logical

a

plain

not

any

be

generating

a

the

important,

best

cardinal

hold

not

properties

the

series

the

part,

my

since

do

is

the

been

of

theory I

it

as

what

progressions,

that

of

and must

as

theory

are

members

of

have

show

to

ordinals

the

most

would

it

the

order

point,

of

that

correct,

were

this

on

with

variance

but

cardinals,

or

have

in

of

which

series

a

notions

logical

theory

Dedekind's

is

as

by

relation,

first), it

or

so,

progression

a

(the

the

general

of

term

Thus

?i.

be

to

four-cornered

term

that

the

that

of

relations,

cardinals;

development

independent

ordinals

assigned

serial

of

first

a

number

relation

cease

may

view

251

the

upon

the

expresses

an

like

independent

wholly

I

r^th,

of

classes

complex

nth

cardinal

the

Thus

Dedekind's

Hence is

and

first

was

Number

of

series,

so.

in

term.

which

the

what

become

done

is

in

that

so

may

as

first

its

of

included

terms

are

Theory

the

itself of

difficult Part,

view, siderations con-

XXXI

CHAPTER

DISTANCE.

244. to

THE

series*,but

Thus

position. remarks

for

Leibniz, in the

with that

I answer,

Relative

measured

that

which

time

quantity

t."

this

In

and

distance

is

closely analogous sign,

and

distinct

from

does

not

have

a

unless,

what

I

term

or

and

of

series

the

*

t

It

Phil.

special

a

that

seen,

of

form

least) stretches

Whether

there

compact

obtained

series are

in

fulfil

op.

Gerhardt's

cit.

from

Vol.

TO,

entirely does

or

(i.e.such

as

p.

404.

be

may

integers,in

" 17.

ed.

is

progressions the

the

by argument.

others, there

from

obtained

they

numerical

series

series

be decided

to

not

l"e distance;

by Meinong, Werfa,

question

a

must

are

most

there

addition

which

particular

any

that

addition, that

relational

of

two),

real numbers

E.g.

of

an

as

prove

linearityare always capable idea, as Meinong rightly points out,

any

they

before, and

not

of

stretch.

their

have

goes

does

and

therefore,

they

have

we

as

called

have

order

of

proves,

capable

And

which

For

ones.

interval," if considered

fact

mere

that

is

interval.

quantity,

relations.

are

or

distances, will be, in.

between

indeed,

There

(theoreticallyat the

that

contain

discrete

rationale

are

to

But

measurement.

the

their

relations, yet

is distance

these

that

in

"

interval.

Archimedes

of

yet they

consist

there

or

stretches, that

axioms

and

have

which

it

or

absolute

as

not

are

in

distance

is

well

as

order

is that

there

mathematics

in

and

situation

there

rather

quantities, or

are

quantity;

quantity,

sequitur;

non

a

its

remark:

the

follows;

inference, is there

their

space

passage,

which

have

Clarke,

with

controversy

time

that

follows;

proportions

or

and

has

by logarithms;

though

In

also

have

things

instance, ratios

are

in relative

believe

who

his

and

space

quantity,

order

before, and

that

of

course

objection,that

the

endowed

things

are

precise definition.

receives

:

"As

goes

seldom

which

supposed essential An emphasis on

is often

which

one

characterizes, generally speaking, those

distance

so:

is

of distance

notion

"

as

the which

Distance

245]

244,

be distance.

shall find that stretches are mathematically and that distances are complicated sufficient, and unimportant,

there must

case

definition of

The

245.

done

has been

What

But

253

we

to beginwith,is distance,

hitherto towards

easy matter. this end is chiefly due to nonno

the Geometry*; somethingalso has been done towards settling in B ut both these cases, there is more definition by Meinong-f-. for concern

Euclidean

numerical

of distance than for itsactual definition.

measurement

distance is by the notion

no

much

means

indefinable.

Let

us

endeavour

theless, Never-

to generalize

distance need not possible.In the first place, of distance always allow us to be asymmetrical ; but the other properties render it so, and we may therefore take it to be so. a distance Secondly, be taken to need not a quantityor a magnitude; althoughit is usually the it find shall its be irrelevant to other to be such, we taking so and in particular to its numerical measurement. Thirdly, properties, there must be onlyone to when distance is taken asymmetrically, term and the converse relation to the which a giventerm has a givendistance, kind. (Itwill be observed givendistance must be a distance of the same firstdefine must kind of and a that we distance, proceedthence to the as

as

distance.) Thus every distance relation;and in respectto such relations it is convenient

generaldefinition of

converse

of

relation

a

of two

distances of

the two

distances

which is thus

onlyone

one

which

a

are

a

to

respectthe

one-one

its ~lth power. Further the relative product kind must be a distance of the same kind. When as

mutuallyconverse,

their

distances

productwill in

fact),and

(theirzero, asymmetrical.Again the productof

among

is not

is

be

identity,

must two

be the

distances

commutative!. If the distances of a kind be magnitudes, they must form a kind of magnitude ic* any two must be equal they must still form a series or unequal. If they are not magnitudes, in the six second of our generated ways, i^. every pair of different have a certain asymmetrical for all distances must relation,the same if Q be this relation, And and finally, pairsexcept as regardssense. R\ Q#a (Rij R" being distances of the kind),then if #, be any other have Hi Rs QR* J?5. All these properties, distance of the kind,we must and we far as I can discover,are independent; so ought to add a each of which belongs the this two : any terms, field, namely propertyof the same for distance of the kind (not necessarily to the field of some both),have a relation which is a distance of the kind. Having now distance is any relation belongingto some defined a kind of distance, a kind of distance;and thus the work of definition seems completed, will be is it The notion of distance, enormouslycomplex. The seen, to those of stretches with sign, but of distances are analogous properties of

a

kind must

be

"

*

See e,g.

Whitehead,

Universal Algebra,Cambridge, 1898, Book

f Op. tit* Section iv, J Tliis is an independentproperty\ consider for instance "maternal

grandfather"and "paternalgrandmother."

vi,

Chap. x.

the difference between

Order

254 mutual

capableof

far less

are

[CHAP,xxxi

of properties of distances are properties

of the above to many corresponding due of proof.The difference is largely

called relational addition,which

I have

requirewhat

stretches

capable

to the fact that stretches

be

can

whereas distances

(not arithmetical) logical elementary way,

in the

added

The

deduction.

is much

the

same

as

relative multiplication. numerical

The

246.

in Part III. It explained

further

two

of distances has

measurement

as we requires,

which,however, do postulates,

Archimedes:

tially parfor its full application,

the definition of

belongto only. These

not

but to certain kinds of distances distances, of

saw,

alreadybeen

the

late postukind, there exists

distances of

are,

a given any of the distance is greater nth first the such that a integern power of linearity: than the second distance;and Du Bois Reymond^spostulate

two

finite

nth root, where n is any the result follows for any integer).When

distance has

Any whence are

satisfied, we

kind

other

an

can

find

and identity,

than,

kind is of the

distance of the

of course, the numerical In the case of series

powers, as characteristics of distances. rationals

distances; thus in the

a

postulates

distance of the

of the distance.

see

can

In

the

real numbers

or

form

some

of series

case

from

measurement

above

generatedfrom

thei*e integers,

gressions pro-

always

are

are

which are themselves, measure again rationals,

them, and these relations are

certain theorems

of distances is apt to be

about

oneor

of the nature

And

is essential to

These

for himself,verifyall the

shall see, in Part V, that these distances have we importancein connection with limits. For numerical measurement

distances.

various

of the rationals

case

indicate relations between

prime,

any

is any real number*. Moreover, any form Rx^ for some value of x. And x is,

which relations,their differences,

one

where R is jff*,

two

x

measure

the reader

various

as

these

generatedin the firstof our six ways, the relation R givethe distances of terms. generating

of the

powers

for

meaning

a

integer(or

and limits,

the

of

some

in rical nume-

feasible than practically

more

that of stretches. the

On

however,whether series unconnected generalquestion, with number for instance spatial and temporalseries are such as to it is difficultto speak positively. contain distances, Some thingsmay be said against this view. In the firstplace, there must be stretches, and these must be magnitudes. It then becomes a sheer assumption which be set up as an axiom that equalstretches correspond must to equal distances. This may, of course, be denied,and we might even seek an 247.

"

"

"

"

*

The

powers

of distances

thus multiplication;

if a and

are

here understood

it have

the

same

in the sense

distance

as

from resulting1

relative

b and c, this distance is the

The postulateof linearity,, whose expression square root of the distance of a and c. in ordinarylanguageis: " every linear quantity can he divided into n equal parts, where n is any integer," will be found in Du Bois Raymond's AllgemeineFuuctionentheorie

(Tubingen,1H82),p.

40.

Distance

245-248]

255

of non-Euclidean Geometry in the denial. We interpretation might coordinates usual the and as the stretches, expressing regard logarithms ratios as expressing of their anharmonic distances; Geometry, hyperbolic curious Herr at least,might thus find a somewhat interpretation. series all who as distances,maintains an regards containing Meinong, with to distance and in general.The stretch analogousprinciple regard distance,he thinks, increases onlyas the logarithmof the stretch. It

that, where

observed

be

may

the distance itself is

since rationals (which is possible, be made

theorycan square

of

distance,as

a

it is.

distance is trulythe

distance. doubtful

there

whether

former

seem

to

a

Thus

powers

measure

know,

I

as

the

numerical pelled com-

all other

the obtainable, which, as a rule,no are

better,at generally

least in

sophy philotheory of them, in that theory,merely by the generatingrelation. There is no logical distances

except

suppose that there finite space of two dimensions and in

far

we

where

shall be

since,in almost

for complication

eschew

of the

For

of interpretation

distances,and

to

a

great,but that the

as

distance.

If.

great as

as

distance is

to the logarithmof the proportional the theoryof progressions, it is usually

It is therefore

mathematics,

the

as

are

distance adds

of

indices of the so

the

adequatefor all the results that

and progressions,

reason,

might, where

the notation

since,outside

But

necessity appears. of

square

regardthe stretch

stretches series, retention

of

conflicts with

measurement to

We

alreadynumerical, the usual

is

the

square instead that the stretch is twice

rational, say distance

by

number

the opposite relations), followingfact. The

one-one

is said to be twice generally,

saw

we

whose

distance

the

are

formallyconvenient

rational

a

to

are

a

in the

distances

elsewhere,

projective a space ; and if there are, they are not mathematically important. We shall see in time and without prePart VI how the theory of space may be developed supposing the which in distances distance; projective Geometryare appear the properties of our space ; in defining not required derivative relations,

exceptin

in Part

and

V

shall

we

see

how

general.And

few

are

a

the functions

of distance with

distance against

it may be remarked endless distances, an that, if every series must regress becomes distance is itself series. This is not, kind of a unavoidable,since every

regardto

series in

as

contain

since the regress is of the logically sible permisthink,a logical objection, introduced by are kind ; but it shows that great complications distances as essential to ever}* series. On the whole, then, it regarding doubtful whether distances in generalexist ; and if theydo, their seems

I

existence

248.

unimportant and

seems

We

have

now

a

source

completedour

of very great complications. review of order, in so far as is

introducingthe difficultiesof continuityand infinity. and transitive relations, that all order involves asymmetrical

without possible We

have

seen

But closed series, such is open. we found, could be and by the fact that, by the mode of their generation, distinguished

that

every series

as

Order

256

arbitrarily.

We

always

upon

converse.

found

ordinals

this

we

logically a

theory

no

saw

which

of

difficulties

continuity.

philosophical in

to

be

devoted.

a

agree

problems

with

Dedekind

series,

this will

we

philosophers be

can

have

and

of

shall have

solved.

But

though the

little

be

able,

one

To

as

distance

of

hope,

this

the

is

outside to

found

usually

we

cardinals,

importance I

and

cardinals

that

agreed

we

we

series,

of

its

called

such

regarding

accomplished, been

every

them.

in

depends and

which

independent

Finally, to

to

relations

relation

series

of

means

extent

equipment,

which If

by

certain

essential

of

type

sometimes

found,

we

sign,

xxxi

selected

asymmetrical

of

applies

ordinals.

this

With the

to

to

not

particular Arithmetic

be

must

asymmetrical

an

defined

be

to

is

all

be

to

reason

Arithmetic.

and

the

may

subsequent

notion

all

saw

we

finite

between

be

always

may

other

difference

The

analysis.

how

term

relations

analyzable,

when

discussing

progressions,

this

term,

asymmetrical

difference

the In

how

that

the

in

appear

first

a

that

saw

and

unanalyzable, must

have

always

they

though

[CHAP,

in

dispose infinity

greatest

problem

Part

of V

PAUT

INFINITY

AND

CONTINUITY.

CHAPTER

THE

249.

WE

XXXIL

CORRELATION

OF

what

SERIES.

been

has

the generally considered fundamental the problem problem of mathematical philosophy I mean, and of infinity continuity. This problem has undergone, through the labours of Weierstrass and Since Cantor, a complete transformation, come

to

now

"

the had

been

But

it has

in

any

Newton

of

time

shown

that

concerned

way

Leibniz, the

discussions

in

sought been

and

branch

with

of

of

of

nature

so-called

the

Calculus

this

infinityand Infinitesimal

not,

is

infinitesimal,and

the

mathematics

continuity

as

that

is

Calculus. of

a

matter

a

large and

it.

fact, most

The

important problem logically prior to that from has been to a great extent continuity,moreover, separated of infinity.It was formerly supposed and herein lay the real strength of

"

of

Kant^s

mathematical

philosophy

reference

to

and

space

suggests) in

some

time, and

presupposed

way

view, the philosophy of space the

Transcendental

the

antinomies

temporal. is called

the

this

cardinal

and

theory of theory that of

theory

arise

types

tlie

numbers

which

the

series

that

to

fact

enables

series of rational same

type.

But

"

a

us

point in

theorems

of

time,

springs

all

that

shown

forms,

two

from

the

already

are

infinity has

logical

the

In the continuity is purelyordinal. ordinal theory of infinity,the problems with

numbers, and

of

give which the

a

proper

deal

to

name

to

this series differs

kind

which

will

alike.

I

a

term

of

the

otters

of

the

every

occupy

shall

case

call

and integers,

of the

from

What

in the

with

is what

that

all series

with

but

geometry

rationals,which

namely progression,

a

What

mathematics.

and

problems in question peculiarly easy is,that

continuity,

Dialectic,and

essentially spatio-

were

has

of

this

In

of

in arithmetic

occur

compact series,arises from this

former

concerned specially

not

are

the

theory the

which

continuity and

certain

makes of

of

the

;

ones) modern

respect, by space The

of

that

to

fluansm

change.

Transcendental

mathematics

of

arithmetic.

number

prior

was

changed by

this

ordinal, of

time

least

at

or

mathematical

arithmetization

pure

motion

essential

an

(as the word

Calculus

the

preceded the

been

has

problems presented, in present in

and

Aesthetic

(at least the

All

that

had

continuity

that

"

us

in

most

of

[CHAP,xxxn

and Continuity Infinity

260

though obtained in arithmetic,have a chapters, following since they are purelyordinal,and involve none application,

far wider

the

only

irrelevant,save

is

shall find it

We infinity.

in which continuity, find that

other

no

no

to possible

give

appeal is made

is continuity

a

and

in

freer

a

extruded

been

in space and to the doctrine

theory

shall

we

And

time.

we

of limits,it is in the definition the infinitesimal

infinite has

the

an

been

allowed

work

it appears that there are differ from those that are finite.

Cantoris

development. From

infinite numbers

respectsin which

two

mathematics,

from

"

unanalyzed VI

Fart

by a strict adherence with the infinitesimal, even to dispense entirely possible Calculus. the foundations of the and of continuity fact that, in proportion It is a singular as 250. has

class,

some

of

mass

involved

shall find that,

the

generaldefinition of

the

to

call "intuition";

Kantians

which prejudice

in

of terms

distinct part of Cantor's contributions to the

importantbut very of

idea which

transfinite cardinals

theory of

the

in

the

say,

idea of the number

the

call AmaM^

Germans

is to

That

numbers.

of properties logical

of the

to both cardinals and ordinals, which applies is,that they do first, rather,theydo not form part of or not obey mathematical induction series of numbers a beginningwith 1 or 0, proceedingin order of

The

"

intermediate in

all numbers containing magnitude, of

two

any

its

terms,

applies

number

of

number

of terms. in

term

second be

always

terms

The

a

series: it

contains

givesthe

the

if he will condescend

find that it

can

part

a

a

what

of

of

essence

an

induction.

whole

we

the

the

be

an

may

call

ordinal

The infinite

the

same

definition

true

infinite collection, and

to philosopher

of

consistingof

respectconstitutes

rather

givesthe definition of

pronouncedby

is,that cardinals,

first

infinite series,or

an

obeying

only to

second,which

of

and

magnitudebetween

mathematical

an

infinite

infinite. The will doubtless

But self-contradictory. plainly

he wiH contradiction, onlybe provedby admittingmathematical induction, to

attempt

to

exhibit the

that he has

merelyestablished a connection with the ordinal infinite. compelledto maintain that the denial of mathematical induction is self-contradictory he has probablyreflected little, ; and as if at all,an this subject, he will do well to examine the matter before pronouncingjudgment And when it is admitted that mathematical induction may be denied without contradiction the supposedantinomies of infinity and continuity and all disappear. one This I shall endeavour to prove in detail in the following chapters.

so

Thus

he will be

,

251. notion

Throughoutthis

which

Part

has hitherto been

we

shall often

have

occasion

for

a

tion mentioned,namely the correlascarcely precedingPart we examined the nature of isolated series, but we scarcely considered the relations between different series. These relations, however, are of an importancewhich philosophers have whollyoverlooked, and mathematicians have but lately of series.

In

the

The

249-251] realized.

by is

longbeen known how much could be done in Geometry homography, which is an example of correlation ; ami it shown by Cantor how importantit is to know whether a series

denumerable,and how it is not

has

nor series,

In the

are, in most

capableof correlation dependentvariable and

series

that

a

mathematical

are.

its

merelycorrelated

cases,

the

dealt generalidea of correlation been adequately presentwork onlythe philosophical aspectsof the subject

relevant.

are

seri"f #" /

Two

relation R

whenever

of the

correlated

as

a

as

series involves

importancewill

only the

simply,and whenever

their

ordinal

same

one-one

a

and

vrrm^

Thus

correlated

of the

and

one

scries may

two

be

series; for correlation

as

type

distinction whose

a

"

correlation

In order to

these distinguish

the correlation of classes as correlation

speakof

of the correlation of series

correlation is. mentioned

as

without

ordinal correlation. it is adjective,

an

Thus to

be

ordinal. Correlated classes will be being not necessarily similar; and similar; correlated series will be called ordlnaJly generatingrelations will be said to have the relation of

understood called

the terms

cardinal number, whereas

same

hereafter. explained

cases, it will be well to

/, and vice

collections are

cltt"sesor

Two

there is

in then their correlates "**',#'

being correlated

also the

be

of

term

beingleft over.

classes without

classes involves

a

relation between

one-one

other,none

as

with

$

precedes #,

x

precedes y'.

there is

the terms

of

term

of ,?,and

terms

such that of

arc

said to be correlated when

are

couplingevery

when, if "r,y be /

similar two

usuallypointedout

variable independent with.

261

of

been

But

of Series

It has

means

has

Correlation

as

likeness. Correlation is

method

a

by which,

when

series whose may be generated. If there be any relation which holds between is P, and any one-one series and #"

will form

have

we

be shown*

relation is RPR.

the

field includes

if P, conversely, there is

a

of the

Between

to

P'

our

and series, original Hence

#RRPRyR.

these two

assume

of the

"rPy.

J^ow it may

series tliere is ordinal

correlation,

In this way a new completeordinal similarity. relation by any one-one original one, is generated shown also be that, series. It can the original similar series, relations of two be the generating domain

is the

field of P, which

is such that P^ *

x

hence is RPR; so asymmetrical, P-series form a series whose generating

relation R9 whose

one-one

term

P be transitive and

the series have

whose

of

any

then the class of terms call $" type as the class of terms "r. For

x"Rx, xPyy and yRy"

that,if

similar series,

same

other term

the correlates of terms

and

may

we

series of the

a

suppose y to be any Then

which

term

some

given,others generatingrelation

series is

one

See

my

article in lldM, Vol. vm,

Xo. 2.

We

252.

understand

now

can

namely that

between

correlation.

In

denote

there is perfectmathematical justexplained series and the series by correlation ; for, if original the

as

happen

Thus

RQR.

-

and original,

we

may

regardthe

R, instead

that

field of Q, which

of the

the terms

is many-one,

one-one,

if it should

But

derivative.

as

shall find P

we

the P-series

Q-seriesor

take either the other

RPR,

the relation

by Q

great importance, and series by series, independent

case

symmetry between the we

distinction of

a

self-sufficientor

the

[CHAP,xxxn

Continuity

and Infinity

262

of

being

will call y,

we

the same term repetition, occurring to its different correlates in the field in different positions corresponding of mathematical This is the ordinary will call p. of P, which case we with such functions which are not linear. It is owing to preoccupation fail to realize the impossibility, mathematicians in an series that mast of the same In term. independentseries,of any recurrence every the letters acquire order by correlation of print, for example, an sentence letter will be repeatedin different of space, and the same with the points of series Here the letters is essentially derivative,for we positions. of relation the letters to order the : this would cannot points space by instead of one letter in several giveus several pointsin thj same position, In fact, if P be a serial relation, and R be a many-one relation positions, will have

whose

there is

order in which

an

domain

is the fieldof

P, and Q

,

serialrelation except that of to P, and thus there is equivalent of

not

this

a

that

reason

inverse functions

genuinelydistinct from before

convention

many by correlation.

They

to eliminate them

The

253.

they

correlation

-one

a

from

notion

istics Q has all the character-

then

RPR

=

implyingdiversity ; but RQR lack of

a

It is for

symmetry.

mathematics, such

in

as

is

sin""1a?,

are

direct functions,and requiresome device or become unambiguous. Series obtained from as

q not

are

was

obtained

will be

above

and genuine series,

called series

it is

highlyimportant points. to similarity relations,

discussions of fundamental

of likenesscorresponds, among

classes. It is defined as follows: Two relations P9 Q are like among when there is a one-one relation S such that the domain of S is the field of P, and

SP$.

=

be extended

may of

Q

a

relation P

This to

as

notion

all relations.

is not We

confined

obtain

an

the class of all relations that

extension

general. By

means

define the

may

which may proceedto a very general subject relation-numbers we can Concerning prove addition and multiplication that hold for of

we

can

like P; and

can

we

transfinite

laws

and ordinals,

of

thus

part of ordinal arithmetic to relations in of likeness we define a finite relation as one can a

completely emancipate ourselves the

relation-number

those of the formal

"

way

are

but relations,

be called relation-arithmetic.

which is not like any proper part of itself a beinga relation which impliesit but is not Moreover

to serial

of properties

likeness

are

proper

part of

to equivalent

from

a

it.

relation In

this

cardinal arithmetic.

in themselves

and interesting

TJie Correlation

252-254] important. One PS

=

Since

254.

propertyis that,If S be one-one domain, the above equationQ SPS

and have the is

=

QS=

to

or

263

curious

field of P for its

SQ

of Series

to equivalent

SP*. correlation of series constitutes most

the

of the mathematical

and since function is a notion which is examplesof functions, it will be well at this pointto say something not often clearly explained, the of nature this notion. In its most general ality concerning form, functiondiffer from

does not

relation,

For

the

well to recall two has

technical terms, which were relation to y, I shall call x the

certain

it will be

present purpose defined in Part

If

I.

x

and y the rdatunij referent^ with regardto the relation in question. If now x be defined as belonging to

a

class contained

some

in the domain

of the relation, then the relation

defines y as a function of x. variable That is to say, an independent is constituted by a collection of terms, each of which be referent can

regardto

in

a

relata,and any

more

relation.

certain

of these

one

each of these terms

Then is

or

one

of its referent,

certain function

a

has

being defined by the relation. Thus father defines a variable be a class contained in that function,providedthe independent of male animals who have or will have propagatedtheir kind ; and the

function

if A

be the

essential is a

father of B" B

i". independentvariable,

an

relation whose

is the

be

is said to

includes the

extension

independentvariable,and

a

of

any term variable.

is

class,and

some

the referent

Then

is any

its function

What

of A.

function

of the

one

responding cor-

relata. this most

But

There

two

are

confine the relations to be considered

may

i.e.such

many-one, we

may

topics.But Logic,where

series have littleimportance, we

with with

is

so

functions.

member

of the class

becomes

a

a

function of

this subjectsee

t These

are

so

often

it is well to

in other

"

a%

which

what

m

true

words, any

is

for

some

fillour

is

x

values of

x

a

is

some

undefined

then proposition

The

a.

uniquewhen

sidered con-

Thus a very important variable f. Let there be

"

be will,in general, On

par-

postponeit for

a containing propositions a"** the phrase any occurs, where placeof "any a" we may put a% where x is an are

in which proposition

*

been

has

exclusive refereiiee to numbers, that

in

well

as

may

and important,

instances of non-numerical

class. Then

secondly,

second

The

alone. consider the firstparticularization

we

class of functions some

a

or

one-one

are

as

uniquerelatunl ;

independentvariable to series.

idea of function

The

every referent

such

to

relevant to our present is very important, and is specially it almost as wholly excludes functions from Symbolic

while

moment

minds

give to

as

confine the

ticularimtion

a

generalidea of a function is of littleuse in mathematics. the function: first, we principal ways of particularizing

given.

This

position proand false for others,

Nos. especially my article in RdM, VoL vm, functions. Part I we called prepositional

2, 6.

and Infinity

264

by analogy with AnalyticGeometry,

equationof

be

called, general

This logicalcurve. AnalyticGeometry.

a

is example,

for

planecurve,

a

might

to include that of

made may, in fact,be

view

[CHAP

what

form

the function is true

values for which

The

Continuity

a

The

function which propositional

is the assemblage and y, and the curve of is a true. pointswhich giveto the variables values that make the proposition the word any is the assertion that a certain A proposition containing

variables

function of two

Thus significant. is

x

is mortal

"

any man is true for all values of

^

mortal

a

is

"x

and

the

number," have

a

*

a'

asserts that

for which

"

is a

x

it is

it is

implies

man

which significant,

functions,such as Proportional that they look like propositions, peculiarity

admissible values.

be called the

may

for all values of the variable for which

is true

function propositional

x

while yet functions, capableof implyingother propositional fact for all false. The neither true nor is,they are propositions

seem

theyare

admissible values of the variable,but value is not

whose variable,

while the variable remains

not

assigned ; and

althoughthey may,

value variable, implythe corresponding

admissible value of the

a

for every of

some

function,yet while the variable remains as a variable propositional they can imply nothing. The question concerning the nature of a of a and generally function as opposedto a proposition, propositional function as opposedto its values,is a difficultone, which can only be of the variable. It is important, of the nature solved by an analysis other

however,

that

observe

to

Chaptervn,

fundamental

more

are

functions,as propositional than

other

functions,or

to relations. For most purposes, it is convenient is if to and the relation, i.e.} y=f(") equivalent

it is convenient to relation,

of these pointshas investigation

the

and

enough

function of Other

has

been

a

are

a

book

and

in

a

number

been

that

R

is

a

that the

of relation.

alreadyundertaken

case

given;but

word

a

is

functions

a

function

functions of the term in

a

But

in Part

be

proposition may

a

cipheris

series, except in the

of

which

I, a

a

afforded

aries. by dictionthe English,and vice both designate.The function of the

function of the word

for which

generallyunique) when languages, terms of the independent variable do purelyexternal

order

book,

it stands.

the relatum becomes

of

the

are

is a library catalogue

there is a relation by which

cases

(or, in the

said to

for

French

of press-mark In allthese

where

and this will be function,

than

illustrate how

of non-numerical

z%r$d,and both a

xRy"

function

variable.

instances

The

Is

fundamental

is more functionality

idea of

as

the

the identify

in

than

even

follows;the reader,however, should remember

in what

done

speakof J?

shown

was

unique

the

referent

not

form

a

from the alphabet. resulting that our specification, variable is to be a series. The dependentvariable is then independent series by correlation, and may be also an independentseries. For a the example, positions occupiedby a material pointat a series of instants 256.

Let

us

now

introduce

the

second

254,

255]

form

a

series

function as

a

Thus At

The

but

;

rule,a

correlation with

by

in virtue of the

of

Series

205

the

of which the}* iastants, are a also of motion, they form, continuity

series independent of all reference to time, geometrical affords an admirable example of the correlation of scenes. time it illustratesa most importantmark by which, when it

motion

the

Correlation

same

is

present,we time is known,

tell that

a

of position

a

can

the

series in not

independent.When the is uniquely determined; particle

material

the

but when

is given,there may be several momenta, or even an position of them, corresponding infinitenumber to the givenposition.(There will be an infinite number of such moments if,as is commonly said,the has been at in rest the position in question.Rext is a loose and particle I but its defer consideration to Part VII.) Thus ambiguousexpression, the relation of the time to the position is not strictly but may one-one, This was be many-one. considered in our a case generalaccount of correlation,as giving rise to dependent series. We inferred,it will

be remembered, that on

R

the

the

same

we relation, correlating

longerhave

RR

fathers

son

series independent

cally mathemati-

are

and level,because if P, Q be their generatingrelations,

this inference failsas

my

correlated

two

soon

R

as

contained

in

need

be

not

infer

P

=

from

RQR

Q

=

But

RPR.

is not

since then we BO strictly one-one, T, where 1"*means identity.For example, myself,though my son's father must be.

and RR RR This illustratesthe fact that,if R be a many-one relation, must in identity, be carefully but :. the latter is contained distinguished the

not

former.

used to form

Hence

whenever

R

is

a

relation,it may

many-one

be

but the series so formed cannot be by correlation, fataJ to an importantpoint,which is absolutely to the For the presentlet us return the relational theory of time*. When of motion. describes a closed curve, or one case a particle the particle is sometimes which has double points, when at rest or during a finite time, then the series of points which it occupies series. But, not is essentially series by correlation, an a independent I remarked is not as above, a curve only obtainable by motion, which can be defined without but is also a purelygeometrical figure, is reference to any supposedmaterial point. When, however, a curve material so defined,it must not contain pointsof rest: the path of a pointwhich sometimes moves, but is sometimes at rest for a finite time, and when considered geometriis different when considered kinematically cally; the point in which there is rest is one, whereas for geometrically in the series. it corresponds to many terms kinematically a

series

independent.This

The

above

instance,a mathematics. *

See

July 1901.

my

case

is

discussion which

of motion

in illustrates,

normallyoccurs

These

functions

article "Js

in position

among

(when they Time

the

and

Space

are

a

non-numerical

functions

functions

absolute

or

of of

relative?

a

pure real

and Infinity

266

conditions fulfilthe following variable) usually and

dependentvariable

the

relation of

function

the

are

This

-one*.

of

functions elliptic

functions,circular and

the

Both

:

independent

numbers, and

classes of

is many

[CHAP,xxxn

Continuity

the

defining rational

covers

case

real variable,and

a

the

In all such greatmajorityof the direct functions of pure mathematics. be variable is a series of numbers, which may cases, the independent

numbers, rationals, integers, please to positive consists also of variable The other class. dependent primes,or any numbers, but the order of these numbers is determined by their relation not by that of of the independent variable, term to the corresponding the numbers forming the dependentvariable themselves. In a large class of functions the two orders happen to coincide ; in others,again, restricted in any

there

where

we

way

"

and

maxima

are

minima

finite

at

the intervals,

two

orders

throughouta finite stretch,then theybecome exactlyopposite If x be the independent throughoutanother finite stretch,and so on. variable,y the dependentvariable,and the constitutive relation be be a function the same number of, Le. many-one, y will,in general, is essentially the y-series Hence by to, several numbers x. correspond series. If,then, we and cannot be taken as an independent correlation, which defined is wish to consider the inverse function, by the converse

coincide

need

relation, we series.

these,which

of

One

certain devices if of

values

dividingthe

has

a

the

seems

still to

are

have

correlation of

important,consists

most

in

value of y into to the same corresponding (say)n different happen) we can distinguish

x

classes,so that (what may

x\ each of which

we

distinct

relation

one-one

to "/, and

is therefore

simplyreversible. This is the usual course, for example,in distinguishing a nd is roots. wherever the It negativesquare positive possible generatingrelation of our originalfunction is formallycapable of exhibition as a disjunction of one-one relations. It is plainthat the of relation formed each of which contains n one-one disjunctive relations, in its domain certain class a w, will,throughoutthe class z/, be an it may relation. Thus n-one happen that the independentvariable be divided into

can

is one-one,

n

I.e. within

relation to defining

a

within classes, each

of

giveny.

each

which

of which the

there is

In such

relation defining

only one

cases, which

are

x

having the

usual

in pure

mathematics, our many-one relation can be made into a disjunction of each of which separately is reversible. In the case of one-one relations, of Riemann .the method complex functions,this is,vwtatw mutandlsr, surfaces.

But

it must

be

remembered clearly

is not

that,where

the y which naturally one-one, appears from the distinct which ordinally y appears as

our

function

dependentvariable

is

independentvariable

in

as

the inverse function. The *

above

I omit

remarks, which

will receive

illustration as

for the present complex variables, which,by lead to complicationsof an distinct kind. entirely

we

proceed,

introducingdimensions,

The

256]

255,

Correlation

267

o/ Series

have shown, I hope,how intimately the correlation of series is associated with the usual mathematical employment of functions. Many other of the

cases

importanceof

correlation will meet

be observed that every denumerable

which

such integers, calls

Cantor

Cantoris

eo.

proceed.It

we

by a one- valued ordered by correlation mce wrsa. series havingthe type of order a classbecomes a The fundamental importanceof correlation to As

transfinite numbers

theoryof

as

class is related

may function to the finiteintegers, and with the

us

will

appear when

we

to the

come

definition of the transfiniteordinals. In connection

266.

concerningthe

it with functions,

of necessity

a

desirable to say

seems

formula

for definition.

A

thing some-

function

after it had ceased to be merelya power, essentially originally, somethingthat could be expressed by a formula. It was usual to start with some variable #, and to say nothingto a containing expression with what to to tacit assumption as sc was be,beyond a usually begin was

that

was

x

kind of number.

Any further limitations upon x were and it was all,from the formula itself; mainlythe desire

some

if derived,

at

to

such limitations which

remove

number.

This

led to the various

has algebraical generalization*

of generalizations been superseded by

now

ordinal treatment, in which all classes of numbers are defined by a more of the integers, and formulae are not relevant to the process. means and where both the independent for the use of functions, Nevertheless, the

dependentvariables

importance.Let formula, in

A

see

us

the classes,

infinite

are

generalsense,

is

a

with functions of If both

a

variables

The

kind of formula which

variable single are

certain

a

or proposition,

or function,containingone properlya prepositional defined class,or variable being any term of some a

without restriction.

has

is its definition.

what

its most

formula

variables,

more

any term is relevant in connection even

two variables. containing belongingto the class "*,

formula

is a

defined,say

more

one

as

as belongingto the class r, the formula is true or false. It is true if every u has to every v the relation expressed by the formula ; otherwise it is false. But if one of the variables, say a?, be defined as

the

other

class ", while the other, y^ is only defined by the formula,then the formula may be regardedas defining y as a function of x. terms If in the class u there are x Let us call the fonnula P^ thai such that there is no term which makes P^ " true proposition,

to belonging

the

*y

We formula,as regardsthose terms, is impossible.

the

that

assume

of #, make

there do

are

the

*

y

Of which

is

a

class every term

P^ proposition entities y9 which

some

so, then

this way

u

P^ correlates is defined an

as

a

to

every function

excellent account

tique,Paris,1896, Part I, Book

II.

therefore

will,for a suitable value If,then, for every term x of u,

of which

true.

make

must

P*y true,and x

of

certain

a

others

which

class of terms

do not y.

In

x.

Mathemawill be found in Couturat, De rittfati

and Infinity

268 But

involves another

meaning of formula in mathematics by the word law. may also be expressed

the usual

element,which

what precisely

say

[CHAP,xxxn

Continuity

is,but it

this element

to

seems

It is difficultto

consist in

certain

a

of the proposition Pxy. In the case of degreeof intensional simplicity that there is no formula for example,it would be said two languages, them, exceptin such cases as Grimm's law. Apart from the connecting is the relation which correlates words in different languages dictionary, method of meaning; but this gives no by which, given a word sameness word in the other. in one language,we can infer the corresponding of calculation. possibility

is absent is the

What other

hand

(sayy In

;/.

will define the

"r), givesthe

=

the

of

case

means,

dependentvariable.

variable. dependent

this classes,

state

to the

should formula a

of all

pairs algebraical

an

relation enable

the

are

to extend

know

to

us

to infinite

has

for enumeration of thingsis essential,

studyof

be

of

the

on

covering "r, of dis-

know

we

case

If functions

It is therefore essential to impossible. and

the

In

formula,

A

enumeration

only languages,

variable and formula,the independent all about the

when

become

the correlation of infinite classes, formula

that the functions of infinite classes,

P^

the which, given"r, the class of terms y satisfying unable to give discover. I am be one which we can

in

one

should

of this condition,and

account logical

I

Its practical psychological. importanceis

highlydoubtful. condition There is,however, a logical though perhaps not quiteidentical with

suspectit of being purely great, but its theoretical

importanceseems

is

there no

least to

some

others.

there

is one

any

relation

a

It

holds

which

follows that,

term term.

of

By

t?,

and

which

which

no

with

it.

Given

any

between

those

two

given any

relation disjunctive

connected

any term

one

not

term

of

above,

two

terms, and

terms

classes of

two

the

terms it

has

belongingto

w, to u

vy at

has

classes are both finite, method, when two correlation (which may be one-one, or many-one,

this

carry out a one-many)which correlates terms of these classes and no others. In this way any set of terms is theoretically function of any other ; and a it is onlythus, for example,that diplomatic made up. But are ciphers we

can

if the number

of terms

variable the independent constituting in this way practically cannot define a function,unless infinite, we the disjunctive the from relation consists of relations developedone other by a law, in which case the formula is merelytransferred to the relation. This amounts to sayingthat the defining relation of a function must not be infinitely complex,or, if it l"e so, must be itselfa function defined by some relation of finite complexity.This condition,though it is itself logical, has again,I think, only psychological in necessity, virtue of which we of a law of can only master the infinite by means The order. discussion of this point, however,would involve a discussion of the relation of infinity to order will be resumed a question which in the class

be

"

which

later, but

we

we

case,

any

may

the

say

values

mathematical remains

in

connection If

series.

that

other

every

IV) when

Part

of

moments a

above

are

so

term complete, no without belonging

rationals

the can

to

integersare

form

shall

see

will the

are

of

but

between

series

series

consecutiveness,

when

only part hereafter.

they of A

are

the

as

ordered series

complete

after the

be

plete com-

have

all

called

cardinal

the

or

the

with

that which

series

a

for

it

series

explained in a complete

together

be

of

them.

incomplete

integers,the the

numbers, selection

from

of

the in

by

correlation

finite

series

may

and be

with

respect Thus

is defined

discussion

the

is

series

is incomplete, this

another.

series

the

series

a

of

term

any

series

to

When

1.

complete

respect

when

and

0

or

may

with

not

complete

a

relation

they

fulfils

a

R

which

transitive,

complete series

was

Thus

term,

is

terms

series; when

A

case. no longer generating relation, but

;

(as terms,

one

mttch

series, sudh

relation

series.

those

of of

the

to

the

the

to

before

come

the

the

IV

either

some

its not

to

seems

respect to the generating relations of the the are an positive numbers incomplete

incomplete with complete series. Thus

Part

x

relation

complete

is

series

series, and

the

but

of

belonging

only

of

one

all the

series, the

a

the rational negative integers and zero, the line. time, or points on a straight Any

positive and such

for

of

is connected

to

converse

is connected Instances

partial.

and

generating

condition

series which

or

or

It

this

notion

x

belong

terms, its

the

to

series. terms

those

Since

terra.

fulfils this A

of

relation

generating one

other

no

consists

series

the

to

has

and

the

relation

term

given,

logical notion

limits, namely

a

which

term

belongs

R

relation

is

is

all formulae.

of

defining

the

them

found;

be

entirelydistinct with

there

of

one

can

essence

intelligently.In

treat

and variables containing useful,give a relation practically

be

other

an

be

R

when

complete

is

the

of

to

289

two

by which, when

the

importance

a

Series

of

position

a

formula

if it is to

must,

There

constitute 267,

that

variables

two

corresponding

yet in

not

are

defining a function between

Cot-relation,

The

257]

256,

of with

is

oee

finite

by powers of progressionsin

whole

transfinite

regarded

the

to

and

part,

integers,as as

the

we

extension

itself; given relation, together with this term respect differences and owing to this fact it has, as we shall find, some important series. be shown, similar it from But by the ordmally can incomplete be rendered complete Logic of Relations, that any incomplete series can distinction and The the mrsd. vwe generating relation, by a cliange in relative series and between therefore, essentially is, complete incomplete of

to

a

a

term

with

to

a

j*ivengenerating relation.

XXXIIL

CHAPTER

NUMBERS.

REAL

258.

concerning numbers,

said

been

r$al numbers real

that

whole

defined

rational

I

part

see

numbers

they

be

cannot

above

greater

about

it

the

the

finite

cardinals

yet

with

the

real number

number,

associated

other

My

rationals

to

real

discussed

Vol.

vi,

have

which in

all

esp.

so

Peano

the

is of

far

as

suggests of

this

me

chapter ; Annakn, 133.

We

the

I

but

it, and

generalized

express

distinct certain

a

than

%

with,

the

are,

comes

Vol.

XLVI,

" 10;

class

of real

a

rational

I shall

Peano,

by near

very

first,that

classes

such

properties commonly

present

it.

explicitlyadvocated Cantor

a

from

is

assigned

opposite theory presents second insuperable. The point the

is

there

the

for

that

seen

which

manner

less

identical

know,

of

but

number,

not

unduly

positiveintegers,nor like

nothing

be

to

already

1, both

rationals

that

ticians mathema-

the

have

my

irrational

certain

apt

are

For

any

between

In

not.

opinion

mathematical

to

p.

do

rational

secondly, that

Math.

to

obviously

in favour

is.

neither

introduces

When

they

numbers

class

is not,

though

next

126-140,

number

with

contend, the

seems

any,

identified

every

with, but

appear

the

Cantor, pp.

with

it

numbers.

really

numbers

I shall

grounds

numbers,

difficulties

Of.

natural

theory

author,

of

*

the

Thus

This

J.

it*.

to

natural

so

numbers.

be

not

associated

number,

number

of

the

it

are

the

irrationals

have

as

there

difference

the

that

is less than

are

ratios

rational

to

think

are

rational

less than

the

chapter.

next

that

if there

that

of

consists

rationals

the

in

suppose

and

learn

definition, however,

generalizationof

originalnotions

relations,which

any

a

to

;

or

they

"

and

real

sense

effected

have modest

whatever

reason

no

in the

series

This

about learns

he

to

numbers,

of

considered

will be

difficulties,which

grave

A

infinite

an

nor

such

limit.

when

relieved

already

learn

something quite different.

irrational

and of

limits

the

as

all, but

now

horror

be

has

to

ordinarily defined,

as

rational

only

to

will

he

at

numbers,

of

assemblage

being a

real

of

series

numbers

reallynot

are

turned

But

rational.

to

is

he

that

be

surprisewill

his

opposed

real numbers The

and

;

is

find

to

all that

surprised,after

be

philosopher may

THE

logical will

be

merely expound

Rimsta

di

Matematica,

Real

259]

258,

Numbers

271

view, and endeavour to show that real numbers, so understood, my own have characteristics. all the requisite It will be observed that the of followingtheoryis independent be introduced

rational

The

259. which

in the next

there

is a

in order of

we we

shall have

the word

reserve

form

magnitude

Such

two.

any

called continuous,must provisionally to

limits,which

will only

chapter.

numbers

between

term

the doctrine of

which series,

in Part

receive another

now

for the

continuous

sense

series in

a

since

name,

which

III

Cantor

I propose to call such series compact** The rational numbers, then, form a compact series. It is to be observed that, in a there are an infinite number of terms between compact series, any two, consecutive the there are and between stretch two terms no terms, any has

it.

given to

be included

(whetherthese

we not) is again a compact series. If now consider any one numberf say r, we can define,by relation to r, four infinite classes of rationals: (1) those le"s than r, (2) those not greater than r, (S) those greater than r9 (4) those not less than r. (2) and (4) differ from (1) and (B) respectively solelyby the fact that or

rational

,

former

the

contain

r,

differences

curious

of

while the latter do

properties.(2)

(1) is identical with

none;

the

But

not.

this fact leads to

last term, while (1) has a rational numbers less than a

has

class of

of

(1),while (2)does not have this characteristic. Similar (3)and (4),but these two classeshave less importance than in (1) and ("). Classes of rationals having in the present case of (1) are called segments. A segment of rationals maythe properties be variable

term

remarks

applyto

defined with

the

as

class of rationals

a

themselves

rationals

which

(i^.

is not

which

yet coextensive

null,nor

contains

but

some

not

all

and which is identical with the class of rationals less than a rationals), of itself, Le. with the class of rationals x such that there term (variable) rational of the is a said class such that x is less than y J. Now we shall y

segments are obtained by the above method, not only from but also from finite or infinite classes of rational^ with singlerationals, that there must be some the proviso, for infinite classes, rational greater find that

than

any member Let it be any inay be defined

This is very simplydone as follows. four classes infinite class of rationals. Then

of the class. finite or

by

relation

to

u"" namely (1) those

less than

every M9 those every ", (4?) of u can be term

(") those less than a variable M, (")those greater than greatertlian a variable u9 Le. those such that for each a it must have found which is smaller than it. If u be a finite class, in this

and

a

and

(3),the latter alone

to (1) and

the

former, in which

had

*

minimum

Such

series

t I shall for extension

to

such

term

axe

;

we

case

the

only a

called by Cantor

former

are

this

(4). singlerational.

maximum

case

to

is reduced

(2) to

I shall therefore

uberall dicht.

confine myself entirelyto simplicity as

alone

Thus

a

is relevant

rationals without

sign.

whatever. positiveor negativepresentsno difficulty u, Part HI, " 61 (Turin,18"9).

Vol. J See Formulttire de Mathtmatiqttt*, " Eightclasses may be defined,but four

are

all that

we

need.

The

that

in future

assume

reduction to that

u

other

has

has

of

term

u

;

no

in addition

I* assume, rationals

the

to

greater than

absence

of

that

is, the

",

any

of

is less than

u

(4), I shall

presentI confine

For the

minimum.

term

(1) and considering

in

and

further, to prevent in considering (2) and (3),

is,that every

that

maximum,

no

class,and

infinite

an

case, I shall assume,

former

our

is

u

[CHAP,xxxra

Continuity

Infinityand

272

w

myselfto (2).and (3), and the

maximum,

a

some

that

assume

existence

existence

the

of

class

of

(3).

these circumstances, the class (2) will be a segment. For (2) consists of all rationals which are less than a variable u ; hence, in the Under

In the (") contains the whole of u. less than which of in is turn since every term some second place, (5") t/, than less other is term of of some term to (2) (2); belongs (2),every less than some term of (") is a fortiori and every term less than some w, since firstplace,

has

u

maximum,

no

(") is identical with the class of less than some term of (2),and is therefore a segment. terms Thus we have the followingconclusion : If u be a single rational,or

and is therefore

a

of

term

a

Hence

(2).

class of rationals all of which

the rationals less than

?/, if

be

u

less than

are a

singleterm,

or

of tf, if tt be a class of terms, always form a segment of rationals. contention is,that a segment of rationals is a real number.

term

My

260.

employed has

far, the method

So

employedin any will depend upon

compact

been

which be may of the theorems

one

follows,some

rationals

the

fact that

the

In what

series.

are

denumerable

a

series.

dependent present the disentanglingof the theorems of the of this fact,and proceedto rationals. properties segments

I

leave

for the

upon Some

segments,

nevertheless less than

a

defined.

capableof being so

of the series

variable term

But

1.

are

How

this fact has led to irrationals the

merelywish

present I not

capableof

a

an

are

more

we

so

defined,are

example, the

segments,

as

shall

pointout

which

rationals

the

same

as

correspondto

see

the

any such definition. in the next chapter.

well-known

beingcomposed of

some

fact that

correlation with rationals.

infinite class of

all the rationals less than

there has

of

to

one-one

classesof rationals defined variable term

as

For

not

less than

called irrationals, are tisually incapableof

segments are are

rationals

*9,'99,'999,etc.,are

other

what

For

consist of the

seen,

Some, it will be found,though

rationals less than

the

haxe

we

as

givenrational.

some

a

fixed rational,then less*than a variable

some

all terms

which rationals,

are

not

definite rational*.

one

There

less than

definable Moreover

and hence the series of segments segments than rationals,

of continuity

higherorder

than

the

seriesin virtue of the relation of whole

and

a

rationals.

part,or

of

Segments form

a

inclusion logical

of them (excluding identity). Any two segments are such that one is whollycontained in the other, and in virtue of this fact they form series. It can be easilyshown a that they form a compact series. What is more remarkable is this : if we applythe above process to the *

Of. Part

I, chap, v,

j". (JO.

Real

260]

259,

series of

Numbers

segments*, forming segments

classes of

273 of

segments by reference

find that

of

the

be

as

identical with the segment of segments defined by some Also every segment defines a segment of segments which

by

to

segments, segments can every segment all segments contained in a certain definite segment Thus of segments defined by a class of segments is always segment

defined

we

infinite class of

an

this term

be left tillwe

must

have

might of

term

some

be defined

can

render properties

two

the

in Cantor's language of segmentsperfect, ; but the explanation

series of

We

These

segments.

segment.

one

class

a

the conditions that

to the doctrine of limits.

come

defined

segments

our

of rationals.

u

u

be rationals less than

have

to

was

all rationals

as

done

had

we

should

inserted

that there

obtained

have

greater than

and this,

and

minimum,

no

we

every w,

If

were

to

what

may be kind, which

called upper segments, as distinguished from the former lower segments. We should then have found may be called

sponding that,corre-

to every upper segment, there is a lower segment which contains all rationals not contained in the upper segment, with the occasional

of exception to

rational. single

a

the

either

the

There

will be

rational not

one

lower

when

belonging

the

case,

segment, upper segment rational. In this greater than a single the corresponding lower segment will consist of all rationals less

than

this

or

be defined

can

Since

upper all rationals as

which singlerational,

there

is

greaterthan

rational

a

rationals less than maximum in

find

question,to

find

cannot

a

be identical with the class of

ever

class of

a

Hence

segment.

lower

segment

not

it is

rationals

having a

in the impossible,

containingall the rationals

case

not

the

segment.

upper

and

Zero

in the

called

infinity may of

case

a

be introduced

as

of

limitingcases

segments,

be of the kind which we the segment must of the kind (") hitherto discussed. It is easy to

zero

(1)above, not

construct

class of rationals such that

some

term

of the class will be less

In this case, the class (1)will contain no terms, will be the null-class. This is the real number zero, which, however,

than any and

any

segment.

class of rationak

given upper segment. But when the upper segment be defined by a singlerational,it will always be possible aU rationals not belongingto the lower segment containing a

cannot

but

be a

two, the

other; and

some

never

can

belongingto to

between

given rational

a

will itself belong to neither

is not

givenrational.

segment,since

a

In order to introduce

a

segment

zero

should have to start with than

a

class which

class of the kind which

as

a

a

null class of rationals*

we

No

Is not

nulL

called ("),we rational is less

of

a

null class of

case,

is

This

is identical with

class

u

every

as

ami thus the class ("),in such a rationals, be introduced. the real number null. Similarly infinity may

term

a

defined

was

the whole

of rationals such

that

rational is contained

no

rational is

in the

class of

If

have

any

greaterthan all u\

then

class of rationals.

we

rationals less than

some

cuid Continuity Infinity

274

again,if we have a class of rationals of which assignedrational,the resultingclass (4) (of

Or

u.

any

[CHAP,xxxin a

is less than

terra

terms

greater than

w) will contain every rational,and will thus be the real number be introduced as extreme infinity.Thus both zero and infinity may

some

terms

to the numbers, but neither is a segment according

the real

among

definition. 261.

rationals.

and

such classes u

Two

be defined

may

same

different classes of

by

many

be

regardedas having the

infiniteclassesu and

common

a

may

v

property. Two lower segment if,givenany

segment as the

given segment

A

there is a

w,

will define

v

greaterthan it,and

v

given any v, there is a u greaterthan it. If each classhas no The classes u and v are this is also a necessary condition.

maximum, then

what

(mtsammmgehorlg).It can be shown, without considering segments,that the relation of beingcoherent is symmetrical of abstraction, should infer, whence by the principle and transitive*, we calls coherent

Cantor

both have to

that to

other

any

be

the

extend

may

a

them

word

coherent to

classes

two

an

lower

a

We

have

now

has

seen

that the usual

and

u

v,

of which

segment, which

one

between

exception.Similar remarks,

one

will stillapplyin this mutattdis,

mutcdu

neither

the preceding from third term, as we see taken to be the segment which both define. We

upper segment, the other include all rationals with at most

defines

relation which

common

This

term.

discussion,may

third term

some

case.

of properties

real numbers

belong

for segments of rationals. There is therefore no mathematical reason such segments from real numbers. It remains to set distinguishing t hen of theories of firstthe the nature current a irrationals, forth, limit,

to

and then

the

which objections

The

Nate.

above

the above

make

theoryseem

contained theoryis virtually

preferable.

in Professor Peano^s

article alreadyreferred to (u Sui Nunieri Irrazionali," lihnsta di Mateit 1"6 this from the from matica^ vi, pp. article, was as well as 140),and "

Formulaire

de

that Mathcmatiqueiti

article,separate definitions of segments (|8,*0) are given,which this

But after distinguished. (p.13S) : Segments so

were

"

remark

real numbers,"

adopt the theory. In real numbers (" 2, No. 5) and of makes it seem as though the two

I

was

led to

the definition of

segments, we

find the

defined differonlyin nomenclature

from

Professor Peano

proceedsfirstto give purelytechnical for distinguishing the two reasons by the notation, namely that the is to be differently etc. of real numbers conducted addition,subtraction, from analogousoperationswhich are to be performedon segments. Hence

it would

contained

appear that the whole of the view I have in this article. At the same time, there is

advocated some

is

lack of

since it appear* from the definition of real numbers that they clearness, are regardedas the limits of classes of rationals, whereas a segment is *

Cf.

Cantor, Mnlk.

Ammlm"

xi,vi, and

72im/"

di

Muttmuitiw,v,

pp.

15K, 15J".

261]

260,

in

Real

sense

no

limit

a

of

class

a

Number*

of

275

rationals.

Also

It

Is

nowhere "

indeed,

from

inferred

that number.

real

than

no

limit of

seem

why than

are

and

by,

extremity, to

reason

at

all.

irrationals to

this

or

be the

be

constructed

"

mathematical.

(which

reasons

done

by

reasons

which,

The ".,

confesses

be in

are

the rather

tha

regarded

he next

as

is

there

have

complete

a

in

clans

although

limit

segments,

given

the

:

whereas

rational

that

less

is

1

whose

commonly

a

fact,

as

number,

real is

of

means

say

to

segment**;

(i#.)

will

taken

as

from

we

rationals

having

not

segments he

:

the

of

both

that

1

the

interpreted

the "

segment

a

limit,

upper

although

Thus

can

must

or

that

suppose

perceive

says

determining,

end,

all,

134)

(p*

longer

no

or

be

of

(ib.)

be

out

be

must

not

he

1),

:

must

assertion

but

again

1 1

say,

the

points is

be

to

no

distinguished

is

number

should

some,

he

is

opposite

and

(which

this

rational I

and

And

2).

where

when

the

rational,

a

fractions

1,

the

numbers

be

can

proper

case,

fractions

determined the

from

fractions,

proper

less

of

read

appears

number

which

(in

V"

proper is

and

1

integer than

real

the

regards

this

class

the

of

number

And

from

differs

of

real

no

"

a

definition

the

a

theory does

not

chapter)

philosophical

XXXIV

CHAPTER

the

mathematical

THE

262.

doctrine

and

some

be

this

doctrine

Calculus, and

that

this

conclusively as method

of limits

examine A

series

which

terms

reduce

have of

one

these

generating relation

having to The

by

the

x

class

of

ar;

and

x

relation

terms

idea

the

classes to

so

of

P

is

defined

is

term

is said

x

as

the

x,

terminated

be

the

segment

segment does

this

But

to

a

one

which

of

a

of

said

class

be

its

definition

limit, consider

of class

any

of

classes

if P

there

x

is

only

and

series be is such

thus

defining term

u

contained

.

as

in

a

every limit.

a

the

obtain

To

in

series

a

there

limit.

terms

termf

no

if the

a

the

determined

segments

when

be

class of

the

terminated,

series has

a

;

between

term

a

this case,

In

the

to

two

and

always possibleto

demands

series.

limit

compact

constitute

not

general definition

in

is

the

this

'

example,

and

which of

numerical a genera],not necessarily

compact,

The

find two

series,then

one

is

is

it is

For

the

between

one

segment

a

of

term

any

there

and

them,

single term.

a

shown,

is- erroneous.

always possibleto

between

term

has

of irrationals.

in which

one

series it is

a

no

as

the

infinitesimals

true

view

a

creation

the

defined

we

in such

But

two.

any

the

applicationto

superseded by

mathematics

such

upon

mathematicians

In emerged as fundamental. general definition of a limit,

more

forth

wholly

rests

some

been

shown

has

that

me,

and

more

first set

its

compact

to

seems

has

I shall

Chapter, then

it

had

modern

But

limits*.

in

presupposed

continuity

thought by

been

philosophersthat

Infinitesimal to

of

treatment

It has

limits.

of

NUMBERS.

IREATIONAL

AND

LIMITS

the

series

the class u will in general, with Then generatedby P. respect to any term not x belonging to it, be divisible into two classes,that whose have to x the relation P (which I shall call the class of terms terms ceding pre-

x), and call

This

*

Methode

it has

is

und

It is

t

vice

class

the

the

versa.

the

"im

that of

whose

have

for

instance,

of

G"sckichte, Berlin, 1883;

perhaps superfluous to explain relation P to

to

every

term

the

x

If

following x).

terms

view,

terms

Cohen, see

that

of w, and

pp. a

term

the

or

relation be

Dew

1,

P

itself

a

Princip

(which of

term

rfer

I shall u"

we

Infinitesimal-

2.

is between

relation

P

two

to

classes

every

term

w, tJ, when

of t",

or

Limits

262-264]

consider all the terms

and

of

Irrational Numbers

277

other than #, and these are stilldivisibleinto which we the above two classes, call x.ar ami TV respectively, may If,now, TT^T be such that,if y be any term preceding is a term there ^

following ",,Le, between

of Tr^pc if

be such

rcvr

between a

u

x and y, then x is a limit of TIVT. Similarly be any term after #, there is a term of TV is a limit of ""#. We now define that x k

z

and

fc,then x if it is a limit of either

x

limit of

that,if

u

Sr^r. It is to be observed that

or

TT^T

have

and that all the limits together form a new limits, many class contained in the series generatedby P. This is the class (or rather the this, by becomes the class) help of certain further assumptions, may

w

which

as the first derivative of the class u. designates 263. Before proceeding further,it may be well to make some character on the subjectof limits, generalremarks of an elementary In the first place, limits belongusually to classes contained in compact

Cantor

series" classes which series in

compact

of which

belong

series in which

some

of terms.

to #, and

nearest

is not

and

is not

series is in the

same

a

terms

are

true

x

all contained

sense, since limits have

obtain P

be

other

as preceding,

of

formingpart

as

In any

follows immediately an

the

may

shall

We

now may all of which irrationals,

are

to

of this

limits this

regardthe "

a

"

be defined

(or precedes)

generally

class of terms

immediatelyfollowing(or

series. In this way, all infinite series which in be defined generally

be) any

one

term

of the

"

264.

u

some

infinite series,without case

term

be any compact has also other limits,

limit may

A

segments,

co-extensive

compact series

case, if

whether

;

of

that

have

other

some

find,limits may not progressions as, for instance,in the are finite integers. we

tenm specified

limit; but this

one

terms

should limits, we

further circumstances.

which

belonging to

two

defined in relation to

onlybeen

which, as we shall see, may arise. series,every term of u is a limit of u as

that every infinite

least

other

no

term

a

will have u series,

an

can

finite,

will lie. Hence

u

theorem

at

be

u

will have

of the

between

have

case

term

a

of

infinite

an

if division,

term

no

add

must

it contains

each of them

and

to

There

?/.

compact series. To series generatedby

a

except itself. In the

u

former

our

may

alwaysbelongsto

in terms interpretation it stands. In the fourth place, if u be as compact series generatedby P, then every

limit of

dependsupon

of

is any term

x

common

generatedby P,

the whole

with

since

shall find,demands

theorem, we

to

revert

or

may

is a terra of ", it is still a

limit unless

this term

and

;

It is

class,providedits of the- series

u

a

all terms a

limit

limit,but it

be finite. Hence

between

all.

limits at

no

For, to

limit of

a

have

can

??"# will both

7r"# and

it is

the

be identical with

case,

and if it contained,

of consisting

class

place,no

number

is

u

limit of the class third

extreme

an

as

a question. In the second place,

to the class u

not

x

may,

proceedto

dependupon

the

series of finite and

various

limits.

We

trans-

arithmetical theories of shall find that,in the

and Infinity

2T8

involve

there

axiom, for which

an

convenience;

which

to

there

the theoryof real numbers

; and of which objections logical the preceding Chapteris whollyindependent.

grave in

inventors,they all

arguments, either of philosophical

no

are

of mathematical

or necessity

their

given by

been

theyhave

in which

form

exact

[CHAP,xxxrv

Continuity

theories of irrationals could

Arithmetical

means

to

shall find in Part

we

for space and is required continuity for which realize the logical reasons

of

irrationalsis irrationals

was

in the

continuous

become

and

mathematicians;

among sense

that numbers

of them

of order.

notion

the

they dependessentially upon

since

be treated

not

usual

no

other

important theory of

past, the definition of imperatively necessary. considerations. This commonlyeffected by geometrical

be

appliedmust definition

onlyby

the

In

yieldanything but

space is to

to

II,

now

that

It is very arithmetical

for procedurewas, however, highly illogical;

numbers

It is

time. an

given

in Part

sense

VI

are

defined independently there possible,

were

arithmetical entities

as

would

and

of application the numbers tautologies,

if

but

none

geometrical

a

be, properlyspeaking,no

definition

the

;

if the

define.

pretendedto

The

such

braical alge-

in which irrationals were introduced as the roots definition, of algebraic equationshaving no rational roots, was liable to similar since it remained that such to be shown equations have objections,

roots;

this method

moreover

numbers, which have

do not

are

infinitesimal

an

in continuity

Geometry. And

in

any

pass from

assumption,to

it is irrationals,

of the real proportion

Cantor's

case,

so-called

onlyyieldthe

will

sense,

if it is

Arithmetic

or

in the

algebraic

numbers, and

requiredby

sense

without possible, from to Analysis,

further any rationals to

be done. advance how this can to show logical of number duction with the exception of the introgeneralizations of imaginaries, which must be independently effected are all

The

a

"

"

form necessary consequences of the admission that the natural numbers I n the terms have two kinds of a progression. every progression

relations,the

and the generalanalogueof positive constituting the other that of rational numbers. The rational negativeintegers, form

numbers aUe

one

denumerable compact series; and segments of

a

as compact series,

which

is continuous

assumptionof theydo

There I will

are

not

progression.But follow without

a

preceding Chapter,form

sense.

in the

limits

denumer-

;

and

Thus

a

series

all follows from

present Chapter we in this sense,

we

have

the to

shall find

assumption.

new

similar theories of irrational numbers.

that of Dedekind*.

Although rational

there is alwaysa *

on

several somewhat

begin with

266.

in the

saw

irrationalsas based

examine

that

a

we

in the strictest

a

numbers

third, yet there

are

are

many

uud irratiouale Zahten,2nd Xtetigfoit

such

that, between

ways

of

any two, a ll dividing rational

e"L,Brunswick,

1892.

Limits

264-266] numbers

of

last term.

For

example,all

exists

there

a

after which

and

is a number Thus

always

extend

When

between

numbers

the

point of

then

there

lies

any one

whollyafter

comes

the second has

axiom

which

by

its position in

the

maximum.

no

Dedekind

defines

"

classessuch that

class is to the left of every

one

exists

division of all

which

are

line (op.dt. p" 11) : straight If all the pointsof a line can be divided into two

every

that

introduced, numbers,but there

two

of the continuity "

and

one

pointof the other clans, point which bringsabout this

only one

points into

classes,this section of the line into

two

parts.""

two

an requires

If aU

of Dedekind^s

This axiom

266.

emendation

worded,and is,however,rather loosely

by the suggested

derivation of irrationalnumbers.

no pointsof a line are divided into two classes, pointis left the section. If all be meant to exclude ti*epointreprerepresent senting continuous the section, the axiom characterizes series, no longer

the to

applies equallyto be held to

line,but

as apply,

to all the

axiom The series, e.g. the series of integers. of the the division, not to all the points regards

all

and distributed compact series,

pointsformingsome

throughoutthe line,but consistingonly of of the line.

this emendation

When

If, from out

by

these numbers

minimum, while

no

to

number

A

is thus defined

classes of which

any two

demand

to

number, since all the old numbers

new

a

number

a

between

can

we

be

Continuityseems

number, which

new

other,and the firsthas

must

numbers, without exception,

this section.

irrational number.

an

onlyis there

not

other.

correspondto

classified. This is series,

the

classes must

the two

between

but

the

classes may

comes

should

term

some

over

rational

after

come

lies between

and the second has

firstterm

no

class

their squares are greateror lew than ", be arrangedin a single in which Keries, definite section,before which comes of the one clashes,

of both

All the terms

the

279

be classifiedaccordingas

may

are

Numbers

such that all numbers of one classes, the other class, and no rational number

while yet the firstclass has classes,

two

a

Irrational

into two

all numbers

no

and

to

form

a

compact

into two

manner

the

new

but series,

the

series is original

one

of

terms

and

new

made,

is

a

the

series

in DedekhnTs

which the sense

points

throughout

the

divided

in

lies no

term

of

then series, original of the word.

the self-evidence upon emendation, however, destroysentirely as alone Dedekind relies {p, 11) for the proof of his axiom to the

missible ad-

be chosen

can

always be

can

the

becomes

axiom

some series,

between portions, only one term of

continuous

portionof

is distributed

series which

if this

previousseries; and Dedekind's

the

among

a

The which

applied

line. straight

Another

somewhat

less

emendation complicated

gives,I think, what Dedekind

meant

to

state

may

be made, which

in his axiom,

A

series,

when, and only when, in DedekiruTs sense say, is continuous may be divided into two without of the series, if oH the terms exception,

we

[CHAP,xxxiv

Continuity

and Infinity

280

such that the whole of the first class precedesthe whole of classes, either the first class the second,then,however the division be effected, last term, This term, which has

at

comes

be used,in Dedekind^s such

that

as

class and

such

are

excluded

self-evidencein such

classes, may

then

is both class*

second

a

;

is not

there

In discrete series, of the first last term

while in compact series it sometimes continuity,

that the first class division) every possible first has class term. the last Both these cases no

the above

by

both.

never

for

and

last term

no

of the two

one

there integers, of the

happens (though not has

first term, but

a

to define the section.

rationals,where

the

as

end of

one

manner,

of finite

first term

a

class has

the second

or

a

axiom, either

an

I cannot

But

axiom.

see

any

appliedto numbers

as

or

vestigeof applied

as

to space.

let

the

be

to the

the

other

rightof no

series

maximum

have

we

are

such

from

has

one

by

certain

and

since general,

the nature is

supposing is wholly

minimum

no

of series in

true

one

assume

the

many

of order.

And,

such a number at all? It must postulate that the algebraical and geometricalproblems,, which not here be brought into intended to solve,must

be

existence

argued,because,as

a

sense

we

of irrationals

problems. The

negativeand

classes of which

two

demanded

for

we

to

it

The

account.

is not

have

we

without possible

should

remembered

This

It is not

righthave

reason

of which

continuity in

seen,

then

Why

other,and ?

discrete.

are

What

between position

a

What

arises is this:

numbers?

of such

existence

that there must

as

to

return

us

that first question

The

the moment, the generalproblem of continuity, Dedekind's definition of irrational numbers.

for leaving aside,

267.

*r

grows

equationtf3

"

from

0 to

has,

in the

past, been

52

0 must

have

",

=

a?

"

"

a

rational irthe

inferred

root, it

and increases,

was

is first

does x " ; so changescontinuously, positive ; if x hence #*" 2 must assume the value 0 in passing from negative to positive. Or again, it was a arguedthat the diagonalof unit square has evidently is that such that and and definite this .r2 0. 2 length"r, length precise But such arguments were that is show t o number. x a powerless truly They might equallywell be regardedas showing the inadequacyof nuEibers to Algebra and Geometry. The present theoryis designed to prove the arithmetical existence of irrationals. In its design, it is but the the execution to t heories short to fall seems preferable previous ; of tlie design. then

"

"

I^et It is two

examine

in detail the definition of

"J% by Dedekind's method.

fact that,althougha rational number singular rational single numbers, two classes of rational a

defined

*

us

"o

that

no

rational number

=

lies between

lies between

any be may all ot though

numbers

them,

If the series contains a is a progression, it is only true proper part which general,not without exception,that the firstclass must have a last term.

fw

266, one

Limits and

267] class

least of these classes must if not,

could

we

together,and between

insert

the two

the two

may

between the

them.

arrange

last term

having a

Let

infinite class to be denumerable, of numbers

We

that

which This

For nearest

were

wouM

one

when

of the terms

some

at

ooe

of terms.

hypothesis,But

all or

be of

one

in

a

series

without reachingit, class,

continually approachingthe other

and without

the

It is evident

oppositekinds

classes, contraryto

281

infinitenumber

an

of

number

new

a

the classes is infinite, we of terms

consist of

pick out

Numbers

all of the other.

higherthan

are

Irrational

us, for the

moment, suppose our then obtain a denumerable series

to the one but continually class, a", all belonging approaching of the other class. Then class. Let B be a fixed number

other

,

between

and

o"

there is

B

alwaysanother

rational number;

but this

may be chosen to be another of the a\ say 0^+^ ; and since the series of ajs is infinite, do not necessarily obtain,in this way, any number not we the series of a's. In the definitionof

to belonging

ffk is also infinite.

between

number

6OT+g, or In fact,a^p

or

and im, for suitable values of p and q, either is else liesbetween o"+J, and an^,p+1 or between fi^-j^ and o"

alwaysliesbetween

is obtained which both

0*8 and

the

the series of irrationals, denumerable, any rational

Moreover, if the ZTs also be

lies between

while convergent. For, let the o*s increase,

the #*s are

the 6's diminish.

Then

and bm* By successive steps,no term all the 6's and all the ok Nevertheless, o"

6tt

o"

"

and

i"

diminish,and continually is less than a continually either, a"+1

"

therefore fln-t-i "n" which is less than this number number. Moreover diminishing "

for if 6a

had

On

"

a

limit

classes. Hence

the two

Thus

number.

the

the number

",

att+1

a*s and

"

Vs

o" 4-

are

both

"/8 would

less finally

o" becomes

limit

diminishes without

;

He between finally than

any

assigned

convergent. Since,moreover,

number their difference may be made less than any assigned ", they haTe the same limit,if they have any. But this limit cannot be a rational number, since it lies between all the a*s and all the i's. Such seems to be the

a?

Thus

Hie are

existence of irrationals. For

argument for the

V8-l-l,3*-8a7-l

=

example,if

0.

=

*

successive that

such

while

convergentsto the continued all the the

odd odd

diminish. continually

and

the

even

series,if they have defined But

as

+0370 ~~g~3I

convergentsare less than all the convergentscontinually grow, and

ones

next

fraction 1

Moreover

the

difference between

even

the

vergents, coneven

the

odd

both Thus diminishes. convergent continually limit,and this limit is limit,have the same a

V"

the

existence of

a

limit,in this

of this In the beginning

case,

Chapter,we

is

a evidently saw

sheer

sumption. as-

that the existence

of

Continuity

and Infinity

282 limit demands

a

create

the limit

by

a

limit should

diminish

consecutive

all the cfs

But here,it indefinitely. is known

which

terms

is

would the

distance of

onlythe

Moreover indefinitely.

differ theycontinually

Hence

bn.

less than

are

diminish

to

part. To

is to be found

the distance from

It is essential that

error. logical

therefore be

Kmit

of the series whose

means

limit forms

the

which

largerseries of

a

[CHAP,xxxrv

less and

less

be the limit of the

a\ n may be, b^ This cannot that limit cfs. the all and a for bn+l liesbetween bn prove of the cfs or be would i t not it existed, but exists*, onlythat,if any one bn. But

from

b\ to

cannot

whatever

be

exist,but may

irrationals are

Thus

yet any other rational number.

nor

proved

not

fictions to describe the relations

merelyconvenient

of the flnsand Vs.

theoryof

The

268.

of Dedekind.

similar to that of terms than

a^7 a*,

state of

,

such that 2" an, for all values of n, is less infinite is presented, This case e.g., by an however .

.

.

In this

,

method,

as

thingsas

found in Dedekind^s

we

be

must

be defined by

may

a%

terms

many Cantor

supposedto of it.

means

theory:

can

the

series of rational

some

limits,

in either of the above theoryof irrationals, following objections.(1) No proofis obtained

arithmetical

forms,is liable to the it of

exist

This is the

cannot

Thus

take,

we

pointsout*, the

prove the existence of irrational numbers as their onlyprove that,ifthere is a limit,it must be irrational.

numbers

accept

.

created by the summation, but order that "

alreadyin

from

.

3*1416.

less than

limit is not

but

a",

In

fraction S'14159

The

decimal. remains

,

.

given number.

some

same

.

irrationals is somewhat concerning have a series Weierstrass's theory, we

Weierstrass

the

of

existence

and

rational

continuitydifferent

of

axiom

rational numbers;

other than

for such

an

axiom

numbers,

from we

that

have

as

unless

satisfied

yet

seen

we

by no

Granting the

existence of irrationals, they are the series of rational numbers whose

ground. (") merely limits not defined,by specified, Unless they are the series in they are. independentlypostulated, be have known to a limit; and a knowledge of the questioncannot irrational number which is a limit is presupposedin the proof that it is a limit Thus, althoughwithout any appeal to Geometry, any of an infinite series givenirrational number can be specified by means of rational numbers, yet, from rational numbers alone, no proof can be obtained that there

are

niust be

new

provedfrom

Another

a

to objection

irrational numbers

and

at

all,and their existence

independent postulate.

the above

theoryis

that it supposes rationals

and

irrationals to form part of one and the same series generated by relations of greaterand less. This raises the same kind of difficultiesas found to result, in Part II,from the notion that integers we are greater *

in

Mamwhfaltigkmttlekre,p.

2"

I

quote VTeierstrasss theoryfrom

Rkfr ftl/genieim Btolx,rof/fxitftgpu Aritkmetik,i.

the account

Limits

267-269] less than

or

and

that or rationals,

relations essentially

are

Given

relations.

Irrational

rational!

some

between

Numbers

but integers,

283

integers.Rationals irrationals not such are

are

infinite series of

rational^ there may be two integerswhose relation is a rational which limits the series,or there such pair of integers. The entitypostulated the limit, as may be no this latter

in

an

case,

the

series which

the

limit is not,

it a

is

of kind as the terms longer of the same is supposedto limit; for each of them is,while relation between two geneous integer^.Of such heterono

terms, it is difficult to suppose

less; and

of greater and

less,from

and

an

the

definition for the

new

a

which

in

fact,the

series of

they

have

can

relations

relation of greater has to receive springs,

constitutive

rationals

of two

case

that

of irrationals, or

a

rational and

definition is,that an irrational is greaterthan a the irrational limits a series containing terms greater

irrational. This

rational,when

given rational. But what is really given here is a relation of the given rational to a class of rationals, namely the relation of belongingto the segment defined by the serieswhose limit is the given the

than

in the

irrational. And

greaterthan

when

other

the

of two

case

its

one irrationals,

series defining

is defined

contains

terms

to

be

greater

series of the other condition which a defining any terms the that the contains to to saying amounts one segment corresponding of the

than

"

TTiese to the other. part the segment corresponding proper of two definitions define a relation quite different from the inequality as

a

relation of inclusion. Thus the irrationals rationals, namely the logical of rationals, form of the series cannot but new terms corresponding part to the rationals must

Such

terms, as

we

singleseries can be constructed. last chapter, found in segments; but are

be found before in the

saw

the theories of Dedekind

a

and Weierstrass leave them

stillto seek.

philosophically theoryof Cantor, though not expressed, to the with all the requisite lends itself more clearness, speaking, easily which I and is to advocate, designed prme interpretation specially the existence of limits. He remarks* his the existence in that, theory, he strongly of the limit is a strictly demonstrable ; and proposition the involved in attempting to deduce emphasizesthe logicalerror The

269.

existence

Cantor the

are

of the

starts same

series. Each

limit

by as

the

from

series whose

what considering

what

I have

called

of these fundamental

limit it is

calls fundamental

he

(16., p. J"2)f. series (which

contained in a larger progressions) series is to be whollyascendingor

whollydescending.Two such series are circumstances horig)under the following *

called coherent

(zu"mmenge-

:

"

Op. dt.3 p. 24. theory of irrationals will be found in op. tit.,p. 23, and in Stofo, Vortesungenuber cMgemeine Arithmelik, i, 7. 1 shall follow,to begin with, a later to me clearer;this forms " 10 in an article contained in Math. account, which seems f Cantor's

Annalfn, XLVI, and

in Rivixta di

Matemalica,

v.

284

(1) If both

alwaysa

(3)

after any

and descending,

are

the other,and precedes

definition ; and

and descending,

there is of most

which

term

one

the

wholly

one

is between

remarks

in symmetrical,

is

In

that it is transitive.

shows

Cantor

the above

which

other

being coherent

relation of

The

of either there is

before any term

the

series.

fundamental

two

of either there is

term

;

the ascending,

is

one

[CHAP,xxxiv

;

of the other

term

If

and ascending,

are

If both

alwaysa

Continuity

of the other

term

(%)

and Infinity

is

extracted,Cantor

are

virtue of the

the article from

dealingwith

more

the definition of irrationals. But the above general to understand the theory of of coherent series will help us

general topicsthan account

irrationals. This

set forth

that, given any number

such

follows in the

as

series of rationals is defined

fundamental

A

there

e,

small,any two have

terms

of three kinds

one

lies between

all the terms b is to

number

be defined

and

"

"

differences from

of

sequent sub-

beingmentioned,the

(3)

p ;

from

fundamental

by the

be

absolute

terms

may

real

A

/".

"

e,

be

onwards,

term

some

negativenumber

certain

a

series must

onwards, all the

term

some

Such

e.

"

onwards,will all be less than

term

be lessthan

may

+

number positive

certain

a

series

finite number

a

of whose

(1) Any number

:

values of the terms, from some whatever " may be ; (") from

greaterthan

denumerable

a

That

e.

difference which

a

Mannichfaltig-

is to say, given any number e, however both after certain of the series which term come a

exceed

terms

as

most

at

are

in the series the absolute values

terms

of

theoryis

and is said in the series,

and in the third to be zero, in the second to be positive, real numbers, be negative. To define the addition,etc.)of these new

first case

to

observe

we

that,if

the series whose

a,,

aK'be the vth

yth term

is av -j-av'or

av

series;while if the real number not

(a*/av) also defines

zero,

numbers

defined

a

av'or

"

fundamental

of two

terms

av x

defined

fundamental

If

the series

"

bxb' and b'/b

Hence respectively.

greaterand lessamong b"b' that b means

negative all terms "

further that This

may

be

fundamental

*

The

alone.

b' is

which

have

been

b"V

are

means

means

Cantor

the whole

6

equal, b' 0 ;

"

6

"

"

"

6' is

remarks

one

and

the

the

same

may be rational. merable remark that a denurational number

to the definition ; hence in series, according

symbol (aj)denotes

V9

that

defined. already

in part,by justified, formally terms

by

b',b

b +

of the numbers

one

all

defined

the definitions of

define that 4"6'

positive;and

in these definitions

series whose a

We

realnumbers. "

proceedto

we

(aK)*is

b, V be the real

(#"),(a/),the real numbers (a*+ a/),(a* a/),(avx av')and ("//av) are defined to be

by

mental funda-

a

the series

by

series.

series,

av'is also

series whose

vth term

is

constructing

is a*9 not

this term

Limits and

270]

269,

Irrational

the differences #"-#"',by which rational we

placeof a/

in

a

define b

can

There

that

to show

put

may

fixed

some

But the consequence that aixl that for the following follow, reason.

does not

a

-

285

is defined, we

6-i'

for all values of

absolutely nothingin

is

Numbers

the above

is the real number

v.

definition of the real number

defined

by a fundamental series whose self-evident is, onlyreason why this seems that the definition by limits is unconsciously present,making us think the limit of a series whose terms are all equal that,since .a Is plainly therefore be the real number must to a, a defined by such a series. Cantor insists I think that his method Since, however, as rightly, of limits, is independent the be deduced from to which, on contrary,are it (pp.24 to weigh with us. 5),we must not allow this preconception the preconception, if I am And not mistaken,is in fact erroneous. is nothing in the definitions above enumerated There to show that a real number and a rational number be either can ever equalor unequal, terms

a

equalto

all

are

The

a.

"

"

"

and

there

also

we

by

for

reasons

a

Lima, V

proud of

is

Cantor

the

=

show

that

with the

as

number

(bv)is

from this

by

by

fundamental

a

cannot

this will

onlybe

limit of

a

true

definesa

seen, there

from

What

this position prois nothing

real number, and

a

is true, and

in the

real

case

terms

above

are

all

series

series whose

where

equal to

(av)and terms

are

whose

(a,)has

either does not But

has

what

a

if b be the

;

if bv be the real all

equalto

limit is b.

as Cantor infer, supposes (p."4),that Lim

real number.

a

have

we

series of real numbers

series of rationals

is it

theoryrenders

nected theorem, is this : Conreal number, namely that defined

fundamental

a

fundamental

a

we

rationals

series whose

defined

defined

there is a

a

his

be subtracted

can

rational

every

fundamental

real number

case

But,

that

supposedproof is fallacious. mathematical advantagesof the

all the

then

a

h be the real number

6.

=

the

hence

by

rational

Hence

00

supposedfact

demonstrable. strictly to

supposing the contrary.

rejectthe proposition (p.24) that,if fundamental then series (""),

must

defined

strong

very

are

in all

is

in

;

identical with

never

The no

series of

fundamental

a

exists ;

af

rational limit.

exist,or is rational

cases

number, which

a

""

But

any

rational. 270.

Thus

provingthat

to

sum

up

what

we

the

define

to

such

other.

rules of

of principle

series both This

the real number

as

greaterand

two

no

on

Cantor's

theory: By

have the relation of being series may and transitive, Cantor this relation is symmetrical

coherent,and that shows, by the help of the term, and

said

fundamental

two

that assumed),

has been

have

some

is

one

one

tacitly third

term, when our series consist of rationals, We then define can which both determine.

operationfor real numbers, and

less between

(which

relation to

abstraction

them.

But

the

relations

of

equal,

of abstraction leaves the principle

and Continuity Infinity

286

us

in doubt

One

thing, however,

partof any seriescontaining rationals, while the real numbers relationsbetween integers,

certain. They cannot form

seems

for the

are ratiorjals

and the constitutiverelation in virtue of which rationals

not so;

are

form

a

seriesis defined solely by

so that theyare relations,

real

numbers,or between

to what

as

are. really

to what the realnumbers

as

xxxiv [CHAP,

the

same

real and

a

real numbers may

of the

means

be,we

between integers

relationcannot

hold between two

rational number.

a

find that

which

In this doubt

as segmentsof rationals,

all the requirements fulfil laid down chapter, preceding and also those derived from the principle in Cantor's definition, of abstraction. Hence there is no logical groundfor distinguishing ments segof rationalsfrom real numbers. If theyare to be distinguished immediate intuition, it must be in virtue of some or of some new wholly

defined in the

axiom,such

as, that all seriesof rationals must

would be fatalto the uniform

have

a

limit. But this

of Arithmetic development

which Peano from the five premisses

has found

and

Analysis

and would be sufficient,

of those who have invented the arithmetical to the spirit wholly contrary the contrary, of irrationals,llie above theory, on no new requires theory

axiom,for if there and

it

are

there rationals,

what

removes

seems,

same

allthat is required of irrationals,

serieswith precisely to introduce a new the parallel superfluous I conclude,then,that an mathematical properties. irrational rationals which does not

is a segmentof actually a

be segmentsof rationals;

a mathematically, whollyunnecessary

if segments willdo since, complication, it seems

must

have

a

limit; while

real number which would be

segmentwhich does have real number

defined by

commonlyidentifiedwith a rational is a rational limit ; and this applies, e.g.,to the

a a

fundamental series of rationals whose

terms

which was set forth positively in the theory and to which, after examining the current theories of Chapter, preceding back. The greater to we are irrationals, againbrought partof it applies but some of the uses of fundamental series, compactseriesin general;

are

as

all equal.This is the

we

shallsee

distancesor in

our

of hereafter, presuppose either numerical measurement or that a denumerable compactseriesis contained stretches,

seriesin

a

certain manner*.

The whole of

to it, however, applies

the rationalsare as any compactseries obtained from a progression obtained from the integers; and hence no propertyof numbers is

involved beyondthe fact that *

See

theyform

a

Chapterxxxvi.

progression.

CHAPTER

CANTOR'S

271.

FIRST

notion

THE

XXXV.

DEFINITION

of

OF

continuity has

been

CONTINUITY.

treated

as by philosophers, incapable of analysis. They have said many though it were that everything discrete things about it, includingthe Hegelian dictum a

rule, as

and

is also continuous

verm*.

This

remark,

being an exemplification combining opposites,has been tamely But as to what repeated by all his followers. they meant by continuity and silence ; only discreteness,they preserved a discreet and continuous be could evident, that whatever thing was one not they did luean relevant to mathematics, or to the philosophy of space and time. In the last chapter of Part to call a III, we agreed provisionally if it had

continuous

usually

Leibniz

satisfied

sufficient until there

was

order

of

a

to

higher

order

we

the

to

in

Book

the

would

surmise, before

the

kind

Euclid that

of

numbers,

which

Cantor.

of

Cantor, that

time since

the

of

the

rational

to

call

was

and

compactness

kind seen

in Part in

consists

as

to

;

and

having

belong

term

a

to avoid

But

space, and

higher

a

incom-

proof set forth in had continuityof

which, nevertheless,

The

III.

continuity. to

space

numbers,

Nevertheless

of discover}'

is the

which

probablethat

was

continuity defined

of this

again speak

it "

generally thought

of

"

than

agreed

been

definition

This

two.

any

have

possible. For, ever a discovery of Geometry of

rational

have

and

continuity is

tenth

have

between

term

a

f,

as

of

revolutionarydiscoveries

the

reason

mensurables the

habit

Hegel's usual

of

series

r*c^

kind

which

between

belongs any

confusion, I shall that

other

kind

treated,

was

a*

of

two, never tinuity, con-

Cantor

exempted from that religiousdogma, conceptualanalysiswhich is requisiteto its comprehension. Indeed it that often held to show, especiallyby philosophers, subjectwas any into elements. Cantor it not matter validlyanalyzable possessing was that this view is mistaken, by a precisedefinition of the kind has shown remarks];,as

a

kind

*

Logic, Wallace's

t

Phil

of

Translation,

Werke, Gerhardt's

Berlin, 1901,

p. 188;

ed., VoL

p. M8.

| Mannwhfaltigkeitstehrv, p.

28-

Werke,

11, p. 515.

was

v, p. 201.

Hat

cf.

Cassirer, Leibniz

[CHAP,xxxv

and Continuity Infinity

288

if it is to which must belongto space. This definition, continuity urges*,be effected without of space, must, as he rightly be explanatory in his final definition, We find,accordingly, only any appealto space. be fullyexemplified in ordinal notions of a generalkind, which can the kind The proofthat the notion so defined is precisely Arithmetic. t o Part VI. Cantor be postponed to space, must of continuity belonging which the of earlier is not purely has given his definition in two forms, ordinal,but involves also either number or quantity.In the present, chapter,I wish to translate this earlier definition into languageas and then to show how series which simpleand untechnical as possible, in Arithmetic, and generally in the occur in this sense continuous are The definition will later be whatever. given theoryof any progression in the following Chapter.

of

In order that

272.

a

have

these terms

Both

enchainee)f

perfectand

be

characteristics: it must bien

be continuous,it must

seriesshould

cohesive

(zusammenhangend, technical meaning requiring

a

.

the latter.

I shall beginwith considerable explanation.

series a (1) Speaking popularly, it contains

when

Cantor, is

finite gaps.

no

ic

follows :

as

The

call T

We

a

have two

has

or cohesive,-

is

cohesion, definition, given by precise as

cohesive collection of

if for points,

pointst and t'of T, for a number " given in advance and as in several ways, a finite number there are always, small as we please, of such that the distances to ttl9tj^ ys"... belonging T, pointstl9"*,..."",

any two

tjf are

all less than

e."J

condition,it will be

This

It is not

reference to distance.

should consist of numbers,

nor

that

the

series.

e

should

be

an

the

if it be or relation,

transitive

;

and

the whole

may forms part. But have

we

are

even

must, in order

in Part

asymmetrical

some

having a

certain

given term, we may substitute onlypart of such a series, completeseriesof which our series to give any meaning to cohesion,

a

measurable. somethingnumerically

measurement,

there

having no minimum, kind presentedby

of the terms

How

be done without necessary, and what can stage. It is through this condition that and

and

if the series be

substitute the stretch in the

we

series in which

a

field of

whole

transitive relation to asymmetrical stretch for distance

essential

distance of the arbitrary

series be

If the

has

necessary that the collection considered that e should be a number. All that is

necessary is,that the collection should be of Archimedes distances obeyingthe axiom and

seen,

III, become

far this condition

it,I shall show our

at

discussions of

relevant to

the

a

is

later

quantity

discussion of

continuity. *

Ada

Math,

t Ada

Math.

I see

The

words

FommMre

n, p. 403. ny pp. 405, 406; in several ways

*c

de

Mannichfaltigkeitslehre, p. 31. seem superfluous.Tlieyare omitted by Vivanti Mathtmatique*,Vol. i, vi, " 1, No. 22. "

:

Cantor's

272]

271,

If the distances

Archimedes,there numerical there is

rational

some

For

finitenumber

that

obey the

are

of

axiom

of incapable

finite

a

numbers, and

the

series

is

not necessarily

let 8,d be two distances; let them be such that,for any In this case, if S be the distance ", and ", nS is less than d .

tt\ it is plainthat the condition of cohesion

d be the distance

be satisfied. Such

cannot

what seems actually occur, and paradoxical in certain cohesive terms by merelyinterpolating the of series of rationak is cohesive; example, segments cases

"

"

be created

they can

For

series.

these segments Add contained in them.

have

when

and

them

289

others amoiig them. In this case^ kind with either the analogy of the requisite

real

the

series do not

our

among of some

in terms

longeran

or

cohesive.

stretches in

or

are

measure no

firstdefinition of Continuity

segments,i.e. completed

rational

limits,the limits

to the series what

now

may

not

are

be called the

the

segments having rational limits together with their limits. These are new terms, formingpart of the same series, of the relation and have whole former But since they terms. part to the the difference between a segment and the corresponding now completed of

segment consists series consist of

infinite number

an

and the fails,

Archimedes

all other differences in the

of rationak.

series is not

new

satisfied by real

rational numbers

or

is to be extended

to

non-numerical

are

than

where

e,

e

the axiom

of

to have

minimum

is

no

and

it is necessary, if cohesion series, that,when any unit distance ;

there should be distances whose selected, is any

Thus

cohesive.

condition that distances in the series

The

is

while singlerational,

a

rational number.

numerical

is less

measure

For, if there be

a

minimum

make

distances "a, ^2... lessthan this minimum, our we distance, there must not which is contrary to the definition of cohesion. And but there must to distances in general, be no only be no minimum cannot

to distances from

minimum must

be

compact,i.e.

It must cohesive. where

is

a

made

must

any have

giventerm. a

term

Hence

between

every cohesive series

any two.

supposed,however, that

every compact series is the series formed of 0 and 2 Consider,for example, "/n, be

not

"

n

my term

are

any between

less than

1.

such integers

that

m

is less than

Here

n+

0 cannot two, but the distance from any Hencfe the series,though compact, is not

however, series, complete,being part its distances are measured. which of rationak, by means different. We completeseries,the conditions are somewhat

series of a

accordingas there are (a) If there are distances,and equal distances it may happen that, though the equalstretches, two distinguish

distances from

cases,

term

some

never

become

be hesive. co-

only of the

is not

This

there

In must

distances.

or

are

not

do

not

to correspond

less than

series is some

compact, the

finitedistance.

presentedby magnitudes,if we were to accept Meinong'sopinionthat the distance of any finite magnitudefrom zero is always infinite (op.cte. p. 84), It is presentedby numbers, if we

This

case

would

be

and Infinity

290 distances

measure

(as there

are

[CHAP,xxxv

Continuity for

reasons

many

doing)by logaiy.

distances,the series is not

cohesive, there If are no (6) distances, compact. though but onlystretches, then, assuming the axiom of Archimedes,any stretch of n. value Hence, dividing will be less than ne, for a suitable

completeand

it is

the stretch into

there is assume

into

n

be made less than way of provingthat all can of linearity either the axiom (that any stretch can

or equalparts),

n

least of these will be less than

at

parts,one

e,

no

the effectthat

but complicated

more

stretch d

a

be divided into

can

and less than

all

is in almost

a

cases

condition

numbers

conditions

the

to

or

impliescompactness,

but

integer?i

Thus

distinct from has

Compactnessis purelyserial,while cohesion of numerical

we

be divided to

is

may

a

and compactness redundant. superfluous cohesion

unless

completecompact pleteness togetherrender com-

axioms

these two

be cohesive; but

series must

dj (n

But

parts,each of which

n

1 ),whatever

"

e.

generalaxiom,

more

of Archimedes,

and the axiom

this axiom

With

a

dj (n 4- 1)

greaterthan be.

regardto

in this case, with

Thus

we

see

that

compactness.

essential reference to

measurement.

Cohesion

impliescohesion,except

compactness never

completeseries of rationals or real numbers. 273. by a perfectseries is more (%) To explainwhat is meant it coincides with its first derivative*. is w hen difficult. A series perfect the notion of the derivatives examine this definition, To explain must we of a limiting-point and this demands of a of a series"!", an explanation the terms of a series are of two kinds,those series, Speakinggenerally, and those which he calls limitingcalls isolated points, which Cantor points. A finiteseries has onlyisolated points; an infinite series must define at least one limiting-point, though this need not belongto the series. A limiting-point of a series is defined by Cantor to be a term such that, in any interval containing the term, there are infinite an of terms number of the series (ih. p. 343). The definition is given in in the

terms

The The

sole

of the

case

of the

pointson

The

so

Peano

on.

class of real numbers

and let

x

be

the firstderivative of the

first derivative of the first derivative is called the

and derivative, a

essential reference to space. of the original term series.

no

limiting-point may or may not be a of all limiting-points is called assemblage

series.

of

line,but it has

a

a

as

real number

givesthe

definition of the firstderivative

follows : Let

(which may

or

u

be

class of real numbers, not be a u) such that the

may

a

lower limit of the absolute values of the differences of other than

is

second

then the class of terms

x

from

terms

of

u

this condition satisfying definition is virtually identical with u\. that of Cantor, but it bringsout more of the the connection explicitly derivative with limits. A series, it consists of when then, is perfect, x

zero

;

is the firstderivative of

*

Acta

Math,

t Formulate,

$

This

n,

+ Ih. pp. 341-4.

p. 405.

Vol. n, No.

3

(1890), " 71, 1*0

and

4'0.

Cantor's

272-274]

firstdefinition of Continuity

291

its firstderivative ; i.e.when all its pointsare the same terms as exactly and all its limiting-points limiting-points, belongto it But with regardto the latter point, 274. namely,that ail limitingthe series must is necessary. belongto it,some pointsof explanation the of for series rational numbers. Take, example, Every rational is the limit of

number

rationals

contained

are

series of rational

some

in their firstderivative.

seriesof rationals which

do not

chapterthat theydo preceding of rationals which

numbers, and thus the

have

have

have

not

limit have

a

a

limit at all. Hence

the rationals should the letter of the definition,

But this is not the what believes,

which certain conditions, fulfilling a limit. vergency, must which have no rational limit as

have

having a

may

he

Hence

regardsthose

numbers. it is

when

take this view; and

impossible for

theorem

regardthe real numbers

we

to

limits,it

This perfection*.

is necessary to

modification

must

we

as

therefore

rationals ; and therefore the of its firstderivative.

fact,the first derivative of the rational numbers But

series of rationals

irrational limit, and

contain all the terms

series of rationals does not In

irrationals,

that every series erroneous, be called the conditions of con-

havingan to belonging the series of

limit not

series. perfect

a

in connection with

saw

we

form

compelledto regardas

were

we

Cantor,as

case.

all series

limit,and therefore,by

rational

a

regardsthose agreedin the

as

we limit,

rational

a

But

as

when

is held to be the real

segments of rationals,

deny the

we

modify Cantoris

existence-

definition of

examine.

now

when all its pointsare say is,that a series is perfect and when series further, being chosen out of our limiting-points, any What

must

we

first series,if this

new

series is of the sort which

defining limit,then it a

make

To

has actually

this statement

we precise,

limit

a

must

is

usually regardedas

to belonging

examine

what

our

firstseries.

are

the

tions condi-

In the

of denuroerable considered as defining a limit. case usually set been forth. a nd have to already They come series, theyare simple, all of terms however the series distance small, our that,givenany this, ",

definite term, say the wth, are such that any two of them This statement, difference whose absolute value is less than ".

after

some

have

a

Le~ it is not purely or it will be seen, involves either number quantity, the ordinal. It is a curious fact that,though supposedcondition for

limit cannot, purelyordinal terms, the limit of existence

the

of

always be

can

a

defined in

Cantor's fundamental

relation P, of the

or

alwaysthe

presentmethod, be stated in

denumerable

if there be series,

a

I shall

one,

distinguish

and compact series into progressions

earlier terms relation P

have

(whereP

to

later

is the

ones

always the

generatingrelation

and regressions ara compact series in which the said progressions

This point is 1900, p. 167. *

a

our

purelyordinal terms.

series in

accordingas regressions,

by

ablydisctissed by Co^turat, Ifawc

de M"

ei d"

Morotey March,

and Infinity

292

compact series is further assumed

contained).The term

is then the limit of

x

Continuity

a

if every progression,

[CHAP,xxxv to be

complete. A the progression

of

term

relation P, while every term which has to x the relation P of the progression. This definition^ term also has this relation to some similar will applyto a definition and ordinal a it will be seen, is purely ; has to

the

x

regression. Let of

examine

us

limit to

a

a

what

next

the

are

usual conditions for the existence

series. When

non-denumerable

to

come

we

examine

to be restricted to series,we shall find it inconvenient will it and therefore be well to consider other series denumerable series, series contained in our denumerable if at once. Here, of course, any

non-numerical

there will be a corresponding seriesfulfilsthe conditions for a limit, larger in our definition of a limiting-point largerseries. And the upper or if there is one, lower limit of the whole or part of our larger series, may of a progression But be defined exactly or as in the case a regression.

generalconditions it will be the

assumes

existence

observed

of such

existence

Cantoris

limit cannot in

definition of

which

are

our a

such

greatimportanceof Cantor's fundamental of segments will,however, throw some

down,

largerseries. limiting-point into

a

points. This series.

this

lighton

that any class of terms in a series segment,and that this segment sometimes can, but sometimes

We

matter.

defines

laid

be turned

cannot

there

be

method

The

a

in

saw

Chapterxxxin

cannot, be defined

term. by a single

is its upper limit

and if this term

the

a

point, and

a

conditions under

definition of the illustratesthe

that

of

series contained

reference to denumerable

exceptby And

for the

segment

;

defined* then

was

When does not

it

can

be

so

belongto

it is also the

upper

defined,this

term

the class by which

limit of that class.

But when

the segment has no upper limit,then the class by which the segment was defined also has no upper limit. In all cases, however and this is one of the chief virtues of segments the segment defined by "

"

infinite class which

an

segments defined or

not

by

the

the class has

an

define

terms

has

in which

alwayshave

the

class is

no upper limit is the upper limit of the several members of the class. Thus, whether

upper

limit^the

segments which

that is,that provided, contained has terms coming one

"

the

its various

compact

series

after all terms

of

the class. We

can

express, without

now

assumingthe

existence of limits in

this is not demonstrable,what is meant by a series containing its first derivative. When any class of terms is contained in a compact the which are series, conditions commonly said to insure the existence

cases

of an

where

though theydo not insure this,do insure upper limit to the class, the class limit to of segments defined by the several members upper

an

of the class. And

concerningwhat class

M

of terms

as

the regardslower limits,

same

holds proposition

called upper segments. Hence define : A we may when formingthe whole or part of a series is perfect

we

Cantor's

275]

274,

of

each of the terms in ", and

firstdefinition of Continuity

is the upper

u

or

lower limit of

293

class coutaii"ed

some

be any class contained in t/, and the lower segments defined by the several members of v have an upper limit, the upper or this limiting segments have a lower limit, segment is one of those that

when, if

be defined

v

singleterm of te, Le. have a term of " for their This definition, it most be admitted, respectively. upper or is more than Cantors,but it is free from the unjustifiaWe complicated assumptionof the existence of limits. We in what is perhapsless may repeatthe definition of perfection Given series*and difficultlanguage. any any class of terms u contained there are an upper and a lower segment corresponding in this series, to of u. Any infinite set of terms # being chosen out of w, every term there are certain conditions which are commonly said to insure that v has an upper limit,which, it is admitted,may belongneither to w, nor

can

by

a

lower limit

to the series in which

is contained.

u

these conditions do insure,

What

however, is that the

class of lower segments corresponding to r has an If the series is perfect, v will have an upper limit whenever

upper limit. class of segments has one, and this upper limit of v the corresponding of u. The definition of perfection will be a term that this requires hold

should

in

and

lower

and limits,

the

necessitated the

the question concerning

above

I shall

of

of

tained con-

v

logically priorto

own

belong.

with individual irrational numbers;

is general.Wherever that the principle

there perfect,

it

is to

That

formation.

series that the perfect say, it is onlyby presupposing series. We have to be the derivative of the imperfect case

existence

the rational numbers its

has

portance, imphilosophical

some

while its firstderivative Is imperfect,

the first derivative is

this is the

limits,which

repeatthe arguments againstassuming the

series is

a

existence

is one complication,

of limits in the class of series to which Where

for any class

u.

As

275.

for upper

both

can

be shown that

seen already

it is easy

to

the derivative contains

see

term

a

series,that term is the limit of some original denumerable series formingan integral part of the first series. If this nition serieswith a limit have the generalterm wordingthe defiOH, then is there not to applyonlyto series of numbers alwaysa so as such however distance small, for any specified definite number e, m,

belongingto

-not

the

"

"

that,if

n

is

greater than

my

the distance between

an+p

and

o"

is less

this it is inferred be. From may that the series (an) has a limit,and it is shown cases, that, m many series the (a")was this limit cannot belongto the series out of which

than

e, whatever

But

chosen. be

positive integer p

the inference that there is

supportedeither

limit, or When

by

some

the term

shown easily

to

by

axiom which

be the

a

limit is

It precarious.

previousknowledgeof the term the necessitating

existence of such

may

is the

which a

term.

is the limit is independently known, it may it is not known, it cannot limit. But when

be be

and Continuity Infinity

294

Such

introduce

we

axiom

some

by Dedekind, but

is introduced

axiom

an

all,unless

exist at

provedto

[CHAP,xxxv

we

of

continuity.

that his axiom shows that two

saw

of abstraction, which The principle unsatisfactory. is fully satisfied by segments. coherent serieshave somethingin common, "

is

in

And

the constitutive relation of the terms

belongingto

not

this

the series is

belongingto

is that of the

which

cases, among

some

it rationals,

that

seems

hold between

series cannot imperfect

any

that the existence of limits so series,

whollyimpossible.For

a

limit must

not

have

in a series of which the series which it limits forms position constitutive relation of which the limit, some as part,and this requires An be well as the terms limited,must capable. independentcomplete certain

a

rationals, cannot, in fact,have any limiting-points to it. For, if R be the constitutive relation,and two not belonging terms a, ", have the relation J?,any third term c, which has this relation and therefore both, of the terms ", 5, belongs to either, its converse or have the must to the same series as a and b. But the limit,if it exists,

series,such

the

as

constitutive relation to the terms the

which it limits ; hence it must Hence

completeseries to which

belong

series which

they belong. any it is to only part of some belonging completeseries;and a completeseries which is not perfectis one in which the limits defined in the usual way, but not belongingto the do not exist at all. Hence, in any completeseries, either some series,

to

has actual

not limiting-points

definable limits do not exist,or the series contains its first derivative. In order to render the arbitrariness of assumingthe existence of limits stillmore

evident,let

endeavour

us

to

set up

an

of

axiom

tinuity con-

than Dedekind's. shall find that it can irreproachable perfect impunity. in a series continually differ less and number of positions We

more

stillbe denied with When

a

lessfrom each other,and

known

are

to be all on

side of

one

given

some

exist (so our there must axiom position, might run) some positionto which theyapproximate that distance can be specified so no indefinitely, small that theywill not approachnearer than by this distance. If so be admitted, it will follow that all imperfect this axiom whose series, firstderivatives are perfect, and are to presuppose these first derivatives, be regarded the consequences them. Let us examine as selections from axiom of denying our in the case of a series of numbers. In this next to all the terms might suppose, the position case, the unwary is belongingto them, would be (say)p, where p On" but not On "

greater than if

our

Titus nearer

above

for

e,

series is

pf

"

aU

"n

suitable value

compact, there is

is less than

the

denial

a

a's than was

not

p

is : There

a

"n,

p

direct,and

is a term

term

e, whatever

between

whatever

n

is, contrary to the

illustratesthe fallacieswhich axiom

"

of

the

in this

to which

fact

n

But

and p e, say p'. Thus be. p' is may p

"

hypothesis.But

that

it seemed

subjectare hard approachas

the a's

be.

may

to near

correct

avoid. as

the

we

The like.

Cantors

275] denial

The

the

The

specify,say

there

p,

to

beyond

all the of

limit),none

distance

finite

introduces

the

of

a**s

have

we

be

term

a

specified, p

may

be

the

limit

This

but

;

is

if

of

the

of

against of

limits

to

be

as

in

This

above.

Having

its

to

various

showing, are

now

and

his to

is

at

p

things, well

a^

always is

later

its

series exact

more

of

which

than

strong the

shown

be

perfection

general in

fatal

show

to

be,

to

must

be

reasons

existence

exist, is modified

established.

as

the those

these

the

to

continuity* I

definition,and

points

n

when

may

n

c

necessary; logicallyun-

favour, and

earlier definition

ordinal

whatever

(except

thus

will, in future, be regarded Cantor's

a

terms

then, if

",

whatever

superfluous,and

of

definition

of

is

admitted,

not

e,

which

there

series

found

is

segments,

otherwise

cannot

some

-

designed

axiom

a

than

e,

certain

a

irrationals

that

are

be

rational

distance of

so

term

self-contradictory.The

not

in

reasons

ttou

than

more

as

is less

we

greater

no

a

by

distance

greater than

to

has

finite

no

small

a

"

can

curious,

no

Cantor's

possible,the

required.

an

"

they

analyzed

examine

e

is

opposed

are

conclusion

portions if

p

and

is

is

than

term

irrationals

mathematically

where

cases

of

term

also, whatever

introduction

state

that

so

finally, any

Hence,

rejected;

proceed

so

that

rationals, there

it.

The

a\

specified, an

is also

it

is

there

series

0*$

there

When

all the

irrationals,as

of

#ls.

things, though

as

theory

p

rational)

state

admission

be

but

odd

this

which

to

a*

-

finite

a

series of rational

case,

our

the

exceeds.

chosen

be

can

where

p

of

case

distance, while

to

term

no

that

this

approach indefinitely, as

after

p

such

the

In

beyond

this

into

is

at

words, whatever

e,

in

nearer

them,

a's

indefinitelyapproaching if

finite

term

the

symmetry which

to

term

is true

limit.

approach

all

beyond

term

every

This

a

Every

from

finite distance

other

205

a's, but

There

In

specify (except

a's

e.

been:

the

to

finite distance

at

we

the

have

rational

no

#*s

nearest

like.

be.

a\s, but

the

nearest

term

we

some

may

have

which

numbers

as

is

a^

a

should

near

as

whatever

e,

is

denial

approach

a's

than

There

was:

distance.

first definitionof Continuity

shall

application of of

various

numbers,

portions

XXXVI.

CHAPTER

CONTINUITY*.

ORDINAL

276.

of

definition

THE

chapter was, points, some measurable magnitudes.

we

as

least

reference

two

ordinal

notion

is free

this

all elements

from

definition,as

well

led

continuity

Cantor

seems a

purely

a

definition

I shall orderf be suggested. may

which

examine

now

.

which

others

ceding prein at

numerically

or

like

construct

to

the

demanded,

it

;

in

to

extraneous

as

examined

numbers,

either

to

Nevertheless

this has

and

;

we continuity which not purely ordinal saw,

shall find

We

quantity are excluded, there theorems of great importance, especially as regards fundamental are of definition series, which, with except that suggested ordinal any falsej Cantor, remain indemonstrable, and are presumably sometimes to be given, are fact from which of Cantor's the merits definition,now a

that, so

long

all references

as

and

number

to

"

apparent. 277. follows.

We

numbers

than

call

type marks. (1) It we

is

we

The

integers. (%) between

proved

any

two,

i.e. the

that

these

three

presented by correspondence,between which

the

This

main

shall

and f

is

virtue

with

any

of

the

in

say

Math.

"

in

the

with de

this, will

use

of

that

same

much

they the

with

is It

are

finite term

a

is then of

define

is

there

is

three

properties,in

to

say, these

terms, and

later

mathematical

series

of

suhject as

Metaphysique

suitable

a

which

completely having

earlier

following

et de

of what

type

Couturat's

Morale, March, I said

type

a

one-one

later

to

ones

induction, which

this M.

the

iu the

are

denumerarticle,

1900.

"

Sur

1 agree

preceding chapter,

found.

be

Artna.lsn} XLVI.

RdM,

Math.

fact

this article,in which

| Mathematical found

series

two

in

terms

in

as

rational

magnitude.

the

(iibercdldicht).

compact

rationals, that

correspond to established by the

applicable in

the

is

the

(3) There

last term.

or

characteristics

* The present chapter deals la definition da Contiuu/' Revue

in

first

series

that

from

by

correspondence

one-one

no

define

we

is,by taking its

different

earlier terms

ones.

is

a

has

series

order

be

must

type

artide" is

of

order

their

in

1,

this

denumerable, that

give

can

type

later

presented by

of series

less than

them

in his

continuum

series of

A

(which, however,

given),

the

and

0

17.

of the

from

(" 9)

start

greater

This

order

definition

Cantoris

vn,

Annakn,

proofs of such 3, XEVL,

"

11.

theorems

as

are

not

already

well

known

will be

276,

277]

able.

Thus

Ordinal all series which

similar. ordinally

are

fundamental show

been

207

denumerable,endless*,and compact, proceed{"10) to the consideration of

are

now

series contained

has

(as

We

Continuity

in any

one-dimensional

alreadyexplained)what

We

series Jf.

is meant

by callingtwo

fundamental limit of

limit

series coherent,and we give an ordinal definition of the fundamental the series, namely,in the case of a progression, after the whole progression, but every term before the limit

a

comes

before

comes

of

term

some

definition for the limit of series can

have

with a corresponding progression; We regression. prove that no fundamental

a

than

more

the

limit,and that, if

one

fundamental

a

series

limit,this is also the limit of all coherent series; also that two fundamental series, of which one is part of the other,are coherent. Any has

a

of M

term a

is the limit of

which

term principal

of M,

fundamental

some

If all the terms

.

of M

is called

series in M

terms, M principal

are

is

called condensed

If every fundamental series in M in (insichdicht). itself limit in Jf, M is called clewed (abgeschlossen)f. If M is both closed and condensed in itself, it is perfect.All these properties, if they has

a

belong to Jf, belong to these

With

advance

we preparations,

("11).

continuum

real numbers

from

perfect type.

But

series which

any

" be the

Let

of the series to which

1, both inclusive. Then #, as this alone does not characterize 8.

0 to

within itselfa propertyof containing

the

rationals

the

belong,in

^-series there

of the

such

one-dimensional

(") contains

between

terms

In

which

this

any

show

last term, this would could take it away from or

condition

the

condition Cantor

between

know, is

we

It has

a

further

type 17, to which terms

two

any the

series which

(1)

following

3f

is

a

is

perfect,

there

of which

series S

are

of M.

necessary to add the other properties that S is of the type 17. For if S had a first not

be also the firstor

last term

of M

Sy and the remainingserieswould

("),but would

have

(%) togetherwith (1)

proves that any

M.

belongthe

Hence 77-series.

denumerable

a

it is definition,

to required

are

to

:

terms

two

series of the

that

way the

of

continuum

itself

within

a

terms

are

definition of the continuum A

similar ordinally

last to the definition of the

at

type

is

seriesM

first or

no

insures

that

;

S

the above satisfying

a

we

stillsatisfy

last term; is

hence and

compact

the

series.

nally conditions is ordi-

similar to the number-continuum, Le. the real numbers from 0 to 1, both inclusive ; and hence it follows that the above definition includes included in his the same class of series as those that were precisely former definition.

does not

He

ordinal,and it might us

see

*

t Not Part IV,

having neither

a

to be confounded

that his

definition is

new

whether be doubted, at firstsight,

for ourselves whether I.e.

assert

notions any extra-ordinal

beginningnor

an

are

it is

so.

purely Let

contained in it,

end.

with the elementary

sense

of

a

closed series discussed in

Continuity

Infinityand

298

only point as

The

278.

collection is to be

any doubt could arise is with To be a denumerable denumerable.

which

to

being

condition of

the

regardto

collection whose

a

[CHAP,xxxvi

terms

all the terms

are

far, is purelyordinal.

This notion, so progression.

But

in

of any ordinally similar or be capableof two orders, in one

of the rationals

supposed,that terms formingthe series must while theyform a compact series, discover whether

To

not

or

a

given set of

the

case

series,the of which

ordinal conditions;

purelyordinal.

itself is

some

they form a progression. is capableof these two

terms

than

other

orders,will in generaldemand the notion

in the other

of

Now

theless, never-

know, from

we

the

of all such seriesto the series of rationals (which involves only similarity that no such series is perfect.But it remains to be seen ordinal ideas),

whether

result from

of the rationals which distance.

know,

We

but perfect*,

be

Such

as

we

here is

proof,however,

a

a

and progression,

series in which

being a

For

easilygiven.

compact series S

denumerable

there

order

the

in

take

in this order call them

in

series

fundamental

a

progression and

in the

in

u

;

fundamental

our

for the

first of these

the

order in

u

succeeds,in $, every will be

S, the

which

hence this fundamental

of

xn

series has

no

in

u.

us

endless, compact

series

to state

terms

of the

lower)limit

S,

limit in S.

series of

is a term

form

a

onlybe to

It must

of

construct

have

the

an same

xn

any term of S series #, and

theorem

all the

that

a

Segments of

a

theoryof

denumerable,

mental segments defined by funda-

and perfectseries, is

former

a

between

segment whose

kind,which type as S, and

this

rational segments,are a series of the same in the whole series of segments in the requiredmanner. ordinal definitionof the continuum is complete. 279,

also

are

is,therefore, perfect purelyordinal.

segments there

of "

can

The

no

construct

These

series in S.

the

predecessor's

always infinite.

fundamental

our

and our difficulty, matter simply. Given

segments enables

other

the second

as

which

Hence

From

there is

a

limit in S, for each term

a

be

pointonwards

form

first in

the

of

our

the third term

denumerable endless series cannot this

#19

is

this way, we of which terms

precedesit

term

some

they

in S

say "ra, as

have

series cannot

term

surpassed by

term,

in S

-successors

successors,

series in

This

one

successors

of

which,in the

Proceedingin

series v.

in S.

as

which

finitenumber

a

has

of

number

common

ascendingfundamental

has

term

therefore

u^

successors

Take

This

v.

there is

the terms

Startingwith

u.

be this order,which we will call a?0, there must Take the first such order S, follows this term. in

special properties

series can of fact,that no denumerable purelyordinal proofof this theorem.

matter

a

want

the

to appealing

prove this without

can

we

may are

any

upper

two

(or

be called

contained Hence

the

be

above defined can as supposedthat continuity in Arithmetic, from integers exemplified, by the devious course not

and thence ratioimls, *

to real numbers. Ada

Matkematica^11,

On

the

p. 409.

the integers contrary,

Ordinal

278-280] themselves

be made

can

infinite classes of

plan.

and integers,

the smallest in t?, u

identical,but smaller

(n

be

terms (n + l)***

is to

arranged

firstn

possible following

of

is less than

u

and

u

that which different,

are

first. This

come

in

terms

all

the

on

the smallest number

first. If the

comes

the

l)"*term

+

let them

of which

v9

299

illustrate continuity. Consider

to

classesuy

Of two

Continuity

series has

t"

has

are

the

first term,

a

but no last term. namely, the whole class of the integers, Any completed the of is continuous series, the reader can h owever, a a s series, segment for himself. The denumerable see easily compact series contained in it is composed of those infinite classes which contain all numbers greater those all than some i.e. number but finite of number, a containing Thus classes of finite integers numbers. alone suffice to generatecontinuous series. 280.

The As

above

it will be observed,dependsupon prodefinition, gressions the of discreteness, it seems are progressions very essence

that paradoxical

should

we

requirethem

after all,as it is certain that

defining continuity*.And,

in

peoplehave not

in the

pastassociated any

the definition we adopt is,in some idea with the word continuity^ precise in Cantor's enumerated arbitrary.Series having the properties degree, would be called continuous,but so would many others definition generally which his definition excludes. it will be a valuable inquiry In any case to ask what can be done by compact serieswithout progressions. Let u be any endless compact series, relation is P, whose generating of any and concerningwhich nothingfurther is known. Then, by means in ", we any class of terms denote by U the class of all lower

term

or

define

can

may

precedinga v is a every term first term, and every term no

segments

In

".

segment of

u.

Let

us

lower segment, it u. contained in t*, not null,

be well to repeat,is a class t* of terms not coextensive with n9 and such that

and

a

of

v

the

A

has

converse

followinga

v

is

a

no

last term, and has v case, when is called

z?, v

an

upper segment. prove that every segment consists either some term of ", or of all the terms preceding single (orfollowing) It is then

a

variable term

of

easy to

of

class of terms

some

u

;

and

that every single term, lower segment in this

every class of terms, defines an upper and a manner. Then, if V denote the class of upper segments, it is easy to U and V are again endless compact series,whose prove that both two relation is that of whole or part; while if u has one or generating and

segmentsaccording the consideration of segments

ends,so have U and F, though the end-terms to the definition.

*

to

Mr

Whitehead

Cantor's.

limit,and

A

If

we

has shown

series

proceedto

now

that

the series has

a

first and

the when

is continuous a

are

not

simplerdefinition is equivalent following(1)every segment, upper or lower,has a

last term

(2)a

;

d enumerable

compact series fa

,

contained

in it in such

two

of

in

terms

defininga

our

a

way

that there

series. original

denumerable

series.

are

In this

terms

of this latter series between

are definition, progressions

relevant

any

only

and Infinity

300 in U

of ITs

class whatever

which, in all contained

no

be

it must

remembered,

are

all classescontained

Hence

in "*.

of all the members

sum logical

cases, is the

which,

members

segment of TPs defined by any alwaysbe defined by a single"7,which,if the last term, is the upper limit of the class, and

can

has

class is infiniteand

[CHAP,xxxvi

shall find that the

we (U say),

V

or

Continuity

of the class "

all themselves in U

classes

and

having no last distinct proposition)

also (what is a upper limit in "7; and all classes contained in U and having no first term have a lower limit where the lower limit is the logical zero or in U, except in the case have

term

an

null-class; and

productof all alwaysthe logical it limits. Thus by adding to U

the lower limit is

classes composingthe class which we null-class,

in which

shall be

that U

insure

closed series.

a

is

There

a

the the

sense

itself, namely,this : every term of U is the chosen class contained in "7,for every term is suitably

is condensed

U

in

upper limit of a the upper limit of the segment of 6ns which it defines ; and every term of U is a lower limit of the class of those CTs of which it is a proper no proof,so far at least as I have been part. But there is absolutely able to

that discover,

fundamentalseries. limit of

any

every

There

term

of U is the upper

is

a

no

priori

always be

class should

reason

also the

or

lower limit of

why, in any limit of

a

a

the series,

fundamental

of series of the types to prerogative which rationals and real numbers belong. In our present respectively though our series is,in the above generalsense, condensed case, at least, there seems for supposing in itself, its terms to be all of them no reason the 'seriesmay not limits of fundamental series, and in this special sense

series ; this seems,

be condensed

fact,to be

a

in itself.

It is instructive to examine

281.

of U

in

to such

segments as

can

the result of

be defined

the confining

by fundamental

series.

terms

In this

it is well to

their in addition to consider, upper and lower segments, as supplements, theymay be called,of which I shall shortlygive the definition. Let a compact series r be given,generatedby a transitive If asymmetricalrelation P, and let u be any fundamental series in ". case

earlier terms

of

u

have

to

later

ones

the relation P, I shall call

u

a

be w progression ; if the relation P, I shall call u a regression.If now in r, w defines, have already seen, as we any class whatever contained four other classes in v, namely (1) the class of terms before every w, which I shall call WTT ; (")the class of terms after w, which I shall every

call

WTT

;

(8) the class of terms

before

some

22?,which

I shall call

TTZO

;

classes The (4) th" class of terms after some w" which I shall call TTW. and (3)and (4)are lower and upper segments respectively ; the classes (1) *

The

definitionof the logical of the members in a form of a class of classes, sum It is as follows : Let w be a class involving finitode, is,1 believe,due to Peauo. of classes; then the logical of the members sum of w is the class of terms js such that there is some class belongingto w, to which See x F"rmulaire, Vol. u, belongs. Fart I (1897), No, 401.

not

Ordinal

281]

280,

supplementsto (4)

(") are

segments. supplemental of

WTT

and

thus

But

when

term.

zrir is a infinite, has

has

ID

a no

301

and (3) respectively, has

w

I shall call them

upper limit,this is the firstterm

an

segment,since no upper segment has a first upper limit,then, whether w be finite or

Similar

segment.

last term, this

a

is not

ZPTT

and

When

,

Continuity

remarks

belongsneither to

TTW

apply to

lower limits.

If

w

to ZPTT, but all other terms

nor

belongto one or other class ; if w has no last term, all terms of v belongto TTW or zrir. Similar remarks applyto WTT and TTW. Applying these generaldefinitions to the cases and regressions, of progressions for shall find the classes that, a we progression, only (%) and (3) are onlythe classes (1) and (4). The question important; for a regression, where a beginsor a regressionends is quite unimportant. progression has no Since a progression last term, and a regression first term, no the segment defined by either,togetherwith its supplement, contains of Whether in term and v. progressions regressions t? have limits every of decidingfrom the no always,sometimes, or never, there seems way given premisses.I have not been able to discover an instance of a but I cannot have limits, find any proof compact series where they never instance is impossible. that such an Proceedingnow to classes of segments, as we proceededbefore to our class [7,we have here four such classes to consider,namely: (1)The class of whose is the class UTT defined by some terms M, I77T, each regression the terms of v which come before all the terms of some in i.e., regression of all the classes inr class ITTT, consisting defined by prov ; (") the gressions of

z1

(3) the class TTV, whose (4) the class tw, whose progression; u

terms

;

terms

are

where

ini*"

is

if

term

and had case

no

way

(4), have

and

of

any

contained

v

so proving,

Each

and progression

limit in v, but there is arises in the givenseriesv.

no

ever

When condensed

we

come

in

to examine

pair would of

(3),or (")

have

which regression way

common

a

coherent,

were

whether discovering

the lour classes thus

whether obtain

themselves,we

a

no

compact series.

know, that (1) and

I

as

terms.

common a

far

some

is some u "?r, where class of classes, for its terms are

regression. Each of these four classes is a Each of the four is itself a classescontained in v. are There

is

u

the most

this

defined

curious results.

are

Every

series in any one of the four classes has a limit,but not in the series of which its terms are composed,and conversely, necessarily

fundamental

every

term

of

our

four

classes is the

limit of

a

fundamental

class to which of a series contained in the same necessarily is in of term belongs.The state limiting things, fact, as follows

but series, the

of each not

:

in Every progression

Every

VTT

or

in z"5ror progression

TTV

has

?np has

a

limit in

TJT.

a

limit in

TTV.

in VTT or w has a limit in w. Every regression in t?5ror 5h? has a limit in VTT. Every regression in Every term of "DTT is the limit of a regression

z*?r

and of

one

in

wt?.

and Infinity

302

Every

of

term

Eveiy term in

Every

limit

xxxvi

in VTT and of one regression in VTT and of a progression

of

one

and

of

one

in ?rr".

a

of

iro

is the

of

TTV

is the limit of

TrtJ.

term

in

is the limit of

m

[CHAR

Continuity

in progression

a

VTT

TTV.

in VTT or TTV ; regressions in VTT or TTV ; is identical with the class of limits of regressions rnr in VTT or TTV ; the class of limits of progressions TTZ? is identical with in TTV or VTT. ifz?is identical with the class of limits of progressions four classes has a kind of one-sided perfection; Thus each of our on one side,the other two on the other. of the four are perfect two of the four classes that it is wholly But I cannot prove of any one perfect.We might attempt the combination of UTT and TTU, and also -of whose generating form one For mr and TTV together and TTV. series, vir and wiU relation is stillwhole and part. This series will be perfect, in itself. But and of regressions contain the limits alike of progressions there be if for and u this series may not be compact ; any progression limit in v (a case in z", which both have the same which,as u regression

Hence

we

wr

know,

is identicalwith the class of limits of

in

occurs

of the series formed

consecutive terms the

will contain

will belongto both

v

series is compact,we

made

it

is not

Although we are

and

an

can

show

that

TTZ?

TTU

to

or

that

show

cannot

we perfect,

series which

of

limit, while

common

of

all other terms

then compact series),

some

it is it may

and

TTU

and

U'TT will be

VTT

for together,

will not

contain

neither.

Hence

it, but when

perfect ; and when not

wV

be compact.

our

have

we

And

a

hardlybe called continuous. original compact series t", there prove that,in our of progressions coherent with a givenprogression

compact can can

infinitenumber

having no term in common one regressioncoherent

with with

it,we

cannot

prove

that there is

nor can we given progression; in v has a limit,or that or regression prove that any progression is limit of of W e a v cannot a or progression regression. any term that such and u' that u TTU wV, regression are prove any progression that and differ of nor UTT term Tra a v. yet Nor, by only single may that in can we has limit in a finally, ZJTT single p rogression any prove wr, with similar propositions concerningthe other three classes VTT TTV, m. At least, unable to discover any way of proving I am any of these theorems,though in the absence of instances of the falsity of some of them it seems not improbable that these may be demonstrable. it is the fact"- as it seems If to be that, startingonly from a

even

a

=

,

"

compact series,so we

see

how

many

fundamental

of

the

is the

usual

theorems

are

indemonstrable,

dependenceof Cantoris ordinal

theory

upon the condition that the compact series from which we start is to be denumerable. As soon this .assumption is made, it becomes as easy to prove all those of the above propositions which hold concerningthe

types ^ and

#

This respectively.

is

a

fact which

is

of obviously

con-

Ordinal

282]

281,

Continuity

.siderable importance philosophical ; and it is with that clearly

out

not

assumed

282.

The

are

dwelt

have

I

long upon

so

303

a

view of

bringingit

compact series which

to be denumerable.

which

remark

we

made

that

just now,

two

compact

to form one which sometimes has consecutive series may be combined to continuity defined by as terms, is rather curious,and applies equally

Segments of rationale form a continuous series,and so with their limits) completedsegments (ie.segments together ; but two togetherform a series which is not compact, and therefore

do

Cantor.

continuous. that of

It is

series should

continuous

a

terms

new

notions,make

between

the old

continuity so interpolation merelyby This should,according to the usual continuous. It might be suggested the

to be

cease ones.

series stillmore

our

not

usual idea of

the

certainly contraryto

the

be calledcontinuous unless a series cannot that,philosophically speaking, it is

i.e.contains complete^

havingto

the

If

converse.

giventerm we

a

add this

with all the terms together transitive relation or its specified asymmetrical certain term

a

the condition,

series of

segmentsof rationaleh

completewith regardto the.relation by which we have hitherto since it does not consist of all classes of regardedit as generated, rationals to which a givensegment has the relation of whole and paxt, not

and

each

of which

contains all terms

less than

one

any

of its terms

"

segments. But every series completed with regardto some is complete relation, simpleor complex. This is mathematical standpoint, need not, from the reason why completeness since it can in the definition of continuity, be mentioned alwaysbe relation. insured by a suitable choice of the generating in what Cantor's definitionof continuity We have now consists, seen this condition is also satisfied by

a

the definition may that, while instances fulfilling ordinal the only be found in Arithmetic,the definition itself is purely not the is a denumerable compact series. Whether datum required or and

we

have

seen

"

thoughtto be the denoted by the word, most similar to what has hitherto been vaguely be acknowledged and the stepsleading to it,must the definitionitself, and generalization. to be a triumphof analysis Before enteringupon the philosophical questionsraised by the

kind of series which

Cantor

defines as

continuous

continuum, it will be well to continue remarkable theorems,by examining next ordinal numbers.

concerned,we to

the

two

this has been

discuss the

continuity.

problemswith

of Cantor's

most

which

this Part

continuity ; it

is

now

is

time

Only infinity. concerning a we adequately position of and problems infinity philosophical has

to

shall accomplished,

closelyallied

review

his transfinite cardinal and

hitherto considered only

consider what4 mathematics

when to

have

Of

our

is

say

be in

XXXVII

CHAPTER

CARDINALS.

TRANSFINITE

to

283.

THE

mathematical

begin

with

Cantor.

and

contrives

infinity,has

abandoned

is

dependent

cupboard,

its

on

without

infinite, results numbers,

are

it is necessary far

more

they

are

which

to

seenis

me

transfinite

with

defined

be

to

Thus

distinguished. a

We

which, by collection the

from

This,

call the

This

order

in which

it will

is the

a

be

is

seen,

Math.

the

to

finite

concerning In this

theory,

ordinals, which than

as

are

when

"

Anmfen,

are

followed

XLVI,

"

1.

also

the

finite

finite

called pozvers,

be

may

cardinals, leaving it

and

the

transfinite

are

following definition-}-. number

the

of

M

nature

of

that

general

is deduced

its diverse

idea

from

elements

the and

given."

merely

in

are

facultyof thought,

definition.

true

order

they

the

cardinal

active

tgkeHi"krG* t

gives

or

our

include

respects the

by abstracting from

spoken of, not

*

power

to

as

Cantor

of

means

M9

the

that

previously the order I shall begin philosophicallycorrect

cardinals,which

first

the

Mathematics,

transfinite

are

wholly Speaking

"

place so investigatedin what in

and

it

cardinals*. transfinite

The

284.

was

its

that

extending,

proved

they order

same

alone

be

to

the

of

out

shown

of all numbers.

properties when

Following

finite.

be

can

of

is

upon

separatelyof cardinals

their

in

they

it

has

Cantor

day.

of

branch

new

deduction,

true necessarily

treat

to

diverse

of

a

cannot

by denying

light

said

possible,

as

skeletons, it

in the

established

it

skeleton

the

course

infinityall depend

of

sense

no

this

other

vanished

while

which,

in

and

correctness

contradictions

supposed

in

it

world.

the

brought

has

many

has

Cantor

metaphor, which, by mere

in

like

Indeed,

dealings with

facing

emboldened

been

has

He

skeleton.

a

before

away

be

almost

may

Calculus, though

few

as

cowardly policy,and

this

cupboard.

it

hide

to

Infinitesimal

The

wholly dispense with

infinity

of

theory

It

Math.

a

phrase indicating what presupposes

AnwleH)

XLVI,

that

but

not

be

collection

every

in

is to

the

Mannich-

has

Cardinal Tram-finite

284]

283,

such

some

property

of independent

say,

might

we

feel

that

as

the nature

indicated

of its terms

"

305

a

property, that

and of their order

;

by Cantor to that every collection has a number. a primitiveproposition therefore consistent in givinga specification of number which formal

He

IN

is not

a

definition. of abstraction, however, of the principle we can give,a" in Fart II, a formal definition of cardinal numbers. This

By we

ing, depend-

In fact, add, onlyupon their number. be a primitive idea,and it is,in his theory,

tempted to

is taken

number

it* to

means,

saw

is givenby Cantor method, in essentials,

We

informal definition.

there is

similar when either with

and

one

have

alreadyseen

onlyone

that,if

two

classes be called

couplesevery term of is symother,then similarity metrical

relation which

one-one

a

after the above

immediately

of the

term

and transitive, and is reflexivefor all classes. A one-one relation, be defined without any reference to number, it should be observed, can A relation follows : is one-one as when, if x has the relation to y, and x

y' from y^ then it follows that x' does not have the relation to y^ nor reference to number ; and x to y'. In this there is no also is therefore free from such reference. the definition of similarity is reflexive, transitive, be and symmetrical, Since similarity it can of into the relation and its converse, and product a many -one analyzed

differs from

a?,

indicates at least one

several,a certain

if there be

or,

of similar

cardinal number a

and classes,

the cardinal number

entityas

of

class,taken

class. This

of

a

the whole

class with

a

of these

one

as

a

the

relation is that of

many-one

we givenclass,

decide to

the identify

class of classes similar to

has, singleentity,

as

the

the contradiction set forth in Fart

the

given proof of the

to method, however, is philosophically subject

The

from resulting

definite

one

of abstraction shows,all the properties requiredof principle number.

call the

we properties, may

In order to fix upon

of its terms.

class to the number

number

propertyof similar classes. This property,

common

a

cardinal

the doubt

I, Chapterx.*

of a class. In this way we obtain a definition of the cardinal number Since similarity is reflexive for classes, every class has a cardinal number. It

might

be

this definition would

thought that

only apply to

finite

classes, since,to prove that aU terms of one class are correlated with all of another, completeenumeration might be thought necessary. This, however, is not the case, as may be seen at once by substituting any for aH

"

a

word

concerned.

relation R

xRy Here

;

and

which

is

w]iere preferable generally

are

there is some classes w, v are similar when one-one such that,if x be any w, there is some term y of v such that x' of u such that x'Ry'. term if y' be any f, there is some Two

there is

no

need

givenline

axe

completeenumeration, but onlyof and any v. the pointson For example, lines througha given pointand meeting

wliatever of

propositions concerning any a

infinite classes

similar to

u

the *

See

Appendix.

Infimtyand

306

for any pointon the and line throughthe given point,

given line ;

the one

meeting the line.

given line determines

one

classes

are

Thus

where

our

propositionabout do not

any need enumeration.

286.

Let

given line determines

and

only throughthe given point only one point on the given one

line

any and

need

infinite,we

general

some

establish

of either class to

term

[CHAP,xxxvii

Continuity

but similarity,

in order to prove that every (orany) class has a cardinal number, we need only the observation that any term about of any class is identical with itself. No other generalproposition for the reflexive property of similarity. the terms of a class is requisite we

us

any

has been said

repeat what

propertiesof cardinal numbers. since I should merely these properties, chief

the

examine

now

giveproofsof

I shall not

And

by

of

first their relations Considering

Cantor.

if there be two sets of classeswhich are may observe that, set have any common and no two of the one similar in pairs, part,nor of all the the then classes other of logicalsum set, yet any two of the of all the classes of the other set. set is similar to the logical sum one to

classes,we

infinite classes.

greater than

is

part

of

uy then

u

a

to

that

is said to be less than

r

u

v

with Incompatible

We

cannot

whether

doubtful

part

no

holds also classes, class

a

of

u

is similar to

v

of

is said to be

but

w,

the number In this case, also, be proved that,if there It can

v.

u.

part of v which is similar similar*. Thus are equal,greater,and less are all and the last two each other, all transitive, asymmetrical. is similar to

which

and

of

of

number

class z", when a which is similar to

of u

of finite

case

cardinal

Again, the

that

part of

there is a of

in the

familiar proposition,

This

prove can

we

at

prove

^,

and

a

all

simply

and

at

all" that

"

it

seems

of two

more

less

or

different cardinal

greaterand the other lessf. It is to be observed in the that the definition of greater contains a condition not required be of i t number is of finite If the sufficient cardinals. v finite, case numbers

be

mast

one

But among transfinite proper part of u should be similar to v. cardinals this is not sufficient. For the generaldefinition of greater^ therefore,both parts are necessary. This difference between finite that

a

and

transfinite cardinals results from

and

infinite, namely that

it

always has

a

when

the

definingdifference

the

of

number

a

part which is similar to a part (and therefore

proper

number

class is not

finite, the whole; that is,

infinite class contains

every of parts)havingthe

of finite

an

infinite number

itself. Certain

of cases particular this proposition have longbeen known, and have been regardedas constituting in the a contradiction notion of infinite number. Leibniz,for since number be can doubled, the example,pointsoutj that, every *

de*

PP*

Bernstein

Schroder's

theorem

fometWM, Paris,1898, Note

as

; far

I, and

proofs see

Borel,Le$on$ sur

la tMorie

Zennelo, GottingerNackrickten, 1001,

*ML"-" oo. '

t Cantor's me

and

same

to be

grounds

for

holdingthat

valid. They depend upon

well-ordered

relation.

See

J Gerharat'sed. I, p.

the

this is

so

are

vague,

and

do

not

appear

postulatethat every class is the field of Cantor, JfcrfA. Annalen, XLVI, note to $ 2.

some

to

TrawjiniteCar"nah

284-286]

is the

of numbers

number

he deduces that there is

same

this propertyof generalize

contradictory, was,

far

so

thingas

numbers, whence

even

infinite number.

infinite collections, and

know, Bolzano**

I

as

well

induction,as

due to Cantor

are contradictory,

of

the finite cardinals

when of the proposition,

mathematical

the number

as

such

no

307

to

cardinals f.

acquirea

more

the

defined

are

the demonstration

as

and

examiningthis

intimate

treat

But

Dedekind.

first to

it

by

not

as

strict

that

The

proof of

means

it is not

itself proposition

the transfinite among may be taken as the definition of to all of them, and to for it is a property belonging Before

The

cardinal numbers, of the finite

none

however, we propertyfurther, the

acquaintancewith

other

must

of properties

cardinal numbers. I

286.

come

to the

now

arithmetical properties of cardinals, strictly

addition,multiplication, etc.J. The addition of numbers is when theyare transfinite, defined, exactlyas it was defined in the case

i". their

namely by

of finite numbers, of the

logical

of the numbers

sum

addition. logical

classes which

of two

sum

of

means

of the two

have

no

classes. This

The

number is the

term

common

be extended

can

by

of classes; for an infinite number steps to any finite number the sum of their numbers, if no classes, forminga class of classes, have

term, is stillthe

common

any

of any

logicalsum

the

definable.

For

of their

class of

finite classes,

two

three

of

sums

number

or

and associative laws stillhold,i"?.we + b

a

as

b +

The

of two multiplication

and

N

be

to form

of N

element

a

notion

productof the of a couplein

Let

be

is the

u

u

classes. This a

defined

If

of M

mutative com-

we

wish

If M

with

of all such

the number N.

Cantor:

by

element

any

and

of M

numbers

to

any

couples

avoid

the

then

;

the

ft terms

contain

term

common

is thus

couple(m, n) ;

logically

stillhave

(a + b)4- c.

=

two

and

"

defined,the

(b+ c)

combine

can

so

of

definition, we may substitute the following!: in number ; let each of these classes belonging class of classes, a

a

to

+

a

numbers

classes,we

two

and

a

is infinite,

or

numbers,

sum logical

cessive suc-

ab

and

;

let

is the number

no

of these

two

of the

classes have

sum logical

and definition is still purelylogical,

v

an

of all these

avoids the notion

of

defined obeysthe commutative,associative, so couple. Multiplication

and distributive laws,i"

we

have

of cardinals, even multiplication rules of Arithmetic. all the elementary transfinite, satisfy addition

Hence

*

Paradoxien

and

de*

Dedekind., Was find J Cantor, Math. Annafen, xxxv, 4.

axe

und

solkn

wan

XLVI,

"

3 ;

die Zakten

Whitehead,

f

No.

64.

American

Jmmal

"f Jfalfc,

No. 4.

" Vivanti,Tkeorie

" 2" No.

these

UnendKcken, " 21.

t See

VoL

when

des

Ensembles,FormuMre

de

Mtdktmt*Kqw*y Vol.

i,

Part n,

Infinityand

308

Continuity

[CHAP*xxxvn

definition of powers of a number (ab)is also effected logically (ib." 4). For this purpose, Cantor first defines what he calls a covering {Bekgimg)of one class N by another M. This is a law by which,to The

and only one element 772 of Jf, but every element n of N is joinedone elements of N. That is, the same element m may be joinedto many domain w hose includes a N9 and Belegiingis a many-one relation, of M. correlates with the terms of N alwaysterms number of terms in M, b the number in Ar, then the number

If

which

relations is defined to be ab.

many-one

numbers, this definition agrees with

numbers, indices In the

from

which

related to

are

related to

chosen

has

Cantor

a

in all

of b

but finite,

b is

transfinite

there is

rest

cases

get

we

thingsany

6 is the

number

"

of

collection of b terms

some

a

class

classes.

at

time

a

a

"

b is transfinite,

when b

class of

of b terms,

out

number

still true

case

all such

be formed

those

related to

are

get in each

we

can

when

Now

specifythe

to

case,

alwaysgreater than

difficultieswhen

when generally,

more

two.

proof that 2" is

however, leads to

each

of classes that

when

For

terms.

given,the

are

Hence

b terms, and

of combinations

familiar theorem

of two

one

enough,in

of the

is the number

the number

that,for finite

see

i.e. properties,

of the two

one

one

of the

out

"

Hence

=

it is

Hence

the other. terms

"

of all such

deduced 2, ab is capableof a simplerdefinition, of ways in which ", "6 will be the number

=

related each to

be

can

a

If

the above.

b terms

or

have

where

case

stillthe usual

It is easy to the usual one.

be the

a

a

proof which,

all

classes, or,

in which

all the

themselves

of b*. terms single The definitions of multiplication quire given by Cantor and Yivanti reof factors in a product should be finite; and that the number and it necessary to give a new this makes independentdefinition of

sets chosen out

of the b terms

are

powers, if the exponent is allowed to be infinite. Mr A. N. Whiteheadf which is free from this restriction, has givena definition of multiplication therefore allows powers to be defined in the ordinary way as products. He has also found proofsof the fonnal laws when the number

and

of summands, brackets,or factors is infinite. The is in

as

follows

ways,

number of terms a

AT be

Let

Choose

of terms

no classes,

two

of which

have

a

product

any terms

out, in every

in this class is defined to be the

in the various

finite number

of

respectively a, ", See

classes which

u^ v9

7 terms.

Chapterxun,

are

infra.

w

is easy to see be the members

Then

one

term

t American

productof

members

members, it

usual definition. Let

*

class of

a

and onlyone term possible way, one each of the classes composing fc. By doingthis in all possible called the multiplicative class of AT. The we get a class of classes,

common.

from

:

definition of

the numbers

of AT. Where

k has

that this agrees with the of Ar,and let them have

can

Journal

be chosen

out

of

u

of Mathematics,loc. cit.

in

287]

286,

Cardinals Tran"finite

ways : for every way there are ft ways of and for every way of choosingone term out

choosingone

a

are

of

ways

7

choosingone usual

choosing one

term

The

sense.

of

out

of

out

each, when

there

can

definitions

in its

of

means

deal further

good

finite and transfinite

applyto

alike,and, as we see, the formal laws of Arithmetic differ from finite ones, however, both finite integers their relation to the classes of which they are the

regardto the

of properties

have, in

numbers

are

a

,

of r, there afty ways of

is understood multiplication be carried

tr

out

one

class is an importantnotion, by multiplicative

All the above

287.

and

u

of

out

term

has carried it.

Cantor

than

of

Hence

w.

transfinite cardinal Arithmetic

which

309

all finite

classes of the

integers

still hold. in the

Trans-

of propiTties

numbers, and also in Classes of

themselves. integers

the numbers fact,very different properties a" according

part at least transfinite. transfinite cardinals,some are Among important, particularly the number of finite numbers, and the number of the continuum. especially The of finite numbers, it is plain,is not itself a number for the class finitenumber finite number; is similar to the class even conclusion finitenumber, which is a part of itself. Or again the same are

proved by

be

may

define

to

serves

in

are

or

consider

Hebrew

Aleph by a$.

with

it

denote

the suffix 0

Cantor

transfinite cardinals. cit. "

until the

which principle

a

"

number

for

;

proves results from

The

the

number

of

denotes

by

convenient

more

this is the

also

ordinal

more

a

Cantor

it will be

us

that

This

chapter.

next

numbers, then, is transfinite. This

finite

to

induction

numbers, but which, being of

finite

nature, I shall not the

mathematical

least of all the

theorems following

(loc.

6): Every

(A}

others

transfinite collection contains

parts whose

as

is a0.

number

Every transfinite collection which

(B) is

number

a0,

also has

the

number

part of

is

whose

one

"v

finitecollection is similar to any proper part of itself. of Every transfinite collection is similar to some proper part No

(C) (D) itself*. From

these theorems

the number

said to

be

it follows that

of finite numbers.

no

is less than

transfinite number

Collections which

have

this number

are

count

such

wth.

This

to denurnerable,because it is always possible there such collection, of a sense that,given any term

in the collections,

finite number

is

some

is

merely another

collection have

a

way

one-one

such

n

of

saying that

as

that of the finite numbers.

primes,the *

Theorems

induction,or

the

given term

all the

correlation with

equivalent

the

the

sayingthat

is

to

that

number

a

denumerable

of the collection is the

require tliat the finite should else they become tautologous. D

of

the finite numbers, which

It is easy to see perfectsquares, or any other

0 and

terms

is the

that

the

even

again same

numbers,

class of finite numbers be defined

by mathematical

Infinityand

310

having

class in order

such

any

will form

maximum,

no

denumerable

a

series.

will be

there magnitude,

of

[CHAP.XXXYII

Continuity

For, arranging finite number

a

of

terms, say 71, before any given term, which will thus be the (n + remarkable is"that all the rationals,and What is more term.

equationsof

of

all real roots

degree and

finite

a

numbers), form (Le.all algebraic even

an

smallest

in the

denumerable

are

the

be

can

those with smaller

order in which

precedethose with

And

seen, by arrangingthem easily

largersum,

of numerator

sum

of those with

and

precedethose

smaller numerators

and

nator denomithose

equal sums,

larger

with

Thus

ones.

1, 1/2,2, 1/3,3, 1/4,2/3,3/2,4, 1/5 This

is

with a beginningand series, and will have in this series,

discrete

a

will

number

occur

In the other All

denumerable

different

we

be

(ifthere

all

class of them all have

which

predecessor.For the finite numbers; but Cantor's

grounds for

if it had

know

we

he

has

number

are

prove

which

See

Aetti

t See

this

well-ordered

rfetitxchwi

Matenuttir"(t9 II,pp. 16-5-7. is open to question. J Acttt Math. II,p. 308.

cardinal

}|Math. See

seem

open question. the most a^, important proved that this number an

that

has

it is

al"" a hope which, though that ;

this number

IT. probable

As

to

but

the it

is a,, though there the definition of cq

Mafhtmutictt,u, pp. 300, #13, ;*2(".

Jtthrextterickt der

Cantor's

5

are

remain

But

other than

Cantor

whether

doubtful rendered

diate imme-

no

last finitenumber.

no

not

be the last of

"

reasons

*

to

every

they do

But

to

to

last of

has

so

long cherished it,remains unfulfilled. He has shown "of the continuum is 2*"||a most curious theorem

still remain

must

and

have

his assertion that the cardinals

hopes

Cantor

by

except the

all after it.

presentthis must

there is

infinite series

an

asserted

are

this would

one,

of the continuum.

a0",and

at

however

a0,

supposedthat

successor,

that there is

("f the transfinite numbers

is the number is not

number

predecessor ; for example, a0 itself has

that for the insufficient, so 288.

difficult.

more

of them

every one immediate

any numbers

has

immediate

an

an

;

contrary,there is

is,such that

last)has

a

a

be

not

transfinite cardinals

The

f.

that well-ordered,

the

On

erery rational finite number of predecessors.

cardinal

same

it must

But

appear. greaterthan a0.

numbers

of such

series have

the

...

end

no

proofis rather

the

cases

theymay

number

be

series.

the series

get

no

efficients co-

terms, where n is a finite number, That transfinite ordinal,is still denumerable*. the

rational numbers

with

denumerable

a

even

rational

//-dimensional series of such

the

or

with

l)th

See

MntheMtitiker-Vn-ehiigung 13 1892; assertion

that

there is

Chap, xun, itifra. " /i.p. 404. a1 is the

greatest transfinite

no

number

next

Attwifetty XLVI, " 4, note. Couturat, Ik /'/iiJJwi MutMtnatique, Paris, 1896, p. ("55.

allegedt"yCantor

for

the identifying

second

power

with

that

Jtirixta di

of the

after "0. The

ground

continuum

is,

that every infinitelinear collection of pointshas either the first power, or that of the it would seem contJMuam., whence to follow that the power must of the continuum be the next after the first. (Math. Annalen, 23, 488 also Math, Ada see vn.) But ; p.

287-289] whole

of the

and

TramfiniteCardinal*

is better

which

It must

supposedthat

by merelyadding one On

cardinals. It is known and

can

obtain

a

by adding any finitenumber

even

isequalto its double

also that in the

;

transfinite

of two numbers its square. The sum is these classes of equalto the greaterof the all transfinite cardinals

whether

transfinitecardinal

new

equalto known

matter

a

we

presumablydifferent class of

a

is

have discussed the tramfinite ordinals.

or

crf .

cannot disturb the transfinite puny weapons that in the case of ct0 and a certain class of trans-

finite cardinals, a number 00

this cardinals,

we

it,or

to

contrary, such

the

of transfmite

postponeduntil

be

not

succession

311

case

of

number cardinals, a

is

the former

to belonging

Wong

to

It is not

numbers.

two

one

both

or

of these

classes*. be asked

It may

289.

In what

:

respectdo the finite and transfinite

togetherform a singleseries ? Is not the series of finite without the possibility of extending its numbers completein itself, relation? define of If the series of integers we by means generating which is the generatingrelation of differing the method by one the series is to be considered as a progression natural when most then, it must be confessed,the finite integersform a completeseries, of adding terras to them. But if,as is and there is no possibility the in of series consider the we as theory cardinals, arising appropriate that of with and the whole correlation which cla"sesof by part among cardinals

"

"

be

integerscan

then asserted,

we

this relation does extend

that

see

There are an infinitenumber of infiniteclasses beyondfinitenumbers. in which any given finite class is contained ; and thus, by correlation that of any one with these,the number of the givenfiniteclass precedes in which all there is any other sense form a single I leave undecided ; transfinite, series,

of the infinite classes. Whether finite and integers, the

above

error

in two

our

attention

the inference

one

sufficient to

as

a

that

singleseries,if

be the

must

show

it

greater. But

there were

is

no

known

it is now

time

logical that

of

to turn

to the transfinite ordinals. seems

precarious.Consider, for example,th" following-

somewhat

the stretch determined by two terms consists either of compact series, of one of terms, or, when infinitenumber term the two term** coincide, only,and in

analogy: never

by

be

regardingthem cardinals

any

an

would

sense

of

other

a

a

finite number

of terms

other than

But finitestretches

one.

arc

presented

types of

series, e.g. progressions* that the number of the continuum

is 2*" results very simplyfrom the of Chapter xxxvi, that infiniteclasses of finite integersform a continuous proposition of of allclasses of finiteintegers series. The number is 2** (mfe mpra),and the number

The theorem

finite classes is OQ. the subtraction

the number

Hence

tlienumber

latter number, *

of as

number

greaterthan %;

:$*"is therefore

therefore is QI it would prove that the number of infinitecksses of finite integersis tbe same To

types of series that we

is 2"* for of all infinite classes of finite integers

diminish any

of the continuum.

be sufficient to show as

the number

of "0 does not

shall

see

that

can

in the next

Of. Whitehead, loc. tit. pp.

this number

be formed

of all the finite

chapter,is at.

integers;for the

XXXVIIL

CHAPTER

ORDINALS.

TRANSFINITE

290.

tran^fimte ordinals

THE

remarkable

and do

the

obey

not

for

rate

any

of

transfinite

the

belong so

on.

The

still,classes

less than

ordinals,also,

cardinal

be

may

from

of

n

be taken

may

*c

nth"

and

types

as

to

by "rath,"

The

the

is

which

ordinals

class

second

in

terms

which

is

It

relation

This

logically prior to

is

is this

this

to

as

so

of

that

apply

;"

terms

n

notion, is the

n

sense,

not

sense,

the

example,

ordinal

an

In

it*.

Cantor

for

series:

serial

row.

a

generalized by

of

finite

The

induction.

"a

mean

class of serial relations.

a

all ordinals

of

called

are

mathematical

to

conceived

popular language, n

name

is ""

collection

infinite

an

cardinals.

number

relation

some

number

distinct

all

is

number

cardinal

of

that

every

quite

cardinal,or

those

part, by

or, in

a

is therefore

transfinite

class,there

certain

latter,they

the

corresponding to ^ are called the third class,and ordinal numbers are essentiallyclasses of series,or better of generating relations of series ; they are defined, for the

ordinals;

ordinal

one

any

or

of

most

of

series whose

to

For

interesting

more

Unlike

arithmetic

their

ordinals, although the

as

same

law, and

commutative

possible,even

cardinals.

transfinite

elementary arithmetic.

different from at

the

than

if

are,

expressed infinite

to

series.

Let

291. ordinal *

of

the

numbers,

new

I call classes

from and

certain

definition

Cantor's

be shown," he

to

now

The infers.... arises

with

the

of

second

class

of

numbers*!*.

It is

which

begin

us

and

says,

in what

in

(1)

of

the

positivereal

(AnzahT) of

of the units

*

i

Cf.

such

posited into

a

the

successive

whole.

numbers

whole

of

units

is the

Part

definitions

of

series

1, 2, 3, which

positings,andfor

Thus

sections,

...

are

expression both

the formation

"" 231, IV, Chap, xxix, MannichfaltigMtelehre," 11, pp. 32, 33. mpra,

the

natural

absolutelyendless

repeated positingand combination regarded a* equal ; the number v

finite amount

led to

are

we

obtained

are

way

of numbers,

series

"how

232.

the

real i",...

supposed prefor

a

bination com-

of finite

Ordinal* Transfinite

291]

290,

numbers

real whole

alreadybeen

has

rests

formed

addition of

the

on

;

among a

of the class

v

Thus

them.

greatest number

imagininga

in

that

whole

the

new

however

the

certain finite amount numbers

each

of

of units to follow the

here

which

we

numbers **

v

same

way

as

v

againwe may and

in its natural

law

It is

whole.)

a

as

co

numbers

come

to

+

no

a

numbers

of

which

but

that

a

the is

"

be called

z". is to

v,

By allowingfurther additions

v.

obtain,by the help

we

o"

further numbers

number, we greatest

which

order

to permissible

even

limit,towards

of the number positing

call 2o",and CD

greatest

no

be

expresses the combination

follows all the the

be the

is to

imaginea

new

firstafter all

one,

previous

v.

function logical

The

and there is infinite,

(1) is givenby its

of formation,the principle thej"r*"

Since

(Anzahi)of

amount

it would contradictory

of units into

integerwhich

greaterthan of

The

tend, if by this nothing else is understood

v

first

the

(1) is

newly created number

of the

think

which

number

a

to speak of of the class (1), there is yet nothingobjectionable number, which we will call a", which is to express

collection

(In

of succession.

unit to

a

I call this moment, which, as we shall see essential part in the formation of the higher

also playsan immediately, the firstprinciple of formation. integers, numbers possible

813

which has

the two

givenus

numbers

"a

and """

different from the first of formation ; I call it the evidently principle second principle and define it more exactly offormationof real integers, follows of real have If determinate defined succession as we : integers, any this which of there second is means no greatestnumber, by among of formation is created,which is regardedas number a new principle is

limit of those

the than

of

the

as

next

number

greater

all of them."

The that

numbers, i.e. is defined

two

of principles

ordinal number

an

their

will be made

formation is

merely

generatingrelations.

type

a

Thus

if

we

or

clearer by

considering

class of series, or

have

any

rather

series which

has

last term, every part of such a series which be defined as all can series will have of the the terms certain term a up to and including no

a

last term.

different

But

since

type from

any

the such

series itself has

part

or

last term, it is of

no

segment

of itself.

Hence

a

the

be different the series as a whole must representing and must be a from that representing any such segment of itself, has series last the since no number having no immediate predecessor, of the or term. Thus is simplythe name of the class ptr"gre^h)^ co

ordinal

number

generatingrelations of series of this class. The formation,in short,is that by which we define a

having no "t

which

last term. is obtained

second certain

of principle

type of series

the ordinals preceding Considering any ordinal as segments representing by the second principle

and Infinity

314 of

series

a

limit

ordinal

a, the

by represented

segments; and

such

of

Continuity

as

we

they have (provided

no

a

itself representsthe limit

segments alwayshave a when the original series

before,the

saw

maximum),

[CHAP,xxxvm

even

has none*. In order to define is

evident,the

a

of principle

transfmite

class among

a

succession

is

ordinals (of which, as he calls introduces what

Cantor infinite).

(Hemmungsprincip)^. According to this consist onlyof those whose upwards,form a series of the first power, I.e.a series

limitation

the second clasfsof ordinals is to principle, from predecessors, have

one-one

a

the

whose terms, in a suitable is "0, or one relation to the finite integers.It is then shown

number

cardinal

whose

power,

1

order^ that

class of ordinals a*s a cardinal number, of the second is and further the very next cardinal cc0 (p.35),

or

whole,is different from after

number

to "o results

What

(p.37).

ae

the

from clearly

establish

a

or finite,

"

simplyinfinite series,or it between reciprocal correspondence

unique and

say, any part of of the first power,

is to

That

cardinal number

next

be any

If M

of the second class of numbers^ power of M be taken, then either the collection

M if any infinite portion be considered as a can

Mf

the

following proposition (p.38) :

well-defined collection of the and

by

is meant

a

collection of the second

or

of

the

second

;

and

is

to possible

and

M

power

hence

M'?

is either

there

is

no

the first and second.

power between Before 292.

ordinals,it will

proceedingto

the

addition,multiplication, etc., of

be well to take the above

far as possible* as propositions, out of their mathematical dress,and to state, in ordinarylanguage, As for the ordinal "", this is simplythe exactlywhat it is theymean. class for the of have We name generatingrelations of progressions. is how is defined .it which has series first seen : a progression a a term* tion. inducand a Jxirni next after each term, and which ol)eys mathematical

By mathematical of

a

where

induction

if it has progression, w

denotes

a

last term, has

the clash of series

every part which has show (what is indeed

no

itself we

last term

of consisting

obvious) that

no

that

j"art

every

finiteordinal number

some

is itself a

show

can

n

terms

in order

progression ; also

finite ordinal will

; we

/*,.

while can

representa

Now definite class of series, are a progressions progression. perfectly of shows the attraction that and there is some principle entityta which all of them have a relation which they have to nothingelse for all progressions similar (Le. have a one-one relation are ordinally "

*

Oil the

" 1" analogous,in xxxm,

the

segments of well-ordered

series see

Cantor's article in Math.

It is

important to observe that the ordinals above explained are their genesis,to the real numbers considered as segments (vide('hap. xupra). Here, as there, the existence of "o is not open to question when

is adopted,whereas segment-theory and implausible, t Mwnid"atigtetMkkrr, p. 34.

is indemonstrable

on

any

other theory the

existence-theorem

291-293]

TramfiniteOrdinals

such that earlier terms

correlated with earlier ernes, and

are

later with

ordinal

later),and

is symmetrical,transitive, similarity and (among This entity, to which the principle of abstraction

series)reflexive.

be taken

points, may series

315

be the

to

type

class of serial

or

since relations,

belong to more one type of series. The type to is what Cantor calls ", Mathematical progressions belong,then, induction,startingfrom any finite ordinal,can since reach never ",

no

than

can

which

is not

of the class of finiteordinals.

member

Indeed,we may define the finite ordinals or cardinals" and where series are concerned,this the definition best those which,startingfrom 0 or 1, can be seems as co

a

"

reached

mathematical

by

be taken

as

axiom

an

immediate

10,937,is

successor,

can

of

most

us

That

There

as principle

ordinal

is to

for

point,a to

seem

infinite is

reason

that every number principle that any assignednumber, that the

number

has say,

is assigned

a

use

mentioned

not

kind

no

of

in the Arithmetic error logical

supposingthat

the

in

of

our

using the is there

principle appliesto

the

on

word

to

suppose whose

one

subjectas

the

philosophers may

be in

that the distinction between

a

aff

Most

season.

the finite a"d

meaning is immediatelyevident, and they needed. though no precisedefinitions were

the fact is,that the distinction of the finite from

But

It

concerning10,987 say, every proposition of mathematical induction,which, as

the

is therefore

reason

a

or

of them

as

course,

to

the definition of finitude,

all cardinal numbers.

this

At

prove

is not therefore, principle,

definition of the class of finite numbers, nor

a

of

shadow

can

remember, was

can

childhood.

the

we

proved without

be

virtue of the

of finite"provided,

finite number.

This

but postulate,

or

that,in

is to be observed an

induction.

the infinite is

only been brought to lightby modern 0 and 1 are capable of logical definition, We has a successor. that every number and it can be shown logically mathematical define finite numbers either by the fact that can now reach from 0 or 1 in Dedekind's language, induction can them, starting that theyform the chain of 0 or 1 or by the fact that theyare the

easy, and has mathematicians. The numbers

by

means

no

"

"

proper part of them has the same shown These two conditions may be easily to be number as the whole. the finite from the distinguish equivalent.But they alone precisely of collections such

numbers

and infinite, or

any

that

discussion of

no

them which neglects infinity

must

be

more

less frivolous.

we

terms

regardto numbers of the following make remark. for some always,except possibly With

293, may is

the field of their among order.

more

than

one

rank, age, wealth, or men

But

a

A

class other

collection of

two

than or

"",

more

very largeinfinite collections, serial relation. Men may be arrangedby order: all these relations in alphabetical

generate series,and when

the second

collection is

placesmankind in a different orders give one and all possible finite, each

Infinityand

316 the

number of

certain

a

finite number

this is series,

all series which

is to say,

That

of the collection.

of terms

the

to corresponding

that

namely

ordinal number,

same

[CHAP,xxxvin

Continuity

are

be formed

can

similar. ordinally

cardinal

With

infinite

infinite collection of terms

quitedifferent. An

which

capableof different orders may belong, in its various orders,to in one that the rationals, have already different types. We seen

is

quite order,

beginningor end, while in another order different types; These are series of entirely they form a progression. extends to all infinite series. The ordinal type and the same possibility of two consecutive terms, of a series is not changedby the interchange in virtue of mathematical induction,by any finite nor, consequently, number The generalprinciple of such interchanges. is,that the type of a series is not changedby what may be called a permutation. That is,if P be a serial relation by which the terms of u are ordered,R a form

a

RPR

is

no

and whose

domain

relation whose

one-one

then

series with

compact

serial relation

a

relations whose

of the

field is ", and which

type

same

of the

are

domain

converse

type

same

",

all serial

and

P;

as

both

are

P,

as

of

are

rearrangement not reducible to a permutation,the type, in general,is changed. Consider, for example, the natural numbers, first in their natural order,and then in the order in their natural in which " comes then all the highernumbers first, form order,and last of all 1. In the first order,the natural numbers second, they fonn a progressiontogether.with a progression ; in the last induction no In the second fonn, mathematical term. a longer and of which there of hold are ", applies ; propositions every subsequent the

above

But

RPR.

form

finite number, but

of 1.

not

series of the kind the

type

that

of any

every second

collection

any class of ordinal

numbers

means

upon

that collections,

an

infinite

found,

the

collection was original

which

will have a

defined

the series,

as

part,by

types

collection The

infinite collection

this

sponds corre-

Hence

all the

can

possibility

propertyof

correlation with

one-one

shown

class*.

givendenumerable

an

has

order which

second

the fundamental

part of a

be

the second is

;

Cantor

generatingrelations.

of such different typesdepends

be

may

one

any

of different

given an

of the

number

mental type of any funda-

its limit.

be

can

is the

Chapterxxxvi

with series together

denumerable

arrangedby

first form

considered in

we

of well-ordered series in which be

a

The

assignedordinal

to

the

such

by

can

infinite

always

the whole.

If

becomes correlation,

series

similar to the whole : the remainingterms, if added ordinally after all the terms of the infinite part, will then make the whole

a

differentfrom ordinally *

Acte

t The

Math,

n,

what

it wasf.

p. 394.

remainingterras, if they be finite in number, will often not alter the type if added at the beginning;but if they be infinite, they will in generalalter it even then. This will soon be more fullyexplained.

293,

294]

Ordinak Transjimtc

317

of ordinals to that of cardinals may assimilate the theory Two relations follows. will be said to be like when there i* a one-one We

domain

whose relation *ST,

is the field of

is such that the other relation is SPS. Le.

be defined

P may

as

the ordinal number

likenessamong

result from

If P be

(P), and which

well-ordered

relation, well-orderedseries, a the class of relationslike generates

which

one

of them

one

m

a

of P.

Thus

ordinal numbers

relationsas cardinals from

similarity among

classes. understand the rulesfor the addition and multiplication of transfiniteordinals. Both operations the associative, obey We

294.

now

can

law.

but not the commutative the form

onlyin

7

distributivelaw is true,in

The

(a+ /3)

general,

71* -f-7$,

"

That addition does not obey the multipliers*. be Take for example o" 4 1 and seen. easily may first denotes a progression The followed by a singleterm : 1 -I-"", and its limit,which is this is the type presented by a progression Hence "" + 1 is a differentordinal different from a simpleprogression. But 1 -f o" denotes a progression from GJ. precededby a single term, Hence 1 -f " and this is again a progression. ", but 1 -f ca does not The numbers of the second class equal6"-f If. are, in fact,of two kinds,(1) those which have an immediate predecessor, ($) those which where

a

-f/8,a, $

are

law

commutative

=

have

Numbers

none.

such

If predecessor.

immediate

number, the

"".", c*.$,... ""3,^...a)1*.,, have

have

plainthat

a

terms

but endless;

be added

of these numbers

reappears

numbers,we get a

;

but if

have

representseries which predecessor added at the

the addition of

a

a

number.

new

predecessor representseries which

no

which

o",

any transfinite number

same

be added to any of these with

as

no

to

BO

finite

finitenumber The

end,

have

a

an

numbers

while

end.

those

It is

of a serieswith no end leave it beginning series after an endless one proterminating duces

and therefore a new series, type of order. Thus there terminating rules which simply of addition, about these is nothing mysterious express from the combination of two givenseries. the type of series resulting If a is less than Hence it is easy to obtain the rules of subtraction];. a + % $ y3,the equation a

=

onlyone solution in ",which This givesthe type of series that must a. $ produce/?. But the equation has

alwaysone

and

"

we

may

be

added

representby after

a

to

of the type of a series consisting Mannichfattigkeitslehre, p. 39; a-i^ will be will be the type j8; -ya two parts,, namely a part of the type a followed* by a partof Thus of a series of the type a of seriesof the type y. the type of a series consisting: *

a

series

composedof

t Math.

is of progressions 8. Aniwlen, XLVI., " two

| Mannichfaltigktit"tehre, p. 39.

the type

"

.

2.

will sometimes

have

of solutions.

Thus

solution,and

no

solution at all :

no

other

at

[CHAP,xxxvm

times

infinite number

an

equation

the

|--f has

Continuity

and Infinity

318

di

1

H-

added

of terms

number

no

co

=

at the

togetherwith progression

will producea progression " fact,in the equation" 4- a

a

if

=

representsa before

never

can

a

produce the

type ft.

satisfiedby

For a"

n

-f n 4-

"w

all such

minimum,

+ ?*, where

so

zero

or

equation

finite number.

any

has

an

and

a"y

thus

infinite number

values of possible

is of two

kinds, accordingas

be

N

element

infinite number

an

series of the

be two

substitute

n,

second

in the

of ordinals multiplication

The

295. and

never

"

of

have

value of the difference between principal

sort of

solution,and there may M

is

n

however, the

cases,

can

2

therefore,"

always a unique solution ;

is

*".

=

added

we

seek

a

ft a

which, added to a, will give#, or a number to wliich a may be to giveft. In the first case, provideda is less than ft,there as

number added

a

"*"

therefore

consider the

we

In

type,while ft

that terms

will coalesce with this to form

o"

subtraction

Thus

a.

is

which

o"

=

In this case,

"

G).

=

In

values. and

"

before the second

hand, if

other

the

f 4this will be

evident sufficiently

producea terminatingtype, and On

last term.

a

endless

an represents

a

terminatingtype,it is

of beginning

series

a

formed

of all the terms

(1)any

two

elements

types a

Mn

may

be

no

of solutions.

is defined

and

of the

there

case,

follows*.

as

Let

ft. In N9 in placeof each

type a

and

;

let S

be the

series

of all series

of S

Mn" taken in the followingorder : series Mn are to belong to the same

which

elements which two belong to preserve the order they had in Mn\ have the order which n and n have in N. different series Mnj Mn" are to

type of S dependsonly upon a and /3,and is defined to be their productcc" where a is the multiplicand, and ft the multiplicator. the

Then

It is easy to For

example,"

which

is a

that

see

.

QJ

do products

is the

so progression,

#1, "Zy ez

which is a combination In the former

there

$

=

af

has

"""

,

o"

"v) -""?

But

w.

=

fitJD

G".

/35

" is the

fv)

"""

law.

type

"""

but not a single progressions, progression. is onlyone term, el9 which has no immediate are

two,

el and

J\.

of

subtraction,two kinds must three ordinals a, ft,y, such that ft

be ay,

=

no

commutative

of two

the latter there

as division, are

that %

there series,

predecessor ; in Of

type of

alwaysobey the series presented by not

other solution than *

t

"

=

7,

and

Math. AnnateriyXLVI, " 8. ManmchfaMgkeitskhre, p.

we

40.

distinguished *f. If then the equation

may

therefore

denote

294-296] 7

Tramfimte

by y9/a*. even

or

the

But

equation /8

Ordinal"

319

fa, if soluble

=

roots; of which, however, smallest root is denoted by "*//".

This

smallest.

have several

all,may

at

of infinity

an

is

one

always the

of ordinals is the process of representing series of a Multiplication each series as a single series beingtaken as a whole,and praserving series, its placein the series of series. Division,on the other hand, is the series into a series of series,without up a single process of splitting

alteringthe order

of its terms.

may

The

primes.

theoryof

necessary for us to go into itf. 296. Every rational integral or

not

number

Division,as

have

portance im-

some

is

plain,is only which it is not possible but it is primes is interesting,

types of series ; those with

some

called

be

these processes

with dimensions.

in connection with possible

Both

of the second

class,even

function exponential

of

is

a"

a

such numbers as a***,of*,etc., all that of denumemble supposed types of such series are For form. the a capable example, type iy, which the in of order rationals is repi*esent$ inagnitude",whollyincapableof But

occur*.

it must

expressionin

of

terms

ordinal numlwr.

term

series,Le. such

have

the

as

Such

o".

The

There

II.

If F' is

called

by

Cantor

is reserved for well-ordered

a

:

first term. if F

possesses erne or more of F\ then there is

terms

terms a

term

which

f

immediatelyfollows F\ so that there is no term F before/'and after all terms of F'. functions of a" and finite ordinals only,to the exclusion possible F

an

| following properties

after all the

come

number

not

two

part of jP,and

a

type is

a

ordinal

is in the series F

I.

when

be

not

which

of of

-

All

other

types such

though is

the

as

converse

of

that of the rationals, representwell-ordered series, there In every well-ordered series, does not hold.

if there be after any given term, except the last term it alwayscontains parts which ; and providedthe series is infinite, after a progressionhas A which comes term next progressions. term

a

one are

next

the type of the segment formed other is called the second species.The

and predecessor,

immediate

no

is predecessors

of what

and the predecessors,

have

immediate

their

are predecessors

types of the

of its terms

segments formed

of

said to be of the first species.

in a .ft has changed his notation in regardto multiplication : formerly, the oppositeowler is z"d nmltiplicator, ft the multiplicand; now, I have altered tlie adopted. In followingolder works, except in actual quotations, order to that now adopted. *

a

was

t

J "

Cantor

the

See Mannichfaltigkeitslehre, p. 40. On the exponential function,see Math. Math.

IIMath. which series

is

Annalen, XLVJ, "

Atwafeji,XLIX,

to equivalent

(exceptof

course

it: the

" a

Annakn,

xux,

""

1B-2Q.

9.

definition may be replacedby the following, series is well-ordered if every class contained in tfce

12.

The

has null-class)

a

first term.

Infinityand

320

consideration of series which

The

297.

of well-ordered series. Thus

case

function of

the

all functions of a", since

77 has

whereas

[CHAP,xxxvm

well-ordered is important,

not

are

to Arithmetic the results have far less affinity

though the

Continuity

firstterm, and

no

type

is not

77

all functions of

in

as expressible

representseries with

o"

than

represent series

a"

a

first term,

a

in

immediate which has an successor, again is not the every term and of series and zero with 17, Even the positive negative case integers of since has in this series terms be expressed no cannot o", beginning. which

defines for this purpose

Cantor the

of

type have

we

(ib."7). regression

a

is relative to

seen,

*"",which

serial type

a

definition of

The

some

as

as progression,

a

aliorelative

one-one

be taken

may

Pf.

When

P

with respectto P is a regression this progression generatesa progression, with respectto P, and its type,considered as generated by P; is denoted by *o". Thus the whole series of negative and positiveintegersis be divided anywhere into two Such a series can of the type *"" + a).

generatedby progressions,

relations; but

regard to one Such a progressions.

converse

in

any combination of series is completely defined, by the methods of Part IV,

itis not reducible relation,

a

to

aliorelative; the field of P

one-one

finite positive power

of P

relation disjunctive ; and asymmetrical

the series consists of all terms

its converse

or

of series

to

as expressed

of

function

a

of Part

methods

the

or

co

*

IV

or

Thus

the

type

example,to

for

with i.e.

that,

with

But

both

numbers, i.e.

the series of

f An

aliorelative is See

end

to

;

behaviour

a

u.

series form

our

of

Their in

the

can

continuum

progression.

a

rationals with

chief characteristics i.e. every themselves,

which regression

term

gression, pro-

denumerable,

are

form

belongto

regressions ;

has

no

the

by omittingall

yet this series is

no

a

respectto limits.

and progressions

series obtained

relation which

Schroder, Al$d"ra

a

necessary, reference to

that by specifying this definition applies also,

characteristics

real numbers;

a

be

defined

condensed

theyare

the

of

relation, they

limit of certain these

bringin

semi-continuum,i.e.the

case, the

a interval, or progression

contained. irrational

:

is the

to

add that the rationals

another

(1) that

and

cannot

type usuallybe a

series with

be used for definition.

can

respectare

of them

to Pierce.

must

to

limits

(")in any

from

We

respect whether, in this

regardto in this

calls the

Cantor

its ends cut off.

doubt

term

what

beginningor

no

will

terms

of the series of rationals is not

compact,and has

it is

where

either type completely,

our

regardto which the the behaviour of our specify

to

but

both, it

or

a

;

in other relation,

some

I

by

to define

are

we

giventerm

a

the

;

is transitive

is

having this relation The class togetherwith the giventerm. transfinite ordinal type may always be

to any corresponding

thus defined if

some

"

P

:

that of P

is identical with

u

follows

as

not

limit

is

series

of

rationals

denumerable.

have to itself. This term

Logik der Relative, p. 131.

is due

297-299]

TramjiniteOrdinah

it would

Thus

nationals

form relation,

From of

by

with which series,

we

which

by

nothing

terms, with

the

wkieh

to

,

The

reference

to

the correlation

began the discussions of Part V.

we

the order

type

Until

of

of the

For

it is

and rationals,

bringin

we

only

hetice the

other relation than

some

magnitude among

the type distinguish

to

f

the importance of clearly

see

be defined.

can

type

generatingrelations.

compact series whose

of correlation that the

means

that

reference to two

progression.

a

the last remark,

continuum, is

of endless

define tlie

cannot

we

belong,without

type 77 is that another

that

seem

S21

nationals

of the rationals from

arises,there of the

that

irrationals.

CD

consideration of ordinals not

The

298. shows

ordinals in

clearlythat

suggestedat

the

beginningof

and serialrelations,

to this view

generalare this chapter "

Cantor

for in the article in the Mathemaiwche

types of order,not article(Math. Annalen^ XLIX, " 12),he

always as

them

to

well-ordered

series.

to functions

more

kinds of numbers.

by presented

to

functions

be considered classes

as

But

other to

of I

as

"

types

or

of

adheres; apparently Annolei^ Vol. XLVI, he speaksof as numbers, and in the following restricts ordinal numbers definitely now

writings,he confined himself

of "", which bear many td more analogies These in fact,types of order which are,

little resemblance

very

himself

his earlier

series of finite and

cardinal.

some

In

as expressible

transfinite cardinals which

types of order, as

have

we

now

familiar be

may

begin with have

seen,

numbers.

while to repeat the definitions of generalnotiom If P, Q be involved in terms of what may be called relation-arithmetic*. It is worth

299.

two

relations such that there is a

field of P tike.

which

is such that Q

If the

.Fs rebitfon-number.

P -f-Q is defined to be P term

other

relation S whose domain

is the

SPS, then P and Q are said to be I denote by AP, is called class of relations like P, which

and

The

one-one

or

=

fields of P and Q have Q or the relation which

terms, holds between any

no

common

and any term of the field of "" and between P + Q is "not equalto #4- P. Again XP-f X#

of the field of P

Thus

terms.

defined

as

X

(P

4-

Q ). For

the

summation

of

an

infinite number

no

is of

aliorelative whose field is composed of relations an require whose fields are mutuallyexclusive. Let P be such a relation,p its that /; is a class of relations. Then 2$? is to denote either one so field, of the relations of the class p or the relation of any term belonging to the relation Q of the class p to a term belonging to the field of some field of another relation R (ofthe class p) to which Q has the relation P. and p a class of serial relations, Zpp will be the (IfP be a serial relation, series of the various relation of the sum generatedby terras generating define the sum of p taken in the order generatedby P.) We may

relations, we

*

Of. Pfcrt

IV, Chap, xxix,

"231.

and Infinity

322

[CHAP,xxxvm

Continuity

of the relation-numbers of the various terms of p as the relation-number relation-number, of 2j#?. If all the terms of p have the same say a,

relation-number of P,

if 0 be the

and

generallythe

which

laws

three formal

/3 will be defined

be the

to

it is easy

this way, hold of

2pp. Proceedingin

of

relation-number

x

a

to

prove

well-ordered

series,

namely: (0 4- 7)

a

The

proofsare

(Amer.

discovered

those

analogousto closely

very

for cardinal numbers

head

"*j8 -h

=

Journal

method no they differ by the infinite product of relation-numbers, or an defining fact that

by

White-

Mr

of Math., Vol. xxiv); but has yet been discovered for of

even

ordinal

numbers. be observed

It is to

300.

that

is

method

of the above

merit

the

point in which Cantoris work leaves somethingto be desired. As this is an important I shall are matter, and one in which philosophers apt to be sceptical, here repeat the argument in outline. It may be shown, to beginwith,

that it allows

that

doubt

no

all terms

finite class embraces

no

existence-theorems

to

as

from the fact that,since 0 is

:

this

a

"

with results,

a

little care,

a

of numbers

cardinal number, the number

finite number is w + 1. a n Further, if n be a up to and including finite number finite number, n + 1 is a new different from all its predecessors. Hence

finite cardinals

ordinal number obtain

Oo

mere

contained in the fieldof and is impliedby

We

may

the class of serialrelations such

as

^

(inthe mathematical

exist

rearrangements of the series of finite

all ordinals of Cantoris second class.

ordinal number

therefore the

and progression,

a

and the cardinal number

o"

sense). Hence, by we

form

one

of

sayingthat

them,

to say that

u

now

cardinals, define the

that,if u be

has

class

a

implies

successors

of terms ; has "o terms or a finitenumber and it is easy to show that the series of ordinals of the first and second classesin order of magnitudeis of this type. Hence the existence of ^ u

proved;and ^ is defined to be the whose generating relation is of the type

is

G*J

and

aa

and

so

proved similarly

:

such

that, if

u

number o"1%

of terms

Hence

we

can

in

advance

to *"" and on, and even aw, whose existence will be the of relation of "" type generating

be

a

class contained

in the

to series,

series

a

say that

can

to

be series

a u

has

is equivalent to sayingthat u is finiteor has, for a suitable finitevalue of n, a" terms. This process givesus a one-one correlation of ordinals with cardinals : it is evident that,by extendingthe process, make each cardinal which can can we belong to a well-ordered series to one and onlyone ordinal. Cantor assumes correspond axiom that as an

successors

every classis the field of

cardinals

can

some

be correlated with

well-ordered series, and deduces that all ordinals by the above method. This

299-302]

TransfimteOrdinal"

assumptionseems has

one

no

ordered

to

numbers

in view of the fact that unwarranted,especially

me

yet succeeded

series.

do

We

323

in

not

class of $"* terms

arranginga know

that of

any

in

well-

a

different cardinal

two

be

the greater,ajnd it may be that 8* is neither less than and which may be wiled greaternor ccj 0% and their successor*, well-ordered cardinals because theyapply to well-ordered classes. must

one

There

301.

is a

the type of as regards difficulty

the whole .seriesof

ordinal numbers.

It is easy to prove that even* segment of this series is well-ordered, and it is natural to suppose that the whole .seriesi* also

well-ordered.

numbers,

If so, its type would

for the

have to be the greatestof all ordinal a given ordinal forra"in order of

series whose

magnitude,a a

ordinals less than

type is the givenordinal. But there cannot be greatestordinal number, because every ordinal is increased by the

addition

of

this

From

1.

it*, infers that of

contradiction,M. two

cardinals,it is not

different that

covered dis-

Burali-Forti,who

of as ordinals,

different

two

one greater and the necessary other less. In this,however, he consciously contradicts a theorem of

which

Cantor's with

all

But

there

affirms the

possible care, is another

and

should

f. opposite

I have

be

examined

thih theorem

have

failed to find any flaw in the proof J. M. BuraliForties argument, which premiss in

capableof denial,and that is,that the series of all appears to me more ordinal numbers is well-ordered. This does not follow from the fart that

all its segments

since,so far

as

I

are

well-ordered,and must, I think, be rejected,

of proof. In this way, it m-ould know, it is incapable

questioncan be avoided. of the successive derivatives We return to the subject 302. may now This forms one of a series, discussed in Chapterxxxvi. of alreadybriefly of those ordinals which are functions the most interesting applications method of defining of GJ, and may even be used as an independent them. first its from series derivative is We have alreadyseen P, a how, the contradiction in

seem,

obtained|.The class of its

first derivative of

P, which

is denoted

limitingpoints. P", the second derivative

by P^

is the

of P\ consists of

of Py and so on. Every infinite collection has limiting-points is the limit of the finite at least one : for example,a? limiting-point of finite order P". derivative define ordinals. By induction we can any p*"^1 vanishes ; if this happens of points, If P* consists of a finite number the

finite number

for any *

"Una

questions sui

Palermo, Vol. t Theorem

J

1 have

detected,in

z",

P

numeri

transfmiti,"Rmdic"Hti

to

the rth

del rirn"fo Mfttematfa"

(1897). off 13 of Cantor's article in Math. AHm/tw3 reproducedthe proof in symbolicform, in which RdM, Vol. vni, Prop.5.47 of my article.

di

xi

K

Vol.

avoid this assumption; but the necessary

xux.

errors

" What follows is extracted from AHa Mttth, u9 pp. 341-3tfO. i.e. that a series has a that all deniable limits exist-, simplicity I have shown in Chapter XXXYI corresponding segments have one. t^o as

and

is said to be of the 1st genus

circumlocution

are

easily

more

I shall

assume

limit whenever how

to state

i**tiresome*

for the

results

[CHAP,xxxvni

and Continuity Infinity

324

species.But

it

happen that

mav

all finite derivatives may have

in

form

common

have

no

P*

vanishes,and in this

collection which is defined

a

pointswhich

points.The

common

all

It is to be

P".

as

case

the definition of "y. requiring whatever finiteinteger A term x belongs be, x belongs to P8* if, v may It is to be observed that, though Pf may contain pointsnot to P*. belongingto P, yet subsequentderivatives introduce no new points.

observed that P* is thus defined without

This illustrates the creative nature

a

derivative of

of

rather of limits, or

it may yieldnew is first applied, further terms. That is,there is

segments: when it giveno applications between

of the method

terms, but later intrinsic difference

an

been,or may have been,obtained as the and one not so obtainable. Every series other series,

series which

.some

has

contains its firstderivative is itself the derivative of

which

number

of other series*. The

by the

determined is

each term

various terms

partof

successive of

a

an

infinite

like the segments derivatives,

form regression,

a

series in which

each of its predecessors is the ; hence P", if it exists,

lower limit of allthe derivativesof finite order. From

P* it is easy to go constructed in which any be actually

Series can etc. % P1*-2, finite or transfiniteof the second derivative, assigned on

to P"

+

vanish.

When

none

is the first to class, of the finitederivatives vanishes, P is said to be of It must not be inferred, however, that P is not

the second genus. On the denumerable*

the firstderivative of the rationals is contrary,

that all its derivatives are so number-continuum,which is perfect, identical with itself;yet the rationals, as we know, are denumerable. P is alwaysdenumerable,if v be finite or of But when P" vanishes,

the

the second class. The

to the theory of derivatives is of great importance

real functions

f,where

induction

any

to

it

enables practically

us

to extend

ordinal of the second class. But

for

theoryof

mathematical it philosophy,

of it than is contained in the above unnecessary to say more remarks and in those of Chapterxxxvi. the first Popularly speaking, seems

derivative consists of all

points in

whose

infinite an neighbourhood number of terms of the collectionare heapedup ; and subsequent tives derivaof concentration in give,as it were, differentdegrees bourhood any neighThus it is easy to see why derivatives are relevant to be continuous, collection must be as concentrated as a : to continuity in every neighbourhood possible containing any terms of the collection. modes of expression But such popular of the precision are incapable which belongs to Cantor's terminology. *

Vol. u, Part m, " 71, 4-8, Makhematiqu"t, t See Dini, Theorie der Funetionen,Leipzig,1892; esp. Chap, Translator's preface. Formulwre

de

xni

and

CHAPTER

INFINITESIMAL

THE

THK

303.

Infinitesimal

differential and it

of, the The

infinitesimal

himself

who,

"

have

held

Calculus the

neither

be

his

he

hindered

zero,

of

been,

had

"

since

ever

the

rise

left

are

less than

are

made

limits,and

He

this

appears

aside, the

justifiedpractically by the fact

is

it gives

ideas, upon

crude.

extremely

as

metaphysicalsubtleties

tion*. observa-

of

those

regard his

him

in

the

Calculus and

dx

fictions, but

mathematical

finite, nor

nor

tion implica-

or

of mathematics.

thinking of Dynamics, his belief from him discoveringthat the

was

doctrine

invention

own

described

which

to

errors

infinitesimal the

of

if

that,

When

on

only

the

retained

I have

allusion to,

has

Calculus

for

name

condition. Leibniz disgraceful supposed,should have been competent

have

account

the

of

such

as

of this branch

part

only approximate, but

is

traditional

is the

somewhat

a

would

one

give a correct can topic,which

that

in any

invented, in

to

to

Calculus

theory philosophical

subject was

CALCULUS.

integralcalculus together,and shall shortlysee, there is no we

although,as

;

XXXIX.

actual resbi

dy as really

as

representingthe units to which, in his philosophy,infinite division in his mathematical was expositionsof the supposed to leadf- And subject,he avoided giving careful proofs, contenting himself with the enumeration other of rules ". At times, it is true, he definitely rejectsinfinitesimals how, of

without

the

the

respect, Newton foundation

Of.

Gerhardt's t

Calculus

space

and

Mathentatical ed

n"

yet be

exact, and

to preferable

is

of the

continuityof *

could

in the

time

Work*, Gerkardt's

Math.

Work*,

t See

Math.

Work*, Gerhardt's

in

not

Leibniz:

his

doctrine CantorV

of

ed.

vi, pp.

ed., Vol.

iv,

they give 1)1-93;

pp.

n,

235, 247,

v, pp.

p.

905.

220

to

by

.show means

approximate. In this Lemmas [jgive the true limits,and, assuming the

sense,

ed.

Gerhardt's

Work*, (Jerhardt's ed., (Marburg, 1902), pp. 206-7, |jPHnripia, Part I3 Section i. Phil.

failed

valid

proofs

JRW/.

H""**,

p. 282,

See

" Kg.

he

the results obtained infinitesimals,

of

use

Calculus

valid";but philosophically

as

253.

ff. Cf.

Cassirer,

LeiMz'

and Infinity

32(5

Continuity

[CHAP,xxxix

magnitudesare concerned. But spatio-temporal Newton ignorantof the fact that his Lemma* entirely was, of course, the appeal ; moreover, depend upon the modern theoryof continuity to time and change,which appears in the word fluxion,and to space, which appears in the Lemmas, was wholly unnecessary, and served merely to hide the fact that no definition of continuityhad been Leibniz avoided this error, seems highlydoubtful ; given. Whether of the it ih at any rate certain that, in his first publishedaccount of the tangent he defined the differential coefficientby means Calculus, of its rules

to

as

as speculation

direction to

the

on by his emphasis

And

curve.

a

far

so

a

wrong

all mathematicians

misled

Calculus, which

the

to

he gave infinitesimal,

perhaps,of De exception, to the present day. It is only down Morgan), and all philosophers have providedthe in the last thirty or fortyyears that mathematicians of the Calculus;and foundations for a philosophy mathematical requisite these foundations,as is natural,are as sophers, philoyet little known among such works the in France*. on subject, Philosophical except und selm G"chichte'f) are Coheres Prineipder Infimtesimalmethade as vitiated,as regardsthe constructive theory,by an undue mysticism, inherited from

(with

Weierstrass

before

to Kant, and leading

the

such results

the identification of

as

magnitudewith the extensive infinitesimal{. I shall examine of the infinitesimal, which is essential in the next chapter the conception theories of the Calculus hitherto propounded. For to all philosophical I am the present, only concerned to givethe constructive theory as it intensive

results from modern

of

continuous

a

mathematics,

differential coefficientdependsessentially upon the notion

The

304.

function of

continuous

a

The

variable.

notion

to be

in the purelyordinal ; on the contrary,it is applicable, first instance,only to series of numbers, and thence,by extension, to which But series in measureable. distances or stretches are numerically

defined is

not

firstof all

we

We

define

mitst

have

continuotis function.

(Chap,xxxii.)what is meant by a function of a what is meant by a continuous variable (Chap,xxxvi.). If is one-valued, and is onlyordered by correlation with the

alreadyseen

and variable, the function

'

then, when variable, whether

a

the

the variable is

function

is -continuous

continuous,there is no ;

for such

sense

in

asking

series bv correlation is

a

similar to its prototype. But alwaysordinally and the field of the function has

an

order

continuous

both

of correlation, it may independent

values of the

the

are

when, as where the variable classes of numbers, the function

in function,

series in the

or

the order obtained

order. independent

SeeCoutnwt,

/*

t

Berlin,1883.

The

*

^;".rit. p.

lo,

not

happenthat

form by correlation,

When

the function is said to be continuous interval, *

may

they do

in that

so

a

in any

"interval. The

passim. f/w^mXathewatfyHi!, historical part of tins work, it should

be said, is admirable

The

304]

303,

InfimtennialCalculm

327

definitions of continuous and discontinuous functions,where precise and f(x) are x numerical,are given by Dini * as follows. The variable

both pendent inde-

is considered to consist of the real

numbers, or of all the real numbers in a certain interval ; f("\ in the interval considered, is to be one-valued, at the end-points of the interval, even ami is to be also composedof real numbers. We then have the following definitions* the function being defined for the interval between a and /?,and a being x

real number

some

the value

has

but

small

as

in this interval.

call f(x) continuous for

We

"

f(a\ as

x

if for every

exists

difference

the

f(a

6)"f(a)

+

words,/*^)is continuous

is

in the

equal to

cr,

different from

0,

a

are

",

a,

=

rightand

different

less than numerically

numericallyless than

point x

limit of its values to the

if the

it

positivenumber

0, such that,for all values of S which

from

in which

point a,

number positive

there please,

we

in the

or

a"

=

In

"r.

where it has the value left of

a

is the

e,

other

f(a^ and

same,

/(a)."

is discontinuous for "r value Again,,/'{#*) ", if,for anyf positive of "r, there is no value of " such that, for all corresponding positive values of 8 which are less numerically than e, f(a 4- S)"f(a) is always "

=

less than values

in other

cr;

f(a

words,f(x) is discontinuous for

h) of f(x)

4-

to the left of a, the

to the

one

theyhave such, these

are

rightof

and the values

the

o, when

=

f(a

"

k) of f(x)

and the other,have no determinate limits,or, if different on the two sides of a ; or, if they are

they differfrom the value

the same,

a,

x

f(a\ which

the

function has in the

pointa" These be

must

definitions of the

somewhat are confessed,

introduce any say that values

as

and discontinuity of a function,it continuity

without simplification

function

a

to impossible loss of rigour. Roughly,we raay the neighbourhoodof #, when its

complicated ; but in

is continuous

it seems

approachesa approach the value /"(a),and both to left and right. But the notion of

it

their limit

function is a somewhat

more

notion complicated

have the

f(a) limit

than that of

a

for

of

a

limit in

A function of a with which we have been hitherto concerned. general, it approachesany given generalkind will have no limit as perfectly have it limit should order that In x as a approachesa from the point.

left,it is necessary

and

sufficient

that, if any number

"

be mentioned*

to a, but less than a, near sufficiently the value of the function will differ by less than " ; in popular language, does not make jumps as ^ approachesa from the left. any sudden similar circumstances,f (x) will have a limit as it approachesa Under need not be when both exist, from the right.But these two limits,even equal either to each other,or to f(a), the vahte of the function when two

any

*

Op.

t The

values of

tit.

" 30,

(ierman

f(x)^when

x

is

pp. -W, .51. has (nat the Italian)

every instead of a"y,

but this is a

slip.

and Continuity [CHAP,xxxix Infinity

328 x

stated*

a

determinate finite limit may

be thus

:

the values of y to the

In order that

"

a

condition for precise

The

a.

=

to the

(forinstance

should right)

left of

rightor

have

a

and

determinate

it is necessary and sufficientthat, for every number there should be a positive number "r,

a

finite number finite

limit,

small positive arbitrarily e, such

that the difference

the value #"+, of y for ,r a-h", and the value to the value a + S of a-, should be numerically corresponds

between

#"n-e-2/a+s ya+s, which

=

than 0 and less than e." " which is greater thus the limit of a function,and instead of defining

"r, for every

less than

possible, whole class of a whether it exists,to define generally discussing to the classof limits of y limits f- In this method, a number * belongs small,y will a, however for x~a, if,within any interval containing for example, to z than by any givendifference. Thus, approachnearer 1 to + 1 (both from sin l/o?, zero, will take every value as x approaches small. Thus zero, however in every finiteinterval containing inclusive) It is

then

-

-1

interval from

the for

x

method

This

0.

=

this case, the class of limits that the class of limits always has the advantage

forms, in

+1

to

the limit as the only member of the easy to define member. in case this class should happento have onlyone class of limits, and more general. at once simpler This method seems continuous function, of 305. Being now agreedas to the meaning a of the attack the question can the limit of a function, we and It was formerly derivative of a function,or differentialcoefficient. It is then

exists.

of*

is

known

now

but this functions could be differentiated,

all continuous

supposedthat

be

to

Some

erroneous.

can

everywhere, be differentiated tial differen-

others everywhere everywhereexcept in one point, an others contain the left, on none the right,but sometimes on cannot which in they in any finite interval, of points, infinite number of pointsthey number greater be differentiated, though in an infinitely have

others

a

most

the

others lastly-andthese are properly differentiated, But at all, general class" cannot be differentiatedanywhere

be

can

conditions under

function may

a

importanceto

are

of

not

concern greatly

some

which

us

here

the ;

be

differentiated, though they

of space philosophy

and in any case,

we

and of motion, need

must

first know

differentialis. If

ha*

/(#)

it may

then

a

be

function which is finite and

happen

definitelimit

ii,

Heft

i

what

"

a

,

continuous

,,

at the

a

,

point ,r,

that the fraction

as

"

to zero. approaches

^^ Band

the

1899),esp. (Leipzig,

pp. 20-22.

If this does

happen,the

The

304-307]

Calculus Infinitesimal

limit is denoted

and by/'(.r), point.r. If,that

in the

f(x)

that,given any number

by

is called the derivative

differentialof

or

is to say, there be some number z swh however small, if B be any number less than

number

??, but

then \f(x positive,

than let"s

c, then

z

home ;:

e

329

" j$) /(JT)]/ " S differs from is the derivative of f(x) in the pointx. If the -

then f(x) has no derivative at the point questiondoes "not exi.st, If f(x) be not continuous at this point, the limit does not exist ; the limit may or may not exist. f (,r)be continuous,

limit in ,r.

if

The

306.

onlypointwhich it is importantto notice at presentix, that there is no implication of the infinitesimalin this definition. The S is alwaysfinite, number and in the definition of the limit there is nothing to imply the contrary. In fact,\f(x+B) -/(*r)]/S, regarded G. when S The limit of a a function of ", is whollyindeterminate as function for a given value of the independent ha\e variable is,as we notion different the from value its said value of for the seen, an entirely number. and the two may or may not be the same variable, independent =

In the have

present case, the limit may Thus

meaning.

no

Calculus,and

be definite, but the value for S

it is the doctrine of limits that

0

"

can

underlies the

pretendeduse of the infinitesimal. This is the and it is onlypointof philosophic importancein the presentsubject, onlyto elicit this pointthat I have draggedthe reader throughso much not

any

mathematics. account, it examining the infinitesimal on its own remains to define the definite integral, and to show that this,too, does involve the infinitesimal. The not indefinite integral, which is the of the differentia], is of no converse mere importanceto us ; but the has an independent be briefly which definite integral must definition, Before

307.

examined. Just

the derivative of

as

function is the limit of

a

is the limit of definite integral

defined

follows

as

:

Let

finite in the interval n

any

by portions

The

sum*.

be

f(x)

a

function

definite

of the

means

and

this interval xn^.^

"

...

be

integral may

is one-valued

which

ft(both inclusive).Divide (n 1) points*r15 jr2,

to

a

a

the NO fraction,

a

into

and denote

In each of $n the n intervals ^ "? *rH-i. "r2, a, j^ these intervals, S*, take any one of the values,say^/^), which j\**') this value by the interval 8^. in this interval,and multiply assumes

by 8j,"J,

"

"

"

...

...

n

the

form

Now

2

sum

f("*)Ss.

This

sum

will

always be

finite.

If

1

now,

*

as

/*

this increases,

one

deiinite limit,

differs littlein different modern 'Hie definition of the deiinite integral

('p.Dini, op. fit.

"" 41"58; definition the

tends to

sum

"" 178-181

;

drr Kiiryklopiidie as

inverse

the limit of of

a

a

snm

Jordan, four* wttheiwitixchen is

derivative,but

brought back by Cauchy.

See

d'Anahpc, Vol.

was

banished

references

works,

(Paris,18")% Chap, i, The 2, " 31.

Wixxenxt'fuiftett, n, A.

consonant

more

i

with

Leibniz's views

by Bernoulli

and

in the last-mentioned

than

Euler,

place.

tliat

and

a-

only

be

may

f(*r)

from

308.

and

sum

whose

limit

is

If

limit;

other is

mere

the

The are

above

they

be

may

spaces, is

involved can

later the

be

important

to

since

For

infinitesimal

in

it

has

do

to

But which its

limit, but

the

limit of

does

of

nothing

not

tive deriva-

the

to

if

fact.

this do

with

infinite

the

its

"

it

that

each

reach

cannot

with

true,

limit.

instance

another

is of

series

their

to

the itself

sum

its

definition

the

in

meaningless.

Any

is, that

therefore,

involves

and

limits,

to

only

motions

and the on

that

own

account.

under

Dynamics. and

geometrical

it is time

I

shall to

it

have

make

is

As

head, to

something a

that

measured. the

the

dynamical

critical

tinuity, con-

magnitudes, be

can

this

and

evident

measurable

distances

or

included and

integrated,

present, its

stretches are

Geometry

assumption

But

division,

limits

of

ordinal.

numerically

any

and

multiplication definitions

the

purely

in which

and to

involve

Unlike

all series

times,

they

rendered

be

once

the

the

term

equal

but

sum.

occur

becomes

last

rule

merely

being,

extended

differentiated

stage.

infinite

and

a

magnitude

agree.

term

indirectly

only

the

no

general

calculus,

arithmetical.

applicable in

has a

;

not

involves

limits.

cannot

at

therefore

Since

one

which

sum

general

and

have

itself

actually attaining

as

have

we

and

the

case,

it limits

definitions,

essentially they

and

integrable

actually attained,

limit

in

The

has

the

have

series

this

may

integral

and

with

connection

infinite

integral

not

integral

finite, and

are

infinite,

descends

infinitesimal

infinitesimal,

the

the

series

which

definite

integral

regarded

series

series

so-called

The

be

accident.

the

sum.

be in

which

always

or

is terms

a

only

definite

the

suppose

but

not

in

The and

would ;

is

there

All

definite

to

infinite

a

to

and

is

vided (pro-

sufficiently

definite

the

infinitesimal,

of

the

were

respect

a

ascends

belong

xxxix

chosen

for

limit, f(tr)

definition.

the

limit

intervals

must

sum

always

this

such

no

called

is

derivative,

the

this

infinitesimal

the is*

this

so,

we

of

be

Thus

of

nor

the

number

would

limit

is

number

assigned

any

one

there

about

strictly

finite.

the

If

case

infinite

the

onlv

the

make

to

neither

this

be

intervals

the

however

than

"

j9.

to

in

As

remark

then

n)

a

less

are

[CHAP,

Continuity

ft.

to

a

all

of

from

interval, and

its

that

values

great

is

in

chosen

only of

and

Infinity

330

culus Cal-

axioms functions

to

say

examination

at

a

of

CHAPTER

INFINITESIMAL

THE

309. the

UNTIL

XL.

AND

THE

times,it was

recent

IMPROPER

INFINITE.

believed universally

that

continuity,

and the definite integral, all involved actual infinitesimals, derivative,

i.e. that

if the

even

definitions of these

be

could

notions

formallyfreed

from

of the infinitesimal, explicitmention yet, where the definition* the actual infinitesimal must This belief is applied, always be found. The definitions which been given in have now generallyabandoned. and this previouschapters do not in any way imply the infinitesimal,

mathematicallyuseless. In the present appears to have become chapter,I shall first give a definition of the infinitesimal,and then

notion

the

examine

arises.

regardedas

than

any

number

a

finite number

Calculus, the

or

which

a

ball thrown

highestpoint of its course, line and the next a point,etc., etc. all precise. The dx and dyy as we saw

ball is at the

theory has

between

distance

points "

by

none

of

in the

last

fraction

a

highest point

is

a

very

consecutive a

they afford

instances"

most

there no

between these

notions

and

on

at

are

nothing

denominator

time

VII

Part

there

is every

point

a

are chapter,

The

is

precisedefinition

deny.

of

The

time.

there

to

reason

find,

shall

we

such

no

that

points presupposes

which

view

or

during which a complex notion, involving

in

that developed,

been

It has

ssero, is le""

not

dir

numerator

all.

at

philosophictheory of motion;

this

with

its

at

But

the

distance

the

fraction whose

a

itself

is not

rest

whole

when

limit of

dyjdx is the

finite,but

are

critical

a

dy of the vertically upwards is at

been

It has

magnitude.

or

during

time

magnitude which, though

at the

at all :

by

continuityimpliesthe infinitesimaL has, in general,been very vaguelv defined.

infinitesimal

been

rest

I shall end

of the belief that

discussion The

this notion

where

cases

what

secutive con-

are

And

so

is meant

the infinitesimal. There

310. renders

the

far

is,so

infinitesimal

assumed arbitrarily had what

been Cantor

taken

to

be

calls the

I

as

a

know, only

one

which precisedefinition,

purelyrelative notion, to

be finite.

infinitesimal

as

When,

correlative

instead, we

finite,the

to

thing some-

regard what

correlative

notion

The improper infinite (Uneigrntikh-Unendliehex).

is

and Infinity

332

definitionof the relation in of

is question

obtained

by denyingthe axiom matical obtained by denying mathe-

Archimedes,just as the transfinite was induction.

thev magnitudes,

are

If P, Q be any two numbers, or any two measurable said to be finite with respectto each other when,

if P be the lesser, there exists than

Q.

Archimedes

an

greater

the axiom

integerconstitutes

and the definition of relative finitude.

is

that nP

such finite integer n

a

existence of such

The

[CHAP.XL

Continuity

of

It will be observed

numbers that it presupposes the definition of absolute finitude among definition which, as we have seen, dependsupon two points, a (1) the "

of 0 with or simplicity, the logical of mathematical notion of the null-class; (") the principle The notion of relative finitude is plainly distinct from that induction. of absolute finitude. The latter appliesonly to numbers, classes and the former appliesto any kind of measurable whereas divisibilities, which both are divisibilities, or magnitude. Any two numbers, classes, does not absolutely finite are also relatively finite;but the converse For example,o" and o" 2, an inch and a foot,a day and a year, hold. finite pairs, are relatively though all three consist of terms which are infinite. absolutely The definition of the infinitesimal and the improperinfinite is then follows. If P, Q be two numbers, or two measurable magnitudesof as the same kind,and if,n beingany finiteinteger whatever, nP is always connection

of 1

with

the

notion logical

of

.

less than

Q, then P is infinitesimal with respectto Q, and Q is infinite with respectto P. With regardto numbers, these relative terms are then Q for not required P is absolutely if,in the case supposed, finite, ; is

infinite; while if it were for Q to be absolutely absolutely possible P would be -absolutely infinitesimal we finite, a case, however, which shall see reason in future to regardas impossible. Hence I shall assume that P and Q are not numbers, but are magnitudesof a kind of which least at measurable. It should be observed that, are some, numerically of of the axiom is the only way Archimedes as regardsmagnitudes, not but the infinite also. Of a magnidefining, tude onlythe infinitesimal, not numerically measurable,there is nothingto be said exceptthat "

,

it is

greaterthan

of its

some

cannot propositions infinity

greaterthan all others infinite. it is

Finitude

onlyby

relation

be obtained.

of its

and

kind,and lessthan others

infinity are

to

Even

kind, there is no

numbers

;

reason

for

numerical essentially that these

but from

if there be

terms

can

such

magnitude regardingit as a

notions, and be

appliedto

other entities. 311.

The are

than

next to

questionto be discussed is,What be found ?

Although

there

are

instances far fewer

of infinitesi

instance*

there are yet some that are important. To formerly supposed, begin\\ith,if we have been rightin regardingdivisibility a as tude, magniis it whole containinga finite of any plainthat the divisibility number of simpleparts is infinitesimal as compared with one containing was

an

The

311]

310,

and Infinitesimal

infinite number.

The

infinite whole

every whatever

But

of

Improper Infinite 333

partsbeingtaken

the

as

measure,

will be

finite number

instance.

number

the

greater than n times every finite whole, be. This is therefore a perfectly clear may

?/

it must

be

not

wholes, of which

supposedthat

the ratio of the divisibilities

least is transfinite, be measured by one can the ratio of the cardinal numbers of their .simple parts. There are two reasons why this cannot be done. The iirst is, that two tran.sfinite of two

cardinals do

ay9

7

=

any relation

of ratio is effected

analogousto strictly

by

ratio

of mathematical

means

bears

induction.

transfinite cardinals a, bv the 7 expressed certain resemblance to integral and 3$ ratios,

a

be used to define other ratios. The

to finite ratios.

measured

other

But ratios

why

reason

defined

indeed,

;

of two

relation

The

have

not

the definition

at

=

equation 78 may

very similar infinitedivisibilities must not be so

not

are

transfinite numbers

is,that the whole must alwayshave by the than more divisibility part (providedthe remainingpart is not infinitesimal), though it may have the same transfinite number. relatively like ordinals, In short, divisibilities, are equal,so longas the wholes are when and onlywhen the cardinal numbers of the wholes are the finite, the but notion of magnitudeof divisibility is distinct from that same ; of cardinal

number, and separatesitself visiblv as

soon

we

as

come

to

infinitewholes. infinite wholes

less divisible infinitely the other. Consider,for example,the lengthof a finite straight of the square upon that straight and the area line ; or the length line and the lengthof the whole straight line of finite straight

Two than line

of

a

which or

it forms

the

may

part (exceptin

rational numbers

pointson

and

finite part of

a

be such

a

that

finite spaces) ; the

and all are The

are

magnitudesof

line obtainable

one

and

the

infinite divisibilities; but

pointson

a

limited

portionof

an

or

a

and

area

real numbers;

and the total collection of construction, All these

is

one

the

or

Staudfs

by von

pointson

the

a

volume

collection

;

of

quadrilateral part*.

said finite

kind, namely divisibilities,

same

they are

of many

line obtainable

by

different orders. the

quadrilateral

respectto the said portion infinitesimal\ with respectto any ; this portionis ordinally bounded area ; any bounded area is ordinally infinitesimal with respectto is any bounded volume ; and any bounded volume (exceptin finitespaces) In all these cases, the infinitesimal with respectto all space. ordinally

construction form

word

a

collection which

is injimtesimal

used

is infinitesimal with

accordingto strictly

of Archimedes. obtained from the axiom s omewhat from infinitesimals unimportant, that measurement

essentially dependsupon

be extended cannot, in general, which have justbeen reasons

*

See

Part

VI, Chap. XLV.

the above

What a

makes

definition,

these

various

mathematical

the axiom

of

is, standpoint, Archimedes, and

of transfinitenumbers, for the by means of explained.Hence two divisibilities,

f See Part VI,

Chap. XLVII, "387.

ami Infinity

334 which

is infinitesimal with

one

as

regardedusually

respectto the other, are

regardthem

and to

magnitude;

different kinds of

[CHAP.XL

Continuity of the

as

kind

same

All of them, however, correctness. givesno advantagesave philosophic trates and the series of them well illusinstances of infinitesimals, are strictly the relativity of the term infinitesimal. method of comparingcertain magnitudes, An interesting analogous with of those of collections to the divisibilitiesof any infinite points, continuous stretches is given by Stolz*,and a very similar but more matical generalmethod is givenby Cantorf These methods are too matheto be fully explainedhere,but the gist of Stolz's method may be briefly explained.Let a collection of pointsx' be contained in some .

of parts, n the interval into any number of parts,and so divide each of these parts again into any number and let the successive divisions be .so effected that all partsbecome

finite interval

and on

;

in time

all the

each stage,add together the mth stage, let the

number 8. At any assigned parts that contain pointsof .r'. At

less than

be Sm,

sum resulting

but

Divide

to b.

a

Then

increase

cannot

Sm

must

we

shall have

it.

L

b

=

Hence

limit L.

approacha "

a

;

diminish this sum,

divisions may subsequent the

as

If

jr

is

of divisions

number

compact throughoutthe interval,

finite derivative

if any

increases,

of

vanishes,L

x

"

0.

bears an analogyto a obviously integral ; but no be identified with for But cannot L, L the existence of are required the divisibility for some ; compact series,e.g. that of rationals,are L

conditions

definite

others,e.g. the continuum, but givethe

less divisible than

value

same

of L. The

312.

evident peculiarly to possible

numerical

infinitesimals were

in which

case

is that of

compact series. In this

however, it is

case,

"

have

infinitesimal, as we

is another

segment

be

be no infinitesimal segments]:, provided prove that there can measurement be possible the at all and if it be not possible, it is seen, is not definable. In the first place, contained between two different terms is always

evident that the segment divisible; for since there is infinitely there

formerlysupposedto

d

contain

can

between a

a

and

c,

finite number

class of terms

in

term

a

and

between

c so

Thus

on.

of terms.

any

But

two

a

and

",

terminated

no

segments defined by

Chapterxxxrv)have

term. no limiting may (as we saw In this case, however,provided the segment does not consist of a single term other term some b, and therefore an infinite a, it will contain

a

number

of terms.

pointis to define be added to form

*

Thus

all segmentsare

a segment by placing a

Math,

new

divisible. infinitely

of segments. Two multiples

segment

;

equal to the

and if the two

Awutlen, 23, "Uel"er

ehien

zu

eiuer

were

terminated

one

at the

segments can

new

/". "Ueber

I

See

unei*dliche Ihieare

Peaiio,#mW"

di

No. ". Punktmajmigfidtigkeiten/'

Mttenmtiw, Vol.

n,

pp. 58-62.

one

is said

Puuktmeuge gehorigeu

"

t

next

end of the other

equal,the

uneiidlicheu

The

and Infinitesimal

311-313]

The

to be double

of each

of them.

this process cannot defined by Professor Peano

contained in

if the two

employed. the logical sum

as

segments

Their

is

segments obtained

in the respectively

Having defined this sum, we segment. Hence we can define the

a

nated, termi-

not

are

in this case,

sum,

of all the

segments contained

by adding two segments to be added*.

multipleof

Impn^er Infinite335

be

terminated

finite

But

the

two

define

can

anv

class of terms

finite

of sum multipleof our segment, ir. the logical multiples.If, with respect to all greater segments, our segment obeysthe axiom of Archimedes,then this new class will contain some

all its finite

that

all terms be

after the

come

infinitesimal with

will question case,

each

respect to any

fail to contain

other.

segment. But if our segment other segment, then the class in our

pointsof this other segment.

some

all transfinite

that

it is shown

originof

of multiples

our

segment

are

it follows that the class formed

Hence

In this

equal logical

by the of our which be called the multiples segment, may infinite multipleof our segment, must be a non-terminated segment, Each for a terminated segment is alwaysincreased by beingdoubled.

to

sum

of all finite

"

of these

Professor Peano results," so of

the usual notion

segment

segment. And

a

be

cannot

concludes,** is in contradiction with

rendered

from the fact that the infinitesimal

finite

by

of any it cannot

means

I conclude,with Cantor, that multiplication, finite in magnitudes (p.6"). But I think an

infinite actually be

an

element

"

strongerconclusion there is,corseen responding compact series, to every segment, a segment of segments, and that this is always terminated by its defining segment ; further that the numerical of of segments that of measurement as segments is exactlythe some the result above to simplesegments ; whence, by applying segments of since none of them can be segments,we obtain a definite contradiction, unterminated,and an infinitesimal one cannot be terminated. of the rational or the real numbers, the completeknowIn the case ledge is warranted.

which

For

we

we

possess

have

even

that,in

concerningthem

infinitesimals demonstrable.

A

renders the non-existence

rational

number

is the

ratio

of

of two

and any such ratio is finite, A real number other than finite integers, of the rationais of series is a segment zero ; hence if 'x be a real number than

other

zero,

there

is

class u, not

a

if y is a w, and z is less than #, z is is x. Hence which every real number and all rationais are rationais,

number

speak of new

Consequentlyif it

is finite.

an

null, of rationais such

that,

to the segment "r, ije. belongs is a class conother than zero taining

finite;consequently every real in possible,

were

infinitesimal numbers, it would

have

to

be in

any

sense,

to

radically

some

sense.

313.

I

come

now

to

a

gladlysay nothing I mean, of functions. infinitesimality "

*

very the On

difficultquestion, on

questionof this

Loc. eit. p.

the

orders

which of

I would

and infinity

questionthe greatestauthorities

61, No.

9.

and Infinity

336 divided:

are

that

these

form occur,

function

f(x) whose limit,as

that,for as

,r

Reymond, Stolz,and many others, maintaining nitesimal class of magnitudes,in Which actual infia special holds stronglythat the whole theory while Cantor consider To as a simply as possible, put the matter Bois

Du

is erroneous*.

There

approacheszero.

is

zero, approaches

*r

a, the

finite real number

some

[CHAP.XL

Continuity

be

can

It

zero.

has f(z')lx*

ratio

happen

may

finite limit

a

such number, but

only one

there

number, may be called the the order of smallness of order to which f(x) becomes infinitesimal, or But for some there functions,e.g. I/log.?', f(jr)as x approacheszero. If a be any finite real number, the limit of is no such number a. ciently as x I/"r*log.r, approacheszero, is infinite. That is,when x is suffiand may be made largerthan any small,l/jMogxT is very large, be

may

Then

none.

be.

may

a

is such

if there

by making

assignednumber finite number

a,

a

small sufficiently

x

Hence,

shall need

express the order of smallness of

(say)e~llx as

1

is

end

no

to

log (log*r),for

and

Thus

on.

so

all in any class,and

example,is

of

have

we

class

one

new

of these

the succession

which

are one

infinitely great numbers jc

that

that of

:

of

1/log.r,

of which hierarchy of magnitudes,

infinitesimal with class

than

to

And

zero. approaches

orders of smallness

smaller infinitely

whole

a

this whatever

and

express the order of smallness of infinitesimal number, which may

to

it is necessary to invent a l/Iog*r, l"e denoted by \jg. Similarly we there

"

respectto all

formed

only is

of

in any higher all the finite real

numbers. finds

Cantor development,

this

In

vicious

a

circle ; and

though

the

is in the right. He it would seem that Cantor question is difficult, be introduced unless we that such magnitudescannot cit.) (lot*, objects

have

similar

to

present

think

to

reason

that

case,

that there

concerninglimits; and definite contradictions

supposed infinitesimals. for them

even

such

are

we

If there

magnitudes. The maintains

Cantor

point is

that, in the

be

proved concerningthe infinitesimal numbers j, then

may

were

should have g

1/(logx

af) =

.

0

exceed |. And he shows that even continuous, ultimately have an and uniformlygrowing functions may differentiable, entirely such : that, in fact,for some ambiguous order of smallness or infinity must

functions,this order oscillatestatween

accordingto the

manner

in which

the

infinite and

limit is

infinitesimal value*,

approached. Hence

we

may, I think,conclude that these infinitesimalsare mathematical fictions. And this may be reinforced by the consideration that, if there were infinitesimal numbers, there would

number-continuum, *

See

l"u Bois

which

we

have

be

infinitesimal

justseen

to

be

segments

of

impossible.

Functionentkeorie Reymoud, Attfffwutine (1882),p. 270 ff.; i (Leipzig, 18"5), Section ix, Anhaug; Cantor,

Fart AUymebtts Aritknietik, fit Matrnuttica, 104-4$. v, pp.

the

St

Thus

314

to

we

as

regards

which

infinite

of

no

example,

compared

to

regarded

numerical

between

is

what

so

there

closely

regarded

be

to

"

that

saw

we

may

unimportant independent.

is

are

possible, or

Cantor

as

connected

that

again

compared

of

volume

a

has

and the

upon extended

in

orders

of

smallness

we

as

because

numbers, Numerical

area.

Archimedes,

of

And

compact

finally

infinitesimals.

genuine

"

functions

conclude

is

a

"

conception,

of

very which

restricted

infinity

and and

therefore

mathematically continuity

not

are

infinitesimal,

The

and

series,

"

as

are

saw,

kind,

axiom

of

volumes

to

numbers.

segments

certain

in

traiihfinite

an

tesimal, infini-

the

are

another of

means

this

infinitesimal

lines

as

other

there

occur

infinitesimals,

by

even

infinitesimal

no

does

wholes

being

that

saw

straight

magnitudes

dependent

wholly

extended

be

cannot

we

fact,

in

We

of

of

finitesimal, in-

thrnt,

of

meaning,

mathematics,

cases

as

length,

a

absolute

these

and

term,

capable

not

an

bounded

genuine

is

and

area

measurement, and

by

in

and

mathematics

is

the

divisibilities

or

337

concerning

relative

a

finitude.

of

polygons,

comparison an

from

useless

such

But

polyhedra.

it has

it

lengths

of

areas

is

it

sense,

where

completely

for

always

But

said

divisibilities,

than

Infinite

Improper

been

that

absolute

term.

though instances"

the

the

has

with,

indistinguishable

is

meaning

as

begin other

in

and

what

up

to

see,

relative

a

sum

magnitudes

are

than

Infinitesimal

The

314]

313,

are

very

alike

XLI.

CHAPTER

ARGUMENTS

PHILOSOPHICAL THE

315,

have

WK has

to

INFINITESIMAL.

completedour

now

concerning the

say And

CONCERNING

review of what

summary

continuous, the

matics mathe-

infinite,and

the

had treated of previousphilosophers these topics, we might leave the discussion,and -applyour doctrines the paradoxicalopinion that what For I hold to space and time. demonstrated is true. be mathematically As, however, almost all can disagreewith this opinion,and as many have written philosophers infinitesimal.

elaborate

here, if

in

arguments

no

favour

expounded,it will be necessary and types of opposingtheories, in which

I differfrom

of

views

different

from

those

above

the principal controversially the points defend,as far as possible,

examine

to to

of purpose, the work useful,not onlybecause it deals alreadyreferred to will be specially

Cohen

with explicitly

standard

this

For

writers.

present theme, but also because, largelyowing to historical excellence, certain very important mathematical errors,

its

it

which who

but for

to

appears

have 316.

our

not

And

we

dy[dr is not

the

notation

dx

similar to

point,modern Leibniz.

that

and

dy

has

/' (x)

Newton's

saw

mathematics

The

philosophers

at first hand*.

example,Mr

SpiiKB*,ai"l

that

of

differential

are

Hence, in modem

notation

f (x\

it

the

finitesim in-

nothing in themselves, works on the Calculus,

the

may

involves

latter form be

suggests

observed,is

more

is due to the fact that, on y, and its similarity mathematics is more in harmony with Newton than

Leibniz

is not

a

nowhere

replaceddy\dx^since

employed the

in infinitesimals; Newton, his fluxion

its definition

of

fraction.

a

notions.

errorieous

For

modern

other

the differential appearedas a philosophicall exposition, of the doctrine of limits. Indeed, unimportant application its traditional importance, it would have deserved even scarcely

and

*

led astray

contain, have

the above

The

with

to

acquaintancewith

an

In

mention.

this

me

a

on

fraction.

Latta,in his Leibniz,"Mind,

the "

form

other

Those

ultimate

article "On N. S. No

dyjdx because he believed that asserts hand, definitely he ratios," says,

the Relations of the 31

**

with

Philosophyof

315-318] which

vanish quantities

towards

limits

but

Philosophical arguments, etc. which

the

limit do always converge, and any givendifference*."" when

But the

turn

we

dy treated

truly the ratios of ultimate quantities,

not

are

such

to

339

ratios of

without quantitiesdecreasing

to which

theyapproachnearer

works

as

dx

find the

Cohen's, we

by

than

and

real infinitesimals, as as the intensively separateentities, is composed(pp. 14, "8, 144, 147), real elements of which the continuum view

The

as

that

the

thought open brought up

by

Calculus

question;at

to

kind of

what

derived Cohen

be

can

Let

us

as see

are

self-evident for ourselves

urgedin its favour.

grounds Many arguments in by most writers from

317.

arguments whatever

rate, no

discuss the Calculus.

who philosophers

most

any

This view is certainly assumed

supportit.

to

requiresinfinitesimals is apparentlynot

favour

of

space

and

the

view

motion

in

questionare

arguments which

"

(pp. 34, 37), though he admits that the differentialcan be obtained from numbers alone,which however, followingKant, he regardsas implyingtime (pp. "0, "1). Since the to

extent

some

of analysis

space and

countenances

is stillto come,

motion

1 shall confine

myselffor from purelynumerical far as possible extract

present to such arguments as can be derived For the sake of definiteness, instances. I shall as

the

the

to opinions

318.

be controverted

from Cohen.

begins(p. 1) by assertingthat

Cohen

the

infinitesimal is not which

is

it belongsrather : purelylogical I imagine, distinguished, by the fact that it

pure intuitions the

opposedto would

take

us

well

as

as

the

This categories.

underlies the

which philosophy too

far from

our

Kantian

theme

problem of the to Epistemology, depeiKls upon the opinionis wholly

presentwork

to discuss it

;

but

it

here,and I mention

of the work to explainthe phraseology are we examining. chiefly the view that the infinitesimal calculus Cohen proceedsat once to reject of limits. mathematics from the method be independently derived by can This method, he says (p.1), consists in the notion that the elementary conceptionof equalitymust be completedby the exact notion of the is presupof equality limit. Thus in the first placethe conception posed.... of limits the method second in the place, Again, presupposes of of magnitude....But in the presupposed the conception conception time presupposed. magnitude the limitingmagnitude it. at the same The equality which is defined in the elementarydoctrine of magnitude pays no attention to these limitingmagnitudes. For it, magnitudes count as equal if and althoughtheir difference consists in a limiting be of equality must magnitude. Hence the elementaryconception

it

"

"

this is the

*

notion

PrinripiafBk

of the

quoted

in the

text.

of limits" not

i, Lemma portions of it

i, Section

highly important;though

method

so

much

as completed

is Scholium The whole xi, Scholium. the than less free from error are passage

and Infinity

340

Continmty

[CHAP.XLI

by the exact conceptionof the limit. Equalityis to be regardedas an earlier stage of the limitingrelation*."1 I have 319. are quoted this passage in full,because its errors liable in this question. of those to which non -mathematicians are typical relevance to limits. I imaginethat has no In the firstplace, equality circle and the inscribed polygon, such cases has in mind Cohen a as where we cannot say that the circle is equal to any of the polygons, arithmetical instance, but only that it is their limit ; or, to take an

corrected

convergentseries whose

a

there is much

is

sum

unnecessary considered 6"

by progressionsand these are by the finite type presented

the usual

"

example,the

in all cases

equality.Yet

kind of

no certainly

series %

ordinal numbers.

limit of the

the

as

are

of

have

we

"

the

many limit

a

There

limits

togetherwith

ordinals

togetherwith #?

"

where

cases

instances

there

absolutely simplestinstance

The complications.

is

in all such

and adventitious,

is irrelevant and

that

\/%. But

or

TT

is here

defined

are

series of the

a

Consider,for

"o.

beingcapableof

n

all

71

finite integral positive

of the

successive terms

and

thus

late terms

to

seem

we

Cohen

misled

has

the

Here

"

the

of the series "

" .

let

But

But

and

"

and

2

as

the

any

examine

us

here

magnitude, assigned between 2 and the quality

sort of extended

a

type

same

difference between less than

series becomes

have

series is of the

limit of the series.

before,9, is the

before,and here,as this is what

values.

this.

In the first

n

it depends upon place, we are

have

distances which

unnecessary

to

the fact that are

rationals

againrationals. But

and limits,

that stretches

are

% is the limit of " consideringstretches,

are

a

series in

know

we

which

that distances

equallyeffective.

because

no

rational

Now

comes

n

between

2 and all terms

of the series 2

the precisely r

in which

sense

n

co

is the limit of the finite integers. And

a

I.e.is similar progression,

its limit to be 2.

The

",

fact that

the

successive

assignedstretch up but has nothingto

to

we

having a irrelevant

circumstance, or

to

52 may

follows from

be made the

know

we

advance,differ little

series in which

stretches up

2, which

as

forms

"

to the series of finite integers, that

fact that the terms,

depends either upon our which is a fortuitous and distance,

from

it is onlybecause 2

the

upon

less than

notion

do with

there is

of

a

any

limit,

equality.And whenever our series which limit is part of a series which is a function of o", the stretch from any term to the limit is alwaysinfinitein the onlysense in which such series have infinite stretches ; and in a the stretch very real sense

is to have

a

*

")r ratio:

the

German

fe Grenzverhititnus.

Philosophical argument,

318-321] smaller

no

grows

the cardinal number have

We

i.s involved

seen

in

approach the limit,for

we

as

of its terms

remain

fullyalreadyin

so

etc.

limits,that

it

both

the ordinal and

constant

what

seems

341

sense, and

how

to

unnecessary

say

subjecthere?. Magnitude is undoubtedlythat intended by Cohen, that limited must be magnitudes. Every progression which

far,magnitude much

this

on

is certainly not involved

in the seme, which the limit and the terms

a

series which

the

is

function

a

has progression,

Every endless

a

of

e",

and

limit,whatever

in which

there

be the nature

may

forms

part of

terms

are

after

of the terms.

ever segmentsof a compact series has a limit,whatof the in all of series. Now course compact may series we have magnitudes,namely the divisibilitiesof stretches; but of segments, it is not of these that we find the limit. Even in the case the limit is an actual segment, not the magnitude of a segment ; and that they is what relevant is only that the segments are not classes, But the distinction of quantities and magnitudesis,of are quantities. series of

lie the nature

whollyforeignto

course,

320.

But

we

now

order of ideas.

Cohen's

come

to

a

The

greater error.

conceptionof

magnitude,Cohen says, which is presupposedin limits,in turn presupposes as limitingmagnitudes. By limitingmagnitudes, appears the

from

context, he

means

the infinitesimals,

I suppose, between the terms of a to be, that the kinds of seems

series and

ultimate

its limit.

What

lead to

magnitude which

difference*, he

means

limits

are

have infinitesimals. and that,in compact series, must we compact series, in this opinionis mistaken. Limits, we have justseen, Every point need not be limits of magnitudes; segments of a compact series, as we saw

in

any These to

be infinitesimal; and limits do not cannot preceding chapter, are compact. imply that the series in which they occur way is it pointshave been so fullyproved alreadythat unnecessary

in the

dwell upon

them.

duce that limits introcrowning mistake is the supposition we as equality, a new meaning of equality.Among magnitudes, in Part III,has an absolutely saw rigidand uniquemeaning : it applies magnitude. and means that they have the game only to quantities, Ls simply is meant There here what is no of : question approximation numbers absolute logical (which Cohen of magnitude. Among identity such thing as equality. there is no probablyregardsas magnitudes), There is identity, by and there is the relation which is usuallyexpressed relation This 6. the sign of equality, " in the equation x B as about Arithmetic, had puzzledthose who endeavoured to philosophize 321.

But

the

=

until it

was

equationis of two

or

a

more

of the term When one explainedby Professor Peano*. composed singlenumber, while the other is an expression the class that fact numbers, the equationexpresses the

*

See

e.g. /"r.

di Mat.

VH,

p. #".

[CHAP.XLI

and Continuity Infinity

342

contains only one term, which is the single expression definition again is number the other side of the equation. This on : there is nothingwhatever approximatein it,and it is absolutely rigid of any modification by infinitesimals. I imaginethat what incapable follows. In forming a differential Cohen means as may be expressed

defined

by

the

consider coefficient, we y -f but

In

dy.

in the Calculus.

not

terms

may their differenceis

when

dX) x

and

their

since difference,

which

There

I

x

-f dx

as magnitudes

the

dx

y

dy. There

x

approachas real number dx and no

near

as

correction whatever

that

As

the

have

absolute

This

zero.

equalto

and

Ay, but are

the derivative of

where

cases

which

such

no

There

-f A#.

x

view,

standing misunder-

a

in the Calculus

are

for

zero

Az//A#

be

can

made

to

This single Aa? and A#. by diminishing and denote by dyjdx\ but it is not a fraction,

of the

element

classesof terms chosen out of 322.

real infinitesimals

will not

dy are nothingbut typographical partsof

limits;the onlynew

ways their ratio

like

we

choose to

we

in

to

equal, defining is unity,

as

of

allow

we

count

finitedifferences A#

are

real number

one

equalwhen

There

ratios of finitedifferences, A^/A#,and

exists,there is

would

fact, two

others y and

two

Cohen's, depends upon

the Calculus.

will make view,however elementary,

no

-j-dx

unity,but

ratio

to equivalent

and

x

when

But

zero.

-f

in

are,

dx" and

x

is different from

dx

of limits and

and

x

be said to be

will have

suggestas

and

x

Arithmetic, elementary

equality.Two or

numbers

two

the regards

of

notion

introduced a

one

symbol. There

is

doctrine

of

the

equalityby

consideration of infinite

is the

series. of the

natiare

told (p.15) are we infinitesimal,

the inextensive,is differential, or

and the differentialis intensive,

as regarded

identified with

be

to

the

of Kant^s

the embodiment

of Kant) This view (inso far as it is independent reality. I must from Leibniz ; but to me, confess,it approval destituteof all justification. seems It is to be observed that dx and "%, if we allow that theyare entities at aD, are not to be identified with terms of our single nor series, yet with differences between consecutive but infinite number must be alwaysstretches containingan terms, of terms, or distances tinction disHere stretches. to such a corresponding

categoryof is

quoted with

must

be made

have

onlymeasurable

must

to correspond

between

series in which

and

series of numbers

we

of distances or stretches. The latter is the case and time. Here dx and dy are not pointsor instants, which alone space would be trulyinexteasive;they are primarilynumbers, and hence to preposterous case

of

infinitesimal stretches

assign

a

to velocity "

a

distances

"

ratio to two

numerical

point and

or

instant.

an

for it would

or points,

But

dx

and

"

as

dy

be

in the cannot

representthe distances of consecutive points, nor yet the stretch formed two consecutive points. Against this we have, in the first place, the by

generalground the precludes

that

our

series must

idea of consecutive

be

terms.

regardedas To

evade

compact, which are this, if we

322]

321,

Philosophical argument^

etc.

843

would not distances, dealingwith a series in which there are onlystretches, for be impossible to that there : are say alwaysan infinite number of intermediate pointsexcept when the siretch consists of a finite number would be a mere of terms tautology.But when there is distance,it finitesi might be said that the distance of two terms may be finite or inand that,as regards infinitesimal distances, the stretch is not consists but of finitenumber of terms. a This beingallowed compact, for the moment, our dx and dy may be made to be the distances of consecutive points, else the stretches composed of consecutive points. or the distance of consecutive But now points, supposingfor examplethat both are on one to be a constant, which would line,would seem straight We cannot givedy\d^x "1. suppose, in cases where x and y are both function and the the Calculus requires, continuous, as y is one-valued, that x and x + dx are consecutive,but not y and y + dy; for every value of y will be correlated with one and onlyone value of "r, and rice cannot versa ; thus y skipany supposedintermediate values between supposing the y and y 4- dy. Hence, giventhe values of x and y, even to differ from placeto place, distances of consecutive terms the value of dyjdx will be determinate; and any other function y which, for value of x, is equal to j/, will,for that value,have an some equal matical derivative,which is an absurd conclusion. And leavingthese mathe"

evident,from the fact that dy and dx are ratio,that if they be intensive magnitudes,as

arguments,it have

numerical

a

they suggested, is

must

is

be

measurable numerically

it is certainly not effected,

but

ones:

how

to

is

this

This

pointmay ourselves to the fundamental in which case by confining If we both x and y are numbers. regardx and x -f dr as consecutive, and either that must we y+dy are consecutive,or that y suppose that there are finite number of terms between or a theyare identical, measurement

be made

them,

see.

clearer

that

or

there

are

infinite number.

an

If

we

take stretches to

A/, it will follow that dyjdx must be alwayssero, or is absurd. follow that, if which It will even infinite,

dx and

measure

or integral,

y is not x

easy to

and y

next, y

be " 1. As real numbers. positive

Take

constant, dyjdx must are

must

do

so

likewise;for

to

x

for

exampley

passes from

#*, wlfiere

=

number

one

to the

value

of y corresponds one next to if y skippedthe number back to pick it up ; but come

every

Hence of #, and y grows as x grows. of its values,it could never any one

is among the values of y. Hence y that every real number If we be consecutive,and %/tte l. and y + dy must measure by is be when fixed t he distance must not stretches, distances, dy given, y we

know

=

and the distance dx when

but,since x and each

is the

is absurd. =

"1.

y

are

the

x same

is

1,y given. Now if x number, dx and dy must

distance to the next if we Similarly, Hence

-

number:

take for y

the admission

a

therefore

=

1, dyjdx be

8

;

equal,since

dyjdx 1, -

function,we decreasing

of consecutive numbers

-

which

shall find

is fatal to the

Infinityand

344

Calculus;and

since the Calculus

The

*r

and y

"variables.*"

discuss at the

that

notion

there

a

How

in time

Change

is

a

numbers

is embodied

topicwhich

is reinforced in

calling

shall have

we

to

has, undoubtedly, greatlyinfluenced

selves philosophyof the Calculus. Peoplepicturea variable to themoften unconsciously as assuming a series of values, successively might happen in a dynamicalproblem. Thus they might say: x can throughall intermediate pass from #T to arZ9 without passing "

"

as

change,which

stage,but which

later

maintained,the Calculus is

be

be consecutive

must

the idea of continuous

by

must

numbers.

fatal to consecutive

323.

[CHAP.XLI

Continuity

values?

it in this passage, must there not be a next value,which first leaving the value ^? Everythingis conceived on the

And

assumes

on

analogyof motion, in which a point is supposedto pass through all not this view of motion in its path. Whether intermediate positions or any rate it is irrelevant where a pointin the theoryof continuous series is concerned, since do

is correct, I

fundamental time

and

the

of properties

confirm from

such

that

decide

now

path of

views.

our

evident

seems

not

motion

series must For

at

:

be decided before

part,to

my

intensive

be continuous

both

must

return

appealingto

Cohen, I

to

magnitudeis

and series,

the

motion

to

confess,it

must

somethingwholly different alwaysbe

for the latter must

infinitesimal extensive

magnitude: therefore be of the and must magnitudes, in auy kind with them; while intensive magnitudesseem never same the metaphysical smaller than any extensive magnitudes. Thus sense be matically rescued seems, both mathetheory by which infinitesimals are to destitute of groundsin its favour. and philosophically, of 324. We cannot, then, agree with the followingsummary Cohen's theory(p.28) : That element in I may be able to positan and for itself? the instrument of is the desideratum, to which corresponds thoughtreality.This instrument of thoughtmust firstbe set up, in order

smaller than

finite extensive

"

to be able to enter

into that combination

beinggiven,which is

with

intuition,with the

completedin the

of of magnitude. This presupposition

sciousness con-

of intensive principle

is latent in all principles, reality and must therefore be made is independent.This presupposition the meaning of reality and the secret of the concept of the differential" What underlies the we can confusedly agree to, and what, I believe, above or

statement, is,that

terms;

the

Nor

AT can

but these,as

and we

dy

which

agree

science)"can physical not

for sum

every continuum have

we occur

that be

intensive

justseen,

must

will not

consist

that (i.e.

elements

fulfilthe function

in old-fashioned accounts

"this finite"

of

which

of

of the Calculus. is the

objectof

sive thoughtas a sum of those infinitesimal intenis realities, as- a definite integral" (p.144). The definite integral of elements of a continuum, althoughthere are such elements : a sum is not the example,the lengthof a curve, as obtained by integration, of its points, but strictly and only the limit of the lengthsof

322-324]

polygons.

inscribed

of

points

the the

points;

and

other.

There it

not

as

that

the

for

is

was

use

continuity

shown, of

be to

All

in

of its

suppose

the

as

lengths

are

consist

of

last

induction.

as

there of

they

all

finite

a

Part Hence

unnecessary,

belong,

the

to

III,

ratio

to

if

Calculus

erroneous,

of

involve

and

there

does

terms, an

as

each

And

consecutive

infinitesimals

bility divisi-

to

contradictions. be

must

i.e.

number

stretch; ;

of

sum

of

infinite

an

continuum leads

the

to

magnitudes

have

existence

Chapter

regarded

the

345

given

infinitesimal

an

series

every

be which

to

stretches

thing

etc.

can

class

stretches

element

an

that

be

all

such

mathematical must

length.

terminated

no

in

sense

logical

the

and

not

notion

only

its

not

it, and

require

which

two

any

would

The is

stretches,

of

arguments,

curve

itself,

curve

were,

the

Philosophical

mate illegiti-

explaining tradictory. self-con-

XLIL

CHAPTER

325. since

the

word

time

of

unit, and

exclusive

When

characteristic

we

considered

cla"s;

one

in

this assertion

as

having

so

far

class-concept, they whole

asj"ect the to

look

when

conjecture that

terms, of

relation

other

the

their

and

in

Now

I

which

they

be

denying

identityand

indeed

diversityin

a

they I

all

continuous of

the

mathematical

continuum it has

connection,

continuity,as whatever is to

be

order

to

disputes

may

allow 326.

in

another

be

at

seen

In

once

this

about

it

In

no

words the

futile, I

are

time,

of

their

signification but

confining ourselves

way *

with Outlier

common

to

the is the

the not must

habitual that

else. the

to

from

chapter,it discussed. I have quoted state definitelythat this is

themselves, for and

special relation

no

order.

to

everything

to

as

the

of in in

this order

discrete?

this

tion opposi-

fundamental

a

of

philosophy. study of the But beyond this general mathematical meaning of it has

fact that

mathematical

reference

no

meaning

that

philosophic meaning only in since in here question ; and ask philosophers to divest associations

obtained

preconceptions.

with

Cantor's

from

the arithmetical

Logic, " 100, Wallace's

Many

class-concept,

that

"

constitutes

"

but

;

is called

strongly hold

is

members

another, and

compose

collection

be

instance

the fundamental problem Logic perhaps even problem And the it is relevant being fundamental, to certainly of

"

:

an

from

is called

which

"

the

at

number,"

following

must

differentinstances

whole

units.

remember

we

cardinal

"

merely

each

are

compose

must

aspect the

far from

am

of

number,

indistinguishableone

are

maniness, they this

a

these

characteristic,

When

the

to

amounts

cardinal

they

as

sources:

self, or

to

at we magnitude; but in is it Discrete it, magnitude.1" implied and in Hegel, both mean magnitude, quantity

may

two

equalizationof

or

One

that

has

saw,

explicitby abstraction, quantity

made

selfsameness

of

we

as

its immediate

at

Continuous the

philosophers,especially totally unlike that given to it by among

identification

the

look, therefore,

we

borne

continuityhas

Hegel, a meaning Hegel says*: "Quantity,

Thus

Cantor. the

THE

CONTINUUM.

THE

OF

PHILOSOPHY

THE

continuum, Of

the

Translation,

the

we

arithmetical

p. 188.

word,

definition. conflict con-

The

325-327] timmni, M. is

Poincare

nothingbut

PMfosophy of the

justlyremarks*:

"The

collection of individuals

a

Continuum

347 conceived

thus

continuum

arrangedin

a

order,

certain

infinite in number, it is true, but external to each other. This is not in which there is supposedto be, between the the ordinary conception, of the

elements

continuum,

of them, in which

the

sort of intimate

a

point

is not

bond

priorto the

makes

which

whole

a

line,but the line

to

the

is unity in multiplicity, point. Of the famous formula, the continuum alone subsists, the multiplicity the unity has disappeared."" It has always been held to be an the open questionwhether continuum is composed of elements ; and -even when been allowed it has to contain elements, it has been often allegedto be not composedof these. This latter view was maintained even by so stout a supporter of elements in everything Leibniz f. But all these views are as only continua those of space and time. in regardto such The as possible is an objectselected by definition, of arithmetical continuum consisting in

elements at

leabt

virtue

of

the

VI

in Part

arithmetical continuum. theories

in

a

known

be

to

embodied

that

The

of space

constructed

and

spaces

chief

reason

time

has by philosophers,

composed of

continuum

and

afford other

for the elaborate their

been the

elements.

of the

instances

and

which continuity,

The

thesis of the

that spatio-temporal con! inuitymay possibility In this argument, I shall assume as proved the thesis chapter,that the continuityto be discussed does this

In

327.

have

present

be of Cantor's

the

of actual

doxical para-

supposedcontradictions

contradictions. is free from chapteris,that Cantor's continuum be firmlyestablished, before we can thesis,as is evident, must

admission

in

instance,namely the segments of the rational numbers.

one

I shall maintain

been

and definition,

of the not

This aDow

kind.

preceding

involve

the

infinitesimals.

world, nothing is capricious

than capricious

more

most posterity^lack posthumousfame. of judgment is the Eleatic Zeno. Having invented four arguments, all immeasurablysubtle and profound,the grossness of subsequent and his ingeniousjuggler, philosophers pronouncedhim to be a mere years of arguments to be one and all sophisms. After two thousand the continual reinstated,and made refutation,these sophisms were who foundation of a mathematical renaissance,by a German professor, between himself and Zeno. of any connection dreamed probablynever has at last shown bv strictly Weierstrass, banishingall infinitesimals, that we live in an unchanging world, and that the arrow, at every The of its flight, is trulyat rest. moment only point where Zeno did he erred in infer)that, because there was inferring(if probably is no state at one change,therefore the world must be in the same This time as at another. follows,and in by no means consequence

One

*

Revue

f See

de

The

of the

notable

victims

of

Mttaphyxique et de Morale, Vol. i, p. 26. Philosophyof Leibniz, by the present author, Chap.

ix.

and Infinity

348

[CHAP.XLH

Continuity

constructive than ,the ingenious is more professor in mathematics, Greek. Weierstrass, beingable to embody his opinions of common where familiarity with truth eliminates the vulgarprejudices the respectable air of sense, has been able to give to his propositions the lover of reason to if the result is less delightful platitudes ; and

this

point the

than the

German

bold

Zeno^s

it defiance,

Zeno's

arguments

translate them,

instructive to

with

concerned specially

are

they stand,relevant

as therefore,

calculated to appease

more

mankind.

of academic

mass

is at any rate

to

it is

arithmetical

into possible,

as

not

are

But

presentpurpose.

our

far

so

motion, and

language*. -The

328. no

first

motion, for what taken

have

That

place,this

another, and

so

on

motion, and

another

motion. idea of any assigned mere arithmetical form, but it appears an

Consider between an

variable

a

is

which

x

capableof

assignedlimits,say

two

infinitewhole,whose

0

before

course we

to

assume

this

in

turn

and

is

then

far

plausible.

(or rational)values

all real

class of its values

The

1.

less

logically priorto it : for it has

partsare

is

endless regress an be put This argument can

there

infimtum. Hence

in the into

motion

say, whatever

is to

presupposes

ad

of its

reach the middle

must

moves

end."

the

it reaches

"There

argument, that of dichotomy,asserts:

is

parts,

subsist if any of the partsare lacking.Thus the numbers from 0 to 1 presuppose those from 0 to 1/2,these presuppose the numbers and it cannot

from

0 to

1/4,and

in the

regress

so

notion

wholes, real numbers which

appliesto

on.

Hence, it would

of any camiot

infinite

an

infinite whole

be

but without

;

defined,and

series,breaks

there is

seem,

an

such

arithmetical

infinite infinite

continuity,

down.

argument may be met in two ways, either of which, at first of which are but both sufficient, sight,might seem reallynecessary. two kinds of infinite regresses, of which one First,we may distinguish is harmless. kinds of whole, the two Secondly,we may distinguish collective and the distributive, and assert that, in the latter kind, This

parts of equal complexitywith

the whole

logically priorto it. These two pointsmust be separately explained. An infinite regress may be of two kinds. 329. In the objectionable kind,two or more propositions jointo constitute the meaning of some there is one at least whose meaning proposition ; of these constituents, is similarly This form of regress compounded ; and so on ad irifinitum. result* from circular definitions. Such definitions may be commonly *

Not

behiga

reallydid

say

or

derived from the Zenon

Greek scholar,I pretendto no The form of his four mean. article of M. interesting

merely a

are

text

not

first-hand

arguments

as authority

which

to

I shall

what Zeno

employ

is

et les arguments de Noel, Le mouvement de Morale, Vol. i, pp. 107-125. These in any well worthy of consideration, case and as they are, to me, for discussion, their historical correctness is of littleimportance.

d'Elee/'J"svue

arguments

are

de

Mttaphyyiqw

"

et

The

327-330] expandedin

developedfrom

are

defined will reappear,

be

to

example the contain

they is not

part of

identical

an

idea

an

349

is

are

If

part."

idea, then, in

an

definition must

of ideas occurs, wherever the meaning of

said to

similar; and

have

ideas

the

Take

for

idea

same

similar when

are

have

idea may

an

definition is not

a

the

definition will result.

no

peopleare

ideas which

idea,such

an

and

following:"Two

have

they

when

Continuum

analogousto that in which continued fractions quadraticequations. But at every stage the term

manner

a

Philosophyof the

part which

a

objectionable logically ; but if the second placewhere identity

be substituted

;

and

is in question, an proposition which reach a proposition never a

so

Thus

on.

infinite regress

has a definite since we objectionable, meaning. But many infinite regresses are not of this form. If A be whose and A impliesB, a definite, proposition meaning is perfectly have B impliesC, and so on, we infinite regress of a quiteunobjectionable an This depends upon is kind. the fact that implication a syntheticrelation,and that, although,if A be an aggregate of A impliesany proposition which is part of A^ it by no propositions, is part of A. Thus follows that any proposition which A implies means there is no the there was in as logicalnecessity, previouscase, to infinite the before A complete acquiresa meaning. If,then, regress it can be shown that the implication of the parts in the whole, when is

the whole

is

infinite class of

an

Zeno's

regress suggestedby its sting. In

330. wholes

from

order

which such

as

numbers, of

argument

show

is of this latter

dichotomy

will

kind, the have

lost

that

are

this is the case, we must distinguish defined extensionally, i.e. by enumeratingtheir terms,

are

defined

to

relation to

I.e. as intensionally,

the class of terms

having

simply,as a class of class of terms, when it forms a whole, is merelyall a terms havingthe class-relationto a class-concept*.) Now an extensional whole is necessarily extend at least so far as human finite: powers than of finite enumerate number cannot more we a parts belonging given terms. (For

some

given term,

some

or,

more

"

"

to

a

this must of parts be infinite, be known But this is precisely enumeration. what a class-

whole, and if the number

otherwise

by

than

concept effects:

the terms

partsare

of

a

is specified ; and class-concept any the class belongs,or does not belong,to

defined when either

whole whose

a

the

class is

completely

definite individual in

question.

An

and class is part of the whole extension of the class, is logically ; but the extension priorto this extension taken collectively

individual of the

itself is definable and

And

subsists as to

a

without

reference

genuineentityeven

say, of such

a

of

For

say that, though any finite number the impossible established without

these terms

which, again,may proposition *

when

individual, specified any the class contains no terms.

to

is that it is infinite, class,

it has terms, the number a

any

statements, v. precise

be

supra,

to

is not

Part I, Chaps, vi and

"

x.

and Continuity Infinity

350

And

all finite numbers. process of enumerating of the real numbers between 0 and 1. case

whose

is known

meaning

soon

as

as

this is

They form what

know

we

[CHAR XLII

a

the precisely definite class,

is meant

by

real

of the class, members and particular not logically the smaller classes contained in it,are priorto the class. Thus the infinite regress consists merelyin the fact that every segment

number, 0, 1, and

of real these

not

parts are

parts which are logically priorto it, and

harmless. Thus perfectly theoryof denotingand the The

one

which

the

**will

says,

first reach

intensional definition of first argument

second of Zeno^s

331.

be

never

overtaken

point whence

the

the

slower,"it

this argument is with to be concerned

is

When

seen

infinite classes. If

two

this

it appears in Arithmetic. is the most famous: it is

ahead."

it language,

correlation of

one-one

class. With

the swifter,for the pursuer must that the so fugitiveis departed,

by the

translated into arithmetical

a

the tortoise. "The

remain alwaysnecessarily

slower must

as

arguments

Achilles and

concerns

infinite regress is lies in the difficulty

the

solution of the

the

to ZenoV

is made

answer

an

again segments ; but

has

numbers

rational

or

The

Miccen.

Achilles

to

were

of the tortoise would be part tortoise,then the course of that of Achilles; but, since each is at each moment at some point overtake

of

the

his course,

the

simultaneityestablishes Achilles and

of positions

from

this

that

tortoise. Now

of the

it follows

tortoise,in any given time, visits just as many shall conclude it is hoped we does; hence so

the

Achilles

placesas

those

correlation between

one-one

a

"

"

that the tortoise's path should be part of that of impossible Achilles. This point is purelyordinal, and may be illustrated by Arithmetic. Consider, for example,1 H- "r and 2 -f "r, and let x lie

it is

between and

one

0

1, both

and

only one

value

from 0 to 1, the number the number assumed as at

8,

half

while % +

x

inclusive. For of 2 -f "r, and

vice ver#d.

of values assumed

by

" -f x.

But

as many been resolved, as

-f x

by

as

there is x

grows

will be the

started from

3.

at

Hence

1 4- "r

1 4- "r

started from % and ends

values of "

value of 1 -f "r

each

Thus

same

1 and

there should

ends be

This very serious difficulty Cantor ; but as it belongsrather

of 1 -f "r.

as

have seen, by of the infinitethan to that of the continuum, I leave to the philosophy its further discussion to the next chapter.

has

332.

The

is in rest is has

we

third argument is concerned or

always in the

usuallybeen

in motion

the

If thing everyand if what moves a space equalto itself, is immovable." This in its flight arrow "

arrow.

in

instant,the

thought

with

so

deserve serious discussion. To

monstrous

my

mind, I

a

must

paradox

as

confess,it

scarcelyto seems

a

very

plainstatement of a very elementary fact,and its neglecthas, I think, caused the quagmire in which the philosophy of change has long been immersed. In Part VII, I shall set forth a theoryof change which may be called #2o^c, since it allows the justice For the of Zeno's remark.

The

330-333] wish present^I

to divest the remark

namely : be

take

very

are

And

here

may conceptof

variable is a fundamental

change. We shall widelyapplicable tude, plati-

very

1/3,which

or

be inserted

with

difficultiesof logical has for nor

If

1, all the values

0 to

x

it

all absolute

are

concerningvariables.

of

dailylife. Though class, nor a particular

as logic,

it is not the class, of the class, member nor yet the whole class,but any member other class. On the hand, it is not the concept " any member class,"but it is that (or those)which this concept denotes.

it is

alwaysconnected

l/"

as

few words

a

351

variable is a constant."

take all values from

can

definite numbers, such

constants. A

a

variable which

a

can

"

Continuum

of all reference to

importantand value of a Every possible

it is

find that

then

Philosophyof the

some

of the of the On

the

this

I need not now enlarge;enough conception, this subjectin Part I, The usual x in Algebra, been said on number, nor for all numbers, example,does not stand for a particular This may be easily seen yet for the class number. by considering

identity,say

some

does certainly

This

not

what

mean

it would

substituted for j% be a true would

that the though it implies does Nor proposition. for JT the class-concept number substituting

become

result of such it

formed

the various

by

roughlycorrect*. and

each

the

;

of Zeno^s

essence

It denotes

disjunction be taken

may

that

contention

the

as

disjunction ;

simplelogicalfact

This

this

1 to

concept any

the

of the

then the terms

are

constant.

a

add

the

least this view

at

or

values of "

results from

cannot

denote

not

be added.

numbers

is

does

x

cannot

The

of these

constitute

reason,

we

were

substitution

a

what

mean

for

^

concept. For the same number-, to this,too, 1

if,say, 391

is

arrow

to

seems

always

at rest.

333.

But

to applicable

is such

a

Zeno's

argument contains In

continua.

thing as

a

are

which variability

to

belongsto

that all the values of

In the

extend

variable

are

that

there

which

may

take all real values from

are

see

variable.

The are

one

\ve

there

continuous

a

infinitesimals. For rulues once

of

it is

variable

a

finitesi inthe

firmlyrealized

easy to see, by and such values,that their difference is always finite, a

hence

the

of

specially

that

constants, it becomes

two

we values,

When

is

it denies

case general

the

to

it alone.

takingany

these

motion,

denying actual

as

attempt

an

of

case

state of motion.

be taken

variable,it may

the

which

element

an

infinitesimal differences. If

no

be

a

variable

1, then, takingany two of that their difference is finite, tinuous althoughx is a con-

It is true

the

cho^e; but if it had

lower limit to

x

0 to

might have

difference

been, it would

differences is zero, possible

finite ; and in this there is no *

See

shadow

Chap, vni,

been

stillhave

been

finite.

but all possible differences

of contradiction.

esp.

less than

" 1)3.

This

static

and Infinity

352

theoryof

[CHAP.XLII

mathematicians,and

variable is due to the

the

its absence

suppose that continuous changewas impossible which involves infinitesimals and the contradiction state of change,

day led

in Zeno's without

Continuity

a

of

a

to

it is not.

body'sbeingwhere

of the

arguments is that

last of Zeno's

The

334.

him

This

measure.

is

closely analogousto one which I employedin the precedingchapter, againstthose who regarddx and dy as distances of consecutive terms. M. Noel pointsout (loc.tit. p. 116), against It is onlyapplicable, as the previous those who hold to indivisiblesamong stretches, arguments of the infinite refuted bility. divisipartisans being held to have sufficiently We

are

now

to

set of discrete moments

a

suppose

in the fact that at one motion consisting places, in another at another. of these discrete places, one lines composedof the Imaginethree parallel pointsa, b, c, d; a',V, c\ J'; a", 6", c", df' respectively. Suppose the second line,in one all its points to the left by one to move instant, them all one while the third moves place place, to the right. Then althoughthe instant is

moment

or

rather,it is this argument togetherwith

instance

in which

functions

of #, and let dyjdx

the

the

that principle

argument in Zeno's For

if instants

a"

and

indivisible

are

dr

c *

ff

*

/f

^

f

7

.

abed ....

rf y

cr $

....

a" b" c" d!1 ....

put

thus

:

Then

Let

"/,

z

be two

("/"#)=2, which

v-

the value of every derivative must be + 1. form, M. Evellin,who is an advocate of

indivisible stretches, that replies at all*.

o

an

1, dz/dx="l.

=

contradicts To

dy\dx

be

It may

%.

"

is in

^ ^ ^ ^

which

that virtually

body

a

a

by hyp. argument I proved, in the preceding that,if there chapter, consecutive terms, then dyjdx= "1 always; are is

discrete

abed

indivisible, over c',which was c",and is now 6" have must over a", passed during the incontra stant; hence the instant is divisible, This

and

"

V

and

do

not

cross

this is the

each

other

hypothesis all "

instant a is over a",in the next, c is over a'. say is,that at one Nothing has happened between the instants,and to suppose that a!' we

and

can

V

crossed is to

have

the

questionby a covert appealto the of continuityof motion'. This replyis valid,I think,in the case be motion; both time and space may, without positive contradiction, held to be discrete,by adheringstrictly to distances in addition to stretches. Geometry, Kinematics,and Dynamics become false ; but there is no ^reason think them In the to of true. case very good the is of matter since Arithmetic, otherwise, no empiricalquestion existence is involved.

*

Revue

de

beg

And

in

this case,

Mctapky"iqueet

de

as

we

Morale, Vol.

see

from

i, p. 386.

the

above

The

333-336]

Philosophyof the

Continuum

Zeno*s argument argument concerningderivatives,

Numbers

entitieswhose

are

numbers,

and

among be cannot

denied

a

have

We

attacks

the

space, time,

now

this

For

numbers

motion.

or

that Zeno's

seen

the

reason

with

arguments, though they prove

not

the

on

conducted with any

therefore,to make

continuum

new

be called

by

have

not,

far

so

powerfulweapons. generalremarks. Cantor givesthe name of or

few

a

notion to which

The

contradiction. positive

prove that the continuum, as we have become Since his it,contains any contradictions whatever.

with acquainted

day

with

greatdeal,do

very

various forms

the

without

in connection

335.

be

can

sound. absolutely established beyond question; of continuity which occur is

is better discussed in connection continuity

problemof than

nature

353

as

It

more

other

in

know, been only remains,

I

continuum

of

may,

of the

and it dictionary, is open to every one to assert that he himself means somethingquite But different by the continuum. these verbal questionsare purely frivolous. Cantor's merit lies, not in meaning what other peoplemean, what he means but in telling himself almost unique merit,where us an and generally, a continuityis concerned. He has defined,accurately ordinal and notion, free,as we now contradictions, purely see, from sufficient for all Analysis, and This notion was Dynamics. Geometry, in it known not m athematics, was though exactly presupposed existing that was it was what exampled presupposed.And Cantor, by his almost unhas successfully lucidity, analyzedthe extremelycomplex of nature series, spatial by which, as we ^haH see in Part VI, he has revolution in the philosophy of space and motion. rendered possible a salient points in the definition of the continuum The are (1) the

course,

any

name

or

out

"

the doctrine

with

connection

segments. These

long been

had

which

both

continuum

both

is

a

there their not

an

divisible into

elements.

if

But

in the

of Part

sense

has that

no

elements.

it must

cases

is there

for consecutive

what constituting ordinal If

we

a

if are

our no

continuum

stretch

now

see

continuum

at any indivisible,

In this

terms

may

We

be

to

sense

togetherwith called (though

element,then, in this take

consist of at least two

then likewise there distance, of these

as

consecutive

an

Every

of the continuum.

take

we

IV) an

stretches;and elementary

demand

terms

new

antinomy antinomythat the

consist of elements.

does not

relation asymmetrical

serial, so

the

mean

be said,though in different senses. of terms, and the terms, if not consisting

not

are

continuum

scandal,I

open

does and

segments resolves

may

series are

illuminated.

denial of infinitesimal

The

336.

rate

two

becomes subject

of the

that

limits, (") the denial of infinitesimal pointsbeingborne in mind, the whole philosophy of

be

sense,

our

essentially

terms, then there are in which there be one

elementarydistances.

But

no

is

in neither

logical ground for elements. Hie slightest in Part III,from an terms saw we as springs^

the

illegitimate small

distances

III,

Part

are

ones

distances need

not

cause

the

free

from

of

is

logical

continuum,

a

infinite

ones,

exist

at

kind,

and

as

I

we

small

there

lack

are

of

elements

antinomy

able

to

no

smaller

to

greater

in

saw

presuppose

the am

XLII

distance,

as

where

the

Hence

least

all,

not

from

regress

inconvenience. far

do

may

harmless

the

so

is

solved, re-

discover,

is

contradictions.

inquire

to

infinite "

close.

the

Thus

remains the

concerning to

any

and

only

It

all.

stretches

or

wholly

come

at

they

magnitudes,

intensive

regards

as

but

distances

large

And

simple.

large

than

simpler

no

And

induction.

mathematical

are

alike

being

ones:

smaller

of

use

[CHAP.

Continuity

and

Infinity

354

an

whether

inquiry

with

the which

conclusion

same

this

Fifth

holds Part

will

CHAPTER

THE

compelled been

to

of the

In

who while of

one

the

the

the

can

must

argument

did

don't

of

notion

the

in

that not

be

may

know

the

in

it takes the

know

what

what don't

we

hear

of

anything. them.

and

is, it

form; the

on

is

is for

Kantian

supported

therefore

without

this

By

was

argument

telegraph wires, because

near

fact,

In

hence

we

can

prove

conceivable

it remains

But

or

antinomy

view

and

count,

know. ;

except

to

of

know

that

"

first

This

time.

in

number

we

The

is irrelevant,

time

not

whether

essentiallytemporal

an

always happen

should

antinomies,

preceding chapter,

infinite

number.

infinite

done

have

To

question

the

in

actual

an

it.

mathematical

the

resolved

was

schematized

battles

we

we

in

rule, thought

a

as

necessity of

the

that time

unproved. other

\vith the tortoise

continuum

with

to

do

he

gives the

Leibniz, whole

examined

collection

concerned

as

of

and

alarm have

shortly.

And

of

a

seen,

part, which

as

for

made

as

he

Hegel,

a

underlies

"

is

we

cries

the

Achilles.

and

Achilles

"

which

than

is

the

to one-one

This

be

the

perhaps

that

any

zvolfso often

finallycease

contradiction

in connection

scarcely relevant

fundamental

more

contradiction

gives

Pcarmemdes

Plato's ever

difficulties

underlies

which

paradox

the

antinomies

with

infinity.

we

and

;

examined

already been

has

philosophers,Zeno

will be

best

being

of

be

know

not

that

generally

Of

there

infinite,but

that

remains

Of

therefore, this antinomy

prove

they

of Kant.

elements,

that

the

could

not,

with concerned, essentially,

numbers

we

infinity have

to

precise contradictions

question

with

that

time

of

found

be

can

has

mathematics

wish, leaving

I

contradiction

merits

has

the

to

Arithmetic, view

is

supposition

the

concerned

the

chapter,

present

objected

continuum

reduced

we

the

great

not

if

been

have

we

points that there purely philosophical treatment

opportunity

any

infinite

the

of

mathematical

many

exhibit

to

second, which

we

discussions

for

have

the

by

INFINITE.

THE

infinite.

it worth

on

so

whether

inquire

Those

is

into

go

question.

aside, to

OF

previous

our

adequate

no

the

so

PHILOSOPHY

IN

337.

XLIIL

that

here, have when

disturbed. correlation

is, in

fact, the

[CHAP.XLIH

and Continuity Infinity

356

only point on

which

follows I shall

put

mathematical

the

In

turn,. arguments against infinity

most

arguments in and

knowledge;

this will

adapted to

form

a

preventme

from

what

present

our

quotingthem

from any classicopponentsof infinity. the positive finite Let us first recapitulate 338. theoryof the inbriefly the indefinable led. notion have which been as to we Accepting and the notion constituent proposition in which a is a a proposition by ""(a)

of

a

may denote then transform

we proposition^

We

constituent.

can

is any proposition where "$"(x) ""(#), other object fact the that some if at all, from "fr(a\ onlyby differing is what we called a propositioned function. appears in the placeof a ; ""(#) values of x and that ""(#)is true for some It will happen,in general, a

into

variable "r,and consider

a

is true, form what All the values of x, for which "j"(x) function class defined by $(x); thus every propositibnal

false for others. called the

we

defines

and class,

a

the

actual

of the

enumeration

members

of

class

a

enumeration is not necessary for its definition. Again,without we can similar when classes of classes the two two define : similarity u9 v are there is a is

a

relation R such that

one-one

to which

v

there is

has

x

which

the

relation

"

is

x

R? and

has the relation R

"

alwaysimplies there "y is a v" always implies a

"

u

y? Further,R is a one-one xRy, xRz togetheralwaysimply that y is identical with 2, and xRzy yRz togetheralwaysimplythat x is identical with y ; and is identical with y** is defined as meaning "every propositional x function which holds of x also holds of y? We define the cardinal now "

a

u

to

relation if

"fc

of

number and

every

a

class

as

u

class has

a

the

cardinal

function of propositional a

The

definition of

above

the notion

based upon

number,

", if

of its cardinal

member

class of all classes which

a

v

be

since "w

variable.

similar to

are

is similar

z"" is

to

Moreover

u

; a

itself is

u

number, since every class is similar to itself. cardinal number, it should be, observed,is

of

functions,and nowhere involves propositional there is to suppose that there enumeration; consequently no reason will be any difficulty the of classes whose numbers as terms regards be counted in the usual elementaryfashion. cannot Classes can be divided into two kinds,accordingas they are or are not similar to of themselves. In the former case theyare called infinite, proper parts in the latter finite. Again, the number of a class defined by a profunction which is always false is called 0; 1 is defined as positional the number of a class u such that there is a term x, belongingto w, such that "y is a u and y differs from x" is alwaysfake; and if n be any number, n -f 1 is defined as the number of a member that the such a function a?, propositional differs from w-f

1

obtain

number

defines

x"

differs from a

a

class whose

number

is

class

"y n.

is If

which

u

a

u

n

is

if not, not. In this way, startingfrom of numbers, since leads to number n progression ";

any

n-f 1.

It is

easily proved that

all the numbers

has

and

y

finite, 0, we a

new

belongingto

The

337-339]

Philosophyof the Infinite

which progression

the

is to

different; that

from

starts

say, if

1 and

is

numbers

finitenumbers. be

can

n

that the number

But

there

obtained

so

of the

it is

finite numbers

different numbers

numbers,by

different

are

proceedsfrom

7t4-l be not

of this

terms

there is direct

more,

why

term

n

n-fl

1, or

induction

are

1 and

if

;

", and

n

proof fails;and what is the contrary. But since the previous

mathematical

the theorem

proof a

the progression,

proof of

proofdependedupon reason

of mathematical

means

be

and

and

0

all

that

belongingto

not

proof that

the fact that

be

one-one

formal

cannot

number

A

axe

defined is the

capableof

themselves

a

think

to

reason

no

of finite numbers. progression is called infinite.The progression

this

have

cannot

so progression

is

indeed

;

terms

the

in

generatedin

this way and m belongsto this progression,

class of n of its predecessors, a any one with of The terms. correlation m one series of

357

should

induction,there is extend

the

not

slightest

infinite numbers.

to

Infinite

be

like finite ones; by the decimal system expressed, be distinguished of notation,but they can by the classes to which they apply. The finite numbers beingall defined by the above progression, numbers

cannot

has terms, but not any finitenumber is number. This the positive infinite theoryof

if a class

of terms, then it has

u

That

339.

there

an

infinity.

infiniteclasses is so evident that it will scarcely

are

be denied. well to which 1

is

is,or

two

it may be as Since,however,it is capableof formal proof, in the Parmenzdes^ proofis that suggested prove it. A very simple

:

follows.

as

has

hence there is

that 1 is not of numbers form

a

n

from

class which

1 is not

"

a

it be

number

the

2 ; and

a

is

finite

new

Thus

;

prove numbers

n

is not

is

more

infinite.

better

A

fact that,if

n

has

the number and the

Hence

n.

all contained in the class

are induction,

thingswhich are not the number of finitenumbers. is reflexive for classes, of similarity every class has a class of finite numbers

proved

of finite numbers;

number

are

is the number

n

Since the relation

of

the

Being

have

with Being together

that

of finite numbers,no

finitenumbers, by mathematical

that

Then

1.

1 and

Formally,we

number, so

1 is not, the

number

we

number

a

But

Being.

these

that

there is

on.

so

of numbers

n, and

1 to

has

grantedthat

therefore there

the number

of finite numbers. if

Let

Being,and

number

;

therefore

is which, not being finite, the the above, is derived from

number

a

analogousto proof,

from

of numbers

be any finitenumber, the number

0 up

it follows that n is not the number is n + 1, whence n including of numbers. by the correlation of whole Again,it may be proveddirectly, and part,that the number of propositions or conceptsis infinite*. For

to

and

of every term or conceptthere is an it is the idea,but again a term or

every term *

or

conceptis an

Of. Bokano, Paradoxien

tollen die Zahlen

? No.

("(".

idea. des

idea, different from

concept.

There

are

" 13 Unendlicheit,

On

the

that other

of which

hand,

not

tables,and ideas of tables ;

Dedekind,

Was

find und

wa*

;

and Continuity Infinity

358

numbers, and ideas of numbers relation between there is an

Hence

340.

that possibility

The

number

of terms

Zeno's

Achilles for

term

travel

along the

which

is behind

path, the

of all the

of

terms

catch

never

can

with

whole

a

is here in therefore,

sense,

and

if it

:

of

plight ;

a

have,

relating cor-

part. Common-

it must

choose between

I do

and the paradox of Cantor.

the

one

reciprocal correspondence

all the terms

very sorry

a

points

other, the

should

did, we

has

correlated

material

followingthe

one

up

if two

same

also

be

cannot

part

a unique and positions,

simultaneous

oppositeview

strictly follow that,

same

terms.

among

of ideas*

the

that

for if whole

term, it does

one-one

part may have the confessed, shockingto common-sense.

shows ingeniously

shockingconsequence*;

a

and

whole

be

is,it must

there is

only some

are

and

of terms

infinitenumber

Thus

on.

so

but ideas ideas,

and

terms

and

;

[CHAP.XLin

not propose to paradoxof Zeno it face the of in proofs, ought to commit help it,since I consider that, suicide in despair.But I will give the paradoxof Cantor a form resembling Tristram that of Zeno. Shandy,as we know, took two years and lamented that, of the first two days of his life, writingthe history at this rate, material would

he

that

it,so

could

to

come

never

faster than

accumulate

end.

an

Now

he could deal with I maintain

that, if

if his lived for ever, and not wearied of his task, then, even it began,no life had continued as eventfully as part of his biography

had

he

would

have

is

correlative strictly Tristram Shandy. In

of this

cases

formal.

kind,no

care

in rendering is superfluous our the Achilles and

I shall therefore set forth both

Shandy in

strict

of Achilles

;

the

arguments

the Tristram

logical shape.

of the tortoise there is one (1) For every position

I.

show,

be called for convenience

the Achilles,may

to

I shall

paradox,which, as

This

unwritten.

remained

for every

of Achilles there is position

and

and

one

only one onlyone of

the tortoise.

(2)

Hence

number

same

(8)

A

occupiedby Achilles positions series of positions occupied by the

has the

the series of

of terms

as

the

part has fewer terms

than

a

whole

in which

tortoise.

it is contained

and with which it is not coextensive.

a

(4) Hence the series of positions occupiedby the tortoise by Achilles. occupied proper part of the series of positions II.

year the events of a day. series of days and years has no last term. written in the wth year. events of the "*th (lay are

(1) Tristram Shandy writes

in

a

(2) The (8) The (4) Any assigned day is the ;ith,for Hence

(5) * some

It is not

mind

is not

any

assigned day will

a

suitable value of

w.

be written about.

necessary to suppose that the ideas of all terms it is enough that they are entities. ;

exitf,or form

part of

Philosophyof the Infinite

The

339-341] (6) Hence (7) Since

will remain part of the biography

no

there the

happeningand

of

us express both this purpose, let u be

and writing,

these

unwritten.

correlation between

one-one

the part have

Let

variable which

a

times

and

the whole latter,

is

the

the

former

number

same

359

the

times

of

part of the

are

of terms.

as paradoxesas abstractly possible.For

series of any kind, and let x be a take all values in u after a certain value,which we

can

compact

a

be a one-valued function of #, and x a one-valued Let"/(^r) Then function of f(x) ; also let all the values of f(x) belongto u. the the following. arguments are I. Let/XO) be a term preceding 0 ; letf(#)grow as x grows, i.e.if xPxr (where P is the generatingrelation), Pf(x), Further lety*("r) take all values in u intermediate between any two values of f{x). lety(,r) value a of "r, such that 0 P ", we have f(a) a, then If,then, for some values of the series of f(x) will be all terms from /(O) to a9 while that of x will be onlythe terms from 0 to a, which are a part of those from correlation, f(0) to a. Thus to suppose f(a) a is to suppose a one-one will call 0.

=

"

for term, of whole

term

and

part, which Zeno

and

common-sense

nounce pro-

impossible. II.

Let

as uniformly

if

Then such

be

f(x)

a

function which is 0 when

grows, our takes all values after

x

x

does

values,so

identical with

x.

that of the

is

0, and which grows

there is measurement. if f(x) takes all

and 0, so does/*(,r); class of

The

x

in which

series beingone

other.

values of the if at

But

any

is therefore

one

time

the

value

of

x

is

greaterthan that of f(x\ since f(x) grows at a uniform rate,x will alwaysbe greater than f(x). Hence for any assignedvalue of x, the class of values of of

x

were

from a

0

to

Hence

x.

f(x) is a proper part of the values might infer that all the values of f(x)

0 \"

f(x) from we

part of all the values

proper

of

x

;

but

this,as

we

have

seen, is

fallacious. These two may in a

correlative. Both, by reference to of limits. The Achilles proves that two

paradoxesare

be stated in terms

segments, variables

side, approachequalityfrom the same have a common cannot limit; the Tristram ever Shandy proves that and variables which two start from common a term, proceedin the and more, but diverge same more direction, may yet determine the same a segment, because limitingclass (which, however, is not necessarily defined as having terms beyond them). The Achilles segments were which series,

continuous

and deduces a paradox that whole ,and part cannot be similar, ; and the other,startingfrom a platitude, deduces that whole part may is most this be similar. For a it must be confessed, common-sense, assumes

unfortunate 341. must

be

Tristram

state

There

is

of no

things. doubt

which

is the

correct

course.

rejected, being directlycontradicted by since it does not Shandy must be accepted,

The

Achilles

Arithmetic.

The

involve the axiom

and Infinity

360 whole

be similar to

that

the

have

seen, is essential to the

doubtless

cannot

to agreeable

very

[CHAP.XLm

Continuity part. This

the

axiom,

proof of the Achilles ; and it is there is no But common-sense.

we

evidence

and its admission except supposedself-evidence,

for the axiom

as

axiom

an

leads to

is not

but only useless, perfectly precisecontradictions* The its and t here i n against rejection destructive, mathematics, positively of the chief merits of It is one is nothingto be set except prejudice. the result proved. As as to proofsthat theyinstil a certain scepticism of whole and part could be found that the similarity it was soon as not implausible for every fimte whole*,it became provedto be impossible could not be where the for that infinite to suppose wholes, impossibility In fact,as regards proved,there was in fact no such impossibility. dealt with in dailylife in engineering, the numbers astronomy, or accounts,even those of Rockefeller and the Chancellor of the Exchequer the similarity of whole and partis impossible ; and hence the supposition But the supposition is is that it rests alwaysimpossibleeasily explained. the that entertained inductive foundation than better no on by formerly black. that all of Central are men Africa, philosophers It may 342. be worth while, as helpingto explainthe difference between finite and infinite wholes, to point out that whole and part but of terms are capableof two definitions where the whole is finite, where the whole is infinite -f. only one of these,at least practically, such and such individuals, A finitewhole may be taken collectively, as A9 B, C, D, E say. A part of this whole may be obtained by enumeratingsome, but not all,of the terms composing the whole; individual is part of the whole. Neither the and in this way a single axiom

"

"

whole

parts need be taken

its

nor

by extension,i.e. by whole

the

of individuals.

enumeration

and

the parts may Thus know we class-concepts.

but each may classes,

as

be

both

On

be defined

the other

hand,

defined

by intension,i.e. by without enumeration that Englishmen of for is whoever an part Europeans; Englishman is a European, not vice versa. Though this might be established by enumeration,

are

but

it need

not

be

twofold

definition The

When

and disappears,

whole

and

of whole

established.

so

part

and the

we

have

we

by

to

infinitewholes, this

only the

both be

part must

is effected

come

definition

tension. in-

and the definition classes,

of the notions

means

by

of

variable

a

and of is

a

If a be a class-concept, individual of a an logical implication. term relation which we call the classhaving to a that specific If

relation. is

"

an

a

now

b be another

"^ implies

is said to

be

is

a

b"

part of the

class such

that,for all values of #, x then the extension qf a (i.e. the variable x) extension of ij. Here no of enumeration u

individuals is required, and the relation of whole and part has *

The

finite

beinghere

t Of.

" 330.

J

Peano,

See

RMsta

di

defined

by mathematical

Matematiea,vn,

or

to induction,

Formulaire,VoL

avoid

n, Part

no

longer

tautology. I.

The

341-343] that

now

say

and

a

there is term

such

term a y x' of the class

of

Infinite

finite

similar,is

are

such

a

the

where

parts

361 concerned.

were

say that there exists followingconditions: if x be

of the class b such

state

a

b

the fulfilling

relation R

one-one

a

it had

simplemeaning which

that

To

Philosophyof

to

that

some an

a,

if

y* be a b, there is x'Ry Although a is part of ", for the impossibility provedimpossible, xRy

;

that

.

thingscannot

be

to suponly be proved by enumeration, and there is no reason pose enumeration possible.The definition of whole and part without is the key to the whole mystery. The above definition, enumeration due Professor is which is that is to which and necessarily Peano, naturally For infinitewholes. to the example, primes are a proper part applied of the integers^ but this cannot duced It is debe proved by enumeration. if x be a prime,x is a number," and from if x be a number, it does not follow that x is a prime." That the class of primesshould be similar to the class of numbers because we only seems impossible whole and defined enumeration. rid As as we soon imagine by part

could

"

"

ourselves of this idea the

contradiction supposed

vanishes.

as regards" or cto,that neither importantto realize, has a number immediately it. This characteristic they share preceding with all limits, limit of the series is never for a by immediately preceded But series it is term of the in limits. which some sense o" logically any other the to finite ordinal for numbers limits, prior togetherwith o" presentthe formal type of a progression togetherwith its limit. When it is forgotten that G" has no immediate all sorts of contradictions predecessor,

It is very

343.

suppose n to be the last number before finitenumber, and the number of finitenumbers is n -f 1. For

emerge.

n

is

a

to say that

have not

o"

has

precededimmediatelyby one

there

:

is

1, ", 3,

.

.

.

i/,

.

.

.

,

In

which

next

none

then

fact,

to

o".

that, althoughthere peculiarity alwaysone next

the

after any assigned number, there is not Thus there seem to be gaps in the series. We

next

before.

another

of them

any have

Cantoris transfinite numbers is

;

is merelyto say that the finite numbers predecessor Though "*" is precededby all finite numbers, it is

no

last term.

no

o"

is infinite and

has

have

We

last term.

no

the series have

is infinite and series o", a"H-l,6"-h", G" + V, ...which equally This second series comes last term. though whollyafter the first, ...

has

no

immediatelysucceeds. This state of thingsmay, however, be paralleled by very elementaryseries, such as the series whose generalterms are 1 l/i/and " 1/vywhere v whollyafter the may be any finite integer.The second series comes and has a definite first term, namely 1. But there is no term first, What is necessary, of the first series which immediatelyprecedes1, there is

no

one

term

of the firstwhich

G"

"

"

in order that the second

should be

part of

a

the terms

some

series should

series in which

both

series any series which of our series without

can

are

come

after the

contained. be

we

call

an

ordinal

of by omittingsome the order of remaining

obtained

changingthe

If

is that there first,

[CHAP.XLHI

Continuity

and Infinity

362

whose series, the finite and transfinite ordinals all form one the series generatingrelation is that of ordinal whole and part among the various ordinals apply. If v be any finite ordinal,series of to which terms, then

; similarly every series of the type v are ordinal partsof progressions ordinal part. The relation an as type o" -f 1 contains a progression

the

transfinite ordinals all the

mathematical

existence)is

of

sense

not

would

deny o"

which, as

be have

we

this is

affirm that

to

leads at

seen,

to

once

+ l is the

admitted, o"

the series whose

"*",i.e.of

there is

The

344. and series, thus

usual

all series of

are

the be

that

notion

dismissed

the

infinite

such

as

series of

1

integers.

follows. easily

classes,and

self-contradictor

is

remains,however,

groundless.There

as

view And

from integers

infinite numbers, and

to objections

To

a

"

""

cluding ordinals in-

series of

of the

togetherwith the whole up to any finite number all the infinite hierarchyof transfinite numbers Hence

may

themselves.

last finite number

a

(in

"

definite contradictions.

type

terms

of

question,since

to

open

finite and

existence

The

series.

one

by the natural numbers type of order presented

is the

when

belong to

the

thus

and part is transitive anfl asymmetrical,

orduud

a

very

connected with the contradiction discussed in grave difficulty, This

does difficulty large infinite

very

follows.

not

classes.

infinite

the Briefly,

as

such, but

cardinal number, and the number that,if it be a class, when

the number

of terms

"* is greaterthan is easy to as

give an

possible.Such class of all

the

be

are

are

certain

classes

class of all terms, the

Thus propositions.

contain

greatest

no

is found u

state

to

it would

is

which concerning

valid proof that theyhave apparently the

as

greaterthan if a be any number, (what is equivalent),

of u^ or But there

".

of classes contained in

stated

be

difficulty may

given a proof* that there can this proof is examined, it

has

Cantor

the

concern

Chapterx. only certain

as

seem

or classes,

though

as

terms

many

class of all

it

Cantor's

assumptionwhich is not verified in the case of such classes. But when we applythe reasoning of his proof to the in question,we find ourselves met of cases by definite contradictions, which the one discussed in Chapter x is an examplef. The difficulty arises whenever entities to deal of all the class with we absolutely, try with any equallynumerous for the of such a or but class; difficulty of the totality view,one would be tempted to say that the conception of things, of the whole universe- of entities and existents, is in some or and way illegitimate inherently contraryto logic.But it is undesirable to adopt so desperate less as a measure long as hope remains of some proofmust

some

heroic solution. It may *

them

He

has,as

is not

t It at the

be

was

end

to beginwith, that observed, a

of

of

fact,offered

two

that

I discovered

this contradiction ;

matter

cogent. in this way

the class of numbers

Appendix B.

proofs,but

we

is not,

shall find that

a

similar

one

one

is

of

given

The

343-345]

Philosophyof

the

Infinite

363

of those in regardto which difficultiesoccur. one might be supposed, the number of numbers, we should Among finite numbers, if n were

as

have

infer that

to

be

would

number

no

the

or

less than is usually

(ifit be

fact)that

a

well-ordered

up

includinga, where a is infinite, that the number to suppose reason

there is no

The

number

may be the fact

of numbers arises from

contradiction

of individuals is greaterthan

the number

series.

to and

greatestnumber. greatest number, and no

the

there

is o^,,but this is also where 0 is any finite

denumerable

a

is the

of all numbers less than

and

that

so

of finite numbers. peculiarity

includingOe includinga$,

to applicable

and

a,

and

to

of numbers

the number

Thus

greatest of numbers,

But this is a to

up up

ordinal

any

the

was

at all.

n

of numbers

number

ordinal

1

"

of numbers

number

The

n

the number

of numbers.

classes with

discuss the

in which

cases

In the

345.

the

number

there

is

cardinals f.

of the

there difficulty,

no

are

the argument depends upon proofs*, between the one-one correspondence

a

We

ordinals of the second

that,when

saw

series of the

of ordinals

infinite number

causes

contradictions arise.

first of Cantor's

supposedfact that

ordinals and

numbers

it is very hard to deaL Let us first examine there is no greatestcardinal number, and then

which

proofsthat

Cantoris

the

class of

although the

But other

consider the

we

dinal car-

by any ordinal,an type represented

to correspond

which class,

cardinal

one

form

for

"

example,all

non-denumerable

a

collection,

singlecardinal "",. But there is another method of ordinal corresponds in which only one cardinal. to each correlation, This method the series of cardinals itself. In results from considering this series, to a", c^ to o" 4- 1, and so on : there is always a0 corresponds and onlyone ordinal to describe the type of series presented one by the the

to correspond

cardinals from

to

there

is

for

many

terms

a

a

that

For

terms.

and

0 up cardinal

I do

no

series must

be

an

It

ordinal,and

well-ordered series not

have

can

individual,and

must

be

point often overlooked)from every other term be different, because there are instances no individual is absolutely in the and unique, one.

and

But the

two

are

the

in

we

do not,

:

Consider

repeatedat

intervals

we

say

digitsin

repetitionour

series

a

This

individual.

same

the fact that

When

terms

a

class

no

grounds

any

see

that

decimal series is *

as

a

are

most

"

we

for either For

an

nature

of

of

term

every

b, d,

It must each

of

only

the

case

one

point is obscured

e, a,

of

our

where .

.

example,as

theorem

(a

individual: therefore not

are

that

.

,

is

by

series. terms

by presented

where

by correlation;

Mannichfaftigkeitelehre, p. 44. " 300. Chap, xxxvm,

"f-Cf. supra,

so

supposition,

series.

of the of

important

only obtainable

have

can

greater number

a

describe the terms rule,fully

forget the

that

different individual

a

two, and

series ay 6, c, d" a such a series, for

"

be assumed

to

seems

groundsagainstthe latter.

definite

see

every

part I do

my

of them.

one

any

that

there

is

is, the

do

terms

(not one-one) relation wish

for

which

terms

our

compounded of

those

in either of

But

back form

and

series original

of the

the

those

one-many

Hence the

to

series there

these

a

order*.

an

must

we

have

they

either go

must

we

have

which

or correlated,

are

pairs.

order, but

an

terms

to

genuine series

a

series in

have

themselves

not

[CHAP.XLIH

Continuity

and Infinity

364

if

complex

of the is

we

series with terms

correlated

repetition.

no

to a series of individuals, must Hence correspond every ordinal number it may be doubted whether Now each of which differsfrom each other.

discover any series at all : for my part I cannot holds between every pairof terms. relation which transitive asymmetrical Cantor, it is true, regardsit as a law of thought that every definite aggregate can be well-ordered;but I see no ground for this opinion.

all individuals form

a

namely

maximum, formed

by does

terms

be

must

denying^.But are

as

fonn

in this

case

we

there

as

are

in any

cardinals

persuademyself that whose terms other. There

are

a

does

he has

be two

other

but if not, arise.

Thus

increased 346.

more

are

;

it

of two

in this

the

case

see

no

there

definite.

more

to possible

less than, that

is

no

cardinal

*

See

are

of the of the

the

is

impossible;

a

case

which

Vol.

vm,

\ "

Jakrexbtrickt

Chap,

be

producedin

outline.

The

article in which

"0.

higherthan

that

of the

continuum

there of

are

proof

to prove

|j. Let

us

ordinal,see

Burali-Forti,"Una questione sui numeri Also my article in di Palermo, 1897.

del vircolo matematico

xxxviu,

"

.301.

der deulwken

is synonymous

i.e. integers,

cannot

p, 43 note.

Of.

|] Power

cannot

mpra,

maximum

Rendwonti transfiniti/'

RdM,

one

other.

proofsabove referred to"is quitedifferent, and importanton its The^ proofis interesting

will be

powers

Chap. XXXIT,

t On

think.

correlate either

cardinal number

consists of three

there

cannot

to

reason

points: (1) a simpleproof that (2)the remark that this method powers higherthan the first, be to any power, (3)the application can of the method applied

that

different

second of the

The

occurs

for

reasons

to break down.

account, and

own

proof that

a

any in this series I

first proof that

the

there

prove that there is a series, is greater or less than any one

nor

seems

is far

are

equalto, greater than, belongto a singlewell-ordered series,this I cannot see any way of showingthat such

If all terms

that

prove

all

ordinal for each cardinal.

an

classessuch that it is not

of the

part

than

will be neither

and

be

of which

all cardinals

That may

he

cardinals

series

collection of

there

case

of

type

greater (Math. Annalen^ XLVI, " "), I

be the

must

one

perfectlydefinite

doubt whether there legitimately may if Of all cardinals. cardinals course,

that althoughCantor professes

with

impossibleto

then there must well-ordered series,

a

But

ordinal,which

ordinals

many

series it is

a

maximum

a

exceptionf. If the

without

form

not

a

representsthe

ordinal which

that

all terms

ordinals will have

the

allowing this view,

But

with

i. (1892), Muthematiker-Vereinigung, p. 77.

cardinal

number:

the iirstpower

is that

of the finite

Philosophyof

the first of the

examine is

The

346]

345,

above

the

points,and

Infinite

then

365

whether

see

the method

general. really Let

and

m

consider

collection M

a

denumerable

(The

characters

two

some

each of which

is

than

elements is of

remarks

each

are

defined

m

a

or

follows

as

in

w

all

be

series of jETs.

where, for every vice

contains

Then

versa.

at

E0 is

jETs.

therefore

the

every

Hence

jETs

not no

not

are

"

series

a

w.

a

it is less make

the

possible

denumerable, i.e. denumerable

any

all zo's. Or

the

"Ts of

be

series

the

has

not

can

to

the

(bl95S3... ""...),

is an

m,

bn is a

series

of

w9

jE7s

term corresponding

of

contain

can

denumerable,i.e.M

any series

our

chosen,we

Jf, but

i.e.if CL^

terms

(For

manner.

rest

of E*s

the

of the

one

they

take

that

identical with

any such

when

w

denumerable

our

or

:

be

ann

of

us

collection

let E0

one

not

term

one

hence

Eoy and

series of

least

For

bn is different from

n,

a

is not

m^s,the

the

E0, belongingto

term

a

denumerable

1, and

determinate

some

of

is

be rational numbers,

may

let

For

m

is to consist of all

Ep might which insures might be suggested, Then however series our different.)

always find

of

first.

an

E

as respectively greater

M

law

other

and

x$"

collection M

the

an

example,the firstp terms are

the

Then description.

are

is either

x

irrelevant,but logically

follow.)The

power which collection of -"7s,

the dfs

Thus

element

each

each

it is greater than

higher than

a

where

E, where

be considered

may

are

of the above

E

w

when

m

an

easier to

argument

and

m

fixed term.

These

1.

and mutuallyexclusive characters,

collection, "a?19"?"... a?n,...9 and

less than

and

says, be two of elements

w, Cantor

a

denumerable

our

all the

power

-ETs,and

higherthan

the first.

need not stop to examine

We than

that

proof.

We

continuum, which

of the may

the

proceedat

to

once

proof that

there is a power is easilyobtained from the the

higher above

generalproofthat, given any This proof a higher power. It proceeds case. particular

collection whatever,there is a collection of is as

quite as

simpleas

follows.

be

such has

Let

u

that, if -R be relation

the

a

the

the

class,and

any relation

either

R

proof of

to

of 0

consider

the or

to

the

class,every 1.

(Any

class K

of relations class

term

of the

other

pair of terras higher power

u

has a the class'K 1.) Then first place that K the in observe than To the class u. prove this, has certainly not a lower power; for, if x be any u, there will be a relation R of the class K such that every u except x has the relation Relations of this kind, for the R to 0, but ac has this relation to 1.

well

will do

as

various

values

the

terms

as

0

and

correlation with class having a one-one of #, form a Hence K has at least of w, and contained in the class K.

the

Continuity

Infinityand

366

To

u.

as

power

same

has

K

that

prove and

[CHAP*XLm consider

greater power,

a

correlation with u. in K having a one-one any class contained where is some be called RX9 u x the Then any relation of this class may relation define Let with K us now X. a suffix x denotingcorrelation "

by the followingconditions the relation R# to

0, let

for which y of u Then R R to 0.

y

For

every term the relation K

:

have

x

for which

u

has

x

for every term let have the relation to 1, y of ", and is a relation of the

Ry

relation

has the

of

x

is defined for all terms

to 1 ; and

of the relations Rx. Hence, whatever any one as it we in K and of the same take,there class contained power may class and this to therefore K has of K not belonging is always a term ; class K

;

it is not

but

than

higherpower

a

u.

may, to begin with, somewhat eliminatingthe mention of 0 and 1 and

by

of

this relation

have

u

in

class contained

a

is

there

the

and

if k

Thus

contained

class

be

any in u, and

which

may the number

of the

for every

class K

that

as

same

whatever, the of K

the number class.

be any

Each

know

we

the

Thus

of the

form which

relation R

of classes contained

is that of

is omitted

the

which have

We

one-one.

u.

class

a

ku, where k is a variable is reduced

argument

is

^",

in

is the

to

this

the number

:

that

of terms

following.Take

any

(1) that its domain, which properties its converse domain, (")that no two terms of

is correlated

of p

of relata to

ku

in

has the two

will call p, is equalto the have exactly domain term

of

means

itself).Thus

u

logicalproduct

argument

same

by

class contained

of classes contained in any class exceeds belongingto the class*. Another

which

0, that is, it is defined

to

is the

of K

number

relations to them.

when

(includingthe null-class and

u

relation of the

one

is defined

class K

of the relations of the terms

simplifythis argument,

We

347.

to

set of relata.

same

with

a

is

that

in this con-elation. of the

consists of all terms

class contained

said term show

themselves,i.e.the class

Then

in p,

referent; and

at

The

by

least

which

is the have

not

is the domain

which

the

namely

class

class contained

one

do

jR,any

this correlation is

class omitted

domain

of

means

we

the

class the

w

in p

which

relation

R

of the

logical product For, if y be any term of the domain, and therefore diversity. of the converse domain, y belongsto w if it does not belongto the class correlated with yy and does not IxJongto w in the contrarycase. Hence class as the correlate of y ; and this applies is not the same ever w to whatselect Hence the class w is necessarily term omitted in the y we

to

w

and

of R

correlation. above argument, it must dubitable assumption. Yet there

348. no

The

conclusion If

we *

the

assume, The

number

argument

plainlyfalse.

seems as

was

done

in

of classes contained

shows that 2* is

To

"

be are

confessed,appears certain

begin with

cases

the

to contain

in which

class of all terms.

47, that every constituent

in

a

class which

alwaysgreater than

a.

has

the

a

members

of every is 2* ; thus

Philosophyof

The

346-349]

the

Infinite

367

And terms. term, then classes will be only some among since there is, for every term, a class consisting of that conversely, is proposition a

This

terms*.

classes of

objects. Or

can

occur

in

show

only

the

it

of

typesf,and evident

kind,it becomes

while objects,

yet there

than

of them

more

Every object propositions. that

indubitable

seems

there

at least

number

classes of classes. But

among are

class of

and proposition,

as

classes.

some

the

doctrine of

the

some

that there

again,take

some

by

with

same

of classes and

be

objectsmust

are

the

of all objects^; of every

argument would

Cantors

be

adequatelymet

is

case

the notion

admit

we

that

of classes should

analogous case exactly

is the

if

correlation of all terms

one-one

a

number

the

Hence

so

is

only,there

term

if

there

are

fixed

be

as a u as are propositions objects. For, many will be a different proposition for every different value class, x is a u of x ; if,accordingto the doctrine of types,we hold that, for a given uy "

"

has

x

have

to vary

"

is

x

*

a

u

the number

thus

great as that of objects.But

as

than

there

are

of

and affected,

of xy and case

is

for it is true

;

such

to

false of

or

more

by

cases

and

class

x

illustrate the

Cantoris

an

itself.

and

ix

(This both

are

point in

classes which

be not

x

is not

a

be omitted not

doctrine of this *

w

See

This

types;

let

case,

results from

be similar

to

a

us

but

the

true

the theorem v, and

to

a

class,let

from

if

x

with

us

be

but

a a

but

^r;

the class which, the

correlation,

of themselves;

members

itself.

But

w^

is and

as

we

is not

be solved

by is more difficult. propositions class of propositions with the

of Schroder v

this

of

case

correlate every

part of

4"x

in the

one-one,

correlated

question.) Then are

a

is #, but

onlymember

which both class, Chapterx, is a self-contradictory of itself. The in this case, can member contradiction,

In

if

contained

in

the

"

i.e.

of Cantor^ application attemptedcorrelation.

actual

example,if

argument, should

of those

w

not-^ry1

"

in detail the

for classes,

for correlation, to

Then

x.

yet this,being a class,should be correlated with a

of

more

x

of

means

it with

correlate

us

to according

saw

are

is false or accordingas "f"x for every value of x. But "f"x less met by the doctrine of types.

or

ix^ i.e. the class whose

correlate it with

many-one it will serve

there

it differs from

of terms

case

class,let

that

function not propositional

a

It is instructive to examine

argument

is the

be the correlate of "j"x

perhapsbe

may

In the

let

therefore

349.

shows

argument

only some

are propositions

that

hold of x"

correlation

classes of

are propositions. Again, we can easilyprove functions than objects.For suppose a propositional all objects and some functions to have been propositional

more

not

of

we significant, only this of form for propositions must be at least propositions

there

correlation

does

Cantoris

objects,yet

among them

is to remain

u

in order to obtain suitably

possible a?, and

every

if

restricted range

a

and

Bernstein,accordingto which,

part of u, then

u

and

Borel, Lemons xur la Tkforie dex Function* (Paris,1898),p. t See Chapter x. and Appendix B. J For the use of the word objectsee p. o5, note.

102

v

must

ff.

be similar.

and Infinity

368 which proposition But

applyingCantor's

class

of those

w

members

be

we

find that

its

propositions.

some

have

we

omitted

the

but are not logical products, whose logicalproductsthey propositions

which propositions

correlated with

appear to have

we

to propositions

are

class, accordingto the definition of

This

are.

argument,

classes of

of the

[CHAP,xmi

is its logical ; by this means product

relation of all classes of

one-one

a

Continuity

should correlation,

our

logicalproduct,but

own

examining this

on

of the product,we find that it both is and is not a member logical class w whose logical productit is. of Cantor's argument to the doubtful cases Thus the application though I have been unable to find any point in yieldscontradictions, which the argument appears faulty.The only solution I can suggest

greatestnumber

is,to acceptthe conclusion that there is no doctrine of types, and to deny that there

are

concerningall objectsor

since all false, plainly

the

the

propositions

true

any

Yet all propositions.

and

latter,at least, true or false,

are propositions any In this unsatisfactory no common even properties. of the reader. leave the problemto the ingenuity state,I reluctantly To sum 350. saw, to begin up the discussions of this Part : We seems

if

theyhad

at

rate

other

with,that irrationals are

be defined

to

and that limit,

as

in this way

those

segments of rationals

is analysis

able to

dispense to saw possible ordinal kind of the in which define, a purely continuity manner, belongs defined is not self-contradictory to real numbers, and that continuity so We calculus has no need of the found that the differentialand integral forms and that,though some of infinitesimal are admissible, infinitesimal, have

which

axiom special

with any

the most is not

no

of

continuity.We

that

it is

form, that of infinitesimal segments in a compact series, and is in fact impliedby either compactness or continuity, usual

Finallywe discussed the philosophical self-contradictory. questions and infinity, and found that the arguments of concerningcontinuity valid,raise no sort of serious difficulty. Zeno, though largely Having t he twofold the of definition that which infinite, as graspedclearly induction starting from 1, and as cannot be reached by mathematical that which

has

partswhich

definitions which "

and can

do

we

be

may

found

have

the

same

number

as distinguished

all the

that

usual

of terms

ordinal

and

arguments, both

and that fallacious,

as

itself "

cardinal as

to

spectively re-

infinity

definite contradiction

no continuity, certain infinite classes either, although provedconcerning special giverise to hitherto unsolved contradictions. as

to

are

be

It remains of this

applyto

sults space, time, and motion, the three chief rewhich the of infinitesimal discussion, are (1) impossibility

to

segments,(") the definition

of

the

the

consistent

doctrine

of

hope,persuadethe reader that have not been superfluous.

and continuity,

(S) the definition and infinite. These will,I applications

the above

somewhat

lengthydiscussions

PART

SPACE.

VI.

CHAPTER

DIMENSIONS

with

one-dimensional

of

the

from

of

that

and

integers

magnitude, of

most

ordinal.

In

series

set

was

be

kind

was

problems

the

fourth

forth,

raised

problems

which,

these

under

the

ever

since

of

the

that

of

to

and

veloped. dethe

found

was

purely

are

sitions propo-

finite numbers

of

a

Part,

series

by compact

could

series

of

the

examined

we

problems

"

defied

continuity, have

sophers philo-

discussion of thought. The ordinal theories the logicaland

abstract of

combination

a

it

arithmetical

fifth

the

be

one-dimensional

integers form

finite

theory

could

and

of

nature

how,

the

quantity

all the

logicaltheory

series

of

theory

that

In

of

theory

shown

order, namely

account,

own

the

infinityand

dawn

the

was

sign

of

case

general

the

endless

by the

its

on

progression.

a

without

particular

shown

was

concerned

and

Parts, it

numbers

the

been

general logical notions,

arisingin

it

names

led

problems

of

a

Part,

and

called

we

first two

examined

by means proved by assuming

which

the

the

Part,

obtained also

preceding Parts have logical theory of numbers

rational

the

NUMBERS.

the

In

of

third

the

In

order

series.

COMPLEX

indispensableapparatus

finite

of

of

themes, the

main

two

AND

discussions

THE

351.

XLIV.

universallyvalid,of two connected regarded as definitions of the principleswhich, following Cantor, we These two finite,not as applicableto all collections or series. principles of

Arithmetic, and

(1) If

were:

another, (2) as

follows:

term,

successor

of

the

series.

classes and some

the

of

not

induction,

any

These

any

philosophicaltheory

is

all of

purely a

the

of

finite series This

With

one-dimensional

coextensive

terms

as

ordinal, and

to

the

first

property, belongs

to

with,

the

other

be

may

relation, and

one-one

;

htated

having term

every

a

and term

regarded as definitions of finite but as inapplicable respectively, view,

continuity, except classes.

not

of

property, belonging

series.

some

of

same

which

in, but number

principleswe

two

infinityand

notion

the

possessor

progressions or and

contained

generated by

that

of

classes

difficulties to

of

wholly

has

one

is such

the

rejection,as

be

series

A

to

as

the

mathematical

first

to

class

one

then

the

to

this

we a

found, resolves

all the

purely logicaldifficulty result, we completed the

series.

[CHAP.XLIY

Space

372

matics previousdiscussions,largebranches of mathethe of One of remained unmentioned. generalizations have and number, namely complex numbers, has been excluded completely, mention has been made of the imaginary. The whole of Geometry, no in all

But

352.

our

also,has been hitherto foreignto connected.

were

that

Not

to

are

we

this would

theory of complex numbers: we

much

as

region of pure mathematics

in the

to remain

are

be

place as Space,

is called

the mathematical of the

the space

certain affinities to

entities discussed will have

:

of

out

Although this Part

theoryof irrationals. geometrical

a

thoughts. These two omissions i.e. spatial, accepta geometrical,

our

actual

dependenceupon world,but they will be discussed without any logical considered a pure a priori as science, these affinities. Geometry may be studyof actual space. In the latter science,to be conducted by means experimental

or

the

as

an

But As

of

branch

a

this latter

in

it is not

I hold

sense,

of careful

that I wish

sense

it to

be

ments. measure-

discuss

to

it.

deductive, mathematics, Geometry is strictly

pure

questionwhether its premisses define. there exist (in the strict lead to propoinconsistent sets of premisses Many different and even sitions which would be called geometrical, but all such sets have a This element is whollysummed element. common up by the statement Geometry deals

that

questionwhat Geometry,

to

which to

as as

it

I

one

be

which

see, no

called

simple.What course

or

is meant

This

series is indifferent of the

relations

relations

alwayssuch

are

dimension,but have, Series of

agreement. those

of

dimensions

one

far

so

than

more

dimension

will

I shall endeavour

to

present,I shall

Geometryis

the

set

studyof

it will be seen, definition,

causes

of Geometry,since part of the subject-matter

series; but

it does

not

show

that

any

definition of

above

wilful misuse

of

words.

presentusage of

for them

give an

to

to

bear this

which retrospect,

I

believe,however, that it represents mathematicians, though it is not necessary d efinition of their explicit subject.How it has

the correctly

come

one

The

logical dependenceupon actual space. Geometry is,no doubt, somewhat unusual, will produce, especially an philosophers, upon Kantian appearance

The

of

These

present chapter. At

two-dimensional have

such

definition : following

to form a

by

dimension.

one

consequences

series : multiple

dimensions.

more

complexnumbers they constitute complexnumbers

more

of the

the anticipation,

up, by series of two

than

generalpointof

I shall call

dimension

of

only the

the terms.

series of

than

more

actual terms

the

examines

other

explainin the

and

series of

postulates among

generate a

can

of its

with

be

may

the

premissesand to sense)such entities as

indifferent to the choice

be

meaning,may

explainedby

will illustrate also the

a

brief historical

difference between

pure

and

appliedmathematics. 353.

Until

Geometry,i.e. a

the

nineteenth

certain

system

century, Geometry of

meant

deduced propositions

from

Euclidean

premisses

Dimensions

353]

352,

and

Complex Nwzbers

373

supposed to describe the space in which we live. The because (what is no doubt important was pursuedvery largely subject to the engineer)its results were in the existent practically applicable which

were

embodied

world, and be

sure

we

must

be

in themselves

was

be certain

must

we

or

this

that

one

so,

of the truth

able to show

scientific truths.

of

that

to

Either necessary. their own account,

thingswas

two

of the

in order

But

premisseson premisseswould give

other set of

no

with

results consistent

experience.The first of these alternatives was advocated by Kant. The adoptedby the idealists and was especially the position second alternative represents, of empiricists before roughly, whom the non-Euclidean must include Mill). But we period(among raised both to alternatives. the Kantian For w ere view, objections it

necessary to maintain honest peoplefound

was

which

that all the axioms it hard

self-evident

are

a

"

extend to the axiom

view

of

parallels. search clared Hence a arose more axioms, which might be deplausible truths. such axioms a But, though many were suggested, priori the search only led to scepticism. all could sanelybe doubted, and to

for

alternative

second

The

the

"

view

that

other

no

would

axioms

give

experiencecould only be tested by a greater ingly Accordthan fallsto the lot of most philosophers. mathematical ability the test was wanting until Lobatchewskyand Bolyai developed

results consistent

with

their non-Euclidean mathematical

of

could

system. It

all the

with proved,

then

was

demonstration,that

premissesother

cogency

Euclid's

than

of

within give results empirically indistinguishable,

the

limits

Hence

the

empirical

observation,from argument a

"

those of the orthodox

for Euclid

spiritamong

new

was

also

system.

destroyed.But

Geometers.

Having

the

produced investigation

found

that

the

denial

of

self-

led to a different system,which was parallels became and possibly true of the actual world,mathematicians consistent, interested in the developmentof the consequences flowingfrom other a Euclid's. Hence arose less resembling sets of axioms large more or Euclid's axiom

of

number each

of

Geometries, inconsistent,as

self-consistent. internally

a

resemblance

The

each

other, but

Euclid

requiredin

rule, with to

less,and possible graduallygrown their on deductive systems have been and more investigated more it was In this account. formerly own way, Geometry has become (what of pure mathematics, that is to say, a subject a branch called) mistakenly a

suggested set

in which such

and

the such

of

has

axioms

assertions

are

that such

not premisses,

and such consequences

that entities such

the

as

follow from

describe premisses

exist. That is to say, if Euclid's axioms be called J, and P be actually impliedby A, then, in the Geometry which preceded any proposition P Lobatchewsky,

itself would

now-a-days,the geometer would A

A^

and

P

themselves

doubtful.

since asserted,

be

onlyassert And

A

that

he would have

A

was

asserted.

But

impliesP, leaving other sets of axioms,

would the implication* AS... implyingP19 P2... respectively:

belong

Geometry, but

to

[CHAP.XLIV

Space

374 not

Thus propositions.

Al or Pj Geometry

and any of the other actual axioms longer throws any direct lighton

or no

increased

the indirectly,

and analysis from Geometry,has thrown resulting knowledgeof possibilities, Moreover is now it immense proved(what lightupon our actual space. that Kantian the is fatal to every Geometry is rigidly philosophy) deductive,and does not employ any forms of reasoningbut such as applyto Arithmetic and all other deductive sciences. My aim, in what But

actual space.

of

the nature

modern

philosophically Geometry,

and

of space, upon which philosophy the first section of this Part, though

in the questions,

light. In

throws

discussingGeometries

be

I shall

those

is

constitute modern

which

deductions

importantin the to then to proceed mathematics

what in brief outlines, first,

to set forth

follows,will be

shall select for discussion

of pure mathematics, I the most throw lighteither

branches

as

which

onlythose

of mathematical actual space, or upon the nature reasoning. A upon is neither desirable in treatise on non-Euclidean Geometry necessary nor a generalwork such as the present,and will therefore not be found in

chapters. following

the

354. than

Geometry-,

It

dimension.

one

definition to

Geometry

which

generates a transitive

of P

a

forming is

a

of

all the

series.

two-dimensional

generatesa series.

In

relations in the series other

transitive relations asymmetrical

series. may any

and term

Now

every term

relation,which

fields of

instead of

But

start

given one

To

Let

(with possibly one

tion defini-

mere

that

of

dimensions

two

relation P, itself an metrical asym-

Let

all the

field

let each of these class ?^ of terms

generatedby

words,the total field

of

a

P

class

a double forminga the asymmetrical relation P, we

simpleseries is

there

be

a

class of terms

exception) belongsto

the

"25 of which

field of

one

certain class u^ of serialrelations. a That is if x be a is also a term of the field of some relation of the class u^

*r

let

HI

be

to constitute

seems

our

of

onlyone of "*, further

of

startingfrom

the terms.

from

the

of w, be

a

the

relevance

series of

and relations, simpleAeries of asymmetrical simpleseries of terms for its field. Then the

form

have

The

to

transitive asymmetrical

some

Let

A

more

dimensions,and

the fact that

dimensions.

series wa.

have

dualityclosely analogousto

a

I^et there be

follows.

as

two

define

to

series. multiple

a

leads to

dimensions

time

now

will appear from

Geometry. projective Let us begin with arises

is

by

is meant

explainwhat of

said, is the study of series which

we

obtain three

a

the

series. Then

?tj will

be

a

double series.

definition of two-dimensional

dimensions,we

have

only

This

series.

suppose that u$ itself or series, asymmetrical transitive relations. Or, starting with the terms of the three-dimensional series, let any term of a certain and only one class M*I belong to one series (again with one possible

consists of

which exception,

to

of

may

belong to

many

of series)

a

certain class

ii".

Dimensions

353-355]

every term of 11% be and let u^ itself be

of

term

a

and

series

some

of

series ?^ whose of w1? and term

a

terms a-z any

relation,and

so

on.

obtained,be alwaysa terms

?/ -dimensional

terms

Let

:

un^

be

not

of serial

proceeduntil

Then

is

355.

some

be any

If #j

serial relation

any

Before

serial relation

only one

in relations,

series of

a

be

there

Held of ^,

field of

discussed at

class wn_2 I"et this itn

the

field of

the

need

which

follows:

serial relations.

serial relation,Lrn^ say, which of serial relations. Let each

some

definite class

belong to

Let

form

"V-i*

an

series. Or, to give the definition which starts from the be a class of terms, any one of which, ,r,tsay, belongsto

un

field of

the

the definition

obtain

we

way,

let ,r2 be again a serial Proceedingto #3, x49 etc., let .rn-i" however all the relation generatinga simpleseries. Then of the

term

belongingto

"rn

all themselves

are

375

belongingto a class "j of series, series a series, or n^ is a triple

simpleseries. Then three dimensions. Proceedingin this which be given as )i dimensions, may

of

Numbers

Complex

n

general (with exceptions lead

nn^

let

ttn

", be

to

new

a

led to *""_,.

simpleseries.

a

dimensions.

proceedingfurther,M"me be

,rn_3

a

in

av

in which

way class ?/19and

a

term

Let

exactlythe

reach

we

present).

itself belongsto

useful.

observations

above

the

on

definitions may first plac",we justseen that alternative definitions of dimensions suggest themselves,which have a

In the

have

relation

until

analogousto what is called dualityin protective Geometry. far this analogyextends, is a questionwhich discuss cannot we have examined we protectiveGeometry. In the second place,

every

series of

How

dimensions

n

dimensions, but

imply

of

one

series of

a

dimensions.

n

dimensions,the class wn-i if

be less than

m

And

in the other

This i.sa

may

is

an

series of all smaller

1) dimensions

"

(n

class ^^t_TO is a terms method, all possible

n

-dimensional

be established

by

order of each series

the the

simpleseries,preserve

In

on.

general

in

not

of

definition of

of the

//

"

jrn-i

third

the

series is also

form a series together a place,if n be finite, series.

one-dimensional

a

followingrules order

numbers

\) dimensions,and generally, series of (n m) dimensions.

"

w, the

so

does

the second form

is a series of

"

class which

(n In

("i 1) dimensions,and

of

involves

:

In the

class */, which , ternal ?/a, keep the in-

unchanged. In placethat series before

unchanged,and

which

before

Thus in //a, and that after which conies after in w,. "/2 is converted the same into a simpleseries. Apply now process to z^,, and Then by mathematical induction, if n be finite, or be any infinite so on.

comes

oixiinalnumber,

fact,which *

if

n

Cantor be not

has

wrt can was

simpleseries.

a

discovered,for finite numbers

proved, not

greater than

number, this

be converted into

only that

and o",

the

is ate) the cardinal number

cardinal :"pace has the same Acta Math, n, p. 314ff.

number

a

simple series

constituent

can

and

l"e

of the resultant series of

wo

series all have

points as

a

finite

:

This "",

able remark-

by Cantor*,

formed, but that, the

i.r. an

same

cardinal

w-dimensional

portion of

a

line,

See

[CHAP.XLIV

Space

376 has

important bearing on the place,the definition of n

a

the

very fourth

the

case

where

purpose, it is we

only necessary to suppose take,any um will belongto some

may

dimensions

Geometry. be

can

extended

transfinite ordinals.

is "a, the first of the

n

of

foundations

that,whatever

For

In to

this

finite number

m

simpleseries of series wm+1

;

and that the sequence of classesof series so obtained obeys mathematical of dimenThen the number sions and is therefore a progression. induction, is the is

This

co.

of

case

a

does not appear of dimensions,that the number

bringsout, what

case

finite number

from clearly

so

of dimensions

ordinal number.

an

multipleseries,as ways of generating discussion of these various of generatingsimpleseries. The

There

356. there

are

are

very

many

follow since it would ways is not, however, of great importance, will Instances meet the discussion of Part IV, Chapterxxiv. us course

of

will For

the

examination

our

Geometries;

of the various

of giveopportunities

and

this

closely in the ination exam-

definition of dimensions

testing onlyimportantto observe that dimensions,like defined in purelyabstract terms, without any are continuity, our

present,it is

order and

reference to

actual space.

when

Thus

that

say

we

space has three idea which can

to it an merelyattributing only be obtained from space, but we are effecting part of the actual logicalanalysisof space. This will appear more clearlyfrom the of dimensions to complex numbers, to which must we applicability

dimensions, we

turn

now

our

are

not

attention.

was formerly considered a very theoryof imaginaries importantbranch of mathematical philosophy,but it has lost its amination philosophical importanceby ceasing to be controversial. The exof imaginaries led, on the Continent, to the Theory of Functions a subject which, in spiteof its overwhelmingmathematical importance, appeal's to have little interest for the philosopher.But

The

357.

"

ourselves the same among it led to an examination

examination

took

a

more

abstract direction :

of the

of symbolism,the formal principles laws of addition and of a and the general nature multiplication, Hence Calculus. freer spirit towards ordinaryAlgebra,and a arose of regarding the possibility it (likeordinary Geometry) as one species This was of a genus. the guidingspirit of Sir William Hamilton, De Morgan, Jevons and Peirce to whom, as regards the result, though not as regardsthe motive, we must add Boole and Grassmann. Hence the philosophyof imaginariesbecame merged in the far wider and more Universal of interesting problems Algebra*. These in dealt be with cannot, problems opinion, my by startingwith the genus, and askingourselves: what are the essential principles of "

Calculus?

any

*

See

It is

necessary to

Whitehead, Uniwrxal

adopta

more

1898 Algebra,Cambridg-e,

inductive

;

method, and

Book especially

1.

355-358]

Dimensions

the various

examine

and

Complex

Numbers

377

speciesone

The mathematical by one. portionof this task has been admirablyperformedby Mr Whitehead the philo: sophical of a portionis attemptedin the present work. The possibility is deductive Universal Algebra often based upon a supposedprinciple of Form. of it is Thus Permanence for the said, example,that complex numbers must, in virtue of this principle, laws of addition the same obey real numbers obey. But as a matter and multiplication, of fact there as In Universal Algebiu,our such principle. is no symbolsof operation, the hypothesis of any one such as 4- and x are variables, Algebrabeing that these symbols obey certain prescribed rules. In order that such an be it should is important, Algebra necessary that there should be at least one instance in which the suggested rules of operationare verified. ,

But

this restriction does not

even

statement

as

Permanence other

to

of

must Form, therefore,

be

us

make

but properties,

to

regardedas simplya

arithmetical addition

than operations

formal

enable

any generalformal rules of operation.The of the possible principle

all

may

easilybe

have

some

or

mistake

:

all of its

which lack suggested

some operations all of these properties. or 358. Complex numbers first appearedin mathematics through the of number. The principle of this generalizaalgebraical generalization tion class is the following of Given is it some : numbers, requiredthat

numbers

can

should be discovered

equation in class of

one

or

variable,whose

invented which coefficientsare

will render soluble any the said from chosen

this method leads Startingwith positive integers, numbers at once, by means simpleequationsalone, to all rational and positive negative.Equations of finite degreeswill give all the

numbers.

of

numbers, but to obtain transcendent numbers, such algebraic which not of any finite degree. In need equations are as e and TT, we is very inferior to the ariththis respectthe algebraical metical, generalization irrationals latter uniform since the method, by a gives all will whereas the former, strictly the speaking, give only algebraic wise. But with regardto complex numbers, the matter is othernumbers. No arithmetical problem leads to these, and they are wholly incapableof arithmetical definition. But the attempt to solve sucli so-called

eq nations as of numbers,

^4-1

=

0, or

since,in the

.z^-h^r-f 1 whole

=0,

domain

at

once

of real

demands

numbers,

a none

class

new can

be

such cases, the algebraical these equations.To meet satisfy numbers of the equationswhose defined new by means generalization numbers to obey It showed that,assumingthese new roots they were. fell into two the usual laws of multiplication, each of them parts,one number number and fixed of of the real a other the product some real, and was the new could be chosen arbitrarily, kind. This fixed number Numbers 1. thus always taken to be one of the square roots of composedof two partswere called complex numbers, and it was shown that no algebraic operationupon them could lead to any new class of

found

to

"

[CHAP,xuv

Space

378

provedthat any further of the formal laws some lead to numbers disobeying must generalization 'But the algebraical was of Arithmetic*. whollyunable generalization stage)to prove that there are (as it was, in truth,at every previous such entities as those which it postulated. If the said equationshave this is all that roots, then the roots have such and such properties; method algebraical

the of

remarkable,it

is stillmore

What

numbers.

allows

us

was

There

infer.

to

every equationmust contrary,it is quite essential to be able to

nature

the

do

entities which

have

effect that

the

to

demanded properties

the

have

generalization. of 359. The discovery

law

is,however, no root;

a

on

actual

point by the algebraical out

only to be obtained by form of the theory of dimensions. Ordinary complex numbers means which of a certain type, dimensions a series of two happen to occur as real. coefficients the in are roots of equations which Complex numlxra but of a higherorder represent certain type of ^-dimensional series, a here there is no algebraical problemconcerningreal numbers which they are requiredto solve. As a matter of fact, however, the algebraical entities have seen, does not tell us what new our we as generalization, are,

whether

nor

such

entities is

all:

entities at

they are

the

it encourages

moreover

complex numbers whose imaginarypart vanishes is analogous that real numbers. to that of supposing This error are and positive real numbers rational* integral, some rational,some are identical with signless integers.All the above errors having integers the reader will probablybe willing been exposedat length, to admit the corresponding No complex number, then, in the presentcase. error is a real number, but each is a term in some series. It is not multiple while to examine worth the usual two-dimensional complex specially numbers, whose claims,as we have seen, are purelytechnical. I shall therefore proceed at once I shall givefirstthe to systemswith n units. usual purelyformal definitiont, then the logical to this objections view

that

and definition,

then

erroneous

the definition which

different entities, n "?n ea, units,be given; and let each be I^et

or

number, or,

in

special cases,

way let entities a^r unless r ,yand ar =

nuinerical

partsof

where arise, =

a^r and

members

a^

are

"a,

^,0^,

which may

*

a

...

a^x

substitute.

to

call elements may capableof association with any real any rational or any integer.In this rn, which

......

af is

we

arid a^r differsfrom is,if either the numerical or the

That

ag.

Further, let there be set of values

with

I propose

number,

a

l"edifferent, then the wholes

way

of

combininga,^,

"n,

to

form

a

...

an

SeeKtok, AtlgrwrinfArithntetik, n,

t See Stolz,"AiV, n, Section

1,

" i).

.

.

.,

2,^,

different. for each

class whose

new

art^t will IK- such be written as

a./.,,

are

ct^ non-

entity.(The entity.)Then

Section

the

bination, com-

Dimemlom

358-360] is

number

complex

a

terms

component

be

should numbers

a19

of

Oj,

...

the

Complex order.

"th

Numbers

The

379 the

of

arrangement

be essential to not may bination only thing always essential is,that the com-

""#,,

a^,

the definition;but

and

the

a"^rt

...

such

that

c^

insures

a

or

may

difference in any one of the or more in difference the a resulting complex

number. The

360.

pointout

entitywhich

one

any

of real numbers. a

4- ib,b -f la

definition suffers from

above

it does not

defined

complex number

by

a

set

real numbers, #, ",the two complexnumbers it is desirable that such determinate; and

Given two determinate

are

is the

the defect that

of any should appear in the generaldefinition of complex numbers ness the e\ in the above order. But definition are variables,and the

onlydeterminate when the "r*sare specified well as the o*s. Where, as in metrical Geometry or in the Dynamics as of a finite system of particles, there are importantmeanings for the e\ find in the above sense that numbers we are complex important. may But no special associated number the can give us interpretation complex We with a given set of real numbers. might take as the complex the class of all such entities as the above for all possible values number of the er*s be too general to serve a class would our ; but such purposes. A better method to be the following. seems We wish a complex number of the /?th order to be specified by the enumeration of n real numbers in a certain order,i.e.by the numbers a^ where suffix. is But order indicated the the cannot we by "", 0^5 define a complex number a series of n real numbers, because the same as complex number suggested

is

...

real number and

relation whose domain

domain

be the

to

of

consists n

real number

is

and

real numbers the

integers(or, in

one-many

a

whose of

case

r

converse a

complex

The

(say)OT

relations may be defined nition purelyarithmetical defi-

in this way a ^-dimensional series of

and

complex

arrangingall complex numbers

in the

order

of the real numbers

differ

which

which

numbers

are

ctr

of

only

in the

cases.

In order that

complex numbers have any importance,there must of terms selected out assemblages a

a

be different whenever

not

Such one-many

integerr.

results from

n

various

in

need

as

defines

what

complex numbers,

is obtained.

as

and

of infiniteorder, of all the integers) ; for the suffix in "r indicates

correlation with the

order

a,,

consists of the first

number

to

z.er.

different. Thus

are

s

recur,

may

metrical space of

essential

numbers.

the

to

Let

a

n

in the

be

motive

some

Such

of continua.

dimensions,

utility, though

defined

sense

owing to

not

collection of entities

to

the

by for a

Stolz should

considering

motive

circumstance

a

definition,of

be (points)

given,each

exists

which

is

complex of which

lias to each of the entities ^, e"y measurable relation fH a numerically and let each be uniquelydefined by the n relations which it (distance), ...

has to e1? tfs,

...

c,n.

Then

the

complex number

a

will

representone

of

this be

collection of

terms

numbers is

[CHAP.

Space

380

a,

essential of

which

have

late

is

essential

for

reason

my

general and

observe,

definitely

the

this

practically

absent

equally

numbers to

beyond

the

are

a

of

f.

purely

of

what

complex

arithmetical,

dimensions.

Thus and

we

this

numbers

complex

the

But

usual

Arithmetic,

of

domain

considering

the

to not

plurality

consideration

the

postponing

is

themselves

...

for

a

will

en

e^

,

motive

is

applies

our

:

el

there case

what

reference

passed

elements

Thus

the

in

Algebra,

involve

and

entities, collection*.

the

to

numbers but

of

XLIV

was

this

to

stage.

*

el

is

complex t

not

identical

with

1

The

x^+Ox^-f

former

is

a

latter

the

point,

a

....

number. In

geometrical

applications,

it

is

usual

to

consider

only

the

ratio*

a1

:

o.2

:

: ...

as

relevant.

In

this

case,

our

series

has

only

(n

-

1)

dimensions.

nn

CHAPTER

XLV.

PROJECTIVE

foundations

THE

361.

times,

to

Euclideans, which sufficient for

and of

the

denial

the

from

showed

certain

with

those

the

consistent Next

axioms.

nature

in

series,mentioned certain

a

type of

order

an

non-Euclideans

it is

Dedekind

being very recent, shall of

range 362.

Geometries,

Cantor

only work

The

as

on

of

and

this

in

is not

respect

the

necessity

of

work

Italian

which

we

line, to

a

work

of

the

all

the

foundations, while

the

becomes

on

able,

are

of

The

relevant

closed series,on

at

the

present Part, I shall

the

a

fairly

contrary,

and

to discuss differ both

unlike

in the

mutually compatible.

divide

not

fundamental

so

a

of

division

a

each

of

the

indefmables

with

late axioms.

The

three

respect of the indefinables and three

That

previouskinds,they is to say, given a

propositionsconcerninga

certain

number

kinds The

which

kinds in

on,

another,

as

above

greater logicaldifference. the

so

it whenever

mention

respect of the majority of the axioms, but

comparativelyfew

axioms, but

Dedekind

of

and Euclidean, hyperbolic, elliptic,

correspondsto

differ,not in

the

recognizethis division and

course

But

of

applies,generallyspeaking,within

yet

followed

analyticalGeometry.

line.

in

consisten in-

universally recognized,although, as yet present chapter, it has enormously increased the

rule,into

a

it is relevant.

kinds

a

results

been

not

discussions

the

I shall

Geometry,

work

be

to

other

four points

produce

likelyto

and

each

non-

protectiveGeometry.

pure In

though

the

in

see

has

points

of

known

recent

the

time, produced probably almost

stage of Geometry.

advanced

between

long

showed

by

virtue

in

IV.,

relation

all the

of

that

of

work

Part

of

the

came

introduced

been

has, by this

modifications

which

has

work

\.e* that

with

prerequisitesof

introduce

of

also necessary,

the investigatingcareffrlly

given

nor

axioms,

continuity,which

on

subjected,in

the

came

of

Cantor

closed

we

been

have

various

results but

usual

of

that

results,were

Lastly,a great change

the

of

Geometry scrutiny. First

threefold

a

GEOMETRY.

of

above

start,

we

only in respect which

I wish

respect of

the

roughly speaking, certain body of geometrical

are,

of

entities,it is

more

or

[CHAP.XLV

Space

382

result from

different selections

Geometries,

assume

to

be

not

that

there

be

ductions systems of de-

the

discussed. separately

startingwith is, there is a certain class-concept jjoint of which in different Geometries), we

That

the

same

at least two,

are

in

three,or four instances, according

or

points,result from

instances, i.e. further

Further

circumstances.

are

which

differenceswhich logical

commonly developed,agree

as

points as indefinables. {which need

indefinable and

great, and

very different selections lead must

to which

All

as

the

But

indemonstrable. as propositions

of the

take

of the entities we

which less arbitrary

the three great types Where assumptionsin the various cases. special the of Geometry beginto divergeis as regards straightline. Protective Geometry beginswith the whole straightline,i.e. it asserts that any

pointsdetermine

two

by

detenu ined

of

members

other

two

any

in virtue

of

a

If this class be

class.

the

relation

is also determined

pointswhich

certain class of

a

between

two

I shall call symmetrical.What or relation, contrary,beginswith an asymmetrical

relation the

which

is

be called

may

a

ray

;

or

it may

again

a

determiningthe stretch of pointsbetween line in either of takes the straight Geometry,finally, adds

either

distance,which as

second

a

is

a

magnitudes. Thus

three

kinds

of

relation

between

them.

Geometry

take

any

the

two

relations

of

Any

of the

one

senses,

points,namely of stretches

points,the

two

and different indefinables,

differences of axioms.

sense,

Metrical

above

consideration

the

magnitude,or else in regardto the

line with

begin by regardingtwo

points as and

regardedas

points,then this Descriptive Geometry, on

the

have

three,by

responding cor-

suitable

a

axioms, will lead to any requiredEuclidean or non-Euclidean but shall see, is not capableof yielding the first, as we as space ; many second result the from third. the In the as or propositions present choice of

chapter,I

am

simplestform of entities the next

going of

chapterhow

those constituents

chapterthat the class of

axioms

a a

of

axioms

gives the

which

and

I shall call any collection We shall see in projective space.

set

of entities

forming a projective

Euclidean or hyperlx)lic space ; projective it from the polarform as goes, indistinguishable It is defined,like all mathematical entities,solelyby

forminga far

are

relations between

in themselves.

"points" of

straightlines

"points1"have

obtain

to

of the

nature

set

projective Geometry ;

elliptic space.

the formal

that

assume

these satisfying

tfpace from a set .space itself is,so of

to

the

in

a

a

Thus

its we

not by what constituents,

shall

projective space may non-projcctive space.

requisite type

of mutual

see

in the

each So

be

following an

long

infinite as

the

relations,the definition

is satisfied. 363. to which *

class of entities, called points, a Projective Geometry assumes it assigns certain properties*. In the first place, to there are

In wlmt

follows,I

Poxiffiotte. Turin,

subject.

1B08.

am

mainly indebted to Pieri,/ Prirtripndvlla is,in my opinion, the best work on

This

Geottictria di the

present

362,

363]

ProtectiveGeometry

least two

be at

will call ab. is

I.e.there is itself a

different

points,a and b say. These two pointsare class of points, certain a their straight line,which we This class is determined by b and a9 as well as by a and ",

determine

to

order of

no

member

a

and

b involved

of the class.

than

pointother

one

and b

a

;

i.e. any

"

be any such

c

belongs

to

ac

a

class contains

often

of fact it is not of what which

least

belongsto

these tions assumpthen rd and ab

lines

two

what

moment

is

points. This

of

but as a matter thought to requireno explanation, The statement. statement perfectly precise precise

a

is meant

at

With

pointsin common. proceedingfurther,let us consider for a by saying that two points determine a class

is expression

b

point,then ab.

Before meant

(and therefore b)

be any points of ab, pointsof a line determine that line,or two

two

coincide if theyhave

moreover

;

Further,the

if

the class ac, and every point of it follows* that, if c, d

coincide

383

is this :

holds between

There

is

definite relation

certain

a

(J"say)

responding coupleof pointsand one and onlyone corthere Without such definite relation, some points. of The relation K two a class. question pointsdetermining

any

class of could be be

may

no

ultimate

of properties relation

the

between

collinear with

in indefinable,

and

class ab.

We

a

but symmetrical,

will

those

(b and c) which

that

instead of

c

need

we

that

reference to

a

as suggests,

a

of

the

above

derivative

being both

transitive and

relation will be

This

its terms.

are

namely

say,

This

always involve

case

obtain, however,

may

and

points,b given point a. two

which

other

term a

than

simplification

coupleof pointsand a class of points, we might have a relation R between the two pointsa and 6. If R be a symmetrical aliorelative, transitive so far as its being an aliorelativewill pennit(i.e. if aRb and bRc imply aRc, unless a and c line will belong of the straight the above are identical), properties to the class of terms having to a the relation R togetherwith a itself. This view seems results. simplerthan the former,and leads to the same line is derived from a relation of two Since the view that the straight I shall in generaladopt it. Any two pointsa, pointsis the simpler, b

have, then, a relation Rab; while identical,

are

Rnc

;

such

b and

c

a,

have

relation Rac- If Rob and R^ and is identical with both R^

a

differ,R^

those

are

belonging to

transitive relation and or

c

a

of any properties of an disjunction metrical asym-

It is to be observed that the formal

if not, not. relation R

between

relation K

a

etc. These after,

its

converse

the "

e.g. greateror

and all symmetrical aliorelatives,

are

are

less,before transitive

so

beingaliorelativeswill permit. But not all relations of the relation into a transitive asymmetrical type in questionare analyzable which its converse; is of the above type, is not so for diversity, or be generated that the straight line can analyzable.Hence to assume

far

as

their

*

Fieri,op.

tit.

" 1, prop.

25.

[CHAP.XLV

Space

384

by

and

asymmetricalrelation

an

characteristic of

what

its

I shall call

present,such an assumptionwould two indefinables, namely point)and

is

converse

a

assumption,

new

Geometry. Descriptive

be out

of

place.We

relation R

the

or

For

the

have, then,

K*,

No

others

in projective required space. of The next point is the definition of the plane. It is one 364 it other allows unlike a very the merits of projective spaces, space that, this For of the need we definition plane. purpose, simpleand easy there is at least new a axiom, namely : If ", b be two distinct points, one pointnot belongingto ab. Let this be c. Then the planeis the class of points lyingon any line determined by c and any pointx of ab. That is,if x be any pointof aby and y any pointof cj?, then y is a point of the plane cab ; and if y be a point of the planecab, then there is It is to be observed some point x in ab such that y is a pointof ex. that this definition will not applyto the Euclidean or hyperbolic plane,

are

lines may failto intersect. The exclusion of Euclidean axiom f : " If #, 6, c be hyperbolic space results from the following

since in these two and three

V

non-collinear

pointof

a

and bb'J"

other than

ac

By

and points,

means

a

af be

and

of this axiom

pointof

a

c, then we

there is

can

prove

be other than 6 and a

point

common

that

the

r,

to aa'

planecab

is

that,if d, e,f be any planeabc or bac, and generally three non-collinear pointsof abc, the planedefcoincides with the plane also show that any two lines in a planeintersect. abc ; we can We 365. can now proceedto the harmonic range and von Staudfs construction. Given three collinear points"x, b, c take any quadrilateral ab. two points Construct the points of u" v collinear with c but not on these intersection au bv and bu ; join av points,and let the line joiningthem meet ah in d. This construction is called the quadrilateral If we construction. that outside the planeabu there is at now assume least one point, of u and v9 we can prove that the pointd is independent and The is uniquely determined point d is called the by ", b, c. harmonic of c with respectto a and 6, and the four pointsare said to the

same

as

the

.

form the

.

harmonic

The uniqueness J of the above construction range. a proofof which,it should be observed,requires point not in the a

"

is the fundamental of projective planeof the construction)! proposition It relation which may hold between four pointsof a givesa Geometry. and when two which, line, are as given,is one-one regardsthe other "

*

We shall ssee in (3iap.xux that these notions,which here provisionally are of definable classes. undefined,are themselves variable members t Pteri,op. tit," 3, p. 9. I 'Hie proofof the uniqueness of the quadrilateral construction will be found in Geometry, e.g. in Cremona's (Oxford1893),Chap. vm. any text-book of Projective derivable from ||A proof that this propositionrequiresthree dimensions is easily a theorem given by Hilbert,(Irmidlag"ider Geometric,p, 51 (Gauss-Weber schrift, FestLeipzig,1899).

ProjectiveGeometry

363-366]

385

Denoting "r and d are harmonic with respect to a and b" by of the relation are important the following cHaA properties : (1) cHg^d I.e. Hab is symmetrical; (") cH^fi impliesaH^ i.e. impliesdHa^ the the relation of pairsab, cd is symmetrical; (3) cH^l impliesthat two.

c

d

and

different

are

points,i.e. H^

is

aliorelative.

an

propertyis independentof the others,and

has

to

last

This

be introduced

by

an

*.

axiom

Having

the

obtained

different directions.

We

harmonic

range,

proceed in

we

may harmonic

two

relation as a relation regard the points: hence, by keepingone of the pairsfixed, called what is involution. Or we may obtain we an regardthe harmonic relation,as in the symbol cHayd, as a relation between two points, which involves a reference to two others. In this way, regarding #, b, t\ fixed,we obtain three new as pointscZ,"?,f on the line ab by the of these may be used, with two relations cHatd, aH^e, bHaef. Each This and so on. of the previouspoints,to determine fourth point, a leads to what Mobiusf calls a net^ and forms the method by which coordinates. This construction givesalso Klein } introduces protective directions in of defining ratio. harmonic the method These two an which protective pursued Geometry may be developedmust be separately I take shall first. with. the former to begin of the harmonic define an involution. 366. we relation, By means with This consists of all pairsof pointswhich are harmonic conjugates ||.That is to say, if a, b be the two fixed respectto two fixed points involution is composed of all pairsof points#, y such that an points, four points"r,y, "r', If xHaby. yf be given,it may or may not happen that there exist two points a, b such that xH""y and as H"i/. The such points#, 6 constitutes* a certain relation of of finding possibility It is plainthat this relation sometimes holds,for it X) y to #',y. with holds when It is plainalso identical of,y respectively a:, y are of two

may

of paii-s

.

and

y\

then

for if

but not of x and y be identical, ; has shown how, by means the relation is impossible. Pie"ri"

that it sometimes

does not

hold

of certain axioms, this relation of four terms

straightline

into two

and to generate an *

See

Fano,

be used to divide the

segments with respectto any

order of all the

Giontole

may

di

pointson

Matimatiche, Vol.

30;

a

line,

Fieri,op.

two

of its

(Itmust cit*

points,

be borne

" 4, p. 17 and only conclusive

in the Fano has provedthe necessityof the above axiom all the previousaxioms, but not this; by constructinga system satisfyingA simplerbut equivalent axiom is due to him. is of its necessity The discovery

Appendix. manner, one.

that t

our

space contains

at

least

one

Catruf,Section Barycentrixcher

line n,

on

which

there

are

more

than three

points.

Chap. vi.

Geometric^ Anndfen, 4^ 6, 7, 37*, Vorlemngeti Hber nicht-Euklidiache Gottingen,1893, Vol. i, p. 308 ff. ||In what follows,only involutions with real double pointsare in question. " Op. cit. "" 5, 6, 7- Pieri's method was presumablysuggested by von Staudt ('f. Geometiie der Lage, " 16: especially No. 216. J

Math.

[CHAP.XLV

Space

386 in

that, in projective Geometry, the pointsof a line do order is obtained order to l"egin with.) This projective

mind

have

an

not as

follows. Given

367. class of with

any three such that

pointsx

respect to

b and

x

but

pointsy,

not

or

a

Let

c.

of b

to

(a and

*r

(1) and

does

givenby

are

If d is not

on

points y, pointscan

class contains

Fieri

avoid

as

a

here

fulfilled. Three

are

belongto

e, then know r, we

with

or

/;, c*,d be distinct It follows that

we

either

have

must

found, point",

a

relation of

such

an

order

axioms

are

follows.

In of assumptions.) superfluity must pointson a line,we

a

c,

Here

.

be

the

d

the

segment (abc\

belong to the alreadythat d belongs must

segment (bca). (If d coincides,with is therefore excluded from to the segment (bca). This case to

y

IV, Chapter xxiv,

the line "Z",but does not

coincide with

have

We

in Part

saw

we

will result if certain further axioms

and required,

such

and

denote the us segment (abc). Ilien Qac is symbeing fixed)by bQaefv. metrical,

c

as

words, a

Let

impliesaQ^-.

which,

from points,

conjugates

are

call it the

us

also bQ^r

and

y double

This

line,consider the

a

harmonic

other

in

"

is,if any

class considered.

on

each

,r are

involution whose

an

the

belong to

relation

four

a

supposed variable : that

are

y, y ,r is to

some

r, b and

and

pair of

pair*in

are

different pointsa, b, c

bQ^d

or

axiom

of this

virtue have

the

either

axiom, if ", bQacd or cQabd-

Thus

aQ^d.

at least two

^-relationshold between any four distinct collinear points. (2) If and d be a point belonging to both 4, 6, c be distinct collinear points, the segments (fca)and (cab),then d cannot belongto the segment (abc). of three which the That d is, belong,it never belongs segments to may than two. From this and the previous axiom it results that, if to more d

be distinct from

the

a, b and

a

d

a,

belongsto

b and

point,other

a

d

c.

than

pointof the

(Here to

then ,

segments defined by

three

and collinear points, e

c

for the truth of the

not superfluity,

this axiom

that

onlytwo

of

(3) If #, 6, c be distinct /;,of the segment (abc\and point of the segment (o/x;).

then e is a segment (ode), again,the condition that d is to be other than

avoid

and

two

b is

axiom.)

required only

In

terms

of Q,

bQac(land

dQa^e imply """""; that is, Q^ is transitive. saw alreadythat Qac is symmetrical. We can now of this relation, all pointsof the line except a prove that,by means which and c are divided into two classes, we may call (ac\ and (ac)z. states

We

the

pointsin

two

Any

different classes have the fact other

that,if we

than

a

and

the

formal

two

classes, just as

class have

same

not.

c),then

of properties

we

nor bQtufl,

do

sameness

sameness

of

relation Q^,

division into two

The

do not have

the

have

of

in

(6,d, e beingpoints say, Q^ has divides the line into

That

sign,and

two

classes results from

yet dQ^ bQ^.

any

sign divides numbers

is to

into

and positive

negative. The

of Q^, opposite

which

I shall denote

in by Tac9 corresponds

like

PryectiveGeometry

367]

366,

to

manner

difference of

of Qaty but the that

means

but not

d does not

in the

that h and d

fact of

sign. Tac

not

is

belongingto

387

denote

to

the

different segments.

coincide with

segment (abc). Then

a

or

c, that

bTaed may

mere

negation

That

is,bT^d

d lies in the be

taken

line or,

meaning

as

and

It is a relation which has the c. separated by a of o f in Part enumerated as properties separation couples, IV, If five be distinct in xxiv. /;, d, c line, r, #, Chapter points one straight have the following we propertiesof the jT-relation. (1) bTud is to dTaJ), aT^, have one and cT^a^ cT^a, etc. (2) We equivalent the relations three aT^ aT^, aTe^). (3) dT^Jb implies onlyone of are

formal

dTafe

or

eTJ"*.

of By comparing the above properties that T leads to a it will be seen couples,

of

Part IV), I.e. to

series in which

a

is arbitrary,lire

term

scries

(which involves,as

given by

Pieri

follows.

as

the With

with those of

closed series

there is

definition of the in

T

a

separation

(inthe

first term, but

sense

this first

generatingrelation

generalcase, regardto the

three

fixed

natural

of

of

the

points)is

order

abc, a

precedesevery other pointof the line; c precedes every point d not with a or c, i.e. every point d belongingto (ahc)and not coinciding generalpointd precedesa generalpoint e if dQaJ) to the segment (al)c) and eQajfi or ^ dTaJj and eT^c, Le. if d belongs if 6,and d are separated and e to the segment (acd), or by a and c, and It is then shown, that of any two points likewise c and e by a and d. of the line, one precedesthe other,and that the relation is transitive all the pointsof the line acquirean order. and asymmetrical ; hence introduce our we can points, Having now obtained an order among Pieri f givesa form analogous to which to that axiom of continuity, an of Dedekind's axiom, namely: If any segment (abc) be divided into two parts h and fc,such that, with regardto the order abc, every pointof // precedes every point of Ar,while ." and k each contain at least one point,then there must lye in (abc)at least one point x such to A, and every point "r belongs that every pointof (abc)which precedes of (abc)which follows "r belongsto k. It follows from this axiom that and having no last (or first) term infinite class contained in (alx-) every has a limit,which is either a pointof (abc)or r (or a) ; and it is easy to such pointas there can be onlyone h and k are given, prove that,when such that dTa"Jb;

"r

a

in the axiom.

By

means

tetrahedra.

of the Three

and segment, it is easy to define triangles protective which between them pointsdetermine four triangles,

to me) last property affords an instance (almostthe only one known Relatives, of outside relative the Peiree's addition occurs t(dTaee or Algebra This property is the relative sum of 7^ and T1^, if rf,**,and /" be variable. cT'rtc/;" *

This

where

results t

formallyfrom regardingT^ at. " H, p. 7.

Op.

as

the negationof the transitive relation t}^

all the

contain the

without properties

complete planealwayshave

in the

of three

Geometry, namely

our

dimension*.

This

a

space

:

This

of

n

dimensions

by denying precedes, of

or

to the axiom

amounts

is altered,in what

Nothing

to it,and proceeding

of dimensions.

point.

common

a

prove

Only one other axiom A planeand a line not

further axiom*.

any

to required

pointsexcept

common

no

transformations,and

define harmonic

can

we

have

plane,and

the

pointsof

angles. Also

their is

[CHAP.XLV

Space

388

infinite number

an

than

fewer axioms last,in fact,requires

a

of

space

three dimensions + .

I^et

368.

Geometry on

a

us

in which be developed,

may

line,and

examine

all the

direction in which

other

the

resume

now

start

we

do

We

constructions. quadrilateral

successive

these

from

pointsobtainable

protective

three fixed

from

here,

not

points

three as

by

in

the

;

but

axiom developmentwe have been examining,requireany new In restriction in the results obtainable. there is a corresponding its fullest possible to give protective developmentwe Geometry

order must

the results of both directions.

combine

line,let us see to begin with, to one straight Confiningourselves, coordinates. how to construct a net and introduce protective Denoting and d are harmonic "a conjugates by aHfrd, as before,the proposition with respectto b and c,"we can, by the quadrilateral construction,when d determine the this proposition. onlypoint satisfying a, ", c are given, We next construct the pointe for which bH^e^ then f for which dH^f^ In this way we of obtain a progression g for which eH^g, and so on. pointson our line,such that any three consecutive points, togetherwith c,

form

harmonic

a

pointswill

number

these

does not

With

range.

all these

belongto

next

the

pointsobtained

and d'HavCylet e'Haabtf'Hajd'i

a

so

definition of

segments (abc)and

points,beginning with we belong to the progression,

Consider

of

former

our

the

on.

as

a,

may

0, 1, @,

(bca). ft,

...,

We

may Since c

it the number

assignto

follows. We

segment,

a

d' be

Let

have thus

a

new

oo

J.

such that

progression

points,such that any three consecutive pointstogetherwith a form harmonic range, and all belongingto the segments (abc),(cab). To

we can pointslet us assignthe numbers 1/n in order. Similarly construct to the two a progression belonging segments (cab\ (tea),and the them to assign negativeintegers.By proceedingin a similar with any triad of pointsso obtained,we obtain more and manner can more points.The principle adoptedin assigningnumbers to points has no motive which, from our present standpoint, (a principle save

these

*

These

t

developmentswill be Fieri," 12.

J

We

must

not

assignto

further axioms,that Fieri's three

axioms

ordinal relation to

c

found

in

Fieri,loc. tit. S" 8,

the definitenumber

is the limit of

""",since

we

10.

cannot

assume,

without

progression. Indeed, so long as we exclude above mentioned, we do not know, to begin with,that c has any the terms of our progression. c

our

367-369]

ProjectiveGeometry

389

if convenience)is the following :

assignedto p, 9, r be the numbers and be the number to be assigned pointsalreadyconstructed, s to the harmonic of the not conjugate(supposed constructed) previously

three

with respect to gr-point D

n

"""

^

have

qt

"

point of Thus

"

r

=

for each

line a

these

course

assume

however

often

a

rational

denumerable

Whether

or series,

of

In this manner,

1.

"

decide without

cannot

p-point and

the

r-point,then

to

are

we

we

'

find

can

and

one

onlv

one

s

"

obtain

we

the

?

~-

our

line.

our

i)

*-;"

"

r

'

number, positiveor

compact series of pointson points of our line or not, we

all the

are

further axiom.

a

negative*.

endless

collection of the

If

line is to

our

of the

power

be

continuum,

tinuous con-

a

must

we

constructions, pointsnot obtainable by quadrilateral which repeated,

the definition of

with

start is

three

But

given elements. if

like,content we rational space, and introduce an axiom to the effect that all pointsof our line can be obtained from three givenpoints.

as

ourselves with

is very

think,even

I

we optional,

proceedingfurther, it

which

logical error,

space

may,

a

Before

369.

our

by

apt f.

Klein

be

may

well

So

long as

point out

to

be committed, and

to

has

been

a

mitted, com-

above

PierTs three axioms

assumed, our pointshave no order but that which results from the net,whose construction has justbeen explained.Hence such as, starting have from three given points, onlyrational points(i.e.

enumerated

are

not

can coordinates)

rational

be there can points, rational points, nor For

them.

a.

have

no

in which

sense

for

reason

any

limit and

order at all.

an

the

these

can

If there

other

be any

be limits of series of

assigningirrational coordinates

series which

it limits must

both

to

belong

series ; but in this case3 the rational points form the whole to some one other points(ifthere be any) cannot of the series. Hence be assigned as

limits of series of rational

merelyfrom springs form

the habit of

series,without Indeed,justas

a

axiom.

points. The

notion

can

be done

assuming that all the pointsof

statingthis explicitly we

that this

found

that

or

its

a

line

as equivalent

an

have series of rationals properly

they happen to have a rational limit,so series of pointsobtainable by the quadrilateral construction will not have struction, conlimits,qua terms of the series obtained from the quadrilateral except where they happen to have a limit within this series, fore, thereAt this point, i.e. when their coordinates have a rational limit. of virtue in PierPs three it is highlydesirable to introduce axioms, order. We which shall find that,in all the pointsof a line have an

no

limit

except when

natural order co", the

the

Pierfs

*

On

axioms, is the

this

same

subject,see

order of the rational as

that

of their coordinates

Slw Klein, I'ortexungen

where proofs will l"e found, p. 338 if., t e.ff.Op. cit. p. 344.

points,resultingfrom on assigned

nicht-Euktidimhe

Geonwtrie*

the

[CHAP.XLV

Space

390 Thus principle*.

above

of rational

that

that all infinite series

assume

parts of Pieri's series,and that all in order to show limits of rational series,

or

continuityin Cantor's

line has straight

our

onlyto

limits,as

pointshave

either rational

pointsare

have

we

In

sense.

shall assignto non-rational pointsthe irrationalnumbers limit. to the series which such points

Returningnow

370. the

anharnwnic

the

number

pointswhose

ratio of four

f^-3- ?

*

~

It

r

be

can

.

r-qj

coordinates

we

define

are

as

p, 9, r,

that this number

shown

case

corresponding

construction, we quadrilateral

the

to

this

*

is in-

-s

original pointsa, 6, c. It expresses constructions requiredto obtain * when /", the series of quadrilateral relation of the four r are given,and thus expresses a purelyprotective "jr, just points. By the introduction of irrational points,in the manner it follows that any four pointson a line have an anharmonic explained, dependentof

the choice of

PierTs three axioms

be provedwithout possibly

(This cannot

ratio.

three

our

or

ratio is unaltered

anharmonic to them.) The by any equivalent the for linear transformation,i.e.by substituting every point x point B fixed are 8, 4- S), where whose coordinate is (ar-|-/9)/(y"r a, 7, any From this point we such that aS at numbers can /3y is not zero. the beginning of projective last advance to what was formerly Geometry, its name. to which it owes namely the operationof projection, if be shown with respectto harmonic It can be that, p9 r conjugates pointo, and if op" 05, or, o* meet y, *, and /?,y, r, * be joinedto some with respect r #',then p\ r are harmonic conjugates any line in p'9g', show that all anharmonic Hence we can ratios are unaltered to q\ /. if I be any straight line not coplanar by the above operation.Similarly with pgr", and the planesZp,Zy,7r,Is meet any line not coplanarwith / in p\ q',r',/?', these four pointswill have the same ratio anhannonic These facts are as /", gr, r, s. expressedby sayingthat anharmonic ratio is unaltered by projection. From this point \ve can proceedto the assignmentof coordinates to any pointin space f. To begin with a plane, 371. take three points#,-",r not in one and assigncoordinates in the above manner to the points line, straight of "6, tic. Let p be any pointof the planeah\ but not on the line ftr. if Then ab in /",, and bp meets in j?2, and #, y are the or rp meets coordinates of p19 pz respectively, let (,r, y) be the two coordinates of p. In this way all pointsof the plane not on be acquirecoordinates. To avoid this restriction, let us introduce three homogeneous coordinates, follows. Take as three of any four pointsa, 6, r, e in a plane,no some

"

,

which *

are

This

collinear;let has the

and constructions,

device of + See

givingr

one comes

exception that

c

came

-

oo

Ge"mftri*,"

instead of "2 ;

m

in r2,

last in the order of the

firstin Fieri's order.

the coordinate

Pasch, Newr?

be in el9 be meet

meet

ae

This cc

may

be remedied

.

Klein, Math.

Annalru, 37.

ce

meet

ab

quadrilateral by the simple

369-372] in "?3.

Protective Geometry

Assign coordinates

391

the

pointsof AT,ra, ab as before,giving giving0 to ay and oc to 6, and of coordinate *r of any place the single pointof "c,let us introduce the homogeneous coordinates ,r2,,r3, where x Xilxi. If now p be any pointof the planeo"r,let ap meet be in/;,, ip meet era in /?2, and cp meet "" in /"3. Let *r2, "rs be the homogeneous coordinates of pl9 x^ xl those of p2; then "r15 #2 will be thase of /;3*. Hence In we may assignoc^oc^ x3 as homogeneous coordinates of p. like manner four coordinates to any point can we assign homogeneous also assign We of space. coordinates to the lines througha point, can the planes of the or through a line,or all the planesof space, by means anharmonic ratios of lines and planest. It is easy to show that, in and a linear equation a point-coordinates, planehas a linear equation, a point has a linear representsa plane; and that,in plane-coordinates, and linear Thus till a we secure equation, a equationrepresents point. the advantagesof analytical Geometry. From this pointonwards, the interest. and ceases to have philosophic subjectis purelytechnical, It is now time to ask ourselves what portions of the Geometry 372. accustomed are not included in project! In to which we Geometry. are ve the lirst place, the series of pointson a line,being obtained from a to

coordinate 1 to *19 12, e3, and for the other sides. In similarly the

in #Z"

"

is closed relation,

four-term of

three pointsrequires effect of The practical one

no

between easy to

of them

fixed this

is between

ve project! space

exaggerate

in the

of Part

sense

pointsto

be

givenbefore

is,that given only the other two.

and Euclidean

this difference.

three

This ib

a

it

can

is,the

in Part

saw

order

be defined.

pointson

a

line,

definite difference

hyperbolic .space.

or

We

That

IV.

IV

But

it is

that,wherever

there is also the four-term relation,

seriesis

generatedby a two-term of couples, by which we can generatea closed separation series consisting of the same Hence in this respectthe difference terms. does not amount Euclidean and hyperbolic to an inconsistency. spaces besides. contain what project! and more ve contains, something space line is defined that the relation by which the projective We saw straight a

relation of

has the

formal

is transitive and metrical. asymshall ha\e said relation be actually of this form, we

of properties

If the

"P

or

P*

where

P

will to -P,and of three eollinear pointsone open .serieswith resj)ect l"e l"etween the other two. It is to be observed that,where the .straight

an

l)e closed,as in elliptic essentially space, ftetween must excluded where three pointsonly are given. Hence elliptic space, in is not this respect, axioms, but only consistent with the projective contains nothingmore than they do. It is when to the plane that actual inconsistencies arise come we line is taken to be

*

Sec Pasch, toe. cit.

t The

anharmonic

line is that of the four

ratio of four lines

pointsin*which

through

they meet

a

pointor

any

line.

of four

planesthrough

a

between

and

lines

two

any

intersect

of

or

definition

of

render

of

be

discussed

be

in

few

A

projective

of

place

in

saw

would

from

all

generate

duality *paces

the

spaces,

than

that

of

point.

belongs

XLIV,

a

this

smallest series

have

is

of

of

and

from

in

a

of

that

the the

planer

two

point,

even

is

matter

of

cannot

when

shall

we

distance

of

to

a

Such

that

from

to

dimensions those

spaces,

duality.

planes

a

it

duality,

in

required

accordingly,

space as

we

such.

as

is

the

taking

projective

projective

and

is also

before,

as

series

which

dimensions,

of

planes

similarly,

conjecture)

of

of

being,

7Z-dimensional

axioms

to

principle class

non-collinear

three

two

that

other

this

theory

dimensions.

always

additional

which

of

difference

the

But

when

virtue this

Geometry,

dimensions,

than

more

properties series,

of

only

in In

concerning

(;/"!)

number

flows

limitations.

added

n

of

determined

is

though

points

two

passible.

intersection

sub-classes

therefore

?i-dimensional various

two

projective

that

axioms

projective

dimensions,

In

which

space,

the

intersection

the

(though

seem

be

either

metrical

present

three

assumed

inconsistency.

an

also

may

the

and the

Chapter

employs

of

hyperbolic

measurable.

examined

at

in

states,

line,

straight

is

words

space,

not

examine

than

principle

results

they

projective

the

complicated

satisfy

projective

have

we

position

a

373.

the

the

render and

more

distance,

numerically

from

to

is

it

stretches

that

or

addition,

an

until

advantage

This

spaces

called

become

differs

space

nature

more

far

form

intersection

the

to

Euclidean

to

Geometry relation

given,

are

divisibilities

elliptic

to

metrical

in

points

their

these

of

theory

quantitative

a

inapplicable

satisfies

differences

These

elliptic Geometry,

space.

Finally, have

apply

Euclidean

antipodal

the

wholly

form

polar

plane.

a

plane

a

the

projective

the

only

and

line

a

the

considerations

Analogous

planes,

the

Thus

in

XLV

Geometry.

the

in

In

occur.

but

intersect;

plane

a

twice.

axioms.

and

in

not

hyperbolic

or

intersect;

plane

a

does

this

Geometries,

hyperbolic

in

lines

two

any

space,

Euclidean

and

Geometry

protective

projective

In

a

[CHAP.

Space

392

It

Geometry possible

that

projective

general.

Other

make

them

to

to

duality

is

liable

CHAPTER

XLVL

DESCRIPTIVE

374.

Geometry is descriptive Geometry. These protective

from rule,sharplydistinguished

a

as

called

I have

subjectwhich

THE

GEOMETRY.

terms, and

the

term

synonyms.

But

it

the

is not such

one

present chapter.

pointson other

a

line

We

include

to

by projection,and

abc,c

it is

by

the

property that I wish to define the subjectof have that, in protective seen space, three that

a

definite

of them

one

is between

the

simplestpossiblepropositioninvolving between^

segment (abc\

the order

as

ve project! Geometry

in

ve Geometry, requiresfour points,and is as project! ft,c be distinct collinear points,and d is on ac" but

to the

two

used

commonly

Position,1" are

unaltered

such

not

are

The

two.

improper

seems

which

any property introduction of

of

"Geometry

not,

yet coincide

nor

is between

b and

of

the

of

in perspective we triangles

of

generating order

d?

with When

a

we

or

c,

in

follows:

"If

does

belong

not

then, with

#,

regard to

reflect that the definition

construction" which segment (abc) involves the quadrilateral demands, for its proof,a point outside its own plane,and four pairs shall admit

"

ordinal

elementarysense in the

result

of between, on

present chapter,is

in

the

descriptive Geometry, we before,any two pointsdetermine

*

only of

The

the

are

start, a

points between

unaltered

method protective any

the

rate

be introduced

is to

unaltered as

by projection*. before, with points,and

as

points. But this class now is to given points. What

class of the

at

by projection.The

contrary,which

generalnot

In

consists

the

complicated.But

is somewhat

propositionswhich

that

two

present subject is admirably set forth by Pasch, Neu"re

Geometric, Leipzig,

for preferring it to proempiricalpseudo-philosophical reasons 1 jectiveGeometry, however, (see Einleitung and " 1). It is by no means agree carried further, especially as regards the definition of the plane, by Peano,

1882, with

1

whose

Principii"ti Geometria

whole

line

by

Matematicdj "6. iv, p. 51 Riv.

of

pp, -58-62.

n,

its various See

found

n,

pp.

in the

71-75. above

also

1889.

segments,

his article ""ui

see

For

Peano's fondamenti

the

definition

note

in

of the

Rivista

di

della Geometria/'

della retta," Principifondamentali della Geometria Whatever, in the followingpages, is not controversial,

ff.,and Vailati,"Sui

d. Mat.

will be

means

Turin, logwamentttesposti,

sources.

[CHAP.XLVI

Space

394 be understood

except Vailati,in

terms

a

this

on

subject

asymmetricalrelation of two

transitive

condemned

by Peano*, on the of two three points,not only. and is relevant ireven inadequate IV,

is explanation

Vailati's

points; and

of

writer

by any explained

is not

between

by

ground that between is a relation of know from Part This ground,as we the best mathematicians of relations, even But on the subject with the Logic of Relation*. for want, I think,of familiarity go astray, Geometry, we may start In the presentcase, as in that of protective with a relation between a or either with a relation of two points, pair and is equally either method legitimate, leads to and a class of points : far is fiist but the former simpler. Let us examine the same results, method, with

start,in the former

We

375.

Peano, then that of Vailati.

of Pasch and

the method

indefinables, point,

two

be three points, and c is between a and 6, we c /", ab, or belongsto the class of pointsab. Professor say that c is an has. enumerated, with his usual care, the postulates Peano requiredas

and

IxtiiMen.

regardsthe

If

#

is between

c

b) is If

",

such

they are

between

not

so,

b and

Z",it is also between

and

a

the pointsa and /; must be place, there alwaysis a pointbetween them.

In the first

ab"\.

class

and when distinct, If

,

and

a

distinct

b be any two that b is between

a

We

b.

and

a

:

introduce

now

then points,

a

is the

ab

itself (and therefore a

definition.

new

class of all

b'a will be Similarly

i\

pointsc

the

class of

pointsd such that a is between b and d. We then proceedto new If a and b be distinct points, ab must contain at least one postulates. and c is between a and (I,b between a r, d be points, point. If a, Z",. If b and

and r, then b is between a and cL and r, or identical with between a then a'fi,

to

is between c

b and

is between

IxtwecH. are

The

either

a

Peano

c.

and

c

and

d

are

This

d.

between

r, or

identical, or

If b is between

c9

r

and

he i" unable

r

and

a

If r, d /; and

d.

is between

c7,b is

belong rf,or

d

b and rf, then

with regard to postulates to

hence theyare onlyshown to independent: completestraightline (db) is defined as

and

is between

c

in all nine

makes

that confesses!

and

a

be between

c

prove

that all of them

be sufficient, not necessary. b'a and a and ab and b

and

tlmt is,(1) points between which and /; the point a lies; fl'fr; (") the point// ; (#) pointsbetween a and b ; (4) the pointb ; (o) points

between

which

and

a

the

point/;lies.

Concerning this method,

we

very complicated.In the second the phrase"two pointsdetermine

follows:

"There

is

observe

may

we place, a

to

must

class of

begin with

that

it is

that remark, as l)efoiv,

points"must

be

expanded

certain

relation A\ to \\hose domain specific of distinct A" is a many-one and Ix'Iongs points every couple relation, the relatuin, to of a class i s of corresponding couple pointsas referent, a points." In the thin! place,we may observe that the points of the

as

*

Kir. di Mat.

iv,

a

p. ""2.

+

fk

jv,

p. .55 if.

J

fft.p. 62.

374-376]

Descriptive Geometry

onlyacquireorder by

line

and

inclusion. logical of pointsab or b of

a, 6 be

Let

any two points,and consider the class Let r, d be any two distinct pointsof this proper part of ad, or ad is a proper part

a'b.

or

either

Then

class.

relation to the segments which theyterminate, by the relation of whole and part, or

acquireorder

these

that

395

is

ac

a

and ad may be called segments,and we originis a and whose limits belongto ab or b or Here

ac.

whose

ac

that

see

segments

a'b form

in virtue of the transitive asymmetricalrelation of whole

a

series

and part.

By

correlation with these segments, their extremities also acquirean order ; substitute and it is easy to prove that this order is unchangedwhen we for

a

from

ab'. But the order still results, it always must, as any pointof relation of two transitive asymmetrical a terms, and nothing is

gainedby not admittingsuch 376. a

Passing now

to

relation

a

what

between points. immediately called Vailatfs

I have

We simplification.

theory,we

find

the

present theory(which may There ij" in every detail identical with that of Vailati) as follows. relations. certain class,which we will call JT,of transitive asymmetrical very great

state

is not a

Between

pointsthere

any two

is

and

one

relation of the class K.

only one

relation of the class Ar, R is also a relation of this class. such relation R defines a straightline; that is,if a, b be two

be

If R

a

Every pointssuch that aRb, then a belongsto the straightline p. (I use of a relation ; Greek letter to denote the domain corresponding if S be term

cRb

or

;

relation, is the class of terms having the relation S other.) If "J?6,then there is some pointc such that

a

"r

also there i.sa

distinct

have all that

apparatus we We

is not

relations; and

(1)

Further, if a, Jb be any two With this either aRb bRa. or

p, then

require. "

remark

I may

class of relations

we

and

formallythe above definition of the for K itself concerningits members postulates

the

rather

defined.

axioms

some

do well to enumerate

may

class JT, or

we

aRc

thus

that bR d.

pointd such

points belongingto

to

the

the

as

There

is

begin with that

be the

class,I denote

I define the field of

fields of the

a

constituent

its fieldby k\

Then

the

follows.

as

class of relations

a

of the

sum logical

that,if K

requireare

to

A*,whose

field is defined to be the

class point.

(2)

There

If R

be any

(3)

R

is

(4)

R

is a term

(5)

"

R.

(6) (7)

p

=

is at least

an

of K

term

point.

one

have,

we

aliorelative. of K.

(thedomain

Between

any

of

is contained in p. points there is one

R)

two

and

of the class K.

(8)

If a, b be

pointsof

/",then

either uRb

or

bRa.

only one

relation

point,and since by (1) this point JT, and since by (3) all relations of the

there is,by (2), at least

Since has

proofthat

the

sketch first briefly

of the

relation

some

class K

one

class

pointhas

point,to which this one But since It,by (4),is a relation of the the one to which pointis so related is distinct points. Let a, b at least two

the

one

be the

R

But

aRb.

do

we

should

we

relation

one

bRa,

have

not

a

also be

R

with R, together

or

for if

that

by

there is

R

there

pointc

some

there

field of

R

what,

Part

IV,

fall apart into two terms

unless in #t

of

relation

one

one

might

are

and bd.

ac

geometry

inserted here

of

of

more

onlyone

377. is to of are

With

it is

line,while than

be observed

p

not

the

argument /",and

=

field of

the

such

of

that

more

class A'

the

cannot,

point

one

is called

lines have straight

two

that,if ah, cd be

us

its

than

more

than

not

of

two

relations

is

R

the

a

at

same

proved that our axioms are sufficient (7) goes beyond a singleline,but is implythe existence of pointsoutside relation of the

one

above

the

mutual

cla^s AT.

enumeration two

as

in

It is mobt

fundamentals,

of

Peano's

system.

independenceof the

axioms,

it

that

(1) is not properly an axiom, but the assumption indefinable A'. (2) may obviously be denied while all the others our maintained. If (3) be denied, and R be taken to be the symmetrical

relation

of

of /", we

obtain

with Geometry, togetherwith identity projective which is from different projective Geometry,

system, but self-consistent. If (4) be denied,all the rest ; n

generated

impliesthat

but of any portions, after. By (7),if there be

us

implies

series,that is, it does

indefinable,namely A", not

regard to

have

we

by (5), always imply

aRb

R\

=

let

the relation

a

(8), the

By

relation

one

important to ol"serve that,in there is

end.

assures

because it does

singleline,or

;

A", the fields of two of the other, have

Thus a

to

bRc

i.e.the series

Thus

(7) point,while (8) assures

common

for the

a

thus

cRb

aRc,

dRa.

other

field of

Ihte; and

one

line,so

converse

Since R

are

and

aRb

have

are

have

we

R,

=

and R

R

R,

=

RZ

detached

the

class

The

common.

most

the

is the

R*

connected

a

more

or

is before and

one

that

Thus

(6),p is contained in p, aRb that bRc. Applying the same

called

we

a

there

points,and

and b.

did, since

we

series.

a

point c such is a pointd such that has no beginningor

some

in

such

Since, by

compact.

is

R,

to

is

form itself,

Hence

distinct

pointswhich

the

Hence

a

point.

a

two

is asymmetrical.Since

Le. each different, i.e. R is transitive. rt/fc,

than

of the class A~

class AT, it follows that the term

contradicts (3). Thus

aRa, which

have

term, other

some

relation R

between

class K

of the

is

there

that

it follows aliorelatives,

are

is easy to see. But let theygive all the requiredresults.

these axioms

of independence

mutual

The us

[CHAP.XLTI

Space

396

term

indeed

the of

is as regards(7),for onlydifficulty

A',b will not

it has

one

which

have is not

to

a

any

#, which

relation seems

can

if aRb, and of the

to

some

the

class

term

present

be

tained main-

R

is not

A", unless

be not contradictory.

376-379]

Descriptive Geometry

regards(5),we

As

397

deny either that J? is contained in R\ or that Rdeny the former makes our series not compact, to there is no logical objection.The latter,but not the former, is which can be made to satisfy all the other axioms regards angles*,

is contained which

falseas

may

in R.

here laid down.

To

(6)will become

false if

lines have last terms

our

thus

:

the space on the left of a plane,togetherwith this plane, will satisfy all the axioms As it is except (6). regards(7), independent plainly

of all the rest

;

it consists of two

pointsthere

two

there is not

that If

is at least

consider

we

a

one

than

more

parts,(a)the assertion that between any relation of the class A",(b)the assertion such relation between

one

Euclidean

and

given points. hyperbolicspace together,all the

a

two

will be true except (a). If we combine two. different classes A'15A"2 of relations of the above kind, such that A-^fc,, (h) alone will be false.

axioms

Nevertheless it axioms.

plainthat (b) cannot

seems

regards(8),it

As

be

alone is false if

deduced

we

take

from

the

the

for K

other

class of

directions in Euclidean the

lines all have space, in which a set of parallel Thus the necessity of all except one of our axioms

direction.

same

that of this

and is strictly proved, We

378. with

that the above

saw

further,and the

Hence

that every member then

axioms

indefinables

we

of

C

become

is

a

enabled

K

to content

us

indefinable^. statement

But The

in terms

class C of classes of

class of relations

a

parts of If

axioms.

nor

define

can

highlyprobable.

class of relations K.

with dispensealtogether class K were all capableof

of relations.

The

method

indefinable, namely the

one

is

one

a

be any

member

we

may go axioms cerning con-

of the

we

of the

logic

such relations,

our satisfying

and definition,

ourselves

have

axioms.

neither

class C, and

of Ar be the a A", then descriptive space, and every term k is a descriptive point. Here every concept is defined in terms of general be applied method to projective can logical concepts. The same space, I* is

field of

entityexcept the indefinables of logic. This is, indeed, though grammaticallyinconvenient, the true way, notions. Outside logic, to define mathematical speaking, philosophically matics, not required indefinables and primitive are by pure mathepropositions This and should therefore, not be introduced. speaking, strictly subjectwill be resumed in ChapterXLIX. line that of Pasch and The two ways of defining the straight 379, that which have I Peano, and equallylegitimate, justexplained seem or

to

any

mathematical

other

"

"

and

lead

therefore of

no

The

choice consequences. mathematical importance.The two

the

to

same

between

them

is

methods

agree

in

enablingus, in terms of two pointsonly,to define three parts of a the part between and b a line,namely the part before a (#V/), straight and the part after b (ab). This is a point in which descriptive (/*/;), Geometry : there we had, with respect Geometry differsfrom projective *

See

Part

IV, chap.xxiv.

to

a

not

line ab" and these could and 6, onlytwo segments of the straight reference to another be defined without pointc of the line,and

to the

construction. quadrilateral

line straight forming the field of The

the sake this

word

lie

has on

or

or

as

the class of

as

this

The

I

come

ray, since

oppositeray. In passingthroughone point0, it will of pointsto which ray also to the class those pointsof a line through O which

of

name

Those

context now

a

of lines all

side of O.

one

points

relation itself. For

be the

will then

R

sense;

relation #, i.e. to

some

380.

relation #,

a

suggests a

rav. opposite

in

regardedeither

be

may

it will be well to call the relation R of distinction,

number a considering be well to give the O

[CHAP.XLVI

Space

398

on

the other side of O will then

will show

to the

in which

plane. Easy

as

sense

the word

be

the

is used.

it is to define the

plane

the line is not a closed series, protective space, its definition when some rather,when we wish to call coplaiiar pairsof lines which

Pasch* takes the plane, difficulty. indefinable. It is, rather a finite portionof the plane,as a new or I of shall show. now Peano, however, capable definition, as, following axioms. We new First,if p be any need, to begin with, some straightline,there is at least one point not belongingto p. Next, if #, 6, c be three pointsnot in one line,and d be a point straight of be between h and r, e a pointof ad between a and d, then be will meet in a pointy* and e will be between b and f9f between a and c. Again, ac be a pointbetween a and c, then ad and bf a^b^c^d beingas before,if/' will intersect in a pointe between a and d and between b and /*+. We the product(ina geometrical define what may be regarded now as sense) ok is to of a point and a figure.If a be any point,and k any figure, denote the points which lie on the various segments between a and the pointsof k. That is,if /; l"e any pointof Ar,and tr any pointof the segment ap, then x belongsto the class aAr. This definition may be appliedeven when a is a pointof k9 and " is a straightline or part of one. The figure ok will then be the whole line or some continuous of it. Peano now transformations, portion proves, by purelylogical that, if a, 6, c be distinct non-collinear points, a (be) b (ac). This is called the triangle alx\ and is thus whollydetermined by its figiire is three defining also shown that, if jp,q be pointsof the points.It the segment segments 06, ac respectively, pq is wholly contained in the triangle After some abc\ tion. definimore to a new theorems,we come If a be a point,and k any a'k is figure(i.e.class of points), to denote all the pointsbetween which and a liessome pointof A?,that

is a matter do not intersect,

of

some

=?

remarks, the whole shadow of k if a be an illuminated a' (be)will representthe point Thus if ", 6,c be non-collincar points, is,as

Peano

class of

pointsbeyond *

(q".mt. " 2.

fa and

bounded

t

by (ib^ac Kiv.

di Mat.

produced. This

iv, p. f"4.

379-382] enables

us

Descriptive Geometry define the

to

"/;,the

/"r, m,

399

of plane(abc)as consisting and

#"', triangle

lines straight

the

the

a'ftc, 6'ra,r'o",6V'a, c'a'b^ figures a'b'c*. It is then easy to show that any other three pointsof the plane define the same plane,and that the line joiningtwo pointsof a plane lies whollyin the plane. But in placeof the proposition that any two lines in a plane intersect, have a more we complicatedproposition, d If be : 6, namely coplanarpoints n" three of which are a, c, collinear,then w/, be do so.

either

the

lines a/",cd intersect, or

defined Having successfully

381.

Ixl do

or,

so,

or

the

advance plane,we can now axiom : the to we need, begin with, Given any plane,there is at least one point outside the plane. We then define a tetrahedron exactlyas we defined a triangle.But can that two in order to know planes,which have a point in common,

solid Geometry.

have

line in common,

a

are space we this axiom was

pointin is

For

simplythat But

given by

Peano

on

plane,or

the

has a

a

three

dimensions.

line and

here,no

such

shows

axiom, which

new

In

that

the

projective space,

planealwayshave at least one

a

simpleaxiom

holds.

The

following

(foe.cit. p. 74) : If p be a plane,and a a point a point of dp (i.e. point such that the segment ab

pi and b a contains a point of p, side of the planefrom not

to

need

we

dealingwith

common.

this

segment

segment by contains

or,

in

ax

if

contains

the

language,a point on

common

a),then

be any

x

point,either

x

other

lies on

the

point of the plane,or else the plane. By adding to this,finally,

a

point of the have all the apparatusof three-dimensional axiom of continuity, we an Geometry -f*. descriptive 382. Descriptive Geometry, as above defined,applies equallyto of the axioms mentioned Euclidean and to hyperbolic space: none discriminate between these two. Elliptic space, on the contrary,which included in projective was Geometry,is here excluded. It is impossible, rather,it has hitherto provedso, to set up a generalset of axioms or which will lead to a generalGeometry applyingto all three spaces, for at

some

series of

point our

*

axioms

must

lead to either

an

open

or

a

closed

generalGeometry can be constructed this results from to givingdifferent interpretations indefinable* in one being definable in interpretation This will become plainby examiningthe method vice versa.

pointson but symbolically, our symbols,the

another,and

a

a

line.

Such

a

//" and both ca or b'c'a,represents tbe angle between figureb'(c'a)y produced,as may be seen from the definition. t I confine myself as a rule to three dimensions,since a further has extension little theoretic interest. dimensions than two, far more Three are interesting have seen, the greater part of projective as I"eeause, we (reometry i.e. everything the construction is dependent upon impossiblewith less than three quadrilateral be taken construction as an dimensions,unless the uniquenessof the quadrilateral

The

"

"

axiom.

in

[CHAP.XLVI

Space

400 ve Geometry project!

which

to applicable

is made

the

space

abme

I shall call

descriptive space. to descriptive Geometry we applyprotective space, in of the tion construca that the some met required points are we difficulty by Thus in the quadrilateral construction, not exist. giventhree mav We exist all. at can prove as points#, 6, r, the fourth pointd may not

defined,which, for When

383.

of

want

a

better name,

tryto

it is unique,and so with other protective before that,if it exists, propositions the indicated construction since is not become : they hypothetical,

alwayspossible.This has led to the introduction of uhat are called of which it becomes lines and planes), ideal elements (points, by means ideal T hese theorems elements to state our generally. projective possible in numbers have a certain analogyto complex Algebra an analogy close. Before explaining becomes which in analytical Geometry very "

in detail how

these elements

are

introduced,it may

be well to state the

of the points, lines and planes of the process. By means nature logical of which of descriptive Geometry, we define a new set of entities,some lines and have a one-one relation)to our points, planes correspond(i.e. entitiescall These ideal others while not. do we new respectively, find that they have all the properties lines and planes ; and we points, of projective points,lines and planes. Hence they constitute a projecSince our tive space, and all projective propositions apply "o them. of the elements of descriptive ideal elements are defined by means space, ideal elements theorems these are concerning projective propositions actual its not concerning concerningdescriptive pointy space, though of the way in lines and planes. Pasch, who has given the best account which ideal elements are to be defined*,has not perceived (or,at any where even rate,does not state)that no ideal pointis an actual point, holds of it has a one-one relation to an actual point, and that the same lines and planes. This is exactly the same remark as we have had to make concerningrationals, numbers, real numbers, and complex positive numbers, all of which are supposed,by the mathematician, to contain cardinals

the

the

or

of the cardinals

one

identical with the air of 384.

or

actual

an

magic

ordinals,whereas

which

ordinals.

point,line

So or

no

one

here, an

of

them

be

ever

can

ideal element

is

If this be borne

never

in

mincl, plane. expositions disappears.

surrounds the usual

ideal

pointis defined as follows. Consider firstthe class of lines passingthroughsome This class point,called the vertex. of lines is called a sheafof lines (Strahknbiindd).A sheaf so defined has certain properties which can be stated without reference to the Such vertex t. are, for example,the following: Through any point (otherthan the vertex)there is one and onlyone line of the sheaf; and of a sheaf, any two lines of the sheaf are coplanar.All the properties An

all the

*

Op. tit. "" 6-8.

t These

1898),p.

are

8-2.

enumerated

by Killing, Grnndlagen

dvr

Vol. Geonietrie,

n

(Paderborn,

which to

Descriptive Geometry

385J

384,

be stated without

can

certain

follows*.

Let

having

these

For

be any

Z,

m

the

plane.Then

this

reference to the vertex, are

classes of lines

the class intersect.

401

vertex, and

no

simpleconstruction

a

lines in

two

such

planesAly Am

line in

a

such lines,for all possible pointsA outside the

belong

that

two

no

of

be

can

plane,A

one

have

found, to

any

given,as pointnot in class of

The

common.

planeZw,has

the

properties

alluded to, and the word sheafis extended to all classes of lines the sheaf has a vertex ; defined. It is plainthat if Z, m intersect, so to if not, it has none. Thus, in Euclidean space, all the lines parallel

above

a

givenline form

sheaf which

a

has

When

vertex.

no

sheaf has

our

no

of the sheaf. But this must vertex, we define an ideal pointby means for the not be supposedto be really a point: it is merelyanother name so, when

and sheaf itself,

ideal

in which propositions

That

for its vertex. an

actual

two

lines straight having a

in

make

the sheaf

sheaf iit

no

A

is

line and

three planesof which Thus

a

observe

may

to both

common

at least have

two

of

the

followingpoints.

sheaves.

vertex, determine

planealwayshave

a

sheaves

relation to lines and

we

planeuniquelydetermine a sheaf. Two determine a line,namely that alwa)rs

one

vertex

least has

at

one

collinear,

385.

pointsoccur, we pointis simplya sheaf,and

of lines

which joiningthe vertices,

of which

to

are

we

substitute

must

point.

both

sheaves

vertex, if

a

ideal

is,an

Concerning sheaves Any

sheaf has

our

a

lines have

a

sheaf,and

by ideal lines. For this purpose of planes define pencils (axialpencils, Ebenenbusckel).An consists,in

first

the

instance, of

straightline,called the axis.

But

all the in

as

have

so

in protectiveproperties,

some

lines

with, replaceour

are

point.

common

which are lackingto points. In planes, further protective we must, properties,

obtain entities with

sheaves,

plane,unless thev

common

a

Three

we

order to to

axial

planes through the

case

of

begin first

must

a

pencil given

sheaves, it is

figurehas many properties independentof the all belong to certain other classes of axis,and that these properties of pencilis therefore extended. These planes,to which the name Let B of defined as followsf. be two sheaves lines. A, are figures found

that

Let D

be

a

such

a

point not Then

sheaves

A, B. call ABD"

on

the

line

A^ 5, D

(ifthere be one)

determine

common

to the two

a plane,which uniquely

we

those (say).This will be the planecontaining may lines of A and B that pass throughD. Any other point", not in the will determine a different plane ABE, or Q. The class of planeP, and has all planesso obtained,by varyingD or J5,is a pencilof planes, of a pencilhavinga real axis,except those in which the properties the mentioned. axis is explicitly Any two planesP and Q belongingto the determine it. Moreover, in placeof A and B above, pencilcompletely or

*

P

Pasch, op.

tit.

" 5.

f

Pasch, op.

tit.

" 7-

[CHAP.XLYI

Space

402

substitute any other sheaves of lines A\ one P and*Q. (A sheaf belongsto a planewhen plane.)Any two sheaves belongingto both P we

mav

to both B\ belonging

of its lines lies in the

and

Q will

to

serve

and will belongto every pkne of the pencil. of planes, j"encil substitute ideal points, i.e.sheaves Hence we if,in placeof actual points, o f of lines,every pencilof planeshas an axis, consisting a certain define the pencil This of which two collection of sheaves of lines, any

define the

is called

collection of sheaves

ideal pointsand Substituting

386. that

ideal litie*.

an

have

we

made

now

lines for actual

further advance

a

towards

ones,

find

we

projective space. a given plane

ideal line ;

and

only one pointswhich do not Wong to by any ideal line,but three ideal pointsdo not alwaysdetermine a plane. one ideal point,and so Two ideal lines in a planealwayshave a common have a planeand an ideal line. Also two planesalwayshave a common ideal point or and three planes ideal line, alwayshave either a common where is The line. ideal our not strictly a common only point space is There to is in a plane*throughany two regard planes. projective actual point,or through an ideal point and an ideal pointsand one actual line. If there is a planeat all throughthree non-collinear ideal pointsor throughan ideal line and an ideal pointnot on the line,then there is no such plane. such plane; but in some there is onlyone cases class of entities, To new remedy this,we must introduce one more namely ideal planes. The definition of ideal planesfis comparatively simple. If A^ 5, C D an ideal point on the ideal line AB, and be any three ideal points, line DE then has an ideal pointin common the ideal E on AC, with actual there determined be whether an by A^B^ C or not. BCy plane and E any other ideal point Thus if #, C, D be any three ideal points, ideal

Two

pointsdetermine

three

is determined

one

of its ideal

and so do BE, CD. BD, CE intersect,then BC, DE intersect, define the ideal planeBCD we Hence, if 5, C, D be not collinear, as

such that

class of ideal

that

pointsE

which

are

such that

the ideal lines AD,

CE

intersect. the sake of

For

us

repeat this definition in

lines ?and planes, without points, original

our

Given a

let clearness,

three

sheaves

pencilof

common

of lines

there is a sheaf of lines

the class of all sheaves

Weal

planeBCD.

For

purposes, logical

E

D, which be another

to

common

Then

*

j?,C,

let E planes,

the two

the are

use

of

terms

of the word

ideal.

all contained

not

sheaf of lines such of pencils

planesBD,

in

that CE.

this condition is called the satisfying

it is better to

define

the

ideal

line

as

the class of ideal

than as the sheaf itself, pointsassociated with a sheaf of planes, for be, as in projective Geometry,a class of points. f Pasch, op. ciL " 8.

we

wish

a

line to

385-387] The

Descriptive Geometry

403

usual

of planes are our easily properties provedconcerning new ideal planes, that any three of their pointsdetermine them, that as the ideal line joining is wholly two of their ideal points contained in that the new lines them, and so forth. In fact,we find now points, and planesconstitute a protective described space,with all the properties order of points in the preceding a on line, chapter.The elementary with which we has has order and to be disappeared, a new began, the of of Thus all by means separation couples*. projective generated ideal available lines and and wherever becomes our Geometry points, ; to actual ones, we have a corresponding projective planescorrespond the latter. proposition concerning this development 387. I have explained at length, partlybecause of projective it t"hows the very wide applicability Geometry,partly which mathematics because it affords a good instance of the emphasis irrelevant what laysupon relations. To the mathematician,it is wholly his entities are, so longas theyhave relationsof a specified type. It is that an instant is a very differentthingfrom a point; for example, plain, but to the mathematician

as

such there is no

the instants of time and the

pointson

the

a

relevant distinction between

line. So in

our

sheaf of lines

stance, presentininfinite class

an highlycomplexnotion of a of infinite classes" is philosophically dissimilarto the simple very widely be formed,having notion of a point. But since classes of sheaves can lines and relationsto their constituent sheaves that projective the same sheaf of lines in descriptive have to projective a points, planes space w, is It for mathematical purposes, a projective however, not, even point. for mathematical purposes, a pointof descriptive space, and the above shows that descriptive of transformation clearly space is not a species distinctentity.And this is,for philosophy, projective space, but a radically result of the the principal chapter. present of a projective which the remarkable above It is a fact, generation "

that if space demonstrates,

we

remove

from

projective space all the side of a closed quadricf,

a

all the points one on or plane, form a descriptive the remainingpoints space, Euclidean in the first metrical language, the in the ^second. Yet, in ordinary case, hyperbolic while the part of it which is descriptive is projective space is finite, nature of metrical infinite. This illustratesthe comparatively superficial

pointsof

a

notions. *

See

Pasch,op.

t For the

tit. " 9.

definitionof projective

a

surface of the second

order

in (quadric)

a

der Lage (Hanover, 1868),Part H, Lecture v" projective space cf. Reye, G"meine finesthrough not on it such that all straight is closed if there are points A quadric

them

cut

the

quadric.Such pointsare

within the

quadric.

XLVII.

CHAPTER

GEOMETRY.

METRICAL

treated

Euclid

by

subject includes

This

non-Euclidean

or

Geometry,

but

the

with

fallacies.

Metrical

which

have

rule,

a

as

in

only

examined,

assume

connection

differences

from

which

closed

series.

definition

of

assumed

similar

as

adds

further

as

avoidance the

to

develop

Euclid's

to

the

with

which

one,

shall

I

those

one

of

kinds

two

other

or

of

specifications. I shall,

and

in

Euclid

of

form

In

that

another,

or

the

the

the

the

or,

line

with

positions prois not

suitable

a

with

therefore

are

first,

for these

;

depend,

"

parallels,and

postulate of

"

fir^t

latter, the

straight

twenty-sixth

twenty-eighth

all the

case,

require modification

Propositionsafter the

former

hold.

will

seventeenth

one

the

In

descriptiveGeometry.

few

not

ceptions, ex-

be

to

generally. Since

389. a

present,

necessarily assumes

merely

parallels,after

upon

the

question of

terms

difficult

very

and

descriptiveGeometry, mentioning projective Geometry it shows with important metrical points in which

seventh, sixteenth,

a

in

logicallysubsequent

it

for it

twenty-six propositions

assume,

a

descriptive Euclid

The

defined

manner

a

is

Geometry

kinds, to

two

be

requisitegenerality and

the

Euclidean

of

indefinables.

its

cannot,

or

can,

straightforwardly,in

subject

we

and

descriptiveGeometry, is following chapter. For

the

to

is consistent

these

method

its

that

to

as

century.

whether

opposition corresponding

by

projectiveand postpone

author

analyticalGeometry, projectiveand distinguishedfrom

it is any

nineteenth

priorto the

usual

its indefinables

whether

I

by

other

any

the

;

not

non-Euclid,

of

by

or

present chapter is elementary Geometry,

the

subject of

THE

388.

Euclid

still has

reputation for rigour, in

windedness

condoned,

are

with,

a

begin

with

few

of the

the

circles which

the

whole

the first we

virtue

of be

it may

his

worth

while

first

proposition.

There

are

Cf.

told

to

twenty is

Euclid's

Killing,

op.

u,

and

point out,

to

-six

no

Section

to

longbegin

propositions*.

evidence

problems

cit. Vol.

mathematicians,

circumlocution

intersect, and

construct

fails.

with

even

which

his

in

errors

proposition *

popularly,and

are

5.

whatever if

often

they

To that

do

not,

regarded

as

388-390]

Metrical

and existence-theorems,

from

Geometry

405

this

sumption point of view, it is plain,the asthat the circles in question intersect is precisely the same the assumptionthat there is an equilateral as on a given base. triangle And in elliptic line is a closed series,the space, where the straight fails when the lengthof the base exceeds half the length construction line. As regardsthe second and third propositions, of the whole straight there is nothingto be said,exceptthat they are not existence-theorems. The corresponding existence-theorem Le. on any straight line,in either direction from a givenpoint on the line, there is a pointwhose distance from the givenpointis equalto a given distance is equivalent to the the the and is thus second and circle, postulateconcerning priorto With third propositions. regardto the fourth, there is a great deal "

"

be

to

said

indeed

;

Euclid's

proof is

so

bad

he would

that

have

done

this

axiom*. As the issues raised as an proposition both to mathematics and to by this proof are great importance, shall its I forth fallacies set at some philosophy length. fourth proposition The is the first in which Euclid employs 390. of superposition method the method a which, since he will make any he evidently and rightly, since it has no detour to avoid it*f*, dislikes, and child as a juggle. In the validity, strikes every intelligent logical that our not spatial, first place, to speak of motion are triangles implies and can but material. For a point of space w a position, more no than the leopard can change his spots. The motion changeits position of a point of space is a phantom directly to the law of contradictory it is the supposition that a given pointcan be now one : point identity Hence is in the and now another. motion, onlypossible ordinary sense, But in this case to space. to matter, not superposition proves no is by the geometricalproperty. Suppose that the triangleABC

better to

assume

of

y

"

window,

the

and

thermometer; ABC we

to

are

in all

would ABC

also that

if

followed.

have

DEF

are

we

But be

to

littledifficultiesremain.

two

DEF

suppose Euclid as

respects. But and

consists

of

column

the is

by

in

of

mercury the fire. Let us

had

broughtDEF

to

shall be

how

foolish ! I

both

rigidbodies.

In the

ABC*

Well

firstplace and "

such

no

of

Then

equal result

told

;

and

good.

for my it is as

a

apply

DE. directs,and let AB just cover and DEF, before the motion, were conclude that ABC

DEF

to

side AB

course

But

opponent,

certain as this pointis serious empirical philosopher, In the universe. anythingcan be that there are no rigidbodies in the he would second place and if my not an empiricist, opponent were find this objection far more fatal the meaning of rigidity presupposes For matter. of metrical a purely independent spatial equality, logically who

is an

"

"

"

*

This

course

is actually adopted,as

by HUbert, Grundlagen Ganss-Weber t Cf.

der

Geometric

of regardsthe equality

zur (Festschrift 1899),p. 12. Denkmals, Leipzig,

loc. tit. " Killing,

2.

Feier

the der

remaining angles,

Enthullung

des

[CHAP.XLVII

Space

406

by a rigidbody ? portionof time,preserves

is meant

what

Hence

incur

we

which, throughouta

one

all its metrical

fatallyvicious circle if

most

a

It is

tinuous con-

properties unchanged. define

to

attempt

we

which by rigidity.If afty be a material triangle, properties the the space ABC, at another space A'B'C',to occupiesat one time' times be chosen the however two that, say that a/tyis rigidmeans the"triangles ABC, A'B'C' are equalin assigned period), (withinsome all respects.If we are to avoid this conclusion, we must define rigidity We manner. in some may say, for example, whollynon-geometrical of brass. But which is made of steel, or one that a rigidbody means slave eternal brass to mortal to regard as error then it becomes a logical define equalspaces as those which can be occupied by rage ; and if we of metrical the the and same propositions one Geometry rigidbody, metrical

will be

and all false.

one

is used

the word

by geometers, has a in dailylife, different from justas meaning entirely is which in not a mathematics, variable, changes,but is something of change. So it is with the contrary,somethingincapable on usually, motion,

fact is that

The

as

that which

Motion

motion. has

every

converse

is

certain class of

a

it has

each relations,

one-one

point of space for its extension,and also belongingto the class. That is,a referent and

the

relation,in which

the relatum

of which

each

motion

is

a

has

a

one-one

points,and

both

are

of which

point may appear as referent and again as relatum. it has this further characis not this only: on the contrary, teristic, A motion that the metrical properties of any class of referents are identical with those of the corresponding class of relata. This characteristic, togetherwith the other, defines a motion as used in Geometrv, or it defines a motion reflexion ; but this point need not be or rather, a

in which

every

elucidated at present. What is clear is,that a motion presupposes the i n metrical existence, different partsof space, of figures havingthe same and properties,

this

of the

sense

word

relevant to Euclid's

391. the

be

cannot

motion, not of

use

used to define those

Returningnow

(1) DE

"

the line DE

AB,

(") On

This

Eucli"Ts fourth

to on

there is is

material

sense,

it is

which

is

superposition.

of ABC superposition On

usual

the

And properties.

DEF

that we see proposition, assumptions. following

involves the

pointE, on either side of Z",such that about the circle. providedfor by the postulate a

either side of the ray DE, there is a such that the ray DF is equalto the angleBAC. This is required for the sibility posof a triangle DEF such as the enunciation demands, but no

angleEDF axiom to

construct

and must

is

from

a

which an

there I. 4 be

added

point F

this follows

angleEDF is used to

such

be found

in Euclid.

equal proof. Hence to

BAC,

does not

in the

Euclid's that

can

axioms.

DF=A{\

It

now

Hence

the

The occur

present assumption

follows that the

problem, till I. 23,

on

DF

of two possibility

there such

Metrical

300-392]

the enunciation

as triangles

that

prove

is

DEF

demands

equal in all

axiom, namely: With

Geometry

is established.

respectsto

angle at D,

one

407 ,

ABC,

side

one

But

need

we

alongthe

the other side to the is

in order a

to

further

ray DE, and which triangle

right(or left)of DE, there exists a ABC*. This is,in fact, the respectsto the triangle assumptionwhich is concealed in the method of superposition. it finally this assumption, becomes that DEF to prove possible the above conditions and equalin all respects triangle satisfying

equal in all

exact

With is the

to ABC.

The

remark

next

of which

he

concerns

I. 6.

Here

Euclid first employsan

wholly unconscious, though

is

axiom

essential to his

it is very

system, namely: If OA, OB, OC be three rays which meet a straight line not passing and if B be between through0 in A, B, C respectively, is less than the angleAOC. This axiom, it A and C, then the angleA OB will be seen, is not that the line is not

in projective applicable space,

closed series.

a

since it presupposes is to In I. 7, if this proposition

further the axiom : If three nonapplyto hyperbolic require space, we lines in one plane meet two lines in A, B, C; A', B' \ C ', intersecting ; and respectively

and

C'.

Also

sides of

if B

it may

be between

A

and

C

be observed that Euclid which

;

Bf

then

givesno

is between

definition that the

A

'

of the

straight requires sufficient axioms to show that they are series of the kind explained a axiom in Part IV, Chapterxxiv the descriptive must assume ; or else we to the effect that,if A, B, C, D be coplanar of the last chapter, points, the stretches which of is three there to are collinear, no a pointcommon will be AH these assumptions AB, CD, or to AC, BD, or to AD, BC. in L 7, as may be seen found implicit by attemptinga symbolicproof in which no figure is used. In I. 12 it is assumed that a circle Similar remarks apply to I. 16. must if at all. But enough has been said to meet a line in two points, axioms and that his explicit show that Euclid is not faultless, are very of metrical examination insufficient. Let us, then, make an independent Geometry, said to be distinguished 392. Metrical Geometry is usually by the introduction of quantity. It is sufficient for the characterization of between every pair of metrical Geometry to observe that it introduces, points,a relation having certain propertiesin virtue of which it is be given a one-one Le. such that numbers measurable can numerically in question.The of various relations class the the with correspondence class of relations is called distance, and will be regarded, though this is of the properties not strictly necessary, as a class of magnitudes. Some two

line is not

a

line,a notion

closed

a

series.

And

again presupposes

with

regardto

angles,I. 7

*

"

of distance

*

See

are

as

follows.

Pasch, op.

nY.

" 13, Grundsatz

ix.

The

whole

"

is excellent.

(1) Every pairof pointshas Distances

(2)

is

(4?) There The

(6)

There

(7)

If

is

pointson If

a

line,whose straight

be)

(whateverinteger may the

last,are

all

equalf

distances

"

the

next, of A0 from

the

...

next

be are

d.

1 there exist n points, line A^Anj the straight

on

An,

...

from

one

is greaterthan

distinct such

points

that

the

and of An from first,

.

be observed that, if we

It may

393.

the

A^An

two

any

n

from

distinct points.

and A^ A^ A^ given distances,

two

A^ An be

itselfis zero*.

to the distance between

value of n,

some

of each

pointfrom

a

minimum

no

two

are

distance.

maximum

no

d, S be

S, then for

distances

distance.

relations. symmetrical

are

distance of

(5)

(8)

onlyone

and

a

and only two

distinct

one

there line througha givenpoint, givenstraight pointsat a givendistance from the givenpoint.

(3) On

all

[CHAP.XLVII

Space

408

that the whole

the axiom

admit

(1),(4),(5) and (6) belong to part,the properties from the sense of while (") becomes admissible by abstracting stretches, With a stretch. (3),(7) and (8), regardto the remaining properties, there is nothingin descriptive Geometry to show whether or not they belong to stretches. Hence we may, if we choose, regardthese three is

the

greaterthan

regardingstretches,and drop the word distance the simplest altogether.I believe that this represents course, and, as time, there is no regardsactual space, the most correct. At the same relations distinct from contradiction in regardingdistances as new If stretches^. we identifydistance and stretch,what distinguishes metrical from descriptive Geometry is primarilythe three additional axioms namely, the (3), (7) and (8), appliedto a new indefinable, of of a stretch. This is not properly a notion magnitudeof divisibility since it cannot be derived from our original pure mathematics, apparatus of logical notions. On the other hand, distance is not indefinable, being class of with relations one-one a certain assignable properties.On this either is course but point logically permissible, only distance can be as properties

axioms

introduced

into pure mathematics is used in this work. The

above

axioms

numerically measurable

are

If properties. *

See

Part

t Further

of any

the

are, of course,

See

Part IV,

Chap. xxxi.

standard

a

not

form

line form

added

a

later

a

the word

all distances

distance ".

magnitudes,or

distances should

points of

present discussion.

"

be

III, Chap. xxn. propertiesof distance will be

I Stretches the

is that

in which

sense

requiredfor showing that

in term*

that distances should

necessary that is essential

in the strict

even

It is not

relations; all

series with

continuous

are

certain

series,then

on.

properlyrelations ; but this point is irrelevant in

Metrical

392-394] distances do

whether

asked

409

also,in virtue of (8) ; thus all signless real numbers

so

for their required 394. Assuming

be

Geometry

will

measurement.

that

distances

distance and do

not

the need of

line,without straight

stretch

be it may distinct, suffice for generatingorder on the any asymmetricaltransitive relation of are

I think, the usual view of philosophers points. This represents, ; but decide it is by no to whether it tenable view. means representsa easy It might perhapsbe thought that (2) might be dropped,and distance regardedas an asymmetricalrelation. So long as we confine our attention to one But as soon line,this view seems as unobjectionable. that distances fact the lines be consider different on we equal,we may

that the difference of

see

distance,bince there is be

no

sense

between

AB

and

difference between

such

distance

ib not

BA

distances

relevant on

to

different

equalboth and BA, and hence AB and BA be equal,not opposite, must to AB the be made evident by considering distances. And same thing may this consists of For a sphere. certainly pointsat a given distance from have the centre ; and thus pointsat oppositeends of a diameter must the same distance from the centre. Distance,then, is symmetrical ; but it does not follow that the order on line cannot be generatedby a distance. Let A, B be givenpoints and let C, C' be two points on a line, distances AB and less than AB. whose A from If we now on are equal, if CD

lines. Thus

set up the axiom

BCf

a

that either BC

on

or

another

line,CD

be

may

AB, while the other, further shall,I think, after some

BC'

is less than

greater than AB, we axioms, be able to generateorder without any other relation than distance. If A, B) C be three collinear pointssuch that the distances AC,, CB both less than AB, then we shall say that C is between A and B. are If A, B, C' be pointssuch that AC', AB both less than BC', then are shall say that A B and C'. is between If, finally, we A, B, C'r be pointssuch that AB, BC" are both less than AC", we .shallsay that B of a is l"etween A and C". It remains to see whether, as the generation three these be series requires, of one alwayshappens. Let A, B, C any collinear points. First suppose, if possible, that the distances AB, BC, or

CA

equal. This case is not excluded by anything hitherto we therefore,the further axiom that, if AB, BC be require, is not equalto either of them ; and I think it will be prudent of two equal that A C is greater than either. Thus the case

all

are

assumed

;

equal,AC to

is

BC,

assume

distances and A

the three

distances

be the greatest: let this be AC. must AC\ therefore,one But our in virtue of the definition, B will be between A and C

B, BC\

Then

difficultiesare between D

Of

less than either is excluded.

one

and

A

and

C,

B

point,if therefore

E

not

B

at

end.

an

For

shall be between

shall be

be between

less than

AC.

between A

and But

D

we

requirefurther

and

A

and

B, AE

nothing

that, if A

C'; and C.

and assures

With EB

are

us

that

any point be between

regardto

the

less tlian AB, that

EC

first and

is less than

[CHAP.XLVII

Space

410 AC.

this purpose

For

need

we

will be

axiom, which

new

a

justwhat

we

EB be both less than AB, and AB, prove, namely: If AEy to prove Finally, BC be both less than AC\ then EC is less than AC. that, if A be between D and C, and B between A and C, then jB is Here between D and C DA^ AC are less than DC, and JJ5, J?C are

set out

to

less than

For this

than DC.

shall need

we

Moreover

395.

shown, in

Fieri has

stillneed

we

axiom, and then at last our order is evident,is extremelycomplicated.

method

a

admirable

an

less

new

a

the process, as

will be definite. But

Z"C; but nothing proves BD

is less than

BC

Hence

AC.

of

the straight line. defining

memoir*, how

deduce

to

metrical

geometry by takingpoint and motion as the onlyindefinables. In | 390, we objectedto the introduction of motion, as usuallyeffected,on the but Fieri ground that its definition presupposes metrical properties; motion at all,except throughthe objectionby not defining line joiningtwo points assumed concerningit. The straight postulates is the class of pointsthat are unchangedby a motion which leaves the escapes this

the order of the plane, perpendicularity, pointsfixed. The sphere, is logically pointson a line,etc. are easilydefined. This procedure for elementary and is probablythe simplestpossible unimpeachable,

two

But

geometry.

must

we

return

now

of

consideration

the

to

other

suggestedsystems. There

is

method, invented

a

and

Peano",in

line

is defined

to

begin

with

transitive

as

the

domain space in

pointsis called

planes,when the

In

means.

this

and

revived

and

method

which

same

a

it is not

a

line given by Peano||is straight

as

a

this

assume

we

;

certain

all distances

domain, namely

all the

from pointsequidistant

plane,and the intersection of two null,is called

^

given

are

field of

smaller) ; if

converse

the locus of

question;

Frischauf

distances the

are

and (greater

the

by

fundamental, and the straight

continuous,distances will be measurable

same

pointsof the fixed

its

is

class of relations

relation asymmetrical

relation to be have

a

Leibnizt

distance alone

which

by

by

two

non-coincident

line. (The definition of straight line ab is follows : The straight

the class of

pointsx such that any pointy, whose distances from a and b dent are equalto the distances of x from a and ", must be coincirespectively with x.) Leibniz, who invented this method, failed,accordingto line is Couturat,to prove that there are straight that a straight lines, or determined

by

aware,

any two of its points. Peano succeeded in provingeither of these

course

possibleto

introduce

to demonstrate professes

*

MUi

+

Cf.

J

Absolute

"

Acctidrmia

idee di

them

by

them, but his

geotmtria eJementare

xixtema

cow/;

has

not,

panto

e

of

means

proofsare

axioms.

as

I

am

it is of

Frischauf

very informal, and

it

ipotetiro drduttico,Turin, 1809. esp. p. 420.

Johtnnt

Reale detle Srlenze

distanza."

far

but propositions,

Couturat,La LogiquKdt" Leibniz,Paris, 1001, Chap, ix, Geotuetrie nark

so

A uhaitg* JKolyai^ di Torino^1DO2-3, La toe. tit. [| "

Geometria

basata

sulle

Metrical

394-397] is difficultto know

what

the definitions prove

he is

axioms

that,by

a

411

assuming. In

sufficientuse

in which

geometry

Geometry of

any

case,

axioms, it

however,

is possible to

distance is fundamental,and the

straight complicated practically is nevertheless important. desirable ; but its logical possibility It is thus plainthat the straight be independent 396. line must while distance be of the of distance, line. independent straight may both as we symmetricalrelations, Taking can, by a very complicated

construct

a

series of

method

The

line derivative.

axioms, succeed

in

is

so

generatingorder

on

be not

to

as

the

straightline and

the addition and measurement of distances. But this explaining in most spaces*,is logically complication, unnecessary, and is wholly avoided by derivingdistances from stretches. We now start, a* in with transitive relation asymmetrical an Geometry, descriptive by which the straight line is both defined and shown to be a series. We define a" the distance of two pointsA and B the magnitudeof divisibility of the stretch from A to B or B to A is a signless for divisibility magnitude. Divisibility beinga kind of magnitude,any two distances will be equal the sum of the divisibilitiesof or unequal. As with all divisibilities, AB of the classesAB the and and EF is the divisibility of sum logical these classeshave no common EFy provided part. If they have a common part,we substitute for EF a stretch E'F' equalto it and having no in

"

part

in

with

common

difference of the distances AB^

The

AB.

EF

of a stretch CD AB the greater)is the divisibility which, (supposing added logically with EF9 to EF, and having no part in common if stretch AB. follows It at once a that, A9 B" C be produces equalto be between

B

and collinear,

No further axiom

that, if

AB

=

is

A'B,

A and C, AB

for required and

these

For propositions. then

CD=C'D',

and AC

+BC=AC

the

-AB

=

BC.

proposition

AB^CD^A'ff+CI/,

to requireonlythe generalaxiom, applicable

all

we

that the divisibilities, (3),(7),(8)

helpof the axioms

equalsare equal. Thus by that is required for the numerical measurement above,we have everything of all in of distances terms speaking) (theoretically any given and for the proofthat change of unit involves multiplication distance, factor. throughoutby a common in in the sense 397. With regard to magnitude of divisibility, which this is relevant to metrical Geometry,it is importantto realize of

sums

the

that it is

ordinal

an

their fields. much

We

wish

to

say

a stretch as divisibility

divihible than

more

assumed or

volume

of

an

*

not u notion,expressing propertyof relation*,

in this is

a

Chap. xux.

of

stretch.

class of "a* terms

stretch of two

a

inch,and

one

Now,

with discussion)

infinite number The

a

that

a

;

if

we

continuous

and

that are

inches

has

area

ib

an

twice

of as

infinitely

dealing(as

will I"e

space, every stretch,area it is the field as a class,

considered

to it of relations beside that (or those) belonging

known only exceptions*

to

me

are

finite spaces

of two

dimensions.

See

respectof the space

in

[CHAP.XLVH

Space

412

in

habit of

considering.The

are

allowingthe

actual space has made the order of points and not merely relative to intrinsic or essential,

dwell upon

to imagination

appear

we

way

some

of many possible orderingrelations. But this point of view is not in regardto actual space, onlyfrom the fact that the : it arises, logical generatingrelations of actual space have a quitepeculiarconnection

one

and, throughthe continuityof motion, with time. perceptions, of the relations havinga given field of logic, the standpoint one From no and the pointsof actual space, like any other class has any preeminence, of 2a" terms, form, with regardto other sets of generatingrelations, with

our

other

of continuous

sorts

pointsof

of the

indeed

"

Euclidean

a

other

any

dimensions, or

of

finite number

having any formed

spaces

even

o"

continuous

space, dimensions,can be

by attendingto

space

other

generatingrelations. a long distinguish

be

must

a

is

requiresome of the

the

for the

any

a

stretches

two

area

a

is to

stretch,

of the class of

points

similar. ordinally of the relations inequality

the real

are

coordinates

Where

numbers)

magnitudeof

define the

from

quiteeasy to define the exact

equalityor

line with

an

or

involved,not

given stretches.

pointsof

introduced,we

a

relations

required ; for sense

fields are

short one,

the stretch. It is not

or

area

property which whose

stretch from

propertyof the

composingthe We

if it magnitude of divisibility,

this it follows that

From

a may of the coordinates of its end-points its limits or

have

stretch

as

a (I.e.

been the

lation corre-

already

difference

the stretch as (according has ends or not) ; but if this is done, the magnitudesof stretches will less arbitrary more or depend upon the necessarily plan upon which we have coordinates. This is introduced our the course adopted in the which has the merit of making ve theoryof distance a course project! metrical Geometry a logical alone axioms developmentfrom protective be adopted is, to that may (see next chapter).The other course that the relations of any two stretches have either a assume generating t ransitive relation an or symmetrical asymmetricaltransitive (equality), relation or its converse or less).Certain axioms will be required, (greater that if the pointsA^ B, C',1) are is and AC collinear, as, for example, The relations of equal, greaterthan AD, then EC is greaterthan BD*. and be regardedas defined by these axioms, and the less may greater "

property of the ^generatingrelations of those stretches that

common

equalto the

of

a

givenstretch may

said

be defined

generatingrelations.

more infinitely

than divisibility

and integer,

stretches

area, there

n

an

area,

as

a

magnitude of which

in

sense

stretch

a

equal to

alwaysremains

The

the

that, if

great

n

is that importantto observe,in the above discussion, "

CA

*

Stretches

is less than

are

DA.

here regarded as

divisibility area

has

be

any finite given .stretch be removed from an

however

is

an

are

havingsign,so that,if AC

n

may the is

be.

What

is

logical parity greater than

AD,

Metrical

397, 398] of all the orders of which to

the

regard

which

classes

or

of

413

is

cajmblemakes it necessary metrical Geometry deals as

relations, not,

as

is commonly

class of

the

supposed,

class of terms

a

magnitudes with relations

belonging to

Geometry

pointsforming their fields. the straight line is a closed series, elliptic space, where distance the attempt to make independentof stretch leads to still further complications.We now no longerhave the axiom that, if to

In

398.

be collinear, we

A,B,C

recognizetwo

is taken

distance

avoid

however

may

greaterof stretch.

as

the

two

between'

AB

EC-CA

"

;

and

we

have

to

pair of points,which, when We extremelyawkward. to regardthe admittingtwo distances by refusing as properlya distance. This will then be onlya every

fundamental, becomes

distances

If two

have

cannot

distances

are

admitted,one

is

always greaterthan the

a other,except limitingcase, when both are the lower limit of the greater distances and the upper limit of the lesser distances. Further

in

6, cr,d be any four distinct points,the greater of the two distances the ab is always greater than the lesser of the two distances cd. Thus be banished,and only greater whole class of greater distances may if a,

stretches be admitted. We

which which

metrical proceed as follows. Distances are a class of symwhich are magnitudesof one kind,havinga maximum, relations, is a one-one relation whose field is all points, and a minimum, is the distance of any point from itself. Every point on a given

must

given distance other than onlytwo other pointson the

line has two

a

and

pointson the not

now

shall say that

line,we

one

four cases, following mutuallyexclusive :

(1)

If ab

(")

If ab

(3)

If ah

(4)

If be

We

then

"

ac

"

ac

"

ac

"

ac

need

be .

"

be .

ad .

ad" .

of which

ac

"

ac

"

ac

ac

ad

ac.

dc

"

ac.

dc

"

ac.

dc

"

ac.

.

.

.

and

(1) and

"

.

a

the

Vailatfs five axioms

maximum

line. c

minimum

(2) and also (3) and

enumerated

closed series from

or

from

If ", ",c, d be four distinct are by b and d in separated

in Part

(4) are

IV, Chap,xxiv,

of couplesso separation defined. Thus it is possible, thoughby a somewhat complicated process, of line the to generate a closed series of points on symmetrical a by means in order

to

generate

a

the

relation of distance. I

shall not

work

out

in

further detail the

of this consequences to the hypothesis

hypothesisin elliptic space, but proceed at once of the number When that distances are the magnitudesof stretches. dimensions of elliptic exceed* two, the polar form space is merely protective togetherwith the necessary metrical axioms; the space form is a space in which two antipodalpointstogetherhave antipodal the propertiesof a singleprojective point. Neglectingthe latter,to

[CHAP.XLVII

Space

414

apply,I shall confine myself to the polar of pointsdetermines Since this is a protective form. space, every pair of these two two segments on the line joiningthe points. The sum the whole is and therefore with the two points, line, segments, together lines have the same that all It is an axiom constant. completestraight of either segment is a distance between The divisibility divisibility. either may distances are equal, be called the two the two points : when the distance ; when theyare unequal,it will be convenient to call the smaller the distance,except in special problems. The whole theorythen of descriptive proceedsas in the case space. But it is importantto construction and the observe that, in elliptic space, the quadrilateral are priorto distances,and generationof order,beingpriorto stretches, which

in presupposed

are

399.

as

is to

in

sense

Geometry

of

in

three

every

merelyintroduces such

Geometry pointsmeasurable.

which, in The

be understood.

introduces

stretch

The

metrical

all stretches

to the

Geometry.

indefinable.

new

one

quantity,and

make

metrical

far,therefore,metrical

So

axioms, and a

will

similar remarks

a

theoretical actual

A

few words

may

the word discussion,

of application

the

new

series is axioms

as

be useful

measurement

foot-rule is here

only those

but propertiespf pure space which are question, the foot-rule. A the of set of magnitudesis theoin use presupposed relation between them and rdimllymeasurable when there is a one-one all numbers; it is practically measurable when, given any some or magnitude,we can discover,with a certain margin of error, what the is to which our number magnitude has the relation in question. But how we that to discover this is a subsequent are question, presupposing there is such a proposition to }"e discovered, and soluble,if at all, With to be invented in the laboratory. means practical by empirical not

in

then, we

measurement, 400. the

I

come

now

are

to

the

a

not

more

at

all concerned

in the

difficultquestion than

definition of

cussion. present dis-

distance, namely

angle. Here, begin with, we The lines. straight rays, ray may be taken either as an asymmetrical relation,or as the half-line on side of a givenpointon a line. The latter usage is very convenient, one and I shall frequently that employ it. ElementaryGeometry assumes two from the same pointdetermine a certain magnitude, rays starting This them. called the angle between magnitude may, however, be In the first place,we defined in various ways. must observe that, since the rays in a planethrough a point form a closed series, every pair of rays through a pointdefines two stretches of rays. Of these, of both rays, while the other however, one stretch contains the opposites stretch contains the opposites of neither except,indeed,in the one where the two case This case is met rays are each other's opposites. EuclicTs that all right angles are by postulate equal a postulate, must

question

as

deal with

to

not

with

to

whole

"

"

Metrical

398-401] however, which

is

known

now

to

Geometry

415

demonstrable*.

be

Omitting

this

rays may be defined as that stretch of rays through their intersection which is bounded by the two rays and does the

case,

anglebetween the

contain

not

two

oppositeof either,i.e. if ^,

B

be

the rays, and

A" B

the angle is the class of rays C which are opposites, separated A B. B We and from A or by might also,but for an objectionto define the be mentioned shortly, angleas all the pointson such rays. A definition equivalent to this last,but simplerin form, and avoiding of the opposite the mention rays, is the followingf.Let a, b be any two pointsof the rays A^ B, and let c be any point of the stretch ab. Then the class of points of a and b on their c9 for all possible positions is the B. That A and is, every pair angle between respective rays, divides the the into of intersecting of two plane parts: the rays rays partdefined as above is the angle. Or rather,the part .so defined is the of angle as a quantity: the angle as a magnitude is the divisibility this part. But to these latter definitions we shall find fatal objections, their

and

shall find it necessary to adhere

we

of rays. 401.

Thus

is angle,like distance,

the

to

definition

as

a

stretch

but like indefinable, axioms. The new angle between a ray A be defined as above, but may be defined

distance,it requiressome

not

a

new

oppositeA* cannot of the angles between A and 5, B and A' respectively. sum as logical This limiting angleis greaterthan any other at the point,being in fact the whole half of the planeon one side of the straight line A A', If the anglesbetween A and B^ B and A' are equal,each is called a right angle. (That there are such angles,can be proved if we assume lines make Two four angles, which intersecting straight continuity.) are equalin pairs. The order of a collection of rays through a point in a planemay be obtained by correlation with the pointswhere these line,providedthere is any straightline rays intersect a givenstraight But since rays througha pointin a plane which all of them intersect. while the pointson form a closed series, line do not, we a requirea definition seems four-term relation for the former order. The following adequate. Given four rays OA^ OBy OC, OD through a point 0 and in one line in A, By C, D plane,if these all meet a certain straight and A and C are separated by B and D, then OA and OC respectively, this In protective and OD. said to be separatedby OB are space suffices. But in descriptive must providefor other cases. space we Thus if OA, OB, OC meet the givenline,and B is between OA and OC, and

its

the

while

does not

OD

"aid to be

*

See

e.g.

Hilbert,op. t

separated by Killing,op.

OB

"t.

cit. p. 16.

Killing, op.

given line,then OA and OC are again be the and OD. OA' and OR If,finally,

the

meet

cit. ii, p. 169.

Vol.

11, p.

171.

A

strict

proof will

be

found

iw

[CHAP.XLVH

Space

416 and

OA

oppositesof and

OBr,

In virtue of the

the

order

among of our

and

OA

then

OB,

OAr

be

will

obtained rays so line choice of the ABC, the

separatedby OB precedingchapter, dependen unambiguous,i.e. in-

are

of the

axioms descriptive

and

will

all

cover

cases.

those

which, in the case analogous of distance,were numbered (#),(7) and (8). At any given point in a given ray, there must be, in a given plane,two and only two rays, on oppositesides of the given ray (Le. separatedfrom each other make which a given angle with by the given ray and its opposite), But

the

now

we

given ray

axioms

and

;

obey the

anglesmust

to these

numerically measurable,

be

the

solution

appears

to

of

this,by

obtain

any the

congruence

the

axiom

that, with

one

given ray

through

that

have

of

this

of I.

means

this result

For

fresh axiom.

that

triangles.Does

require

angleat

new

point,there

necting con-

as

is

required

axiom

?

Euclid

II, 13, without

the

propositionson only,as we saw,

demand

given point,and

a

angles of

47, II. 1", and

depend upon

we

method

some

a

and

insure that

distances,such

(I.4, 8, 26),which triangles

of

of Archimedes

axioms

axioms, which

must

we

angleswith

of

measure

for the

a

to

in addition

But

of linearity. shall

need

along onlytwo triangles of the given ray),

and

exist two

side

one

givenplanethrough the ray (one on each side it would which are equalin all respectsto a given triangle.Thus seem for in a that no fresh axioms are required angles plane. With 402. regardto the definition of an angle as a portion of other retain this (as in many cases),if we a. plane,it is necessary in

a

to restrict the axiom

somewhat definition, whole

the whole

that

is

greater than

parts 5, C, which togetherconstitute AI and if C be infinitesimal with respect to A^ then B will be equal

part. If

the

to

This

A.

a

has

in

occurs

case

0, ff be any

Let

A

two

a

plane under

points,OP,

two

equalangleswith these

lines 0P,

ffPf

will be

is necessary

as

00'

a

less

this

space

angle than

followingcircumstances. one planeand making in Euclidean or hyperbolic space Off and ; thus the anglebetween

O'OP.

angle

Hence

the

that the whole

regardsthe axiom

Euclidean

In with

the ray 00'*. Then ffP' will not intersect

part of the

the

ffP* lines in

is

answer

07"

restriction

greaterthan

the

sufficient, since,if OP

does, OP

and OrPf may in hyperbolic space, OP adhere to the above definition of if we

is

above

not

and

ffP

will intersect.

intersect

even

shall have

angle,

we

part. makes

then.

But Hence

to hold

that

be less than the part. This, however,is intolerable, and may that the definition in questionmust be rejected.We ever, may, how-

the whole shows

still regardangle as

ff

at as

a

can *

are

not

part

of

The

angle between W

produced

and

stretch

of rays ; for the rays in the angle in the angle at 0. it is only Hence

the

stretch of rays, or as be properly defined.

between

the

the (?P.

rays the magnitudeof such

rays

W,

(fPf is what

a

stretch,that

Euclid

would

an

call the

angle angle

401-404] As

which

Metrical

showing,in results from

a

curious

Geometry

manner,

the above

41*7

the increased

axioms

power of deduction d istances and angles, concerning

that the uniquenessof the quadrilateral construction, may remark which before could not be proved without three dimensions,can now we

be

regardsall constructions in one plane,without any assumptionof pointsoutside that plane. Nothing is easier than to coordinate Geometry. by the methods of elementary prove this proposition Thus althoughprotective Geometry, as an independent science,requires three dimensions,any protective proposition concerningplane figures if be the above axioms can hold,for a two-dimensional metrically proved,

proved,as

space. 403.

of three dimension.**, regardsfigures anglesbetween planer and solid anglescan be defined exactly rectilinearangleswere defined. as fresh axioms will not be required, Moreover for the measurement of such from the deduced be data we alreadypossess. anglescan and remarks With volumes to areas some seem regard necessary. taken classes of points uhen and volumes, like angles, Areas are as taken and divisibilitieswhen as magnitudes. For areas quantities, and of Archimedes and volumes we do not requireafresh the axioms axiom criterion of but we of linearity, apiece to give a requireone their with of that and connect to volumes, i.e. equality equal areas is supplied, distances and angles. Such an axiom as regardsareas, by have the the axiom that two same triangles area, and as congruent axiom concerningtetrahedra, regardsvolumes, by the corresponding But

the

As

existence

demands triangles,

of an

congruent tetrahedra, like that axiom.

For

this

purpose,

of

Pasch*

congruent

gives the

followinggeneralaxiom : If two figuresare congruent, and a new pointbe added to one of them, a new pointcan be added to the other that the two new are so congruent. This axiom allows u* to figures and hence the infer congruent tetrahedra from congruent triangles; measurement

of volumes

smoothly. proceeds

dimensions,a curious fact has to be taken account of, of clockwise of rightand left-handedness, or namely, the disjunction 404.

In three

nature, and descriptive two non-coplanar rays, or between may be defined as follows. Between four non-coplanar order,there is always one pointstaken in an assigned of two oppositerelations,which may be called rightand left. The in Part IV (""""); of these relations have been explained formal properties concerned with their geometrical for the presentI am consequences. volumes become to In the first place, magnitudeswith sign, they cause line have sign when in exactly the way in which distances on a straight since not But in the case of distances, compounded with their sense. all are on one line,we could not thus compound distance and straight for a compound,some more should require, : general sense we generally

and

counter-clockwise.

This

*

fact is itself of

Op. tit. p.

109.

a

than

notion in

tetrahedron

the

This

sign.

volume

the

of

sense

the

of

also, which

the it

that

properties like

who,

fact

all

are

a

But

supposed

result

to

definition

of

leading thing,

if #,

in

the

two

In

the

there at

is

will

It

of is

it to

the

this next

we

those

to

are

that

may

chapter

projective theory

end to

of

who

a

and

discuss distance

brief

left-handedness than

its

and

of

relation

angle.

sense.

fail.

For

thus

;

No in

equal

all

there

however,

merely

of

result

a

the

same

dimensions,

two

form a

to a

the

of

notion

difficulty;

in

stumbling-block.

metrical to

points

necessary.

essential

as

be

same

same

dimension, in

;

motion

review

be

fact,

but

one

senses

confirmation

our

this

In

regard

the

metrically

mysterious,

opposite

right

rather

jump

dimensions.

with

the

conditions

would

dition con-

equal figures to

counter-clockwise

to

In

a

figures

have

sense.

nothing

three

to

equality

theory, they With

opposite be

to

observed)

of

is*

former

our

a'b'e'd'

tetrahedron

a

In

non-coplanar

clockwise

into

equality

amounts

these

to

the

metrically

congruent

must

Kant,

facts

homologous

sudden

a

abed

distances

only

abed,

axiom)

than

are

series

what

Or, be

from

series

the

ourselves

two

opposite tetrahedra,

transition the

in

mind,

hold

metrical

tetrahedra and

with

my

confining

areas.

a', 6', c'9 d!

transform

but

to

would

and

was

must

continuous

a

other.

the

one

gradual point

seems,

d

be

must

(as

portance im-

puzzled

metrical

then

fact,

metrical

puzzling

when

is

O

The

whose

which

BA,

the

as

is this

It

superposition.

only

Not

to

equal

of

no

some

our

c9

and omitted

giving

geometrical

more

puzzling

we

there

figures, the

respects,

the

",

case

motion

of

the

and

AB

motion

(" 390)

but

metrically equal,

supposed

opposite

where

which all

no

the

OXYZ^

figures

two

volume

independent

an

two

be

have

Dynamics.

distinction

this

becomes

it

be

the

if

axes.

in to

would

fact

of

the

on

of

sign according

that

from

since,

other

determinant

the

other

or

seems

is

will

XLVII

contrary,

or

Thus

that

fact

one

one

bacd

between

its definition.

to

from

itself

It

from

motion

essential

of

the

on

momentum

stretches

the

indistinguishable.

has

contemporaries,

his

between

distinction

that

distinction

itself, the

In

sign,

angular

identical.

of

most

metrical.

volumes.

positive points

to

(which

makes

all

different

or

any

signs

for

at)cd as

Z

have

geometrical

same

Jf, F,

gives of

this,

the

is

and

origin

one

tetrahedron

a

abed

has

familiar

the

is

volumes

made

be

can

abed

Here,

all

space,

compound

the

of

supply.

vectors

as

three-dimensional

a

be

such

sense,

senses,

is

[CHAP.

Space

418

Geometry,

projective

leaving

Geometry

aiul

CHAPTER

RELATION

OF

XLVI1I.

METRICAL

DESCRIPTIVE

405.

the

IN

even

or

unavoidable

if not, what

alreadydogmaticallyassumed

questions,but

we

any

? properties

The

?

certain

others,

previous these

to

answers

the critically

examine

to

now

without

either of the

from

novelties does it introduce

expositionhas

are

established

be

deduced

be

questions. First,

implying metrical

without

presuppositions, Secondly,can metrical Geometry or,

to discuss two

wish

descriptiveGeometry

metrical

AND

GEOMETRY.

present chapterI

projectiveand

can

PROJECTIVE

TO

various

possible

answers.

distinction between

The

recent, and

which,

is of we

as

and descriptive Geometry protective ordinal nature. If we adopt the essentially

an

simpler of

is the

saw,

legitimateviews

two

is

that

"

very

view

"

the

is defined

of its two by a certain relation between any points,then in projective Geometry this relation is symmetrical,while in descriptive Geometry it is asymmetrical.Beyond this we have the difference that, in projective Geometry, a line and a plane,two planes, lines in a plane,always intersect,while in descriptive two or Geometry

straightline

the

question whether

this is the

or

case

is left open.

not

But

these

present purpose, and it will very important for our and descriptive be convenient to speak of projective Geometry

differences therefore

are

not

togetheras non-quantitative Geometry. scarcelyopen how

be built up

it may

regard

it

question.

to

considerations. as

mathematics, to

is

Geometry logicalindependenceof non-quantitative

The

without

essential does

and

mathematical

seen,

Chapters

in

whatever

reference

any

and

XLV

to

in

occur

The

treatment.

does

mathematics,

to

cases

many notion

not

not at

which

occur

present

does

XLVI,

quantitative

fact, though philosophers appear

Quantity,in very

have

We

now

occupy

still to in

pure amenable the

place

traditionally assignedto quantityis orders and this notion, we saw, is But the purity present in both kinds of non-quantitativeGeometry. of

the

order

by

so

notion

of order

depends upon excellent

a

has

been

distance"

writer

as

a

much

obscured

belief which,

Meinong,

we

have

by

the

though

seen

to be

belief that

all

it is entertained

false.

Distance

[CHAP.XLVIII

Space

420

that series

to admit quantitative, beingessentially

that

is to admit to

once

which

this view

dependsupon quantity. But

order

distance

depend upon

leads at

endless regress, since distances have an order of magnitude, distances of distances, and would have to be derived from new an

an asymmetricaltransitive relation suffices to positively, generatea series,but does not imply distance. Hence the fact that the pointsof a line form a series does not show that Geometry must and no such presuppositions have metrical presuppositions, appear in the detail of projective or descriptive Geometry. But although non-quantitative 406. Geometry,as it now exists,is plainlyindependentof everythingmetrical,the historical development has tended greatly to obscure this independence.A brief of the subject historicalreview of the subject may be useful in showingthe relation of

so

And

on.

the

to the

modem

more

theorems

be found.

to

are

hardlyany descriptive geometers generally,

in Greek

Euclid,and

In

traditional methods.

more

of the

One

earliest discoveries of

portant im-

an

theorem was the one named after Pascal*. Gradually descriptive which assert pointsto be collinear or found that propositions it was lines to be concurrent, or propositions concerningtangents,polesand and similar matters, were unaltered by projection is,any ; that polars, such property belongingto a plane iigurewould belong also to the shadow of this figure from any point on to any plane. or projection those common All such properties to all conies)were (as,for instance, called projective anharmonic or descriptive. was Among these properties

ratio,which one

anharmonic

lines

through a

four

be

/sinAOD

sin JOB

/ _

with together subject, such

But projection.

be used to found

portionsof

other

Geometry,the the second no

was

what

If

a

used

as

on

if

OA,

OB,

ratio

is

Geometry, descriptive

Cremona's

projective Geometry),

earlystage in

very

be

to

the

called

a

conic.

of development

ratio is unaltered cannot

Geometry. With or descriptive projective

To

be

found.

define it

as

Consider, a

curve

requireproject! ve coordinates,which

of

To introducing.

than

more

two

inscribed

iu

a

define it as

pointswould

imaginarypoints,tor

by

therefore

metrical

independencewill

definition of

method

hexagon be points.

in eollmear

a

lack of

of real and

*

points on

.

great work

definition is itself metrical,and

a

line in not straight

any

four

7^/777^;

a'proofthat anharmonic

degree would

known

is

subjectindependentof

a

same

example,the

for

ratio

be

point, their anharmonic

works (such

recent

this definition will be found at the

,

If J, B, C, D

L/C/ZX

in most

even

1 Cnasless

in

_--_

sin

oc/jD/

and

follows.

as

straightline,their

OC, OD sin

defined

was

if

we

a

curve

of there

meeting

tinction requirethe dis-

confine

ourselves to

the three pairsof opposite sides intersect conic.,

405-407] real

ProtectiveTheory of

pointsthere

But

the definition.

innumerable

are

Metrics

other than

curves

imaginarypointsare,

in

421

conies which

metrical ordinary-

satisfy

Geometry,

pretation imaginarycoordinates,,for which there is no purelygeometricalinterwithout definition againfails. coordinates, our ; thus projective To define a conic as the locus of pointsP for which the anhannonic of

(where A, B, C, D are fixed points)is constant, again involves metrical considerations, so long as we have ratio

PB, PC, PD

PA,

definition of enharmonic ratio. And the .same projective dependence metrical other as Geometry or regards projective appears any upon the traditional order of ideas is theorem, so long as descriptive no

adhered

to.

The

true

founder

he

who

introduced

was

of

-quantitative Geometry

non

the

definition of

is

Staudt*.

von

harmonic

range by means rendered it possible, by

a

construction,and who quadrilateral of this construction,to give projectivedefinitions repetitions

of

the

rational

anhannonic

It

of all

definitions indicate the succession

ratiosf. These

constructions required in order to obtain a fourth point quadrilateral from three given points;thus, though they are numerical, essentially

of

they have no reference whatever to quantity. But there remained one further step,before projective Geometry could be considered complete,

by Fieri.

taken

and this step was whether

all sets of four

whether

any

For

meaning

this purpose, we points of a line.

the

In Klein's account, it remains

collinear pointshave

anhannonic

an

doubtful

ratio,and

assignedto irrational anharmonic ratios. requirea method of generatingorder among all be

can

For, if there be

Klein's method, there

order but that

no

obtained from

regarda pointnot obtained by that method as the limit of a series of pointswhich are so obtained,since the limit and the series which it limits must alwaysboth there will be no series. Hence of assigning one belong to some way have rational cowhich the do not irrational coordinates to ordinates. points for that There of reason is, supposing projective course, no is

no

sense

in which

can

we

metrical reasons, and in any case with a continuous it is well,if possible, to be able to deal projectively there

ai*e

such

space. This but without

but

points;

is effected any

new

there

are

by Fieri,with

indefinable^. has

which

the

Thus

help of at

certain

last the

itselffrom purified

projective Geometry is completed. 407. Projective Geometry,having achieved

new

axioms,

long process by

every metrical

taint

independence, of foreignaggrandisement a career has,however, embarked ; and upon in this we shall,I think,though on the whole favourable,be obliged to make theory of some slightreservations. The so-called projective of projective distance aims at provingthat metrical is merelya branch *

1857,

tteonietrie (ter Lage" Nurnbergv

zur 1847; Beitriige

its

own

Geometric

der

Luge, ib. 1856,

1800.

t This

step, 1 believe,is due

to

Kleiu.

See

Matk.

Annalmi, Vols. rv,

vi,

xxxvn.

[CHAP.XLVHI

Space

422

Geometry, and harmonic

distances

that

merely logarithmsof

are

If this

ratios.

subjectof metrical

theory Geometry, and

correct, there

be

must this subject, distinguished

we chapter, axioms. projective

Let

the

examine

us

is not

a

by which, in

axioms

the

certain

an-

special the preceding

be consequences of this result

in which

manner

is obtained*.

alreadyseen how to assigncoordinates to every point of ratio of define the anharmonic line in projective a space, and how to also how to obtain a projective from have seen any four points. We when ideal an point has a a descriptive space, space. In a descriptive have

We

it is

when (i.e.

real correlative

assignto the real

a

sheaf of lines which

pointthe coordinate which

has

belongsto

a

vertex),we

the ideal

point

this way, the coordinate a projective as space. In Geometry of the two spaces becomes very similar,the difference being real point, that,in projective space, every real set of coordinates givesa of coordinate this holds each whereas,in descriptive onlywithin space,

considered

to belonging

(both of which limits are excluded).Jn what follows, remarks concerning therefore, projective tive space will applyalso to descripis stated. when the contrary expressly space except

certain limits

Let are

us

where ratios of all ranges aocby, line. Let a, f, $, our on #, y variable points

consider the anharmonic

fixed pointsand

fc

the

coordinates of these

points. Then

~

be

"

-

pointz,

we

will be

the

=

the

any

log(").

has one of logarithmof the anharmonic ratio in question essential properties of distance,namely additiveness. If xy^ yz, vz the distances of #, y^ z taken as havingsign, have must we

also

may

(??)-

=

log(") + logK)

have

an-

have

Hence

We

be

y-p

which, since a, ft are constants, points, f be the coordinate of by ("77).If now

(")foS) Thus

I

ratio of the four

denoted conveniently

other

/;

/

f-p/ harmonic

b

a,

log(f")

be

log("77) log(^"),which are two further of distance. From these properties (ofwhich the third follows properties from the other two) it is easy to show that all properties of distances which have no reference to the fixed points", b belongto the logarithm in question. Hence, if the distances of points from a and /;can also be made, by a suitable choice of a and 6, to agree with those derived from the logarithm, shall be able to identify distance with this logarithm, we In this way it is contended metrical Geometry may so be wholly "

*

"

0 and

the

=

"

"

The projectivetheoryof distance and angle is due to Cayley (Sixth Memoir 1859) and to Klein (Math.Annalen, Vols. iv, vi, vn, xxxvn). A fuller Quanti^ upon discussion than the following will be found in my Foundation* of Geometry,Cambridge, 1897, "" 30-38.

407, 408]

Protective Theory of Metric* the

brought under anglesbetween the ideal

us

consider first the

pointsof

distinct from

for

similar

a

theoryappliesto

lines

Let

408.

protective sway; or planes.

423

case

descriptive space.

a

and

where Let

our

protective pointsare

be considered fixed,and that 77 becomes and more x

Let y be moved so Then ft. more as 77 approachesft, log("9) will be will but values assume finite, be assigned. always exceeding any that a

b.

nearlyequal to

may

mathematically expressedby sayingthat,if f be any number other than a and ", then log("/?) is infinite. (Iff be equal to a or " log(fa) and log(gS) are indeterminate; this case will therefore be supposed excluded in what follows.)Hence a and b must be at an infinite distance from every pointexcept each other ; and their distance from each other is indeterminate. Again x and y must not be separated I.e.y must belongto the segment axb^ if we wish the distance by a and Z", and %~ft have the same to be real; for if " a sign,y-a. and 77- ft have if the same also must f a and f ft have different sign,but also and must have different signs a these conft ditions signs, ij 77 ; and amount to the same that the condition must as belong to y the segment curb. Hence if we insist that real two points(i.e.points any which are not merelyideal)are to have a real distance (Le.a distance which is not complex or purelyimaginary), measured by a number and shall to b fulfilthe following conditions : (1) they must a we require be the be ideal pointsto which no real ones correspond ; (2) they must limits of the series of those ideal pointsto which real pointsdo two correspond.These two conditions include all that has been said, For, there is no in the first place, real distance of any pointfrom a or ft; coordinates and be hence a of real points. In the second ft must not place,on one of the two segments defined by a and ", there is a real distance xy however near f or 77 may approachto a or ft; hence a and b the limits of the ideal points to which real ones are correspond.In the last it follows from the that all ideal points third place, proposition of the real two which to ones correspondbelongto one segments ab" real ones and all ideal pointsto which no correspond(excepta and ft This

is

-

"

themselves) belong to conditions which The

are

"

-

"

are

the

other

of the two

segments ab.

When

these

the function log(^) will have all the properties satisfied,

of distance. for a measure required to descriptive above theoryis only applicable space,

for it is

onlythere that we have a distinction between ideal and actual points. transitive in descriptive And begin with an asymmetrical space we line. Before the straight relation by which order is generatedon is to pure projective a applicable space, let developing theorywhich which may be called the little further the above theory, examine us a theoryof distance. descriptive real ones the ideal pointsto which In the first place, correspond, which

for shortness

I shall call proper

points,form

part of the whole

[CHAP.XLvm

Space

424

points,which is closed. The proper pointsare a semii.e.they have all the properties continuous portionof this closed series, of a continuum except that of having two ends. It may happen that ideal point which is not proper, or it may there is only one happen the one In the former that there are purelyideal point case, many. in directions. both This is will be the limit of the proper points of Euclidean space, for in Euclidean space there is only one the case sheaf of lines to which a given line belongsand which has no vertex, Hence the line. in to this of sheaf lines the given parallel namely The function be taken to be identical the pointsa and b must case therefore and is of useless " 77, and log(f77)is then zero for all values But of distance. as a measure by a familiar process of proceeding

series of ideal

this case, obtain the value "" y of elementary distance*. This is the usual measure Geometry should the distance of two pointsin a plane or in space we the

to

limit,we

in this

obtain the usual formula of the or

phrasethat,in

common

that the two

for the

in

can,

ends of

We

case.

Euclidean

a

space, The

line coincide.

here

see

+

fact

x

the

exact

is the

same

;

and for

similarly meaning as

is,of course,

"

oo

,

that the

ideal

pointwhich is not proper, and that this is the limit of proper ideal pointsin both obtain a it is added to the proper ideal points, directions : when we closed continuous seriesof sheaves to which the line in questionbelongs. is found to have a very In this way, a somewhat crypticexpression simpleinterpretation. and this is the case of hyperbolic But it may happen also space" line. In this that there are ideal on a points case, improper many will have two different limits ; these will be the the proper ideal points sheaves of Lohatchewsky's in the two directions. In this case, parallels function log(?77)requires but expresses distance as our no modification,

line has

no

it determines

ends,but that

only one

"

it stands.

The

ideal

pointsa and

b

are

distinct,which

is commonlv

expressed by sayingthat our line lias two real and distinct pointsat infinity. Thus coordinates are in descriptive obtained our space, in which by correlation with those of the derived protective space, it is always certain of to define function coordinates which a our possible protective will fulfil the conditions requiredfor a measure of distance. These conditions may be enumerated foliowsf. (1) Every pair of real as is real and finite, and vanishes pointsis to have a distance whose measure and ?/ onlywhen the two pointscoincide. (") If ,r, ?/, z are collinear, liesbetween of

measure *

See e.g.

oc

and

2,

(5)

xz.

the As

sum

the

Klein, rorfawngw

Vol. i, pp. 151 if. t Of. Whitehead, Universal text

to

distances

on

one

of the ideal

yz

to point corresponding

fiber nwht

Algehra,Bk.

line. straight

of scy and

measures

Euklidiacht

vi,

y

is to be the

approaches

tieometrie,Gottingeu, 1893,

Chap. i.

I confine

myself in

the

409]

408,

ideal

the

ProjectiveTheory of is the

point which

limit of

fixed,the absolute value of the

remains

425

Metrics ideal

proper

points,while

without

of ay is to grow

measure

X

limit. well be

It may

function

of

mathematician

should

we

points possessingthese

desire

define a If the properties. to

that his replies

only objectis amusement, his procedure l)e logically but extremelyfrivolous. He will,however, irreproachable, make this reply. We scarcely have,as a matter of fact,the notion

will

of

asked, however, why

variable

two

stretch, and, in virtue of the general axiom

a

know we magnitudeof divisibility,

some

know, without

do not

But

we

the

stretch

once

these

fulfils the axioms are

if these

two

that the stretch has

specialassumptionto

of Archimedes

and

axioms

belongto

must

assumed, there

not

are

the

no

magnitude. effect,that

that

of distance

measure

But

"of stretch.

measure

is

class has

When linearity.

of

assumed, the above propertiesof the

which properties

become

a

that every

why there

reason

the above four magnitude having a measure possessing characteristics. Thus the descriptive theoryof distance,unless we regard it as purelyfrivolous, does not dispensewith the need of the above remarkable What it does show axioms. and this fact is extremely is that,if stretches are numerically measurable, then theyare measured ratio of the by a constant multipleof the logarithmof the anharmonic with ideal points associated with the ends of the stretch together two in the two ideal pointswhich limit the series of ; or. proper ideal points the latter pairare identical, tbe stretch is measured case by a function to obtained as the limit of the above when the said pair approach identityand the constant factor increases without limit. This is a

be any

should

"

"

curios

most

result,but

it does

obviate the need

not

for the

axioms

follows as metrical Geometry. The conclusion same distinguish regardsmetrical Geometry in a plane or in three dimensions ; but here new introduced, which are irrelevant to the present are complications which

issue,and will therefore

not

be discussed.

importantto realize that the reference to two introduced by the descriptive theory of distance,has It is

nature

of distance

or

stretch itself.

device,but nothingmore.

This

The

reference

fixed ideal no

points,

analoguein

is,in fact,a

the

venient con-

stretch,in descriptive space,is

and in no way requires defined by its end-points, a reference completely to two further ideal points. And as Geometry starts with descriptive to endeavour subsequently the stretch, it would be a needless complication four stretch of in terms to obtain a definition of points. In short, even would if we had a protective theoryof distance in descriptive space, this since the whole protective still be not purelyprotective, space composed of ideal elements space. 409.

is derived from

It remains

projective space.

The

to

examine

theory we

axioms

the have

which

do not hold in

theoryof projective hitherto

projective

distance

m

examined, since it used

not projective; descriptive, theoryfor pure projective corresponding

the distinction of real*and ideal have

we

[CHAP.XLVHI

Space

426

Geometry.

the

examine

to

now

there

Here

are

no

was elements,

ideal elements of the above sort avssociated

and /3 be real and distinct numbers, they a line ; if,therefore, there will bewill be the coordinates of real and distinct points. Hence real points"r, y which will be separated by a and 6, and will have an this there could be no of distance. To objection., imaginarymeasure with

our

but for the fact that This a

is the

real

wish

we

our

to be the

measure

of

measure

a

stretch.

why it is desired that any two real pointsshould have of distance. In order to insure this result in a pure

reason

measure

that projective space, it is necessary of pointsat all,but should

and

a

be

(3 should

not

be the

ordinates co-

conjugatecomplex numbers.

of the logarithmshould It is further necessary that the constant multiple the distance of two real points be a imaginary. We then find that

pure

In a projective inverse cosine*. which is an real measure, since between space, the condition (")of p. 424 introduces complications, The definition of has not, as in descriptive space, a simplemeaning.

alwayshas

a

in this

between

case

is dealt with

by fully

Mr

Whitehead

in his Universal

("206). -Algebra 410.

if such

But

a

function is to l^e

trulyprojective theoryof

give a

geometricalentityto

some

which

and geometrical, properly

distance,it will be our

to

necessary conjugatecomplex numbers

of involutions. This can be done by means and ftcorrespond. ideal in a projective points,yet there are space, there are no

be called ideal

to

find a

Although, what

may

In point-pairs.

Chapter XLV we considered involutions pointson a line,all point-pairs points: of such that #, #' are harmonic with respectto #, b form conjugates "27, involution. In this case, x and x are said to be conjugate;a and b an each self-conjugate, and are called the double points of the involution. are But there are also involutions without real double points. The generaldefinition of an involution may be givenas follows (substituting the relation of x to of for the pair#, x'): An involution of pointsisif ay b be two

with real double

class of

other than identity, whose domain and relation, are same straightline,and which is such that any similar to the corresponding referents is projectively class of

relata.

Such

a

symmetricalone-one

converse

domain

the

relation is either

is a selfan or aliorelative, strictly relative as regardstwo and onlytwo points,namely the double points of the involution. For the line asevery pair of distinct pointson double will be and there one points only one involution: all pointthis to of the two exclude the identity pairs(using expressionso as correlation with some involutions. pointsof the pair)have a one-one Thus involutions may be called ideal that correspond : those point-pairs

*

a

This is the form originally given by Cayley in form is due to Klein. Qualities. The simplerlogarithmic

the

Sixth

Memoir

upon

409-411] to

actual

an

and

Metrics

ideal point-pairshave proper double points:two their respective

by

427

the others elliptic. Thus hyperbolic, in fact a one-one indivisible, being

Two

relation. defined

called

are point-pair is point-pair one

ideal

an

ProtectiveTheory of

auharmonic

an

ratio

improper ideal point-

have an analogous a or improper ideal point-pair, pairs, proper and an which is the measured relation, function obtained as by protective the suppositionthat and above from ct ft are conjugatecomplex function

This

numbers.

If point-pairs.

ideal

be

may

called the

anharmonic

of the

ratio

be fixed and

improper,the other variable and proper, an imaginary multipleof the logarithmof the resulting ratio has the properties of the anharmonic requiredfor a measure the t o distance of the actual point-pair corresponding proper ideal theory of distance. But to point-pair.This givesthe pure protective than a technical development, this theory,as anythingmore there are of descriptive in the case the same as objections space ; i.e.unless there be some there is no magnitude determined by every actual point-pair, the for the which obtain above of reason we distance; measure process by and if there is such a magnitude,then the above process givesmerely the measure, the definition, of the magnitude in question.Thus not two

stretch

one

distance remains

or

such

are

fundamental

a

that the above

method

entity,of

givesa

the

which of

measure

it,but

perties pronot

a

definition*. There

411. metrical becomes Let or

natural

p,q,rhe r

abc points,

of

introducing

Let

planepqr.

qr

line not

a

passingthroughp or q through #, rp through b9pq

pass

Let R^ be the relation which

through c.

simplerway

a

a

fixed

three

in the

and

another

distance in this way projective space, and accompanimentof the introduction of coordinates.

into

notions a

but

is however

holds between

x

and

y when

pointson abc, and or, yq meet on ap^ and let R^ R3 be be regardedas constructed net may defined. Then a Mobius similarly shall have, if xRj/9yR" We relations of the J?,,R^ R$. by repetitions

these

then

are

3cHayz.

We

index is

whose

define the square

can

means qr, and .rS/jy = R^ J2,' ap, then fijJZi'

by

"r

and y

methods, projectiye

pure

define

defined

;

Rf+*

to

whence, since Rf

all rational powers integer,

be defined

can

we negative, now

may

*

On

can

as

it follows that

limits.

can

the

Stand t,

above

Rf provided

is

R1 Hence, if can

define Rf, for

method

ztir Bfitrfige

of

a

be defined if

of

take this relation Rf

if

and

RfR"

mean

as

abc and

on

are

any

which propositions,

these

From

.

may

von

that

n

and

8

be

xr, are

y$

meet

proved

numbers,

der

we

been

defined,and irrational powers be any real number, positive or

be .r

JR,-*with R"?. We identify may distance of any two pointsbetween the

we

introducingimaginaries in projectiveGeometry,

ftwmetrie

Rl

is any

already or positive negative

JR,*have a

of

power Further, if $

of 2.

or negativepower positive

a

point of on

of j?2,or

root

J^afffy i, " 7.

see

which

holds, and

it

find

Thus

on

which

called

be

may

properties

method

sum

up

distance, both yet such

and

defined metrical

Geometry,

the

divisibilityof

of

magnitude

stretches, angles,

dealt

with

axioms, the

obey

many

quadrilateral

the

Thus

dimensions.

to

work.

angles and

It

as

doe*"

of

magnitudes and

projective

pure

;

subject

and

requires

hence,

as

bewildering real

affords, from and

number the

point

unification

their

metrical

of view

of method.

in

but

it is

axioms.

spite of

pure

of

of of

word

but

stretches

hand,

other

of

all metrical

does

with

work and

deduction

to

appearances a

or

the

the

matters.

properties, the

of

out

mathematics, metrical

projective

mathematics,

in and

terribly complicated, Hence

a

belong

not

the

which

certain

metrical

properties

pure

theory

having ;

of

to

Geometry,

definition

importance, and, the

metrical

of

three

distances

independent

subjects belong

adopt

mathematics

from

both

On

of

ness unique-

the

planes,

or

kind

of

used

in

proved

without

require

throughout.

are

relations

one-one

lines

applied these

the

as

science

have

does

magnitudes

usual

proved

we

or

development

another

pure

properties has

points

these

should

part of

a

metrical

the

since

the

distinct

is

be

results

idea

and

Without

cannot

be

supposed,

often

is

as

between

allow

defined

is

sense

idea

all the

such

which

in

But new

indefinable, it does

new

a

suffice divisibility

is, it is true,

distances,

space

not,

mathematician

There

the

be

can

the

linearity.

can

of

purely technical,

This

axioms,

genuinely

a

descriptive Geometry

assumptions, themselves

is

in

relations

new

construction

it introduces

mathematics

pure

this

there

but, since

Geometry,

these

theory

indefinable,

that

and

possible, and

becomes

Geometry

elementary

is

propositions

with

of

this

indemonstrables.

or

it is assumed

metrical

manner;

is

mathematics.

pure

whether

subject, requires

Archimedes

of

usual

metrical

usual

of

areas,

the

of

a

relation

a

properties, which

series,which

etc., and

axioms

the

to

is infinite.

have

projective

space,

indefinables

new

Euclidean

space.

metrical

independent

an

as

belong, properly speaking,

not

it

without

deduced

of

know

not

so-called

projective

in

necessarily possess

do

spaces

to

shall

projective theory

a

do

XLVIH

We

point

actually

sense

I

usual

the

descriptive and

in

this

in

justified. But to a plane or

be

Although

:

and

other

any do

points

distance.

properties

from

a

two

distance, can

of

the

usual

the

have

any

extended

be

can

To

line

of

measure

distance

the

that

the

as

x

defined

so

projective

a

metrical

to

regard

distances

that

distances, except

of

[CHAP.

Space

428

this and of

descriptive contrary,

genuine

fication simpli-

CHAPTER

DEFINITIONS

IN

412.

the

OF

sake of

definitions and

indefinable* But

SPACES.

of different Geometries, I have

convenience,adhered on

the

one

to

the

hand, and

this

distinction between axioms

in pure mathematics, distinction, regardsthe ideas and propositionsof Logic.

the other.

except as

VARIOUS

precedingdiscussions

for the usually,

on

XLIX.

all

variable.

the

This

propositionsstate is,in

mathematics.

fact,the

or

has

postulates validity

no

In pure

matics, mathe-

logicalimplications containinga

definition, or

part

of the

definition,of

stated must flow whollyfrom the implications of Logic, which are prior to those of other branches of propositions mathematics. Logic and the rest of pure mathematics are distinguished from are by the fact that, in it,all the constants appliedmathematics fundamental of some definable in terms notions,which we agreed eight What c onstants. other matics call branches of matheto distinguishes logical which usually from Logic is merelycomplication, takes the form rather complicated of a hypothesisthat the variable belongsto some be denoted by a single class. Such a class will usually symbol ; and the is class in be that the statement question to represented by such and call is to say, what mathematicians is T hat a dejinition. such a symbol

pure

The

and does not make at all, anv part of mathematics entities dealt with but the is statement mathematics, by concerning simplyand solelya statement of a symbolicabbreviation : it is a proposition concerningsymbols,not concerningwhat is symbolized. I do the word other has no of course, to affirm that not definition mean, meaning. All meaning, but only that this is its true mathematical a

definition is

no

is built up by combinations but all its propositions can,

of

certain

number

of

primitive length of the resulting in terms of these primitiveideas ; hence But further,when all definitions are theoretically Logic is superfluous. extended, as it should be, so as to include the generaltheoryof relations, there are, I believe,no except such as primitiveideas in mathematics of Logic. In the previous chaptersof this Part, belongto the domain authors most do, of certain indefinables in Geometry. I have spoken, as

mathematics

ideas,and stated formulae, be explicitly

a

for the

[CHAP.XLIX

Spact

430

concession,and

But

this

two

classes of entities which

was

a

mast

be rectified. In mathematics,

now

internal relations of the

have

logical particular

same

dealingwith one class of entities, namely,with all classes but with a whole class of classes, specified type. And by the type of having internal relations of some such as are denoted by its purelylogical relation I mean a properties, for example Thus and so on. the words one-one, transitive, symmetrical, defined the class of classes called progression by certain logical we type

equivalent.Hence

are

and progression,

never

are

is of any class which far it in deals that finiteArithmetic, so as

internal relations of terms

characteristics of the a

we

found

we

numbers, and not with the terms or classes of which numbers can And when it is realized be asserted, to all progressions. appliesequally that all mathematical ideas,except those of Logic, can be defined,it with

is

there

also that

seen

are

when

Geometry

of

so-called axioms

except those of Logic. The

mathematics

in propositions primitive

no

is considered

as

of

branch

a

Geometry, for example, pure mathematics,are

constitute the science. which in the hypotheticals protasis if,as in appliedmathematics, propositions They would be primitive themselves asserted; but so long as we thev were only assert hypothetical of the form "A B") in which the implies(i.e.propositions there is axioms to assert the reason no supposed appear as protasis, to admit genuine axioms My objectin consequently, nor, protasis, the presentchapteris to execute the purelyformal task imposed by

merely

the

and to considerations, which, without spaces, from these

the positioiib, is

to show chiefly

413.

indefinables and

various- Geometries

with the definition of

of the

some

strict definitions of various-

set forth the

without

more

that such definitions

possible.

are

is any class of entities "uch that there

members

of the

class ; between

and

one

far

which aliorelative, symmetrical its

as

to be proj"erties there is taken,

beingan

at least two

are

distinct members

two

any

is

aliorelative will

of

projective space

A

three dimensions

so

myself object

importantspaces, .since my

(1) Protective Sptweof three dimensions.

only

primitivepro-

I shall content

will follow.

there is

connected,and has

permit,and

one

is transitive

further

enumerated

shortly;whatever such aliorelative may be term a projective space not belongingto the field of the said aliorelative, which field is whollycontained in the and is for called, is and projective shortness,a straight line, space, denoted by 06, if a, b be any two of its terms; every straight line which contains three

any the class

of the

two

terms

ab" then

of the

any class if a be circumstances, of

the

a

at least one

projective space,

there is at least one

to belonging

bb' have

contains

terms

common

class ab, and

ex, where

term

a

x

of

is

v

any

two

that

a

term

any

term,

terms

such

;

if a,

b, c be

does not

belongto projective space not

c

of the

any term

be, b'

part; if d be u,

such

term

other term

of ab

;

under

the

same

of

ac, the classes other than a and

that

d

belongsto

aa'" 6, the

413]

412,

Definitions of

but neither

class uv,

of the

only term yz and

a", then

it may

be

is not

x

proved that

uniquelydetermined

by

term

and such that there

gHytfkand

gHyeh^ then

relation of the

be

four

is

class aby and

and

the

d

431

(under and

these

x

part of

circumstances

and

u

have

d

of the

common

independentof

hence

if y be the

onlyterm

t", and

is

symmetrical

a

denoted,for brevity, by

JcH^d^if y,

to projective space, belonging

the

e

class xdy

g,, h of the class xd for which we have write for shortness yQxoe to express this

two

are

au

x

Z",d;

",

may of the

further terms

be two

the

identical with

the

relation which

one-one

Spaces

part "r, z bu^ x the only term of the

and

av

belongsto

v

of

common

part of

common

nor

u

Various

terms

we

terms

d, y^ e) ;

x,

the relation QXd, whatever

of the

is such

projective space

a

that

and d may be, is transitive of one ; also that,if #, i, cy d be any four distinct terms straight line,two and onlytwo of the propositions aQ^l, aO^c^ aQ^b will hold ; line form

a

class

This is a

at

ae

that

?/, such

term

", c, d, e be any five a1

least

term

one

belongsto

#,

in the

of

a

and

defined in

sense

of

a

" "77

;

projective space, there will in the class cd at least one

the class by.

formal definition of

a projective space of three dimensions. class of entities fulfilsthis definition is a projective space.

Whatever

enclosed in brackets

I have

terms

a.-

it results that the terms

series ; this series is continuous

if a, finally, be in the

space

of projective properties space

these

from

terms

a

passage

in which

no

of properties

new

which venience serves projective onlythe purpose of conspace are introduced, of language. There is a whole class of projective spaces, and

this class has

infinite number

an

of members.

existence-theorem

The

a proved to beginwith, by constructing projective space out of defined in " 360. complex numbers in the purelyarithmetical sense least four that the class of projective We then know spaces has at

be

may

members, since of

which

has

we

at

know least

of four

member.

one

above

arithmetical space.

space

of

under

sub-classes contained In

In the second

descriptive Geometry,in

the* first we place,

which

the

we place,

the

have

terms

of the

it, each the

have

projective projective

In the third descriptive space. have the polarform of elliptic we place, space, which is distinguished of stretches,consistent by the addition of certain metrical properties the definition of projective with, but not impliedby, space; in the have the antipodalform of elliptic fourth place,we Geometry, in of the said which the terms of the projective space are pairsof terms elliptic space may be space. And any number of varieties of projective for the definition obtained by addingproperties not inconsistent with In fact, that all planesare to be red or blue. example,by insisting

space

sheaves

are

of

lines in

the

"

every

class of %"* terms

is series)

a

the fieldof Hence

of the number (i.e.

projective space a

;

for when

two

of

terms

classes are

in

if similar,

the other will be the field of certain relation,

by correlation with

a

projective space,

any

continuous

a

a

one

is

like relation.

class of

""* terms

becomes

itself

is

would

projective space

defined

best

be

the straightlines satisfying substitution the to strictly analogous fields

found

we

desirable

the field of

but

It is other

if there

For

to

be

is

regarded

the terms

drop

latter involve

of

the definition

that

implies and

property which

any

in the definition,we

propertiesused

of the

more

certain

a

be

point

the

and

former,

space, as of mo^t is arbitrarywithin certain limits. complexity,

observe

important to

entities of

whose

for series which

are

to

a

:

the latter.

the former

not

of terms

the

class of relations,since

only the

mention

set

it is convenient class of relations,

a

This

of serial relations a

concerned

conditions.

above

When

IV.

in Part

standpointof

is

of relations

class K

a

as

are

the

definition

where

fundamental

more

fact is,that

The

protectivespace.

a

line-Geometry

as

[CHAP. XLIX

Space

432

implied by

is

make

property

new

metrical

or

since

space,

definition

be

might

Euclidean

may

form

of the Euclidean

account

account

my

414.

that any relation of a

such

productof

is

holds

a

vector

as

such

a

definition

a

is

the limit of certain

have

and

one

of

inappropriate non-Euclidean

the vector between

is any itself is

follow Peano, strictly

-f: the

but

b and

d

;

least

at

integer)of any identity; there

between

any one

term

vector

is

of

a

a

givenvector

term

asymmetrical

a

and

vector, holds c

nth

the

between

of the

whose

or

is the

space of the space of the class is

vector

one-one

class of vectors, defined

the

converse

holds

which

only one

called

vectors, is a vector; if a

n

by

relative and

a

same

6,

that

as

has ;

any assigned if the nth power

then identity, power

is

a

the

given

;

is measured

vectors

*

di

give

which

how

and vectors have one only one symmetrical relation of any two certain class having the foDowing properties relation of any two : the

vector a

form

a

I shall not

nvill be

class,which

relation of the class to

(where

instead

show

A Euclidean #pace of three dimensions. space of least class of terms at two members, and containing

of them

two

two

d, then

which

to

similar to his.

followingcharacteristics

and

give in

axioms.

will be very (2) Euclidean

three dimensions

the

serves

a

or

Calculus. is very appropriateto quaternionsand the vector has been adopted by Peano*, and leads to a very simple

.spaces, but This

model

is considered

in

or

I shall

I shall

This

one

above

the

constructed.

space. Euclidean space

when

c

in

more

one

substitution

ample, question. For exin place of definingthe line by a relation between points,it is relation to a couple certain class define line the as a having a possibleto of points. In such cases, we can onlybe guided by motives of simplicity. It seems scarcelynecessary to give a formal definition of descriptive

of the

placeof the

a

"

Analisi della

by Teoria

real

a

dei

number, positiveor

vettori/*Turin,

1898

negative,and

(Accademia

is such

JReale delle Science

Torino). t

For

the

convenience

corresponds to taken

in the

that

sense

of

in which

reader, it may be well to observe that this relation direction being given distance in a given direction direction. parallellines have the same

of the

having

a

all

"

Definitions of

413, 414]

that the relation of

number, and that the to

vectors

two

three or

relative

not

are

third vector

a

irrational power

an

is the

of the

of

a

;

others,then

by a positive productof

the satisfying

vector

a

of two

of their several

measures

givenbelow

vector

433

always measured

of the

sum

powers of which is a

one

Spaces

relation of the relative

there is

productsof

vectors, no

both

of the

measure

the third vector

relations to of

to itself is

vector

a

Variom

there

;

are

given vectors

definition which'

vectors

if i,J, k be

;

productof powers of one of powers of relative products

relative

all vectors

are

""/"*" The

for explanationhere onlypointscalling

irrational power

of

All rational powers the nth root has an be defined.

can

is,when

Peano* u

of

be

Let

u.

vectors

;

vector

a

definite

are

The

an

vectors.

numbers, be the

v

a

number

the limit of the

vector

into which, multiplied relatively

in the

class t", is

the relations to themselves

", will

have

numbers, and suppose these

all

or

liven

the in

approachesx*

give the

of the measurable

means we

x

M.

correlate

this definition is the

point of

The

zero.

as

v

some

of the relation to itself of the

measure

order obtained among vectors by each has to itself. Thus suppose

the derivative

all tfs and

correlative to

limit of the class

given by

following.Let

belongingto

class of vectors

be the

is said to

XQ

relation subsist between

one-one

let

the class uy when

;

translated into relational language, the

some

and

of rational

relation of two

definition of limits of classes of vectors

class of real

a

the measurable

of

for every vector has an 7*th root, and rath power, which is the mjntli power of the original it does not follow that real powers which are not rational

But

vector

and

vector

a

the notion

are

a

to be

of the vectors

^,

use

to

of the

relation which

x^ *rn, progression "r2, the measures respectively ...

a*^

.

#n,

..

oc

Then

.

..

of be

if x

the limit of #19 #2, there is to be a vector whose relation to x^ , itselfis measured by x" and this is to be the limit of the vectors a19 #,",. ...

...

..

rtn,

.

.

other

.

;

and

thus

point to

vectors.

irrational powers of a vector become definable. be examined relation between is the measurable

This relation meavsures,

in terms

of

The two

elementaryGeometry,the

by the vectors into the cosine productof the two stretches represented it is,in the languageof the calculus of of the angle between them: To say that the extension,the internal* productof the two vectors. relation is measurable which

some

or

means,

in the

that all such relations have is employed,

this statement

relation to

of real numbers

in terms

all of the real numbers

;

a

sense

in

one-one

hence, from the existence of

irrational powers, it follows that all such relations form a continuous series ; to say that the relation of a vector to itself is alwaysmeasured that there exists a section (in DedekindV number means by a positive

sense)of the continuous that vectors

can

have

series of

such that all those relations relations,

to themselves

*

appear

Op. ciL p. 22.

on

one

side of the section

;.

[CHAP.XLIX

Space

434 while it

defines the section is that

to itself.

has identity

the vector

which

the relation which

provedthat

be

can

the definition is,of course, by no means be given of Euclidean space, but it is,I think, the

only one which can simplest.For this of ideas which, being of previouschapters,

This

belongsto

it

also because

and

reason,

order

an

Euclidean, is foreignto the methods essentially I have thought it worth while to insert it here. another 415. As example which may serve to enlargeour ideas. rather the space which is or I shall take the space invented by Clifford, analogousto his surface of zero curvature and finite extent*. formally I shall first briefly explainthe nature of this space, and then proceed to

Spaces of the type

definition.

formal

a

sake of

for the

of dimensions, but

number

in

have

questionmay

any I shall confine simplicity

myselfto two dimensions. In this space, most of the usual Euclidean certain size ; that is not exceeding a hold as regardsfigures properties is two rightangles, and there of the anglesof a triangle to say, the sum in which called all translations, pointstravel motions, which may be are Euclidean

from

the space"is very different respects, begin with, the straightline is a closed

in other

But

along straightlines.

To

space.

and the whole space has series, is

motion

translation ;

a

distanceb from i.e.

all translations

can,

of translations in two

out

serves pre-

motion,

a

as

fixed

have parallels, this space, as in Euclid, we i.e. straight remain distance apart,and can at a constant be simultaneously

lines which

described in

motion

a

linear equations.But

formula.

Euclidean

which

In

directions.

by

place, every

one (i.e.

is never point unaltered)

compounded

space, be

Euclidean

in

fixed

distance unaltered; but

leaves every

never

circular transformation

a

certain

a

In the second

finite area.

a

Thus

also

;

formula

the

if Trk be the

lines straight

can

for distance is

be

represented

quite unlike the straightline,

lengthof the whole

y\ (x yr)be the coordinates of any two points(choosinga ("r, line has a linear equation), then if "o be system in which the straight

and

the in

,

anglebetween the is d, where question cos

r

=

cos

lines x

(x"x)

and the formula for the We

may,

(3)

**

are

0,

is

=

"

anglebetween

Clifford's space of two

'y

(y "y')

in order to lead to these

dimensions there

cos

=

two

cos

two

sin

a)

(#

dimensions.

A

the

pp.

definition. following

Clifford's space

terms, between

Chi the generalsubjectof the spaces of which xxxvn,

x) sin (y"y'\

-

relations of different classes, called

Klein,Math. Amuilm Vol. i, Chap, iv.

,554-505,and

points

two

lines is similarly complicated.

set up results,

class of at least two

a

0, the distance of the

any two

of

two

of which

distance respectively

this i" the simplest example, see Killing,Urundtogen far Geometric,

Definitions of

415]

414,

Various

Spaces

435

and possessing the following direction, direction is a a properties: transitive far its aliorelative will aliorelative, so as symmetrical beingan

and

but permit,

connected

not

the said term

to which

terms

is called

line ; straight

a

term

a

;

has

a

of the space together with all the given relation of direction form what

line straight

no

contains

all the

of the

terms

of the relation of direction to space has any assigned space ; every term other but all of terms not the some space ; no pairof terms has more

than

relation of direction; distances

one

relations

are

class of

a

symmetrical

forming ends, one of which all distances except identity is identity intransitive aliorelatives; are ; of the relation of distance to some every term space has any assigned but not all of the terms of the space ; any giventerm of the space has and direction from two and only two other terms any given distance of the space, unless the givendistance be either end of the series of distances ; in this case, if the givendistance be identity, there is no term havingthis distance and also the givendirection from the given term, but if the distance be the other end of the series, there is one and only term one having the given distance and the given direction from the line have the properties, mentioned giventerm; distances in one straight the order in ChapterXLVII, required for generating terms of one an among line ; the onlymotions, i.e.one-one relations whose domain and straight continuous

a

domain

converse

are

each

series,having two

the

space

in

question and

which

leave all

the corresponding the relata the same those among distances among as referents,are such as consist in combining a given distance,a given

and direction, line of

and

;

some

of the two

one

every such distance in

fixed direction,both directions form

a

of the series constituting a straight

senses

combination one

is

fixed direction with with

taken

singleclosed

a

the relative

to equivalent

suitable

continuous

product

distance in another

some

all possible finally

sense;

series in virtue of mutual

relations. This

I think,the completes,

definition of

a

CliffonTs space of two be

It is to be observed that,in this space, distance cannot

dimensions.

because (1) we have identified with stretch, cannot we generate a closed series of terms

onlytwo dimensions,so of on a line by means

methods*,(")the line is to be closed,so that jective order

on

line straight

the

both

that

reasons

directions

relations ; thus a

line that

we

direction to direction to

*

Mr

W.

"

a

the and

method. descriptive distances

it is onlyafter

an

have

to

cannot

pro-

generate

It is for simile r

be taken

order has been

as

metrical sym-

generatedon

two distinguish senses, which may be associated with and with distances in a given render it asymmetrical, give them signs. It is importantto observe that, when can

E. Johnson

by introducingthe axiom

by

we

that

method

which

come might be overpointedout to me that this difficulty construction by a special uniquenessof the quadrilateral would perhapsbe simplerthan the above.

has

distance

taken

is

order

in

the

the

hands

and

Euclidean take

distance

of

Clifford^

For

this

spacef, reason,

what

has

Enough kind

of

relations

of

terms

throughout

*

t

distance

of

indefinables

relations)

mathematics,

pure

is

was

the

spaces. space and

definition

the

and

terms,

of

only

and

principal

the

specification

which

classes

definable

are

the

This

up.

a

new

even

logical

these of

that

but

These

of

composing

terms

but

out

built

instance

such

important,

definitions,

to

distance,

actual

relations.

of

non-

space.

determination,

geometrical

of

are

his

result,

object

logical which of

the

chapter.

Leipziger If

that

the

to

this

some

of

that

logical

relations

classes

collection

small

{including

present

two

the

a

in

specifying

use

show

to

only

individual

in

the

to

unnecessary in

of

of

purely

in

their

logical

used

elements

calculus holds

certain

hope,

Not

only

require

not

of

the

are

in

do

members

as

and

irrelevant,

space

I

required.

not

are

said,

is

perties pro-

applying

interest it

briefly

of

or

distances

produced,

except

consider

definition

the

possible

always

is

space

indefinables

been

now

simpler

instance

in

has

spaces,

able,

to

an

motion,

call

geometers

give

to

most

method

important

was

order

in

a

in

greatest

been

have

I

adopt

to

it

in

since,

indefinable,

as

Clifford's,

as

But

Geometry.

the

of

necessary,

leave

Lie

and

XLIX

property

some

by

groups*,

results

Klein,

of

those

becomes

it

which

adopted

been

continuous

of

theory

assign

to

transformations

or

has

method

line,

straight

the

spaces,

relations

This

Geometry

different

one-one

unchanged.

of

independent

as

distinguish

to

of

a

[CHAP.

Space

436

I

had

as

dimensions.

1800.

Berichte, defined

distinct

an

from

elliptic stretch,

space

because

of

two

dimensions,

the

protective

1

should

generation

have of

had

order

to

take

fails

in

CHAPTER

THE

416.

IT

L.

CONTINUITY

OF

SPACE.

been

has

commonly supposed by philosophersthat the to be was something incapableof further analysis, the not intellect. as regarded a mystery, critically profane inspectedby In Part V, I asserted that Cantor's continuity is all that we require hi I wish to make dealingwith space. In the present chapter, good this without assertion,in so far as is possible raisingthe questionof absolute and relative position, which I reserve for the next chapter. Let us begin with the continuity of projective have seen We space. that the pointsof descriptive similar to those of a ordinally space are semi-continuous portionof a projective space, namely to the ideal points which have real correlatives. Hence of descriptive the continuity space is of the same fore, kind as that of projective and need not, therespace, be separately considered. But metrical space will requirea new continuityof space

discussion. be observed

It is to

do not

that

Geometries, as they

begin by assuming spaces

fact,.space is,as

Peano

remarks*,

easilydispense. Geometries togetherwith to the

certain

number

of

assumptionthat determine

least

one

introduce any

a

with

with

which

begin by assuming

axioms

which

from

treated

infinite number

an

word

a

are

there

are

at

least two

points,the

straightline,to

that

three

the

is at

least

This

gives us joiningit to

a

fourth

very

drawn

as

begin with the points any two

which

have

given straightline.

be

can

other

pointbelong. Hence we new assumption that there

can

point, class-concept

a

conclusions

points,and

now-a-days, points; in

Geometry

points. So, in projective Geometry, we

class of

of

they and

at

points. We now one poiut not on

point,and

since

there obtain

our previouspoints,we pointson the lines obtain an infinite denuin Hence all. three more can we points seven fuz*ther mcrable series of points and lines,but we a cannot, without three points on than one any assumption,prove that there are more b and d line. Four pointson a line result from the assumption that, if

must

be

"

*

jfiv. di Mat.

Vol. iv, p. o*2.

[CHAP.L

Space

438 with

be harmonic

respectto

a

and

then b and d

c,

distinct.

are

But in

pointson a line,we need the order results*. These further assumptionsfrom which the projective of series pointson our line. assumptionsnecessitate a denumerable Such series of content. be a With chose, we these, if we might constructions ; and if we pointsis obtained by successive quadrilateral line could be obtained chose to define a space in which all pointson a with any three points constructions starting by successive quadrilateral order

of the the

obtain

to

of

infinite number

an

contradiction would

line,no

Such

emerge.

rationals and type of the positive

ordinal

line would

form

compact by assuming

extension,introduced

space would the

zero:

series with

denumerable

a

a

the

that

series of

have

points on end.

one

a

The

points is

tinuous, con-

onlynecessary if our projective space is to possess the usual there is to be a stretch,with that is to say, propertiesif, end and its straight line given,which is to be equalto any given one stretch. With only rational points,this property (which is Euclid^s hold universally. But of the existence of the circle) cannot postulate whether for pure projective our space possesses purposes, it is irrelevant is

metrical

"

of does not possess this property. The axiom forms. be stated in either of the two following or

line

points,and all infinite series of limits ; (2) if all pointsof a line be divided into the other,then either the first one whollyprecedes

limits of series of rational

are

rational

pointshave

of which classes,

two

continuityitself may (1) All pointson a

class has

last term, or the last has a first term, but both do not which results is exactly happen. In the firstof these ways, the continuity is a necessary, Cantor's,but the second,which is Dedekind^s definition, a

condition for Cantoris continuity.Adopting this first sufficient, the rational definition, points,omittingtheir first term, form an endless series; all pointsform a compact denumerable perfectseries ; and between pointsthere is a rational point,which is precisely any two the ordinal definition of continuity")". if a projectivespace is Thus have to have the kind of continuity whieh at all,it must continuity not

a

the real numbers.

belongsto

Let

417.

consider next

the

of continuity

a

metrical

space

;

and,

sake of

for the is here axiom

us

let us take Euclidean space. The question definiteness, for continuityis not usually introduced by an difficult,

more

ad

hoc,but appeal^

distance.

It

to

in result,

sense, from

some

the

axioms

of

alreadyknown to Plato that not all lengthsare commensurable,and a strict proofof this fact is contained in the tenth

book

of Euclid.

of Cantor's are

was

But

The gistof continuity.

commensurable,

as expressed

*

Cf.

this does not

togetherwith

follows. If AB^AC

Fieri,op.

take

very

far in the

the assertion that

the

be two

cit. """,Prop. 1.

us

of postulate

lengthsalong the t See

Part

all

not

the

direction

lengths

circle, may same

V, Chap,

xxxvi.

be

straight

The

417]

416,

Continuity of Space

439

line,it may

happen that,if AB be divided into m equal parts,and AC into n equal parts,then, however and n may be chosen,one of the m parts of AB will not be equal to one of the parts of AC, but will be values of m and ", and less for others ; also lengths greater for some either to equal may be taken along any given line and with any given But this fact by no end-points*. means proves that the pointson not denumerable,since all algebraic a line are numbers are denumerable. Let

us

In

then, what

see,

our

axioms

allow

us

to infer.

Greek

of irrationals, two Geometry there were great sources diagonalof a square and the circumference of a circle. But there could be no knowledgethat these are irrationals of different kinds, the one scendent being measured by an algebraic number, the other by a trannumber. No method for known was constructing general less still for n umberf, assigned algebraic constructingan assigned any

namely,the

transcendent

by

number.

of

means

limits,are

numbers

can

And

far

so

I

as

know, such methods, except

stillwanting. Some be constructed

and algebraic without geometrically

scendent tran-

some

the

of

use

and do not follow any general limits,but the constructions are isolated, plan. Hence, for the present,it cannot be inferred from Euclid's axioms that space has

denumerable.

not

are

Cantor's sense, or that the pointsof space Since the introduction of analytic Geometry,

in continuity

For example, made. equivalent assumptionhas been alwaystacitly it has been assumed that any equationwhich is satisfied by real values

some

to be of the variables will representa figurein space ; and it seems even universally supposedthat to every set of real Cartesian coordinates a

pointmast

were correspond.These assumptions

times,without

any

that

made, until quiterecent

without all,and apparently assumptions.

discussion at

they were

any

sciousness con-

such, it becomes as assumptionsare recognized be introduced must apparentthat,here as in protective space, continuity the the philosophers, we by an axiom ad hoc. But as against may make to satisfy remark. is indubitably Cantor's continuity following sufficient existent all metrical axioms, and the only question whether is, space need have continuity of so high an order. In any case, if measurement is to be theoretically not have a greatercontinuity possible, space must When

than

that

The

put

these

once

of the real numbers. that the

axiom

in the form

which

pointson

a

results from

line form

a

continuous

amending Dedekind,

or

series may l"e in the form

first form, every section of the a a line is definable by a single which is at one end of one of the parts point, has no end. In the second while the other part, producedby the .section,

that

*

line is

A

length

perfectseries.

is not

synonymous But terminated. a essentially a

stretch t For

or

a

In

the

segment, since a length is regarded aa with length is,for present purposes, synonymous with

a

distance.

shortness 1 *""" identifynumbers

with

the

lengthswhich

they

measure.

it completely defines because,unlike the first, preferable and has a limit, ordinal type,every infiniteseriesof points every point the that line has add to is It cohesion*, not limiting point. necessary

form,which the is a

is

in any

are

be true

any

to be quiteimpossible

would

what empirically distinguish

rational space from a continuous to think that space has reason

no

of

axiom

tinuity con-

which I see actual space, is a question be and such questionmust empirical,

regardsour

deciding.For

of

means

as

the

Whether

essential to measurement.

case

which linearity,

of

and

of Archimedes

the axioms

for this results from

a

L [CHAP.

Space

440

it

be called

may

in any space. a higherpower than But

no

there is

case

that

of

the

continuum. The

418.

of the postulate

continuityenables

of

axiom

and circle,

to

us

to

dispensewith

substitute for it the

the

following pair.

line there is a pointwhose distance from a given any straight point on the line is less than a given distance. (2) On any straight line there is a pointwhose distance from a givenpointon or off*the line

(1) On

givendistance. From these two assumptions,together be proved. Since it is the existence of the circle can with continuity, and since from the circle, to deduce continuity not possible, conversely, of analytic much Geometry might be false in a discontinuous space, it to banish the circlefrom our initialassumptions, a distinct advance seems with the above pair-of axioms. and substitute continuity of space, and no There is thus no mystery in the continuity 419.

is

greaterthan

of any

need

a

notions

definable in Arithmetic.

not

is,however,

There

most notion that,in space, the whole a philosophers, among the partsf ; that although can area, or volume every length,

is prior to be divided

indivisibles of which

no are lengths, areas, or volumes,yet entities ai~e composed. Accordingto this view, pointsare mere and onlyvolumes are genuine entities. Volumes not to be fictions, are but as wholes containing regarded as classes of points, parts which are never simple. Some such view as this is,indeed,often put forward of what should be called continuity. This as givingthe very essence

into

there

such

questionis distinct from the questionof absolute and relative position, which

shall

I

discuss in

our positionas relative,

continuous

portionsof

concerned essentially

the

followingchapter. For, if we regard present questionwill arise again concerning This

matter.

with

present question is, in fact,

and continuity,

may

therefore

be

priately appro-

discussed here. The And

where

the

real *

oj

arise in

Arithmetic,whether

essentially composedof terms"

are

f

series which

r

we

come

numbers,

See Part V,

-I?: ^^

Leibniz, Chap.

near

each

to

real

the

continuous

or

real numbers, rationals, integers, of continuity space, as in the case

number

is

a

segment

or

not, etc.

of

infinite class

Chap. xxxv. PML ix.

W"rUe

(Gerhardt), 11, p.

379 ; iv, p. 491 ; also my

Philosophy

417-421]

The

of rationals,and is

no

Continuity of Space

denial that

possible.In this case, various

construct

segment is composed of elements

a

start from

we

infinite wholes.

441

But

the in the

elements

and

of

case

gradually

space,

we

arc

told,it is infinite wholes that are givento begin with; the elements and the inference, are is very rash. we onlyinferred, are This assured,

questionis in is

the

main

of

one

Logic.

Let

us

see

how

the above

view

supported. Those, who

in the

They

deny indivisiblepointsas

constituents

lines of argument by which the difficultiesof continuity and

past,two had

to

of space have had, maintain their denial.

and they had infinity,

the

to their school,they way in which space is presentedin what, according called intuition or sensation or perception. The difficulties of

continuity

and

we as infinity,

line of

in Part

saw

is

V,

are

thingof the

a

past; hence this deny points. As

longeropen difficultto giveit a precise argument,it is extremely form indeed I suspectthat it is impossible. We may take it as agreed of whose existence tliat everything become immediately we aware spatial, is complexand divisible. Thus the empirical in sensation or intuition, of space, is the existence of divisible entities in the investigation premiss, But with certain properties. here it may little be well to make a into the meaning of an empiricalpremiss. digression is 420. An empirical reason which,for some or premiss a proposition existential. and which,we is for no reason, I believe, Having may add, this shall to we find,on examination, proposition, usually agreed accept sets of simpler that it is complex,and that there are more one or from which it may If P be the empirical be deduced. propositions let A be the class of sets of propositions form) (intheir simplest premiss, argument

to

no

those

who

other

regardsthe "

from

which

"considered of P

P may

be deduced

member,

and

let two

"when they imply one equivalent of

infer the truth

we

;

that

member

must

one

set of the

be true.

But

members

of the From

another. class A.

if there

If A are

has

many

class A the

be

truth

only one members

endeavour to find some other we equivalent, all of sets of simplepropositions the empirical premissP/^ impliedby class A''. If now it should happen that the classes A and A' have only of the

class A, not

all

of A are inconsistent with member, and the other members be true. If not, \ve of A\ the common member must the other members the is of This seek a new essence premissP", and so on. empirical one

common

a empiricalpremissis not in any essential sense deduction to arrive wish at which is but our we a proposition premiss, of our are In choosingthe premisses deduction, we only guided by of our and the deducibility empiricalpremiss. logical simplicity 421. Applyingthese remarks to Geometry, we see that the common This desire is due to mistaken. desire for self-evident axioms is entirely

The

induction*.

*

Cf,

Coutumt,

/

the belief that

based

the

Geometry

on

self-evident

with

other

concerningwhat

observation,we observed

from

no

law

the

when similarly,

objectto

those ^except

that

: continuity

that

from

from

return

we

science must

of

be

that

in which

this

existingspace with

concerned

concerned

be

premissesmust

cannot

prove

space live.

not

premisses

no

results; but the

Euclidean

that prove that it is not differ in any

our

so,

coverable dis-

we

we : digression

divisible

space,

that agi*eed are

the

concerned

empirical

alwayswith

questionbefore us is from this that the logical premissesfor the (i.e.the definition of existingspace) may or The divisible entities. our questionwhether

have

infer

to

are

we

cannot

would

continuous

a

divisible entities which we

legitimately

can

will

as premisses, regardsthe continuityof

whether

one

maintained

be

It cannot

continuous,but

be

prove To

No

assume.

"

it is with

so

manner

422.

being

we

possible,

and self-evident,

not

yieldobserved Geometry must approximate to closely permissible

must

can

which

those

no empirical,

of Euclidean are

space

we

as

if

and, premisses,

our

gravitationas

is taken

facts.

premisses. And and

from

prioriscience,

a

properlydeducible if we place it along an empirical study based demanded is legitimately

be

can

an

be

parallelsexcept,of course, on the ground that, true in order it need only be approximately gravitation,

yield observed

actual

of

of

the axiom

which

follow

Geometry

like the law of

others

.see

not equivalent to premisses

objectsto

one

space is

exists,as

all that

that

facts should

of

set

actual

our

the case, it would were believed. But axioms, as Kant

sciences

upon that

of

If this

intuition.

from

to

[CHAP.L

Space

442

with

parts.

The

divisible entities is

the

fullyanswered, in indivisible points,

of negative,by actual Geometry* where, by means live is confrom that in which we a indistinguishable structed. space empirically hitherto The only reasons allegedby philosophers against either such this derived as as are answer were regarding satisfactory, and continuity, from the difficultiesof infinity such as based or were upon

a

certain

the disproved; whether

theory of logical

and difficult,

can

of Part

Whatever

II.

be concerned

be answered

only by complex, we

is

composed of simpleelements towards

the

end

doubts.

our

decision

for

of

;

and

of the

means

then

decided

present purposes,

\\ith

question

former

while classes,

may

the

be

more

discussions logical (" 143), must be

carries

our

The

The

already

with divisible entities is far

and this conclusion

unities.

been

have

chapter.

present question. But in Tart II, t\u" distinguished,

We

namely aggregate* rate

former

latter will be discussed in the next

premissesmay

our

The

relations.

us

a

it does kinds

long way quite

not

of wholes,

identified, at

latter

to

seem

be

any distingu in-

from

whose

addition

unities,on

the

constituents.

predicateor

propositions.Aggregates consist of unite from sense (in presupposed in Arithmetic) they result; contrary,are not reconstituted by the addition of their the

In a

all

unities,one

relatingrelation

term ;

in

at

least is either

aggregates,there

is

a no

predicated such

term.

The

422]

421,

is

what

Now

reallymaintained by pointsis, I imagine,the

of

constituents do is

view

that it

Continuity of Space

not

the

seems

who

deny that space is

view

that space reconstitute it. I do not mean

held by consciously

Before

those

443

all who

make

only view which renders

this discussing

the

is

posed com-

unity,whose

a

to

that

say

denial in

this

but question,

the said denial reasonable.

it is necessary opinion,

make

to

distinction.

a

be

An

an be aggregate may aggregate of unities,and need by no means of The terms. is an whether a simple an aggregate question space of unities of is terms or not aggregate simple mathematically, though

irrelevant ; philosophically, the

difference between by jectivespace defined in

the difference of the two

is illustrated

cases

independent protective space and the proterms of the elements of a descriptive space. For the present,I do not wish to discuss whether pointsare unities or simpleterms, but whether space is or is not an aggregate of points. This questionis one in which confusions are very liable to occur, and have, I think,actually occurred among those who have denied that a space is an aggregate. Relations are, of course, quiteessential to a space, and this has led to the belief that a space i?,not onlyits terms, them. but also the relations relating Here, however, it is easy to see then a space space be the field of a certain class of relations, aggregate; and if relations are essential to the definition of a

that,if is

an

a

space, there must the

The

space.

be

volumes, and

relations essential to there

:

no

distance between

by

means

defined

be

to

having a field which is Geometry will not hold between

class of relations

some

divisible terms spatially

two

is

an

that suggested,

of

a

is

two

straightline joiningtwo

no

surfaces.

Thus,

if

a

space

it does not follow,as relations, unity,but rather,on the contrary,that

class of

space is a aggregate,namely the field of the said class of relations.

it is an

againstany

a

view which

starts

is

from

volumes

or

or surfaces,

And

indeed

anything that

urge, with

Peano*, may and is surfaces, volumes, only to be curves, effected by means of the straight then, the most even line,and requires, of any elaborate developments f. There is,therefore,no possibility except pointsand

the

lines*we straight

distinction between

and reason no logical againstpoints, Geometry without points, therefore take it as We in their favour. reasons strong logical may proved that,if we are to construct any self-consistent theoryof space, definite

hold space to be an is indefinable as a class. must

we

it cannot

jRiv. di Mat.

t Of.

Peano,

Annalen, xxxvi,

through of

a

of its terms, but only by means by enumeration the class-concept point. Space is nothing but the

be defined

of its relation *

aggregate of points,and not a unity which class,since a Space is,in fact,essentially

to

iv, p. 53.

"Sur

where

une

all the points of the

cube.

coiirbe

it is shown area

qui remplittoute

that of

a

a

continuous

une curve

square, or, for that

aire can

be

plane/' Math. made

to

pass

matter, of the volume

extension of

[CHAP.

Space

444

of

the

the

soldier

British

concept

po'mt,

concept

Geometry

unable

is

number

the

of

by

Army-List

the

imitate

to

is

army

the

since

only,

;

British

the

as

L

extension

points the

is

issue

finite, in-

of

a

Space-List. Space, be the of of

number the

of

is

points

points with

of

a

space,

and

whether

cannot

line

number

so

be

far

rendered

as

false

be

of

present certain

number

be

;

but

deal

with

finite

certain evidence

less,

of

to

;

shows,

indubitable.

continuity

which

the

will,

continuous

a

by

only

continuum

number

rationals,

from

are

the in

the

the

than,

propositions,

enlarged

as

is to

less

or

some

and

machinery the

to,

which

in

space

similar

its

Geometry

equal

numbers,

Geometry of

parts

and

a

Arithmetic,

Thus

become

either

indistinguishable

empirically

to

be

ordinally

series

elaborate we

the

number

actual

more

that

a

be

adequate

the

the

form

be

may

existents,

to

axioms,

undoubtedly

in

equal

suitable

will

Geometry

analytical

;

must

If

continuum.

the

accepted

points

if

and

points

of

of

number

the

possible,

composed

is

then,

Cantor,

question

required. among

;

is

is

is, It

is

actual

possible,

but

CHAPTER

LOGICAL

IT

423.

been

time It

impossible.

is

material

points,

maintained do

spatial relations,

and

found

in

them,

This

discussed

absolute

the

what

called

relative

that

asserted

are

follows.

and

holds

theory

hold

to

asserting a between

material

question all

is, like to

of

to

as

mathematics

the

form

"Is

:

Is

that, if tnis Some

arguments

position

portions

of

in

this

of

which

two

Time paper

these

between

The

absolute

which

be

may true

every

which

which

terms,

two

of

a

and

hence

question

and are

this

Space here

In

position pro-

relation

this

assert

culled

be

may

of

will

absolute

are

raised

be or

reprinted.

I

points in

the

found relative?"

in

applies world,

against

space

a

follows,

what

answered

point

actual

the

that

issues

composed be

theories

argument

maintaining

space

on

the

But

discuss.

must

question

*

and

time

a

a

explicitly

spatial relations

that at

ance. appear-

simplest spatial propositions

questions concerning

logicallyinadmissible,

true

so

time

to

issue :

terms,

holds

theory

not

the

which

in

of

be

follows

as

certain

involves the

that

mathematics*.

pure

usually takes

the

between

spatialrelation

its terms,

point,

stated

be

relational

the

this

supposed

world

need

matter

points.

The world

from

the

position.

that

maintain

to

which

their

in

absolute

contradictions

propositions

timelessly

relations

triangular

point,

may

true

are

also

between

position, whereas

of

theory

certain

Apart

theories

there

spatial points ;

holds

a

the

we

essentially

but

change

a

relative

belong only

further

which

with

have, of

z.e.

of

position usually

real, but

is, however,

in

theory

of

do

they

motion,

called

is

account

on

not

are

of

logically

is

points

spatialpoints, which

which

the

philosophers,

among of

relations

spatial

between

is called

relative

opinion

the

capable

spatial points

advocate

who

Those

be

of

theory

the

hold

POINTS.

composed

space

relations

are

This

a

that

not

the

which

spatial relations.

universal

Leibniz, that

have

timelessly

and

AGAINST

almost

an

of

concerned

been

have

ARGUMENTS

has

the

since

ever

LI.

am

itself

in

absolute

composed which

a

actual

the

to

position

of

points

philosophy

concerned

only

self -contradictory ?

negative, the the

Jf?W,

earlier

N.S.,

levant irre-

sole

part

of

my

No.

39

;

is

of

with It is

ground paper, the

later

for

[CHAP.LI

Space

446

.such a space exists in the actual world is removed ; but will be further point,which, being irrelevant to our subject,

denyingthat

this is

a

of the reader. sagacity the absolute theoryare, in my opinion, 424, The arguments against and all fallacious. They are best collected in LotzeV Metaphysic one (|108 ff.).They are there confused with arguments for the subjectivity should have been evident of space distinct question, as an entirely

left entirely to the

"

Critique, appears to have advocated the theoryof absolute position*.Omittingarguments onlybearingon this have the following of Lotze^s arguments against latter point, we summary from the fact that

Kant, in

absolute space. (1) Relations

the

either

only are

or consciousness, ($)

as

in (a) as presentations

said to stand in these relations

real elements

in the

internal states

relating

a

which

are

("109).

being of empty space effects(which belongsto a thing), nor the fact of being presented by us. (2) The

is neither

the

mere

What

beingwhich works of a truth,nor validity the

kind of

beingis

it then?

""109). exactlyalike, yet every pairhave a relation to themselves; but being exactlylike every other pair,the peculiar for all pairs relation should be the same ("111). (4) The being of every point must consist in the fact that it itselffrom every other,and takes up an invariable position distinguishes the beingof space consists in an active to every other. Hence relatively mutual of itsvarious points, which is really interaction an conditioning ("110). a mere fact,they could be (5) If the relations of pointswere cannot move altered,at least in thought; but this is impossible we : holes in is explained pointsor imagine space. This impossibilityeasily theory("110). by a subjective If there real points, either (a) one pointcreates others in are (6) relations to itself, or (/5)it bringsalreadyexisting appropriate points which are indifferent to their natures ("111). into appropriate relations, 425. the (1) All these arguments depend,at bottom,upon the first, is relations. As it of the of the absolute theory essence dogma concerning to deny this dogma, I shall begin by examiningit at some lengthf. Lotze tells us, All relations," in are as a only presentations relating (3)

All

pointsare

"

"

or consciousness,

as

internal states in the real elements in these relations."

say, stand indeed as self-evident, wont

to

anterior *

Of.

t The

whom

I

he

well may;

unless it be philosopher,

Comm"itar, Vaihingfcr,

This

dogma

for I doubt

Plato, who

which,as Lotze

are

regardsas

if there

does not,

we

is

one

or consciously

pp. 189-190.

logicalopinionswhich follow are in the main due to Mr G. E. Moore, to also my first perception of the difficultiesin the relational theoryof owe

space and

time.

Logicalarguments agaimt

423-425]

unconsciously, employ the dogma deny it,therefore,is examine It would

kind

two

of

as

447

part of his system. To

somewhat

less, hardyundertaking.Let us, neverthethe consequences to which the dogma leads us. seem that, if we accept the dogma, we must distinguish relations, in a relating (a) those which are presentations a

and (") those which consciousness, be related.

supposedto

essential

an

Points

These

are

of the elements

internal states

be

but it will identical, ultimately time to treat them as different. Let us begin with be safer in the mean which those in a relating These are consciousness. onlypresentations must we beliefs in propositions presentations, asserting suppose, are

the

between

relations

allowed that there

may

which

terms

beliefsin such

are

related.

appear

beliefs

and onlysuch propositions,

in capableof beingregardedas presentations

seem

be

it must

For

relations have

which

But

their

these beliefs, if the relations believed to hold have no being. in the beliefsthemselves,are necessarily false. If I believe being except A to be ITs father, when this is not the case, my belief is erroneous ; if I believe A

and

be west

of B, when againmistaken. Thus to

westerliness in fact exists only

this firstclass of relations has mind, I am no whatever,and consists merely in a collection of mistaken validity beliefs. The objects concerningwhich the beliefsare entertained are as

in my

matter

a

for the inust

of fact

whollyunrelated ; and pluralimplies diversity,

be

There

erroneous.

indeed there cannot

what

now,

namely which

are

object^

all beliefs in the relation of diversity

cannot

be

even

from

objectdistinct

one

since this would have to have the relation myself, which is impossible.Thus are committed, so we relations goes, to a rigid monism,

But

be

even

of far

to diversity as

me,

this class of

those say of the second class of relations, related reducible to internal states of the apparently shall

we

observed that this class of relations presupposes a of objects and hence involves the relation of plurality (two at least),

? objects

It must

Now diversity. a

be

have

we

it cannot that,if there be diversity,

seen

relation of the firstclass; hence

That

is,the

itself be of the second class.

is different from

fact that A

mere

it must

be

B

must

be reducible to

that, before we can first distinguish must from those of #, we the internal states distinguish before they can have A from B ? i.e.A and B must be different, If it be said that A and B are different states. similar,and precisely internal

of A

states

and B.

But

is it not

evident

of A

are

due to difference of admission

that there

theorythat the are

is not that their diversity evidently the mere internal states,but is priorto it. Thus the internal states of different thingsdestroys are

yet two, it follows

thus

to the

the

to the notion

in the internal states

rigidmonism

Thus

more

of relations is to be found

essence

broughtback

thingsconsist

even

in implied

theory

that the of

one

in these

apparent relations of

thing,which leads

the first type of relation,

of relations

We

states.

propounded by

Lotxe

us

two

again

j

|",in

fact,a

[CHAP.LI

Space

448

theorythat there are no adherents of most logical

been

has

This

relations.

recognised by

the

dogma e.g.Spinozaand Mr Bradley God or the Absolute, who have asserted that there is onlyone thing, to and onlyone namely that ascribing predicates type of proposition, the Absolute. In order to meet this developmentof the above theory and of relations, it will be necessary to examine the doctrine of subject predicate. 426. true or false" so the presenttheory contends Every proposition, from what is a corollary and ascribes a predicate to a subject, The consequences of this doctrine the above there is onlyone subject. so are strange,that I cannot believe theyhave been realised by those For if the who maintain it. The theoryis in fact self-contradictory. then there are predicates Absolute has predicates, ; but the proposition the which is not one "there are predicates presenttheorycan admit that the We cannot merely qualifythe predicates escape by saying be qualified Absolute; for the Absolute cannot by nothing,so that is logically "there the proposition are priorto the propredicates" position "the Absolute has predicates." Thus the theory itselfdemands, without a subject and a predicate its logical as ; prnts^ a proposition if there involves for this proposition be only moreover diversity, even different the this be from one one subject.Again, predicate, must and its predicathe predicate is an entity, since there is a predicate, of the Absolute is a relation between it and the Absolute. Thu." bility which \\as the very proposition to be non-relational turns out to be, after all, and to express a relation which current philosophical relational, would describe as language purelyexternal. For both subjectand what neither ib modified by its relation are predicate simply theyare" To be modified by the relation could onlybe to have to the other. and hence we should be led into an endless other predicate, some regress. the

"

"

"

"

"

*

In

relation

short,no

between

and

A

then

S9

modifies either of its terms.

ever

it is between A

and

that it modifies A and B and

C

terms

D.

different if

is to say that it say that two terms

To

B

that it

holds really which

are

For

if it holds

holds,and between

to

say

different

related would

be

is to say something barren ; related, they perfectly for if theywere different, they would be other,and it would not be the in

terms

not

were

but question,

differentpair, that would

a

eternal

of all terms self-identity

form

a

term

can

the constituents

be unrelated.

The

be modified arises from

notion

that

of

neglectto observe the which alone logical concepts, What is called modification propositions*. and

all

consists

merelyin havingat one time,but not at another,some specific relation to some other specific term ; but the term which sometimes has and sometimes has not the relation in questionmust be unchanged, *

Vol. Mr

See

Mr

viu.

Moore's.

G.

E,

Moore's

Also *upra9

paper

on

"" 47, 48, where

"The

Nature

the view

Judgment," Mind, N.S., adopteddiffers*somewhat from of

Logical arguments againstPoints

425-427]

it would

otherwise

be

not

that term

which

had

449

ceased

the

have

to

relation. The

to Lotze's theoryof relations generalobjection be thus may summed The consist in the theoryimpliesthat all propositions up. of t o and that this ascription is not ascription a predicate a subject, relation. The that the a is either objectionis, predicate something or and the pretended nothing. If nothing,it cannot be predicated, If something,predication collapses. proposition expresses a relation, and a relation of the very kind which the theory was designedto avoid.

Thus

in either

the

case

regardingrelations

for

427.

(") This

space.

I

is

is far easier to

come

of

to the

again of

second

There which consists

of validity

truth

a

are,

there is

no

reason

form. subject-predicate

of Lotzefe

to empty objections character,but it logical

abstract

somewhat

a

disposeof,since it

of things,which

(0)

all reducible to the

as now

peculiarto Lotze. being,no one of the

theorystands condemned, and

depends upon

it says, three

a

and

view

more

less

or

kinds

only three

of

belongsto space. These are (a) the being in activity the power to produce effects; or the ; (7) beingwhich belongsto the contents

presentations.

our

The

this is,that there

is

kind of

being,namely, and only one kind of existence, beingsimpliciter, namely,existence #imBoth and I existence, believe,belong to empty space ; being pliciter* to

answer

onlyone

being alone is relevant to the refutation of the relational theory which Lotze confounds with the above, existence belongsto the question but

"

namely,as to the to explain Lotze^s three

be wel] first

of space. It may or reality subjectivity distinction of beingand existence,and then

the

kinds

of

to return

to

being.

belongsto every conceivable term, to every Being of thought in short to everythingthat can possibly possible object, and all such to in true false, or occur propositions any proposition, counted. If be whatever A be any to themselves. can Being belongs be counted as one, it is plainthat A is something,and that can term therefore that A is. "A is not" must alwaysbe either false or meaningless. which

is that

"

For

if A

is.

unless

Thus

whatever

A may chimeras relations,

they were them.

is

there

impliesthat A

nothing,it could

were

"

is not

A

be,

"

and

IK.

empty sound, it

an

kind, we

anything is to

show

that

the

Numbers,

could of

beings. by

the

A

is not

*

that

be false "

Homeric

gods,

being, for if

about propositions and mention everything, to

make

no

it is.

of the contrary,is"the prerogative on JEjri"tence, To

"

hence

must

spaces all have

four-dimensional a

to be ;

being is denied,and

being is a generalattribute

Thus

be said not

whose

is. it certainly

entities of

not

A

term

a

not

exist is to have

a

relation specific

to

some

onlyamongst

existence

This existence itself does not have, way, which of existential the of weakness the theory judgment "

"

relation, dentally, shows, incia

the

theory,

[CHAP.LI

Space

450

that exists. is concerned with something that is,that every proposition For if this theorywere true, it would stillbe true that existence itself is an be admitted that existence does not exist. and it must entity, Thus

and

this distinction is

anything.For

theory.

The

if essential,

ever

are

we

be

exist must

does not

what

in

theoryseems, existence

of the distinction between neglect

arisen from

have Yet

contradicts the

so

itself leads to non-existential propositions,

of existence

consideration

the

to

deny the

or something,

fact,to

and

being.

existence of it would

be

need the concept of we to deny its existence; and hence meaningless being,as that which belongseven to the non-existent. to Lotze*s three kinds of being,it is sufficiently Returningnow confusions. evident that his views involve hopeless Leibniz here as following (a) The being of things,Lotze thinks is a highly Now activity consists in activity. elsewhere complex notion, But at any rate it is plain which Lotze falsely supposedunanalyzable. both be and exist,in is active must what that, if there be activity, the senses explainedabove. It will also be conceded,I imagine,that from activity. Activity existence is conceptually distinguishable may but can hardlybe synonymous with be a universal mark of what exists, Hence the highlydisputable that Lotze requires existence. proposition this to be active. true exists The must whatever answer proposition the grounds alleged in its favour,(2) in proving lies (1) in disproving the existence of time, which cannot be itselfactive. that activity implies For the moment, however,it may sufficeto pointout that,since existence the supposition that somethingwhich and activity are logically separable, absurd. be logically is not active exists cannot which is Lotze's second kind of being of a truth (") The validity in the first place, kind of beingat all. The phrase, is in reality is no "

"

"

what

ill-chosen "

a

Delation

of

truth

the

truth,and

to

is the truth of

is meant

Now proposition.

"

presupposes

truth,or

a

rather

the truth

consists in proposition

a

the

beingof the

a

of

certain

proposition.And

the same are on exactly level,since regardsbeing,false propositions b e. Thus must is not a kind to be fake a proposition already validity but beingbelongs to valid and invalid propositions of being, alike. is (7) The beingwhich belongsto the contents of our presentations a subjectupon which there exists everywherethe greatest confusion. This kind is described by Lotze as "ein Vorgestelltioerden durch*un$"

as

Lotze

holds presumably

that

in acquires,

what

it intuits

have

if it

form

of Kantianism

some

intuited.

not

were

the

mind

is in

sense,

Some

some

creative

sense

existence which

an

such

that

"

it would

not

theoryis essential to

every the belief, that is,that propositions which are believed solely because the mind is so made that we cannot but believe them

may rests,if I

between

yet am

an

be

not

idea

"

true

to

in virtue

of

our

belief. But

of mistaken,upon neglect

and

its

the whole

the fundamental

theory

distinction

object. Misled by neglectof being,people

428]

427, have

Logical arguments agcdnstPoints

supposedthat

what

does not exist is nothing.Seeingthat numbers, other objects of thought, do not exist outside the

and many relations, mind, they have supposedthat the thoughtsin which

entities

create actually

can

but

few

The

Yet

the distinction is as

But

post and

a

the number

except

sopher philo-

a

idea of

my and

2

think of these

we

objects.Every one

own

difference l"etween

difference between

argument that

existent.

their

the

see

2.

number

the

see

451

post,

a

idea

of the

my in as in the other. case one necessary an requiresthat % should be essentially

2 is mental

in that

it would be particular, and it would be case for 2 to be in two in Tims impossible mind at two times. minds, or one 2 must be in any case an entity, which will have beingeven if it is in no But further, mind*. there are reasons for denyingthat % is created by the thoughtwhich thinks it. For, in this case, there could never be two

thoughtsuntil supposedto be it did

when to

two

thoughtso thoughtswould one

arise,would be

1; there

Adam's

some

cannot

be

not

erroneous.

have

been

the person

so

two, and

the

have

been

And

applyingthe

thought until

one

first thought must

for not

hence what

;

Hence

so.

the number

with

concerned

doctrine

same

thinks

one

some

thinking opinion,

singlethought could precedethis thought. In be recognition, on pain of being mere delusion

a

knowledgemust

be discovered

must

discovered the created the

West

just the

in

Indies,and

Indians.

entitywhich

The

we

same

As

;

number

"

is not

be

regardsthe

metic Arith-

than

numbers

create

more

no

;

Columbus

in which

sense

he

purelymental, but is an be thought of has can

thoughtof. Whatever may its and is of its not a result, being, being a precondition, of.

1

-short,all

being thought

nothing be inferred from the fact of its beingthought of, since it certainly can does not exist in the thought which thinks of it no Hence, finally, of kind such. the o f to as special beingbelongs objects our presentations With this conclusion, of. Lotze's second argument is disposed 428.

introduced

therefore any

other two to

Lotze

he

is

objectof

an

(3) Lotze"s third argument

since Leibniz and

existence of

;

yet

twa

the

same

distances must

this,in the

been

we points,

have

must

their mutual

(though in

All

it.

has

thought,however,

ever great favourite, alike* told,are exactly

a

are

mutual and differ,

in which

sense

relation

he

any

according

even,

seems

as

to

mean

it,

pair must be peculiarto that pair. argument will be found to depend again upon the subjectbe exactlyalike To predicatelogicwhich we have alreadyexamined. not to can of Indiscerniblex in Leibniz's Identity as only mean that there have different predicates. But when once it is recognised is no essential distinction between subjects that it is seen and predicates, any two simple terms simply differ immediately they are two, and this is the sum-total of .their differences. Complex terms, it is true, have differenceswhich can constituents The !"e revealed by analysis. the relation mistaken),

of

every

This

"

"

"

*

Cf.

Frcge,Grundg"etze

der

Arithmetik,p.

xriu.

[CHAP.LI

Space

452 of the But

one

the

be A, B, C, D, while those of the other are A, E, F, G. stillimmediate ferences difdifferences of J5, C, D from ", .F,G are of all mediate be the source differences must and immediate may

to suppose that,if there error logical and predicates, ultimate distinction between subjects subjects an were For of before two differences predicates. could be distinguished by they must alreadybe two; and subjectscan differ as to predicates, is priorto that obtained from thus the immediate diversity diversity in the first be cannot terms two of predicates. distinguished Again, of relation instance by difference of relation to other terms; for difference

differences. Indeed

it is

sheer

a

therefore be the ground of and cannot presupposes two distinct terms, at all, there must their distinctness. Thus if there is to be any diversity and this kind belongsto points* be immediate diversity,

relations to each Thus

in diversity consisting Lotze

in their

urges,

other,but also in their relations to the to be in the

theyseem

of

only,as

They differ not

of relation.

difference

kind subsequent

also the

Again, pointshave

in them. objects

as colours, sounds,or position

same

smells.

mediate simplesmells,have no intrinsic difference save imdifferentrelations to other terms. but have, like points, diversity, of the notion that all pointsare Wherein, then, liesthe plausibility This I notion alike ? due to is, believe,a psychological illusion, exactly know remember it to the fact that we cannot when a we point,so as it is it again. Among meet simultaneously presentedpoints easy to but though we are perpetually distinguish; moving, and thus being we are brought among new points, quiteunable to detect this fact by and we our recogniseplacesonly by the objectsthey contain. senses, Two

colours,or

But this

seems

to be

I

see, in

so

far

as

between

to

as

be

can

of.

aware

memory whether he had

Let

blindness

mere

a

supposing

an

colours,but

bad

he

two

a us

take

saw

ever

seen

one

face any

or

an

would many,

of the

"

immediate

difference which

for faces: he

part there is no

our

on

our

senses

but he would Thus

not

man

at

be

he

points,

constructed

not

are

analogy: Suppose a be able to know,

faces before.

difficulty,

difference between

with any aware

might

a

very moment, whether

be led to

define

in which he saw them, and to suppose it peopleby the rooms that to his lectures, old self-contradictory new peopleshould come or the latter do In to i t will be at so. admitted peoplecease point, least, would lecturers he be that mistaken. And with as by faces,so with to recognise them must be attributed, not to the absence points inability but merelyto our incapacity. of individuality, "

429. ad

(4) Ijotze'sfourth argument

absurdum, by provingthat,on The

being of every

an

endeavour

point,Lotze

to effect a reditctio

theory, pointsmust

contends,must

act. inter-

consist in the

itself from distinguishes every other,and takes up an invariable position to every other. relatively tained Many fallacies are conin this argument. In the first place, is there what may be called fact that

it

is

the absolute

428,

429]

Logicalarguments againstPoints

453

ratiocinated

the

which consists in supposingthat everything fallacy, has to be explained by showingthat it is somethingelse. Thus the being of a point,for Lotze, must be found in its difference from other while,as a matter of fact,its beingis simplyits being. So far points, from beingexplainedby somethingelse,the being of a point is presupposed in all other propositions about it,as e.g. in the proposition that the point differs from other points. Again,the phrasethat the from all other pointsseems to be designed point distinguishes to itself kind of self-assertion, imply some as though the point would not be different unless it chose to differ. This suggestionhelpsout the conclusion, that the relations between pointsare in reality a form of interaction. Lotze, believing he does that activity is essential to as existence,is unable to imagine any other relation between existents than that of interaction. How such a view is, hopelessly inapplicable will appear from an analysis of interaction. Interaction is an enormously and involving, in complexnotion,presupposinga host of other relations, its usual form, the distinction of a thingfrom its qualitiesa distinction criticized. Interaction, dependenton the subject-predicate logicalready to beginwith, is either the simultaneous action of A on B and B on A, the action of the presentstates of A and B conjointly their states or on the next instant. In either case Action generally action. it implies at "

may

be defined

as

thingsat the

more

different thingsat in both

and

to

suppose

relations: the states the

if the

Thus

speak of

in

states

the

thing

in

cause

effect in

of interaction presupposes

lowing the fol-

between diversity ; (5) causality

;

as notion, involving, thing to is shows, six simplerrelations in its analysis, inspection

To

a

This

its states.

a

be the fundamental

producedby such to space.

the

or

simply,it causality certain time, and having

between (1) diversity

relation of

supposedto

same

or

one

same

than

things; (2) of things ; (4) succession ; (3) simultaneity

moment's

of the

thing and

one

a

of

there is onlyone

thingbe

action,rather

notion

states

more

more

or

thingsenduringfor the

or

When

if the

be

cause

one

one

instant. subsequent

order to

In

changingstates.

not

a

effect,transient

is necessary

(6)

presentinstant and

cases, the action is immanent

another.

a

causal relation between

a

relation !

supposition.But

No

wonder

the absurdities

absurdities

are

belong to Lotze,

the on pointsto interactions, plete is to display a comis the type of all relations, simplest problemsof analysis.The relations of

reduce the relations of

ground that interaction in the incapacity pointsare not interactions, any

more

than

before and

after,or diversity,

They are eternal relations of greater and less,are interactions. like the relation of 1 to 2 or of interaction itself to causality. entities, each Points do not assign positions to each other,as though they were have the relations which they have, other's pew-openers : they eternally justlike all other entities. The whole argument, indeed,rests upon an or

absurd

dogma, supportedby

a

false and

scholastic logic.

[CHAP.LI

Space

454

(5) The

430.

fifth

argument

seems

to

it

designedto prove the says, necessary propositions be

space. There are, " mere concerningspace, which show that the nature of space is not a intended to infer that space is an a prioriintuition, fact." We are cannot is given why we imagine holes in and a reason of apriority

Kantian

psychological

what of holes is apparently The impossibility space. of thought. This argument again involves much

Concerningnecessities of thought,the to

lead to

the

whatever

curious result that

we

is called

necessity, cussion. purelylogicaldis-

Kantian

a

theoryseems help believing

cannot

in this case, is something cannot must be false. What we helpbelieving, of our minds. The of space, not as to the nature to the nature as offered is,that there is no space outside our minds ; whence explanation it is to

be

inferred that

mistaken.

Moreover

of

fact,"for the

mere

"mere

we

our

unavoidable

onlypush

one

constitution

beliefs about

stage of

our

space farther back the

minds

remains

are

all

region still

a

fact.

urged by Kant, and adoptedhere by Lotze, theoryof necessity fact. A vicious. Everythingis in a sense mere a appears radically is said to be provedwhen it is deduced from premisses ; but proposition have to be simply and the rule of inference, the premisses, ultimately, certain in fact. Thus assumed. is, a sense, a mere any ultimate premiss of which there On the other hand, there seems to be no true proposition is any sense in sayingthat it might have been false. One might as well is What say that redness might have been a taste and not a colour. is false; and concerning fundamentals, there true,is true ; what is false, is nothingmore to be said. The seems onlylogical meaning of necessity is more A proposition less to be derived from implication. or necessary for it the class is which is of a s a premiss according propositions greater the propositions of logichave the greatest smaller*. In this sense or and those of geometry have a highdegreeof necessity. But necessity, this sense of necessity yieldsno valid argument from our inabilityto holes in imagine space to the conclusion that there cannot reallv be any space at all except in our imaginations. 431. of. If points (6) The last argument may be shortly disposed be independent Lotze I interpret him that we so entities, can argues imaginea new point coming into existence. This point,then, must have the appropriate relations to other points. Either it creates the other pointswith the relations, it merely creates the relations to or alreadyexisting points. Now it must be allowed that,if there be real it is not self-contradictory of them non-existent. to suppose some points, But strictly whatever is self-contradictory no speaking, single proposition The nearest approachwould be No proposition is true," since this i ts truth. But it is not own even implies here, self-contradictory strictly The

"

"

"

*

Cf. G. E. Moore,

"Necessity/*Mhid, N.S., No.

35.

431]

430,

the

deny

to

Logical implication.

they

because of

contradiction the

all

points

among such

and

were

to

have

the

space

come

eternally

have

had

Thus

existence.

depends the

to

as

I

upon

of

nature

of a

space, from

necessarily

appears

difficulties I

am

able

validity points,

a

not

as

entity

this,

Cf.

niy

would

but

point,

would

have it

other

comes

points, views

correct

more

axiom,

but

in

of

is

There the

ultimate

of

when

space

relations

Philosophy

of Leibniz,

Cambridge,

1900,

of

nature

judgment,

all

the

space the

once

absolute

and

that

exclusively

therefore,

reason,

regards

which

assemblage

no

nature

namely, the

not

self-

not

the

almost

admitted,

is

propositions

in

found

derived

been

have

is

points

those

part;

a

contradictions;

geometry

mere

to

is

position

of

be

subject -predicate theory

deny

to

absolute

definite

one

than

smoke*.

of a

used to

to

that

reducible ir-

supposed so

far

as

philosophical as

composed

between

Chap.

of

non-spatial

terms.

*

missing

when

on

as

by

met

has

it

are

The

points.

composed

space

which

involve

like

theory

a

and

With

perceive,

to

of

seem

relational

of

vanish

a

terms

hand,

to

nature

easily

both

previously

an

such

that

points,

as

on

discussion,

that

more

logic.

general

is

adherence

upon

other

the

on

and

as

only

some

because

new

relations

same

difficulties

have

must

above

and

The

infinity depended whole

and

argument

logic,

the

logically inadmissible, contradictory.

create

of

judgment.

from

conclude,

not

of

assertion

point

a

:

denial

the

cepted ac-

law

the

implication

false, namely,

being,

the

the

already existing

had

reason

mutual

as

impossiblle^

Lotze^s

faulty

a

and

to

points

no

reason

it would

already

other

to

same

relations

other

The

455

propositions

upon

for

another;

true

existence,

appropriate would

the

both

if" per

But

into

be

to

for

is

come

and

proposition.

a

seems

rejected

untrue.

fact,

into

is such

proposition

a

obviously

in

of is

we

self-evident,

itself

points

Everywhere

are

Points

against

arguments

x.

LIL

CHAPTER

the

IN

432. minute

textual

or

elsewhere, and it need

that

done

be

not

universal

I wish

As

theory, diametricallyopposed In

this I shall

of

follows.

had

hoped,

in

called

points be

Kant

world

concluded

could that

be

the

obtained

of

two,

space It

Geometry. be

can

The

and

all

theory

Prolegomena.

a

in

of space

forms

I shall

works, the

and

seem) the

discuss

of and

pure

that

that

space

on

this

forms

of

the

the

point), will

not

space

objects must

be

logicalpoint Critique and

Metaphysische Anfangsgriinde

Critique

thing some-

these

mathematics

the

hence

Arithmetic,

essential, from

be

with

Of

pure

of

by experience;

source

will

of

knowledge all deal

time

reasoning

that

from

intuition.

thus is

the

of

of

and

What

priori propositions,

that

than

is the

time

subject;

which

differs from

;

a

mathematics

of form

a

the

experience.

Pre-critical

chafi (which

calls

time

only

experienced by

applicable to *

is

he

admit

otherwise

propositions

subjective,which are

willing

a

is different

to

matics mathe-

propositions cannot, logicalcalculus ; on the

(it would

axioms

mathematics

propositions of

synthetic then

is

Prolegomena)

these

of

means

even

the

not

the

his

deduce

to

is pure

the

that

certain

and

from

was

hence

him*.

transcendental

seeks

Kant

"How

all

that

proved by

axioms,

nature

question:

says,

calls the

(especiallyin

out

becomes

of mathematics.

in which

way

infers

He

deductions

Now

external

there

the

the

require, he

be

may

logic.

first

Kant

contrary, they

employed

from

synthetic.

Leibniz

he

from

mathematics

Starting

are

which

from

space

possible?"

as

derived

of mathematical

it

Kant,

or

nearly

I differ from

in which Kant

what

to

of

more

a

won

point

every

those

to

opinions

Broadly speaking,the

433.

as

the

specialattention

pay

i.e. those

arguments,

theory

defend

to explicitly

necessary

almost

on

are,

has

well

outlines

doctrine,

This

done so

broad

the

only

century, and

a

over

views

my

It is

a

been

has

this

commentary,

discuss.

to

field for

the

acceptance.

monumental

undertake

to

propose

opinions ;

here.

again

that

not

Kant's

Vaihinger's over

doctrine

modified, has held

less

in

notably

Kantian

the

of

of

examination

do

I

chapter

present

SPACE.

OF

THEORY

KANTS

be

der

the

Nafar-

considered.

Kanfs

432-434] of

view, is,that the the

told,make to

all

d

formal

inference which

theoryof space

457

prioriintuitionssupplymethods

logicdoes

(which may figure

geometricalproofs.

The

admit

not

of

these

and

;

of

reasoningand

methods,

are

we

be merelyimagined)essential

course

time

opinionthat

is reinforced

jective space are subendeavours to prove and

where Kant by the antinomies, than forms of experience, anythingmore they must

that,if they be

be

definitely self-contradictory. In the above

outline I have

everythingnot relevant to the of mathematics. The philosophy questionsof chief importanceto us, as are two, namely,(1) are the reasonings regardsthe Kantian theory, omitted

in

Logic ? (2)are any way different from those of Formal there any contradictions in the notions of time and space ? If these two of the Kantian edifice can be pulleddown, we shall have successfully pillars in mathematics

playedthe part of 434.

The

questionof

obscured in Kant's doubted he

for

Samson

day by

the

rightlyperceivedthat

of

nature

several

that the

moment

a

toward"s his

disciples. mathematical

Kant In the first place,

causes.

of propositions

those

reasoningwas

logicare

of mathematics

never

whereas analytic,

synthetic.It

are

has

all other kinds of truth ; as appearedthat logicis justas synthetic but this is a purelyphilosophical which I shall here pass by*. question, In the second place, formal logicwas, in Kant's day, in a very much

since

more

backward

than

state

at

present.

did, that no great advance had that none, therefore, to occur was likely

Kant

stillremained

It

still possibleto hold,

was

been

made

in the

since

future.

as

Aristotle,and The

syllogism

correct logism formally ; and the sylreasoning But for thanks mathematics. was now, certainly inadequate formal is the enriched to mathematical logic by several logicians, mainly of forms of reasoning not reducible to the syllogism f, and by means these all mathematics can be, and largeparts of mathematics actually the In third place, have been,developed to the rules. strictly according it in Kant's day,mathematics inferior to what itself was, logically, very is now. It is perfectly true, for example,that any one who attempts,

without

the

the

type

one

figure,to deduce

of the

use

of

from Euclid's axioms, will find the task did not

deduced correctly natural

from the

Since the to

that

logical proof. fact that

But

the

*

See

"f See

my

say, any

proof

proofswere

found

probably correct singlelogically

that

the

was

supra*

esp.

of the

"

18.

by

the was

something different difference lay in

simply unsound.

many

down

which

whole

Philosophyof Leibniz, "11.

Chap, n

reasoning

indubitable,it

result seemed

fact is,that

proposition

there

laid explicit premisses

mathematical

mathematical

examination,it has been

the

of the

correctness

suppose

is to

reasoning,that

its result from

seventh

; and impossible

exist,in the eighteenthcentury, any

pieceof mathematical author.

Euclid's

On

closer

propositionswhich,

to

[CHAP.LII

Space

458

Kant,

were

undoubted

truths,are

as

a

propositionsfor

false*. A stilllarger classof

of fact

matter

"

demonstrably

Euclid's seventh instance,

deduced from certain prebe rigidly mentioned above can misses, proposition themselves are the premisses but it is quite doubtful whether of mathematical true or false. Thus the supposedpeculiarity reasoning "

has

disappeared.

of Geometry are in any way peculiar reasonings refuted alreadyby the detailed accounts has been, I hope,sufficiently and especially which have been givenof these reasonings, by ChapterXLIX. rules of results follow, We have seen that all geometrical by the mere And from the definitions of the various spaces. as regardsthe logic, opinionthat Arithmetic dependsupon time, this too, I hope,has been deed, of the relation of Arithmetic to Logic. Inanswered by our accounts to be refuted by the simple apart from any detail,it seems have parts, whole and therefore plurality, observation that time must All mathematics, we and part,are priorto any theoryof time. may the actual developmentof have in and assertion of we our proof say of formal logic: the subject is deductible from the primitive propositions these beingadmitted, no further assumptions are required. But admittingthe reasonings of Geometry to be purelyformal,a

belief that the

The

"

"

Kantian

may

stillmaintain

that

an

priori intuition

a

him

assures

that

the space, alone among definitions of possible at any or spaces, is the definition of an existent, other of which relation to existents rate an entityhavingsome spaces

definition of three-dimensional Euclidean

the

do not have.

irrelevant to the philosophy speaking, opinionis,strictly of mathematics, since mathematics is throughoutindifferent to the question whether its entities exist. Kant thought that the actual differentfrom that of logic of mathematics was reasoning ; the suggested and maintains emendation drops this opinion, merelya new primitive to the effect that Euclidean proposition, space is that of the actual world. in any immediate I intuition do believe not Thus, although such primitive I shall not undertake the any guaranteeing proposition, refutation of this opinion. It is enough, for my purpose, to have shown that no such intuition is relevant in any strictly mathematical position. pro435.

This

It remains

concerned with

are

to

discuss the

and infinity

We specially spatio-temporal. since both Part

occur

mathematical

antinomies.

which continuity,

Kant

have

that

iii pure

alreadyseen

Arithmetic.

We

These

supposedto this view

have

seen

is mistaken,

also in

V

in Chapter xui) that the supposedantinomies (especially in their arithmetical form, are and continuity, soluble; infinity remains

*

For

to

prove

the

same

conclusion

example, the propositionthat

be

of it

concerningKant's spatio-temporal

every

continuous

function

can

be

ferentiate dif-

434,

435]

Kanfs

The

third and fourth antinomies

form.

theoryof

459

space

relevant here,since

not

are

.

they

involve causality will be examined. ; only the firsttwo, therefore, First

Antinomy. regardsspace also

Thesis

The

"

:

world

has

beginningin time, and

a

is enclosed within limits." This statement is not concerned with pure time and pure space, but with the thingsin them. such as it is,applies in the first instance to time only,and is The proof,

as

effected by reductio world

has

world

is

ad

absurdum.

"For

the

assume," it says, "that

beginningin time, then an eternityhas passedaway before every given point of time, and consequently an (abgelaufen) series of conditions of the thingsin the world has happened. infinite of a series consists in this, But the infinity that it can be completed never successive of series infinite an by synthesis.Consequently past is impossible, and a beginningof the thingsin the world (Weltreihe) no

which necessary condition of its existence,

a

first to

was

be

proved." This argument is difficultto follow,and suggestsa covert appealto for a first cause. and the supposed causality Neglectingthis necessity

aspectof the argument, it would it fails to againstinfinity, the word

any.

seem

understand

It is supposed "

so

that,like the

of the

use

it would

of the

most

seem

arguments

and class-concept

that the events

"

ceding pre-

ought to be definable by extension,which, if their is obviously number is infinite, not the case. Completionby successive and it is true that to enumeration, roughlyequivalent synthesisseems of an infinite series is practically But the series enumeration impossible. the class of terms the less perfectly be none as definable, having a may It then remains a question, relation to a specified term. as specified a

given event

"

"

with all classes, whether

the class is finite

or

in Part V, that there we saw as alternative, In fact, it would to elicit a contradiction, axiom

that

which

can

every class must have a and for which be refuted,

and in the latter infinite;

is nothingself-contradictory

be necessary to state as an of terms axiom finite number an "

there

are

grounds. It

no

seems,

of later

Kant

causes as regardedby be to cause logically supposed priorto the effect. of conditions, for speaking and for confining This,no doubt, is the reason If the cause the antinomyto events instead of moments. were logically

however,that previousevents ones, and

priorto

that

the

be

are

is

this argument would,I think,be valid ; effect,

find,in Part Thus

the

VII, that

cause

and

effect are

the thesis of the firstantinomy,in

and as false, rejected

the

so

on

far

as

but

we

shall

the

same

level. logical

it

concerns

time, must

argument concerningspace,

since it

depends

time, fallsalso. upon that regarding Antithesis. "The- world has no but is infinite both

limits in vspace, and no beginning, The time and space."" proof of this

respectof of pure time and space, and argues that the infinity assumes proposition This view was these implvevents and thingsto fillthem. as rejected, be and as can disproved, regards regardsspace, in the preceding chapter, in

[CHAP.LII

Space

460 .

irrelevant to our similar arguments ; it is in any case time, by precisely contention,since no proofis offered that time and space are themselves since it depends of proof, infinite. This, in fact,seems incapable upon before any given merelyself-evident a'xiom that there is a moment But no as converse moment, and a pointbeyond any givenpoint. proof the self-evident is valid, we in true. this instance,regard as may, is bounded by and whether matter Whether had a beginning, events philosophyof space and emptv space, are questionswhich, if our decide. time i"e sound, no can argument independentof causality in the world substance Second Antinomy. Them: Every complex consists of simpleparts,and nothingexists anywhereexcept the simple, is composedof simpleparts.'* Here, again,the argument applies or what

the

"

to

things ht extend

may or

and time themselves. We space and time, not to space whether existent it to space and time, and to all collections, obvious

is indeed

It

not.

concerned space and

Instead

time.

that

of

a

the

true is or false, proposition, has no r elation to special the consider we complexsubstance, might

whole and

with purely

part,and

And with any other definable collection. the proofof the proposition this extension, must, I think,be admitted ;

numbers

1 and

between

",

or

and conceptsshould be substituted for substances, relations that between of substances the argument are that, instead

only that

terms

accidental

or

should

(zufaU\g\ we

ourselves

content

with

saying

that

imply terms, and complexityimpliesrelations. No complexthingin the world consists of simpleparts, Antitliexis. and nothing simpleexists in it anywhere." The proof of this proposition, first what is ing interestthe alone of antithesis, as really assumes, of to us, the corresponding property space. "Space,"Kant says, "*does not consist of simple parts, but of spaces."This dogma is regardedas self-evident, though all employment of pointsshows that it is not universallyaccepted.It appeal's to me that the argument of extended as I have just suggested, the thesis, appliesto pure space as to and demonstrates the existence of simplepoints any other collection, which compose space. As the dogma is not argued, we jecture can only conthe groundsupon which The usual argument from it is held. infinite division is probably what, influenced Kant. However parts many divide a space into, these parts are stillspaces, not points. But we relations

"

however

parts we divide the stretch of ratios between 1 and 2 numbers. Thus the argument not single into,the partsare stillstretches, against pointsproves that there are no numbers, and will equallyprove that covert

many

there use

consist of once a

are

of the axiom it must points^

this is

space of

colours

no

denied,we

will lead to

points.

A

of

or

tones.

these

i.e.the axiom finitude,

consist of may

All

admit

some

that

absurdities involve

that,if

finite number no

of

finite number

a

space

a

does

points.When of divisions of

while yet holdingevery space to be composed points, finite space is a whole consisting of simpleparts,

but is

Kant's

436]

435,

of

any

finite

of

the

stretch

not true

number

be

accepted, Kant's

Thus

other

Kant

which

position, this

by

are

far

we

because

that

relations, which to

all

define

and

to

had

served

classes the

from

Geometries. of

a

space indefinables to

can

always in

occur

points

raised

by an geometrical reasoning, peculiar to space and time, Leibniz's

specially concerned

answered

all the space.

therefore shall

no

find, will

Dynamics.

time,

to

Other

series.

relative

and

and

relation

difficulties

us

several

we

such

This

of

a

of

infinity,

in

all

found

adequate,

the

saw

new

the

that

call

and

of

logic

enabled

and

propositions continuity

arithmetically defined, We

the

upon

mathematicians

that

and

exists, namely, space. based

the

in

important

order

applied mathematics,

dimensions

still

capitulate re-

study

is the

arise wherever

is

of

briefly

us

said,

subject

new

was

let

series

kinds

important

entities

and

of

spaces,

the

and

that

us

responding cor-

infinity new

no

jections philosophical ob-

in

the

of

existence

certain

antinomies

realization disproved by the modern discussed no Thus, although we what with incidentally actually exists, we of an usually alleged against the existence

has

common

any

give

such

motion,

and

;

definitions

arguments Since

longer

and

space

any same

been

characteristic.

universal

problems absolute

the

by most philosophers are all capable of being belief in the peculiarity and that Kant's logic,

of

of

gives

Geometry.

amended

answered

raises

preceding chapters

the

and

which

be

the

logical method,

We

time:

being essentiallyarithmetical,

series.

hitherto,

of

numbers,

Geometry,

It is

of

abstract

and

space

V.

matter

are

order.

series

one

these

dimension

it

the

in

involve

Part.

one

because

the

deduce

to

terms

generating least

at

found

We

whose

and

Part

this

than

mathematics, of

of

more

series

a

in

proceeding

having

have

pure methods

solved

real

solved

not

rejected.

concerning absolute both straight line as a

been

is

antinomy

that

serious,

most

be

should

of

thing

same

is applicable in Arithmetic essential postulate of Logic,

properties of properties of

antinomies, which

results

the of series

e.g.

"

"

Before

436.

general

concerning points have

the

the

more,

an

the

speciallyinvolve

not

the

already

been

have

is

antithesis

do

are

Kant's

Kant's

Part.

the

is

appeals,

of

collection

is

thesis,which

while

what

than or

which

the

Exactly

Thus

2.

series, including that

And

antinomies

and

1

461

space

simple parts.

answer

any The

antinomies

continuous

problems.

of

between

speciallyspatial,and applicablehere also. should

theory of

us

reason

the

affirms

sense

for

denying

greatest

this it

;

assistance

existence, there

and in

seems

this

conclusion,

the

philosophy

we

of

VII.

PART

MATTER

AND

MOTION.

CHAPTER

LIIL

MATTER.

THE

437.

nature

regarded

been

always

present work, however, is the

merely of

of motion

laws

like

of

which

system.

reality

usually

of

matter,

and

relevant

a

greatly concern incidentallyin the

that

a

that

matter

space

is

subsequent here

to

There

is

other space,

that

in

evidence

of

the

on

which

in

Thus

that

the

fact is

What

explain;

the

of

matter

is

a

occupying

are

a

among

class-concept,is

among

:

it.

If

we

and

an

to

the

of terms

(4). which

case

And

quite

instants.

points instant,

It

simple. Matter do

or

not

great

but

It

but

step.

four

say,

the

new

instants

is

this,

called

is

may

of

there

believe

entirely

an

relation, expressed by

terms

any

because

occupy

occupy

and

in

Dynamics,

what

we

are,

which

or

also

agree

is matter?

are

on

taking

points

point

agreed

logicalimplication

rather

or

they dealt

have

probably

follow, merely in

terms

indefinable the

who

What

no

"

which

terms

no

fundamental

intransitive,

thodox or-

existence.

VI

Part

But

been

mostly

rational

in

here

both

occupy are

bv

meant

this and

bits

there

the

as

possibilityis

is

exist, there

(1) instants, (") points, (3)

be

of

occurs

are

we

to

appear

which

possible,will

things grounds,

terms

terms

space.

not

Among

points, (4)

us

principles. Those

its actual

to

as

does

are

new

it

classes

:

to

Dynamics,

to

have

they

as

question

as

It

senses.

is

the

decided

we

there

it

of

question, which

space.

therefore believe

must

But

questions so

"

entities

we

all

stage,

points

matter

mean,

of

independently 438.

of

immediate

our

is to

matter

mathematical

of

this

at

us

possible.

non-Newtonian

interesting

as

definition, confined

not

are

branch

a

as

bv

against philosophical arguments raise to logical objections to the objections, like the objections to absolute

vindication

composed

we

concerned

are

subject-matter

What

:

the

discussion

not

with

We

the

endeavour

these

to

be

must

question

considered

Thus

In

that

true

notion

are

world.

the

has

space,

philosophy.

exists?

empirically verified:

matter

need

actual

of

Dynamics its

of

that

with

actually

rational

Geometry, is

problem

introduces

the

of

space,

that

of

than

more

concerned

not

are

are

It

even

cardinal

which

-Euclidean

non

a

analysis

observation

by

as

matter

mathematics,

pure

not

the

with

matter,

we

the

of

nature

of

not to

seeiu$

instants.

not

analysis in

or

is

evident

cannot

at^ asymmetrical that

materiality itself, exist, but

bits

of

Matter

466

Motion

and

[CHAR

Lm

in space. They do not, Jtowever,form the so-called secondaryqualities, the whole of class (4):there are, besides, We and at least colours,which exist in time space, but are not matter. of secondaryqualities, to decide as to the subjectivity not called upon are from matter. How, then, is but at least we must agree that they differ to be defined ? matter exist both

matter

Matter,

we

The

well-worn

a

traditional

told,is a substance,a

are

the

are qualities us.

is

There

439.

and

in time

But predicates.

doctrine

whole

of

thing, a

this

to

answer

this traditional

have

of the notion

of

thing.

are

organic unities,composed of the whole.

in expressed

This

We

are

sometimes

content

alreadyhad

be abandoned. and must false, occasion to argue, is radically other than that of questionedwhether, without it,any sense be made

secondary

cannot

answer

as we subjectand predicate,

can

question.

of which subject,

It may

be

Chapter iv

told that

things

parts expressingthe whole and many the older notion of notion is apt to replace

thinking.The only substance,not, I think,to the advantage of precise kind of unityto which I can attach any precise sense apart from the unity whole of the absolutely composed of parts. But this simple"is that of a form of unitycannot be what is called organic; for if the parts express the and therefore themselves whole or the other parts,they must be complex, contain parts; if the parts have been analyzed as far as possible, they of selves. be simpleterms, incapable expressinganythingexcept themmust distinction is made, in support of organicunities,between A and real division into parts. What is really visible, indianalysis conceptual if told,may be conceptually are we analyzable.This distinction, "

seems conceptualanalysisbe regardedas subjective,

the

All

inadmissible. to

whole

a

is complexity

in the conceptual

but capableof logicalanalysis,

sense

is real in the

to

me

that sense

wholly it is due that

it

of the mind, but only upon the nature dependence upon mind the Where there be can must different elements, distinguish object. alas there elements to distinguish often ! different elements are though, ; does not distinguish.The of a finite space which the mind analysis t han the into pointsit"no more objective analysis (say)of causalityinto and of or of into sameness f time-sequence ground consequent, equality relation to a given magnitude. In every case of analysis, there is a relations of with it whole is only the nature of the consisting parts ; which relations the and different Thus the distinguishes cases. parts notion of an be attributed to must organicwhole in the above sense defective analysis, and cannot be used to explainthings. is falsification, It is also said that analysis that the complex is not the of its constituents to and is sum tiquivalent changedwhen analyzed has

no

into

these. of

measure

has and

the

a

In

doctrine,as

truth,when

this is

what

indefinable

certain

constituents

this

we

is to be

.saw

in Parts

I and

II, there is

analyzedis a unity, A proposition

unity,in virtue

of which

it is

an

assertion;

completelylost by analysisthat no enumeration will restore it,even though itself be mentioned a as so

a

of con-

438-440]

Matter

There

stituent.

is,it

be

must

this fact,for it is difficult not its constituents.

by

all unities

For

Thus

think, be both

and

with

view

a

The

of

occupy

two

It cannot,

different

a

to be

those

the

that of

elementarypropertiesof matter

of colours.

I^et

us

these

examine

characteristic of matter

with

time. Two *pace and moment, placeat the same

same

the

placesat

at the

moments

is

why

from dynamics is sharplydistinguished

fundamental

its connection the

and

are

properties

to definition.

most

occupy

applicable

How

? secondaryqualities

belongingto

and secondaryqualities,

quite different from

sequently con-

logicalclass of and relations, things,predicates, appear secondary qualities belong to the first class.

the

the world

Nevertheless the

and

that

seems

questionremains:

the so-called

onlycla",ses

matter

that

observe

to

thingsexist.

no

of substance

notion

The

as distinguished

concepts; the

replythat

the

of matter.

from distinguished

matter

I

no

the definition

to

be constituted

must

unity. If,therefore,it is maintained

a

must

of

whole

a

proportional concepts,and

exists is

form

in logicaldifficulty

however, it is sufficient

us,

we unities,

thingsare

440.

to believe that

or

nothingthat that

confessed,a grave

propositions

are

467

extension,is not

an

and

the

of

nature

cannot

matter

piececannot

same

it may

two

occupy

given moment, has division of hpace always

is,whatever,at

place. That

same

piecesof

though

moment,

same

lies in the

indivisible pieceof matter:

a

division of any matter implies occupyingthe space, but division of time no are commonly (These proj"erties correspondingimplication.

has

attributed to

I do

matter:

belongto it.) By

these

uish

not

assert

to

since of On

time, but they do

same

the

colour

same

as qualities,

the

view

colour

which

attributes,one

were

not

be

mav

and

be

can

ever what-

once.

Other

pairs

one

place.

in which

from distinguished

was

by the matter whose attribute it was, even similar. I should prefer to say that exactly

the

when

possess

property of matter,

subjectof

the

as

colour

actually

place at

.same

also coexist

hardness, may

piece of

in the

possess the other in many placesat

regardedmatter

do

from distinguished for example: these

else is in

the

they

is

matter properties,

Consider colours space. that colours two so no impenetrability,

that

qualities another

colours

two

were

and

is the same,

the colour

place. The relation is and consists in occupationof the same indirect, place.(I do riot wish to but decide any moot merely questionsas to the secondaryqualities, has

direct relation

no

to

the

to show

the difference between

matter

Thus respectively,)

characterize matter

matter

the

in the

notions

common-sense

and impenetrability

sufficientlyto

its

of these and

whatever

exists in space. Two piecesof matter cannot occupy the and the same cannot time, and one pieceof matter occupy at

the

same

time.

simplepiece of Other

But

matter,

of properties

the latter one

matter

which

property

must

of ih incapable

flow from

the nature

be

to

seem

converse

distinguishit from

places

two

understood

of motion.

else

place

same

analysisor

of

of

a

division.

Every

piece of that

if it exists once, it would seem It either retains its spatialposition, qr

always exist. so continuously,

it must

changes it

that

Both series in space. will follow at

continuous

discussion,which

matical,i.e.they involve the

of possibility

a

stage. They

later

a

The

vacuum.

cannot, I think,be decided

on

therefore

or

will be

none

of matter up the nature may has to whatever applicable class-concept,

simple material

units cannot

two

unit the

occupy

follows.

as

the

point at

same

may

material

(3) Two

pointsor

the

two

;

chosen

two

Matter

a

they agree in having itself seems

at any

to be

moments

positionsat continuous

a

times series.

manner

the

to

collective name

a

one

as

two

the relation of inclusion in

rather

or

two

immediate

same

and

(2) Every material

form

must

generalconcept matter,

material unit.

unit is

moment,

the different,

if

units differ in the

colours

class to the

a

but different;

between

intermediate

belongs

answer

Material

same

be

or

decision is

characteristics: following at point occupiesa spatial any moment;

unit

same

vacuum

the

moment. points at the same occupy in space through time ; its positions persists

the

a

to the

here. suggested

two

cannot

times,as

The

of motion.

sum

(1) A

a

purelykinemotion, but only

questionwhether there is grounds,i.e.no philosophical

of matter

from the nature possible to Science,and properly We

form

are

of

of the so-called laws

none

times

these properties requireconsiderable

always existed,since earlyGreek

controversy has

A

various

at positions

its

itself.

of motion

nature

time:

through persists

matter

[CHAP.Lin

Motion

and

Matter

468

generalconcept for all pieces

It is thus space for all pointsand time for all instants. relation and which time to the peculiar matter from distinguishes space and not any logical difference such as that of subject other qualities, and of matter,

as

substance or predicate, We

441. rational

can

and

attempt

now

Dynamics

attribute. abstract

an

its matter requires

statement logical

be.

to

In

the

first

of what

place,time

be

replacedby a one-dimensional and w-dimensional space may series respectively. Next, it is plainthat the only relevant function of is material to establish a correlation between all moments a point

and

of time and So

soon

have

as

any

pointsof

space, and that this correlation is many-one. the correlation is given,the actual material point ceases to some

importance. relation whose

many-one whose converse obtain

To we

have

a

domain material

Thus

we

domain

replacea material may is a certain one-dimensional

is contained

in

far universe, so

consider

a

sioiialand

the

three-dimensional

that each many-one all the kinematical

generalizedand

series.

kinematical considerations go, to the condition subject

as

class of such relations

that the This

a

and series,

certain three-dimensional

a only logical productof any two relations If we condition insures impenetrability.

to

point by

relation is to conditions

expressedin

series define for

terms

a

of

are a

to

of the class is to be null. add

that

be both

continuous

system

of

the one-dimen-

continuous,and have function, we particles,

material

logicalconstants.

CHAPTER

LIV.

MOTION.

442.

MUCH

has

been

written

possibilityof dispensing and

motion,

other

kindred

questions, of great said.

been

the

Yet

of

will illustrate repose

on

a

the

Under

these

by-ways reform

been

refuted

are

before

the

of

concept a

world

of

Change the

a

some

scholium

unequal

by

the

is the

he

author.

next

whose

path, watchword

of

contains

irrefutable

they

:

they

is time

it

were

the

adopted

I have

former,

the

his

theory,

definitions

and

years, to

own

know,

I

as

his

the

to

far

so

"

usually

factory. unsatis-

be

to

different

Newton

to

hundred

also

occurs

in

above

kind

the

of moments. cornered

moments not

change. of

entity and

and

places

at

between of time.

all moments

time

fact

that

T

series

always

This

is

its bare

constitutes

occupation,

a

falsehood, between

or

a

proposition

(1)

with a

the

in

the

fixed

a

the

two

T'

where

some

Mere this

but

a

threenot

existence

definition.

the

series

continuous

entity, (2)

minimum. on

one

cerning con-

propositions of

entity,and

change

of

series

continuous

occurs

correlated

this entity, another

occupying

71', provided that

when

involves

of

is the

and

T

time

is continuous

continuous thus

a

another

the

Change

Change relation

but

a

of

series

an

only by

other. form

that

to

that

to

Motion

difference, in respect of truth

entity

differ

logicallysubsequent

is

continuous

is the

same

propositions

some

Back

"

two

motion

proposition concerning

of the

must

we

Being

demolition seek

any

demolishing

to

constructs

similar

a

unrefuted, and

time, and

entity,

one

times. a

to

Newton's

matter.

accepted.

place at

by

rule,

he

when

;

easilyseen

himself

confines

with

suggested

theories be

can

attacked

befoi*e

philosophicalliterature

the

:

settled

alternative.

The a

author

an

unexplained.

which

or

latter

a

remarks

little has

which be

be

can

preliminary

several

are

modern

basis, and

circumstances,

this

arguments have

as

as

remain

in

relevant

these

is irrefutable

himself,

exposes

of

discussed

relativityof

the

importance, concerning

the

dogmatic

long he

opponents,

of

truth

common

So

there

the

motion,

of

must questions,speaking logically,

Most

success.

laws

Dynamics,

But

questions.

difficultyand

these

in

Causality

complex problems usually

more

hope

with

the

concerning

all, at

Con-

and

know, exists

for example. This, we pleasure,

sider

there

that

Thus

which

subsist between

does not

at

when

moments

are

may suppose there is a relation between

we

[CHAP.LIV

Motion

and

Matter

470

some

moments,

it does not

existence,and pleasure, existence,and pleasure,

some

exist.

moments,

other moments.

the definition, changesin passingfrom therefore, pleasure This shows that the definition vice versa. existence to non-existence or Usage does not requiresemendation, if it is to accord with usage.

Accordingto

exists.

Thus

we

should say, in the

of

case

changesis an existent of whose particulars always that pleasure, my mind is

what

permit us to speak of changeexcept where is at least a class-concept one or throughout,

On the other hand, if exist. ceases pleasure changeswhen should say at different times, we is of different magnitudes my pleasure the pleasure changesits amount, though we agreedin Part III that not of pleasure, amounts are but only particular capable of pleasure, should say that colour changes,meaning that existence. Similarlywe connection ; though there are different colours at different times in some exist. And shades of colour, can not colour, but only particular and the particulars where both the class-concept are simple, generally, exists series of would at if allow to a continuous a us particulars say, usage the

what

series of

to

changes. Indeed it times,that the class-concept

better to

regard this as unchanginga term which

But

if we

if any

but

of its

change

minds

to exist in it.

ceases

not

their

be

the

constituents

:

they

maintain

say that

If this

it. mind

the

is to expression

of its constituents.

sum

to

latter is the correct

parts be changed. The

subtletyis requiredto

some

seems

regard as thrpughouta given periodof time. of existent say that wholes consisting whole cannot a preserve its identity

only

this,we must exist,or else that

not

change,and

itselfexists

to do

are

parts do

the

of

kind

throughouttime, it

For

alternative,

Thus

peoplesay they changeswhen pleasure

be correct,the mind

if it

were

the

sum

must

of all its

evidently unchanging; if it were sum one time, it would lose its identity former soon constituent ceased as as to exist or a new one a began to exist. Thus if the mind is anything, and if it can change,it must be somethingpersistent of and constant, to which all constituents state have one relation. Personal identity a and the same psychical could be constituted by the persistence all a of this term, to which states have The relation. would fixed person's (and nothing else) a changeof mind would then consist merelyin the fact that these states the

are

not

the

would

be

of its constituents at

same

at all times.

Thus

a

when it has a fixed relation to we may say that a term changes, collection of other terms, each of which exists at some part of time,

while all do not

exist at

the exactly

same

series of moments.

Can

we

that the universe changes ? The universe is a say, with this definition, somewhat all the thingsthat exist at a ambiguousterm : it may mean singlemoment, or all the thingsthat ever have existed or will exist,

442-444] the

or

Motion

qualityof

common

change ;

cannot

in the

whatever

471 In the two

exists.

last,if it be other than

Existence itselfwould not

former

it existence,

it

senses

change.

can

be held to

change,thoughdifferent terms exist for existence is involved in the notion of change as at different times; which commonly employed, appliesonly in virtue of the difference between the thingsthat exist at different times. On the whole, then, shall keep nearest if we to usage we say that the fixed relation, mentioned at the beginning of this paragraph, be that of a simple must to simpleparticulars contained under it. class-concept The notion 443. of changehas been much obscured by the doctrine of substance,by the distinction between a thing's and its external nature It and by the pre-eminence of subject-predicate relations, propositions. has been suppo.sed that a thing could,in some way, be different and yet the same : that thoughpredicates define a thing, yetit may have different at different times. Hence the distinction of the essential and predicates and accidental,

the

(I

number

a

and hope) employed precisely

used

vaguelyand sense,

consciously by

unconsciously by

I do

not

at

which useless distinctions,

of other

The

are

physical Change, in this meta-

the moderns.

all admit.

were

but the scholastics,

of so-called predicates

a

mostlyderived from relations to other terms ; change is due. to the fact that many terms have relations to some ultimately, parts of time which they do not have to others. But every term is eternal,

term

are

timeless,and

immutable

equallyimmutable.

are

It is

merelythe

related to different times that makes time

and what

at

one

to

exist,it cannot

as

0/wr,

and

444.

Thus

the

some

different terms

fact that

And

though a

propositions are

importantpointis the

deny

that

the

that

Waverley'sadventures

stories in the

Raschid.

I should

not

1,001

a

Nightsoccupy

the

others false.

without

time

It is hard

can.

occupiedthe time

exist*

to the times

relation of terms

and to existence. Can terra occupy a occupy, ? At first sight, is temptedto say that it one existing

are

cease may be counted

can

and

true

they to

what

term

which entity,

to be ; it is stillan

concerningwhich

parts of time

to

the difference between

exists at another.

cease

have

relations it may

the

^

of the

'45,

Bradley,that these say, with Mr the contrary,I should give them on

or

al

periodof Harun times

are

*

parts of real time;

not

a

in the Christian Era. But I should say that the event* position when In the sense existed. Nevertheless, that they never real, exists at to

be

a

a

time, there is

combination

of

This

may

analyzedinto

an

separaterelations

"A

be shown is now"

without*any tense, we shall have if A did not exist possibleeven separablefrom existence,a term not

exist,even

if there

are

as

and

existence

follows. A

"

to hold

then

to

;

If "A

for if

a

term

the time

and

exists now"

exists,"where that "A

not

are

reducible

relation,not triangular

ultimate

definite

spectively. recan

exists is used

is then"

occupationof

is a

logically time

l"e

it does time at which a occupy may other times when it does exist. But, on the

and

Matter

472

[CHAP.LIV

Motion

A exists""1constitute the very is then11 and "A theory in question, meaning of "A existed then,"and therefore,when these two propositions This can existed then. have only be avoided by are true, A must into a combination of analyzing A exists now denyingthe possibility of a time, of two-term relations ; and hence non-existentialoccupation different from the existential kind of if possible at all,is radically occupation. "

"

"

It should

observed, however, that the above

be

discussion has

a

theme. irrelevant to our interest,and is strictly merely philosophical For existence, being a constant term, need not be mentioned, from a the moments occupiedby a term. mathematical pointof view,in defining from the fact that arises From the mathematical pointof view, change but not all of some true functions which are there are prepositional that is time, and if these involve existence,

of

moments

which

with

mathematics

as

such need not

further

a

point

itself.

concern

applyingthese remarks to motion, we must examine Here againwe seem the difficultidea of occupyinga placeat a time. is If be to there relation. irreducible triangular motion, we to have an and of relation into the a not analyze must place occupation occupation and the For a moving particle essence of a time. places, occupies many If at different times. of motion lies in the fact that they are occupied is is here" and "A into "A A is here now11 were now,""it analyzable is into "A there" is there then1' is analyzable would follow that "A could A is then,11 If all these propositions and we were independent, combine them differently could,from "A is now11 and "A is there,11 : we Before

446.

"

**

infer

"

A

point.

is there

here

avoid

to

"

to

now

"

A\

relation between But

place.

we

know

is suggestedanalysis

The

determined

now," which a

relation

is

material

a

If

we

are

is reduce "A we may is now.11 Thus have we a place

terms,

complexconcept,A^s occupationof

this

tion merelyto substitute another equivalentproposifor the one which it professes to explain. But mathematically, the whole requisite conclusion is that, in relation to a given term which t here is between and time. a con-elation a a place occupies place, a 446.

this

a

be false, if A

therefore inadmissible.

of three

of this occupation

this time and

to

We

seems

can

now

consider the nature

of

motion, which

need not,

A simpleunit of matter, we agreed, think,cause any great difficulty. can only occupy one placeat one time. Thus if A be a material point, I

**

A

*

is here

excludes

"

A

is there

now,11but not " A is here then.11 Thus any given moment has a unique relation, not direct, but via A, to a single place,whose occupationby A is at the given moment ; but there now

need not be

a

uniquerelation of a given placeto placemay fillseveral times.

occupationof the an can

interval be

containingthe

at any assigned,

is a place,

moment

when

given moment

moment

A

otherwise

within which

is at rest.

A

a

A

giventime, since than

interval A

moment

such

moment

when

as

an

the that

end-point

is in the this cannot

same

be

444-447]

Motion

done is a moment at

when A is in

moments neighbouring

such

A occupies motion,provided

either side. A

on

but all have the said intervals,

from rest to motion transition of that,by the occupation and times between places

a

;

473

as

place

there

are

of

end-term,is one

an

Motion consistsin the fact

viceversa.

or

when

moment

moment

some

time,a correlation is established when different times, throughout any period at place

a

correlatedwith differentplaces, are short, there is motion ; when differenttimes, however short, all correlated some throughout are period however

with the We

same

may

there is rest. place, now proceedto state

that terms,remembering logical relationsof alltimes to

one

one-dimensional seriest to

material particles are terms

of

a

in abstract

replaced by many-

or of all terms of places,

some

some

doctrine of motion

our

continuous

a

continuous

three-dimensional

seriess. Motion consistsbroadly in the con-elationof differentterms of with differentterms of domain

its converse

a

is at rest

of itsend-terms rest and

motion.

A

material

term of s for single which is at rest particle

term single

of

It isto be taken

t

,y

may be terms of transitionbetween time of momentary rest is givenby any term for which (ifany),

relation jR definesa correlating

as

of

to a material corresponds the interval, with the possible clusion exthroughout a

which the differential coefficient of the motion is continuous if the

a

relationjR which con-elatesall the terms

A

certain intervalwith

which particle

relationR which has

to a corresponds

all time. throughout in

A

s.

t

partof the

zero.

The

motion

is

continuous function.

definitionof motion that itis continuous,

and that furtherit has firstand second differential coefficients.This is new havingno kind entirely assumption, akin to the purpose of giving a subject merely

an

447.

It is to be observed

but serving necessity, rational Dynamics. of

that,in consequence of the denial of the

technical view of and in consequence of the allied purely infinitesimal, the notion of a state the derivativeof a function, must entirely we reject

of differentplaces occupation in Part V. There at differenttimes, to continuity as explained subject consecutive moment or is no transition from placeto place, secutive conno such thingas velocity no exceptin the sense of a real position, The rejection number which is the limit of a certain set of quotients. facts (Le. of velocity and accelerationas physical as properties belonging and not merely realnumbers expressing at each Instant to a movingpoint, of motion.

Motion

in the consistsmerely

limitsof certain ratios) as we involves, statement of the laws of motion strass in the

;

shaJl see,

some

in the difficulties

by Weierthis rejection imperative.

but the reform introduced

calculushas rendered infinitesimal

LV.

CHAPTER

CAUSALITY.

448.

A

are

the

notion

of

Mach, and, that

in

our

Dynamics

discovers

causal

the

find,either

issue between

of the

questionarises as

doubt

that

is

which

The

end of the

at the

into

quite conclusive.

are

acceleration: many resultant acceleration.

producinga

AgnoxtiDynamics I do

But

not

right: the notion of force of Dynamics. the principles

in the

are

of

cause

very

clear statement a very elsewhere, form mathematical practical be no regardsforce v and in this form, there can

for this assertion

reasons

that

a

but

The

to be introduced

ought not

more

or

schools.

the two

school descriptive

the

to

prove real world.

the

about

book

in Professor Ward's

of the

in

and

Naturalism

theory is used descriptive

truths give metaphysical

cannot

view

merelyregisters sequences,

Ward's

in Professor James

the

to

controversyis discussed

This

connections.

manner interesting

cism, in which

one

that it not

opinionmaintain

traditional

upheld the

adhere

while those who is purelydescriptive,

and

Kirchoff*

not.

Pearson, have

country, Karl

own

questionwhether

the

of Dynamics, on principles in the subjector occurs causality

interested in the

who

those

times, among

in recent

controversyhas existed

CREAT

forces Now

are

is the

Force

supposed to

as acceleration,

an

is precedingchapter,

concur

supposed in

pointed

was

mathematical

mere a fiction, fact ; and a component acceleration is doubly number, not a physical for,like the component of any other vector sum, it is not part a fiction, of the resultant, which alone could be supposedto exist Hence force, a

out

a

if it be this

a

cause,

conclusion

what

does

If the

Dynamics. from

is the

effect which

an

suffice to

show

times

some

inferences

between

different times, and

sense

causal.

in occurring world

as *

a

at

What

does

Dynamics

any

to be the

requiresthe whole

datum, and

Vvrlnungen

appear

uber

does

not

a

place. occurs

But in

inferences

*at others

relation

would

be

of

implication a general onlycausality

.such relation is in case

is,that the

of configuration

yield relations

inathwnutixckK

takes

never causality

occurs

involve

must

never

correct, strictly

what

to

impossible.Such events

that

descriptive theorywere at

occurs

of

cause

not

of

the

material

particularsto par-

Phyxik, Leipzig,188.% Vorrede.

448,

449]

Causality

such ticulars,

475

usuallycalled causal. In this respect,there is in interpreting such seeming causation a of particulars difficulty by for On law of as in the particulars appears, example, gravitation. of this difficulty, it will be account at some to treat causation necessary first the meaning to be assigned of to the causation length, examining as by particulars particulars commonly understood, then the meaning which is essential to rational Dynamics, and finally of causality the as regardscomponent acceleration. difficulty of the present chapteris the logical The first subject 449. nature In this subject of causal propositions. there is a considerable difficulty, fact that the due to temporalsuccession is not a relation between events but between moments*. If two events could be successive, directly, only could regardcausation as a relation of succession holdingbetween we A If without regardto the time at which they occur. events two B (where A and B are actual or possible precedes temporalexistents) be a true proposition, reference to any actual part of time, no involving B. then The law of but onlyto temporalsuccession, we say A causes the thingswhich would then consist in asserting that, among causality existent B now, there is alwaysone precedea givenparticular actually have series of events which would necessarily at successive moments precededB then,just as well as B now ; the temporalrelations of B of this series may then be abstracted from all particular to the terms as

are

"

"

times,and asserted per Such

would

that events a

have

be successive.

can

different and

examine A

se.

of

been the account But

complicatedtheory.

more

denied

have

we

as

As

admitted

had

if we causality,

this,we

require let us preliminary,

a

characteristics of the causal relation.

some

causal relation between

two

events, whatever

its nature

may

be,

of time.

reference to constant

It particular parts B now, is impossible that we should have such a proposition as "A causes would merelymean that A exists Such but not then." a proposition but not then, and therefore B will exist at a slightly now subsequent time exist at a slightly subsequentto the moment, though it did not

involves certainly

former

no

the causal relation itself is eternal

But

time.

if A

:

had

existed

existed at the subsequentmoment. any other time, B would have Thus "A causes B" has no reference to constant particular parts of time.

at

Again,neither

A

nor

B

need

exist at

a any moment, B must all Dynamics (as I shall prove

yet,exceptwhen Their

non-existence

Dynamics. decisions as *

or

applied

to

To

exist,though if A should exist

later)we

work

cases,

is,in fact,the mark

demand to policies,

and vice

subsequentmoment,

concrete

take another

See my article in relative ? "

ever

our

versa.

at

In

causal connections

with terms

of what

are

not

;

existents.

is called rational

example: All deliberation and choice,all of causal series whose the validity terms

Mind, N.S., No. 39, "Is

positionin

time

and

space

absolute

do not

and will not

construction

Unless

exist.

causal series,only one valid,the choice were

of two

both

[CHAP.LV dependsupon

the rational choice

For

exist.

Motion

and

Matter

476

of which

be made

can

could have

the to

foundation.

no

valid causal connections,but the The rejectedseries consists of equally Thus all statesexistents. manship, events connected are not to be found among is based upon the method of and all rational conduct of life, the frivolous historical game, in which be if Cleopatra's had been half an nose

as

particular partsof

to

with

connection

other; if

of the terms Thus

both.

is at

if A

this

also

have

longer. existence,

the

kind of

has, none

less,

some

is the is among so existents, If one is also non-existent.

the other is at "

would

essentialreference to

of its terms

one

moment,

one

J9,we

causes

being at

If

it

But

time.

no

non-existent,the other

is

one

inch

have seen, has

causal relation, we

A

the world

discuss what

we

later

a

or

earlier moment. "

2Ts A*s and implies subsequentmoment.""

A's existence

"

impliesRs being at a B These two propositions ; the second,at are impliedby A causes also implies A causes cation. least, B? so that we have here a mutual impliWhether the first also implies"A causes B? is a difficult of time, or two question.Some peoplewould hold that two moments relation between of points space, implyeach other's existence ; yet the moment

"

"

"

these cannot

be said to be causal.

It would

relations. This since But

have

we

that whatever

seem

is not

non-existent

the absence of this characteristic

exist from define

terms

as

"

But

we

admit

we

giventime

terms

may

be

has

time

causal

M'hat

"

exists,

and effect.

cause

which

terms cannot distinguishes and time,we may Excludingspace

which

has

causal relation to

a

some

This definitionexcludes numbers, and all so-called abstract it admits the entities of rational Dynamics,which might

exist,though If

might exist.

existent any term possible

other term. ideas.

which

part of

characteristic of distinguishing

a

that two

seen

exists at any

have

no

(what

is both

reason

to suppose

that

undeniable)that

seems

theydo. whatever

occupiesany

for either we reason effect, a the infinity the circularity of time, and a proofthat,if there are or events at any part of time, there alwayshave been and alwayswill be events. as

the

a

cause

If,moreover, of another

cause

the world

consists of

existents at monadism"

any view

one

and

an

admit that

we

obtain

a

singleexistent

existent J?,which single as independentcausal many

A

in turn

time.

This

leads to

be isolated

can

causes

series

as

absolute

an

C9 then there

are

Leibnizian

and alwaysbeen held to be paradoxical, in the theoryfrom which it springs.Let us, then, an error return to the meaning of causality, and endeavour to avoid the paradox of independent causal series. The 450. B^ proposition A causes is,as it stands,incomplete. The onlymeaning of which it seems capableis A\ existence at any time implies .JTs existence at some future time." It has alwaysbeen a

which

has

to indicate

"

"

449-451]

suppose that cause but as time is assumed to

customary moments

;

be any

Causality

consecutive

must

connection then time

the specify

after

at which

A

existence of A

interval which

an

a

In

is

and

A

B.

at

a

^

causal

A

time

one

independentof

other words, we

interval t such that A^s existence at any time time

completecausal

at any

an

cannot

moments

any two

more

interval between

existed.

consecutive

occupy

compact series,there

a

in order to obtain

asserts that the

the existence of B

effect must

to be

and the interval between

moments, will alwaysbe finite. Thus we proposition,

and

477

implies the particular

"There

assert:

impliesRs

is

existence

^-f tf." This

the measurement of time, and sequently conrequires or temporaldistance, magnitudeof divisibility, to we agreed regardas not a motion of pure mathematics. is effected by means is of distance, measure our proposition the generalization which is requiredfor a purelylogical

involves either

last

which

if

Thus

our

capableof statement

451. the

A

difficultquestionremains

very

problem

is

and monadism.

monism

Can

the

and

regardone

Let

there be

onlybetween subsequentstate?

whole

when

clearlybetween

most

the causal relation hold between

events,or does it hold and

questionwhich,

the

"

stated,discriminates precisely

particular

the whole presentstate of the universe Or can take a middle position, we

connected with one group as causally group of events now time ? at another time, but not with any other events at that other I will illustrate this difficulty particles. by the case of gravitating

law. the parallelogram

for the a

new

sum. a

components

C.

A" B, particles

accelerations in Ay and

cause

by

three

But

not

are

simpleas

its

we

this

these

compositionis

truly addition,

not

resultant is

The

components,and

any

Thus

the effects attributed to B either is

and

C

by

are

never

both

accelerations

two

not

C

and

B

that

parts of the resultant.

as

different from

say

compound

term,

third term

We

means

their

produced,but

produced. This,

we

may

say, is

But the effect taken as one whole. producedby B and C together, which they produce as a whole can onlybe discovered by supposing it would be each to producea separateeffect: if this were not supposed, accelerations whose resultant is the actual to obtain the two impossible has no acceleration. Thus to reach ail antinomy: the whole seem we

resultsfrom the effectsof the parts,but the effects of

effectexcept what the

partsare non-existent. The

examination

of this

will rudelyshake difficulty

our

cherished

prejudices concerningcausation. The laws of motion, we shall find, contradict the received view, and demand a quitedifferent and actually shall find (1) that the far more view.*- In Dynamics, we complicated causal relation holds between the whole

state

of the

events

at three

material universe

at

times,not two

at two;

of the

three

(2) that times

is

In order to providefor

of a causal relation. necessary to the statement in a less conventional this conclusion, let us re-examine causality

spirit.

of events

sufficientnumber events

more

at

at

time

a

events

en+l

more

or

21?e.2 at

moments

be inferred.

can

if we principle,

of the

means

time

a

of moment**,

sufficient number

a new

example,that,by

suppose, for events

one

at

t^...en at

a

and

time

a

tn+1.

arbitrary, except that data,

we

el of the events e^...e" of the events of our time tn+z* Hence by means

choose at

new

a

can

future than

tfn+1 events

ways

of

order that

the

to

past becomes to the

unambiguous inference

an

to

if for any value of 7-, in general impossible.In

But

of er+l events.

out

events

inference

then

er +i""".9

supposedlaw, inference

if,for any value of r, e} +1 " er, then more be inferred at the time "n+2" since there are several

can

choosinger

original we

And

is assured.

times

tr are

may infer ell+1 events

and

en+l9

infer

times

If,then, er+l ^ e^ is tr^.1 after tr, it follows that,from the For at all future times. infer certain events

at

or us

can

we

if the

one

Let

given el

are

tn, then

time

LV

of which, from

in virtue is the principle generally, Causality,

452. a

[CHAR

Motion

and

Matter

478

be

past may

it is possible,

should be reciprocal, i.e.that el events at necessary that the implication inference should be impliedby e.2at t2...en+iat tn+1. But some time tfj to

the

there

were

this condition,namely, that without possible with the others up to "n, the events implying,

el

fails

t"

en+l events

if,for

any value of r, after inferring e^ events at time t^ er for the next inference ?,.+! " er" since, but of is too small to allow the inference. Thus if takes the place er+19 at time

t)l+l. But

this inference

time

at

is

past

even

unambiguousinference to sufficient (1)that any

and

impliedby Since

other

the

conditions

as

somewhat

Another

one

time is to be

of the

;"

+

it is necessary possible, of events

! groups

should

(52)that er e).+l for all values groups; the possibility of such inference, we may

n

=

demands causality

these two

part of

any

soon

be

of

r.

take

satisfied.

complicatedpoint is

the

following.If el e.2...en so cause e^...eK+ly cause on, we have an independent and a return to monadism,though the monad is now causal series, complex, of But this events. result is not a being at each moment group It may happen that only certain groups e" "2. .eu allow necessary. e^-fi, and

en+" and

.

to 0,i+i, and

inference

that

e." e^...^,

en+l

is not

such

a

Thus

group.

simultaneous with ^...^n, and causing0'tt+1.It suppose e\ e'*.,.e\ may and e'ze's...e'n be that e^e3..,eH tf'n+1 form the causal next e)l+1L groups, In this causal no causingen+"" and e'1l+^ respectively. independent way in spiteof particular series will arise, causal sequences. This however remains

of which, so possibility,

mere

a

far

as

I

know,

instance

no

occurs.

Do

the

stillhold between

generalremarks good

the

.succession ?

Must

?

events

There

that consecutive

we

on

the

nature logical

suppose

the causal

of causal

relation to

propositions hold

directly

el *a...$M+1, and are

times

finite interval* of time

merely to imply their temporal For, havingrecognized it has become impossible, necessary to assume

difficulties in this view. are

between

^

and

^, e2 and

ez etc.

Hence

the

length

452]

of

Causality

intervals

these

regard

without

is,

say

the

which

in

moment, is

inferred

be

can

so

at

have

distance

of

distance

suitable

where,

then

m

and

the

events

new

the the

values

first

and

m

in

example, m

N9

=

n

=

to

rational

Thus

system,

the

at

other

any

upon

particles,

N

of

have

shall

we

material

the

it

Psychology,

in

failed

establish

to

For

question.

in

events

nature

obtain

psychologists

it

But

of

remains,

have

law

at

of

pure a

have

system is

yet

as

any

by

particulars

gravitation.

examined

what

the

as

But so-called

as

we

laws

shall

we

to

are

say

to

appears this

capable

is

statement

be

discussion of

independent

an

the

imply

moments

two

any

mathematics,

question

in

that,

assume

This

moment.

particulars the

Dynamics

configurations

language

the

we

since

of

the

circumstances

say,

the

strict

laws.

causal

as

depends

m

of

nature

consisting

system

Here What

impossible

the

upon

material

a

2.

question.

in

of

depend

may

n

of

assigned

"

to

is

groups be

to

any first

the

from

values,

however,

tr.

at

from

events,

whose

T+

times events

m

can

we

relation

causal for

given

events,

to

All

a

valid

be

be:

groups

have

n

chosen

Given still

whose

n

reference

mere

impossible.

to

moment

new

and

m

will

moment

a

we

any

suitably

be

events

till

on

provided

specified,

at

a

relevant.

relation seems

events

and

specified,

thus

becomes

is

statement

other

m

position this

"r,

are

ultimate

and

position,

relative

times

the

Thus

specified,

temporal

to

only

that

be

must

479

motion.

see

the

must

in be

into

chapter.

next

such

concerning involved

configuration

translation

of in

material

such

causation

principles

postponed

until

LVL

CHAPTER

DEFINITION

453.

BKFOUK

which

it results from

as

Let

be

t

be

If R

a

fact that defines

R

above

jR, and

to

i.e.

no

of

of

two

no

a

the

in

a

of

a

expressed in

are

Let

us

particle. the

further

assume

that

material

a

system,

it

is

only

having the propertiesassigned logicalproduct of any two of them

relate

the

the

be at particlescan these relations fulfilling

two

it asserts

impenetrability. For

expresses

relations

matter

converse

material

relations

the

that

of

motion

yet.

as

whose

.y.

motions

class of

such

our

pure

dynamical

Euclidean

is t and

domain

its field.

of t for

last condition

This

is null. that

consider

to

necessary

to

the

.three-dimensional

a

be

to

assume

defines

R

function

define

series,,y

of ingenerability

whole

continuous

a

not

whose

in ,v, then

the

has

order

In

set

many-one

R

will

we

and indestructibility

The

of

terms

logicallanguage

in

continuous

relation

is contained

domain

in

express

new

previouschapters.

series,which

continuous

difficult to

one-dimensional

a

introduce

motion, which

of

laws

brieflyto define

I wish

mathematics,

the

are

some

WORLD.

DYNAMICAL

A

proceedingto

complications of world

OF

to

the

the

same

moment

same

place

same

conditions

at

will

be

point,

same

time.

called

a

A

class

of

kiiiematical motions. these

With

of

definition

definition

right,our

kinematics

us

from

to

inference

454.

amounts

at

events

motions n

the to

the

time

to

the

in

the

wholly

were

condition

new

Nevertheless

one

generalizedform the following: A

at

add

requires for

this at

events

takes

which

condition

is essential

another, without

which

lose its distinctive feature.

such

of

loss

that, given the

relata

saying

the

have any

that

of

statement

kinetic

we

of

class

the of

given times,

ordinary Dynamics without

kinetics.

kinematics

school descriptive

not

A

require is relations

if the

would

to

from would

Dynamics

matical

and

matter;

all that

have

conditions, we

relata at

2, and

motiom of

this

is

the

all times

causalitywhich

are

a

class

various

of

we

kine-

component

determinate.

In

assumption may be made then interestinggenerality. Our assertion there is a certain relation specificmany-one ;t

=

454]

453,

Definition

which

holds

third

time,

relatum

;

of

relation

a

the

class

of

a

of

The

when

S

universes

class

and

uses

will

It it

its

is

to

the

see

in

third to

but

*

principles,

the

of next

causation

In notion

yet

of

they

chapter, a

form

is

and become

large

A

particulars

by

and

particulars,

actual

the

to

the

which

world,

of

infinitesimal,

the

of

law

causality.

into

statements

soon

as

to

the

we

have

of descend

we

shall

we

as

but

definition,

rise

which

laws, i.e.

Dynamics,

laws,

above

gives

set

causal

matter,

Newton's the

are

systems.

as

necessary

are

whole

a

same

general

our

of

of

of

accelerations

in

novelty,

the

apply

same

terms

considered

treat

rejection to

part

contained

radical

the

*$',the

K

relatum.

as

material

the

the

difficulty mentioned

examined.

Dynamics

the

velocities

though

of

to

integrated

an

motion.

introduces

law

not

give

owing

whole

Then

whose

rational

to

different

the

of

times.

universe

i.e.

of

believed

imagine

Kt"

distribution

procedure

Rt

the

be

Kt

R

be

has

t

relation

may

",

same

the

time, which

class

the

We

If let

configuration

-one

many

given

follows.

two

particular

the

way

that,

introduce

laws

the

only

to

a

as

Formally,

to

other

any

two

at

any

let

configuration

of

the

in

t

the

any

versa*.

having

observed to

cannot

general

S

is

time

as

term

motions,

be

the

vice

ordinary

its

the

between

of

respect

the

liberty be

necessary

We

in

is

defines

commonly

and in

only

This

K.

laws

given,

agreeing

there

the

stated and

expresses

t"

holds and

causal

is

differing

if

which

referent

as

t'9

let

be

of

Kt

third

determinate.

is

and

class

Then

and

the

at

motions, t

times

configurations

may

our

whole

motions

particular

given of

ft?,

form

of

Now

times,

time

between

Rt.

t.

two

other

this

one

the

as

time

three

any

and

X

given

481

their

configuration

any

in

any

be

kinetic

tf, t', t", Kt)

at

only

terms

the

at

class

for

If

such

system

is

holding

R.

the

language,

causality

which

relation

relation

the

ordinary

and

configurations

and

configuration

principle

be

is

in

World

Dynamical

a

two

any

referent,

as

the

times, the

between

of

mass.

applicable

to

the

actual

world,

the

specification

of

S

requires

as

LVII.

CHAPTER

455.

THE

attitude

towards

reallyhold, the

ether; The

or

its

whether

there

problem

is

are

remembered

"

second

The

entity.

definite

a

express

is of

state

a

them

be

treated

as

conclusion

yield

of motion

a

is

times

ation

be

must

uniformity motion

its

connecting is

view

is

In

motion

occurring

the

the

will

been

In

in nature

thus

It

becomes

whose told

would

In

this

enable

not

place,if

fourth

is uniform

;

as

it is

can

be

to

us

there

must

"

the

absence

law, in Newton^s of

causal

action

form, from

asserts some

that other

place,equal is the

is

same.

accelei*-

no

discover

what

by which

holds be

and this uniformity independent of all actual motions the descriptionof equal absolute distances in equal absolute first

is

as

definition

motion

no

of

The

to

always significantto

science

place, hence

there

form,

a

first

the

when

laws

necessary

magnitude

us

the

This, however,

motion, and

of

uniform, there the

regard

to

regard

to

customary

configurations. regarded sometimes

first law

third

'impossible.

applicationin Dynamics

us.

times

as

do), it is

defined.

is

fact, they could this

radicallyabsurd.

except

not

uniform.

given

three

of motion

place, unless it does

laws

for the

itself

not

possible to

was

acceleration.

to

the

In

has

It

is forbidden

definition

no

are

a

one

This

times.

(which

that

found

we

physical facts,and

as

first law

second

motions

but

new,

in

of

be remembered

may it

whether,

"

This

distinctlynew.

integratedform

have

In the

nothing

account,

The

equal

as

connecting configuration and

more

456.

say

was

acceleration

as

evident,

of

fractions.

ultimate

an

seek

themselves

that

velocityand

as

than

there

much

limits

otherwise

does

limit, and

particle. It

physical

a

philosophy

inquired whether

that, in discussingderivatives, we

regard

physicistsnow-a-days

of the

virtue

accelerated

an

meaning.

fiction,not

mathematical

mathematical

mere

a

"

they applying to

laws

a

what

is

is that, in

point

calculus, acceleration

is

laws

naive

a

whether

examine

not

those

give

to

moment,

reallyultimate

other

merely

be

first

will

It

the

for

adopt,

Laws.

Newton's

thing to that force scarcelydeny

will

will

chapter

present

MOTION.

OF

LAWS

NEWTON'S

that a

meaning

definition

of

is,

times.

velocityis unchanged piece

no

matter.

As

in it

Newton's

455-458] stands, this law causal action

any If particles.

an

system

then

It tells

us

483

nothing as

to the circumstances

as

importantmeaning may

is velocity

material

in

occur

action,or

But

occurs.

rememberingthat

by

of Motion

wholly confused.

to discover causal

are

we

is

Lazvs

which

under found

be

how

to

and that the only events fiction,

a

the

are

various

(asall the laws

it,

for

that

of its various positions

do) that tacitly there is to be some relation between different configurations, the law that such a relation can tellsus onlyhold between three configurations, For two configurations not between two. and for velocity, are required another for change of velocity, which is what the law asserts to be relevant. Thus in any dynamicalsystem, when the special laws (other the laws of motion) which regulate than the that system are specified, time at be inferred when two can configuration any given tions configuratimes known. at two are given we

second

The

457.

assume

and

third also

givesone configuration. the

second

The

thus

the

impressedforce

the law

u

to

there particles, be

associated

as

to

idea of

new

acceleration

For

we

viass

;

depends upon know

nothing

produceschange of

motion,

mere

a

certain constant

are

with

these

considered

coefficientsare

be

the

tautology.But by relating the configuration, an important law may be follows. In any material system consisting

to

is

which

except that it

might seem

discovered,which

introduce

it stands is worthless.

as

and

of

respectin

force impressed

the

about

law

third laws

of motion

coefficients(masses)m^m,2

and particles respectively;

...mn

when

these

then m^ formingpart of the configuration, is acceleration a certain function of the multiplied by the corresponding function is for all times and all this the same ; momentary configuration It is also a function dependentonlyupon the relative configurations. in another part of space will leacLto the : the same configuration positions same

accelerations.

we rf,

have

#,.

That

as

is,if "rr,#r,

zf be the coordinates of mr at time

"ff (t)etc.,and

This involves the

assumptionthat

j\

=j^ (r)is a function havinga

second

equationinvolves the further ever, assumptionthat ,?,has a firstand second integral.The above, howform of the second law ; in its generalform, is a very specialized

differential coefficientJi\; the

the

function

F

458. of F into other

as

coefficientsthan

the

masses,

and

positions. and allows the analysis is very interesting, of functions each dependingonlyon i^ and one

TJie third law a

vector

particlemr

acceleration of reference

of the

other

involve

may

velocitiesas well

use

MI

sum

and

their

is made

up

relative of

to //?.,,m3 respectively

position.It

asserts

that

the

component accelerations havingspecial and if these components be/K, ...#?*;

Matter

484

corresponding component frl such law

J?12be the

particle mr

other

has

a

that

Mrfn" "W\f\rFor leads to the usual properties of the centre of mass. ^-component of fu, we have w, "u + m, #2, 0, and thus

if

=

reference special

the

Again,

[CHAP.LVII

that the acceleration of any

fm "-fim it asserts

This

Motion

and

of

J\*to

onlybe

can

w,

a

reference

to

the

mass 7/z2,the distance ria, and the direction of the line 12; for these are It is often specified the onlyintrinsic relations of the two particles. as

the third law

part of this

fK

worthy

seems

that

to

the line 12.

upon

the

acceleration is in the direction 12, and

of the dependence included,as specifying Thusj^ is along 12, and

be

f^-(f"(rn^w2, r,2), /21 "(w2, "

and

(#?!raj, r12) "/"

?/Zj

=

m^

(m^ (ft

vn^

/2l1from2

function of r12)is a symmetrical

^

the resultant

=

"

1, both

will

and m(", say

ml

rls).

wi2,

(Wl n/r ,

77^,:: , r12) ,

is analyxable into particle

acceleration of each *

only upon components depending

itself and

this analysis onlyto the statement applies

one

in terms

other

but particle;

of acceleration.

No

when we compare, not configuration is possible and acceleration, analysis At any moment, though the change of but three configurations.

straightline

distance and

acceleration of it would where

towards

=

Thus

such

rr"),

"

,

(ml97/?", ?-12)w, "jf" (7"2, OTJ, r12). "f" ^(wa,

Thus

rls)

(w2, r^ "f"

towards 2, and

measuring/12from 1 have the same sign,and Hence

"

7/z2

"

,

or,

?^

flij,

a

bq

12

is not

due

to

ml

and

m""

alone, yet the

7/Zj consists of components each of which is the same if there were in the field. But only one other particle

finitetime

is in

questionthis is

no

longerthe

case.

The

total

of ml duringa time t is not what it would have position first alone for a time t, then m3 alone and so on. been if 7W2 had operated of total effect of m^ or of m?; and since cannot Thus we speak any there are really effectsof no momentary effectsare fictions, independent The on statement of accelerations is separateparticles 7;?!. by means to be regardedas a mathematical device,not as though there really actual acceleration which is caused in one particle an were by one other. Nhouid And thus we escape the very grave difficulty which we otherwise

changein

the

458,

459]

have

to

Nervtons

Latex

Motion

of

485

meet, namely, that the

component accelerations,not being of the (in general) resultant parts acceleration,would not be actual if

even

allowed

we

459.

The

statement

that

acceleration is

first two

laws

actual fact.

an

contained completely

are

following

in the

In any of that

at any time is independent system, the configuration time and of the configurations at two given times, include in we of the the various configuration masses particles provided composing the system. The third law adds the further fact that the configurationcan be analyzedinto distances and straightlines; the function of the configuration which representsthe acceleration of any is of vector-sum functions a particle containingonly one distance,one a

:

function

straightline,and the

to

two

each

masses

if

moreover,

"

accept the addition

we

third law

spoken of above, each of these functions is which enter into it. But along the joinof the two particles it law, might happen that the accelemtion of //?t would involve of the

triangle1 " #, or the volume

for this The

three laws

the law of is

should

law,we

not have

the usual

as together,

; this gravitation

of the tetrahedron

law

1 2 % 4

expounded,givethe

now

for this the

;

of

but

mas*.

greaterpart of

far

merelytells us that,so

area

and

the centre

of properties

vector

a

gravitation

as

concerned,the above function ("/*!, i/r Mo, 'w)

Wi

=

It should be remembered as

to

the form

of

that

"'a/Via8-

nothingis known,

^, and that

might

we

have

from

e.g.

the laws of motion,

ty

=

if rla

0

"

R.

If

small compared to sensible distances, yjrhad this form, providedR were the world would seem as though there were no action at a distance. It is to be observed

that

the

laws,accordingto the above

first two

explained analysis, merelystate the generalform of the law of causality in Chapter i.v. shall be able,with the this it results that we From and the existence of first assumptionscommonly made as- to continuity and motion second derivatives, to determine a completelywhen the and configuration

velocities at

a

these data ticular,

will enable

us

The

given instant. the

third law and

law of

the

that the momentary properties not upon momentary configuration,

upon and that

of any dependentonlyon the

components each and one givenparticle of

those of the motion so

far

in relation to

as

Newtonian

it deals with the truth the actual

at all,but

togetheradd gravitation

accelerations the

is particle masses

depend only velocities,

momentary and

the

vector-sum

distances

in Dynamicsapplies

of the ether is

an or

world, it is for

mathematicians, the laws of motion laws properly

in j"arthe acceleration at the

of the

other.

whether question

The but in

acceleration

resultant

the

and

given;

are

determine

to

further the

as

given instant

and

and interesting

such

problems

importantone

falsehood of the laws of motion us

the

irrelevant. law of

partsof the definition of

a

For

us,

as

are gravitation

pure not

certain kind of matter.

;

and

Matter

486

By the above

460.

the view

account

is contravened satisfied philosophers

relation embodied between

two

in

function,i.e. of of the repetition

cause.

same

whole

The

(1)

unityof merely that

;

of configurations at

any then rf,

time

any

(a compressedform for form of F

be any

effect is any

2, or the

the case, and states of the whole

between

which

whole

it is not

form

worth scarcely

is

The

law

law

The

might recur followed it. then indeed

If A our

configuration ever

if A

be the as

equation.

one

greatlystrained to

preserving.The far

system

better to

seems

apart

as

is two

cause

please;the

we

at

system is contained that

of any

does tell

us

does

laws

what

successively infinite number

of

an

a

that

an

one

of

formula.

One

But

should

we

connects

need a

any

;

and

three

no an

system

What configurations.

causes

were

comfort, since

requirementsof

infiniteclass of formula

follow

if this

infiniteclass of effectshave each an

is given,

will configurations

Moreover, the

to

the

tion configura-

distance in time

would

recur. actually

of causal

in

will always

that formerly configuration

whose configurations

recurred,so

A

event

of the system at configuration

meet

of

one

to that at another ; the

recurrence

a

causal law

functional relation to means

;

assert

not

be inferred

be two

law does is to assert

by

be

to

its consequences. causal law told us, it would afford cold

infinite number which has

B

without

if A

our

regulatingany

does

by another time,nothingcan

all that

but it

other.

causal

of F.

them, and

;

any time before,after,or could well be more unlike the Nothing

cause.

be followed one

^ before "

to advocate. Thus the on pleasedphilosophers the word cause while preserving worth : it is enough that any two configurations far leas misleading, allow

infer any

(")

must

cause

the interval tz

it has

to say, what us

effect has

and

cause

the times in the

views

to

coordinates

system, at times as coordinate of the system

one

6*, and

since each is givenin effect,

one

seems

C2. The

or

^ and ",, or of the coordinates of the

singleone

as

languageof

meet

effect is

corresponding configuration

the

C

in the

fall between

i may

collection of these

any

of C^

C\ and configurations

regardeach coordinate Thus

from

modern

is contained

system,and

the choice

upon

please.Further

we

The

at time

or

C has coordinates).The as equations many of particles the number and the dynamical

as

system,not

be taken to be the two may

formula

a

derived

dependsonlyupon

laws of the

both.

not

elucidation.

some

times,C19 C2 the following equation: if tl9 1+ be specified self-contained

the

by always necessary,

was

dynamical causation

of

essence

the second

demand

Both

time.

at least since Newton's

usually

events,

first of these is necessitated

The

infinitesimal calculus

theories of the

three

has the

relation,not

constant

a

law

has

(1) in that respects, between

holds

the causal

that

(") in

;

which causality

of

in two

law

causal

a

[CHAP.LVII

Motion

the

our same

this is done

configurations,

461]

460, and

Newton

Laws

s

but for this fact continuous

laws,which 461.

consist in I have

It remains

of

motions

487 to causal

would not be amenable

of the specifications

spokenhitherto of

Motion

formula.

independent systems

of

particle*.

n

whether

to examine

introduced by the fact are any difficulties in the that, dynamicalworld,there are no independent systems short of

the

material

within

universe.

material

a

We

have

system, to

that

seen

any

of the

part

one

effect

no

ascribed,

be

can

whole

the

system;

for any inference as to what will happen to one effect attributed to the action of only traditionally

system is necessary

particle.The

singleparticle

another

is

a

this

but

acceleration ;

(a) component part of the resultant acceleration, (ft)the resultant acceleration itself is not an event, or a physical limit. mathematical fact,but a mere on

a

is not

Hence

be attributed to

nothingcan objectedthat we

whole

the

material

effect is attributable to any part as anythingabout the effect of the whole.

since

such,we

no

know the

know

cannot

motions

of

planets, neglectthe we

system is the whole universe.

solar

that the effectsof this

it may be universe,and that,

particular particles.But

For

example,in

fixed stars

calculating pretendthat the

we

;

then, do right,

what

By

consequently

cannot

feigneduniverse in any

resemble

wav

assume

we

those of the

actual universe ? The

that, if

show

can

to this

answer

J?, while

within

much the

matter, then matter

all the

universes*.

This

is

We

for

assume

we

effect of

a

in particle

a

at

matter

all of

a

much

contain

them

to particlein questionrelatively

the approximately

will be

R

possiblebecause,by

effect of

the

number

a

greater distance

the

third

partialeffects*is possible.Thus

into of fictitiousanalysis calculate

of

motions

this distance

distance

gravitation.We

in the law of

regardsthe

of the

motion

well within the

known.

the

compare

only as differing

of universes than

we

questionis found

a

universe

of

in

same

law, a we

the

can

kind

proximate ap-

part only is is insensible,

which

not say that the effect of the fixed stars that they have no effect per se ; we must say that the in universe in which they exist differslittlefrom that of one must

they do not exist ; and this we are able to prove in the case of to our previous gravitation.Speakingbroadly,we require(recurring small, there should be function "")that,if " be any number, however which

some

such

R

distance

that, recurringto

if previousfunction "/",

our

then direction,

denote differentiationin any

sj"r)rfl" *

,

_

this condition

When

of two

accelerations

Y

v

f

limit

a

;

certain and

*

if

r

"

R.

the difference between satisfied, within a certain region, which particles

pointwithin

hence

it/

is

assuming different distributions from

there

This- i? true

is

-,-

an

the

of matter

at

region,will

upper

a

have

limit to the

not only of rit/ative,

distance an error

the

relative

results from

greaterthan R

assignable upper incurred

of absolute motions.

by

pre-

464.

If

bucket

a

will become

water

bucket

if the

level in in the

be

is

water containing

and

concave

[CHAP.LVIII

Motion

and

Matte?'

490

observes,the

Newton rotated,

But of the bucket. up the sides the water will remain vessel, rotating

mount

left at rest in

a

spiteof the relative rotation. Thus absolute rotation is involved from Foucault's pendulum phenomenon in question.Similarly,

and other similar

of the earth

the rotation experiments,

be demonstrated,

can

heavenlybodies in to us i-elation to which the rotation becomes sensible. But this requires instances be admit that the earth^s rotation is absolute. Simpler may motion dealt the If the of two gravitating case particles. given,such as if theyconstituted these particles, with in Dynamics were whollyrelative, in the line joining the whole universe,could onlymove them, and would But Dynamics teaches that, another. fall into one therefore ultimately not in the line joiningthem, relative velocity if they have initially a focus. centre of gravity as they will describe conies about their common there are in polars, And terms if acceleration be expressed generally, in the

of

squares

so long inexplicable

are

requireabsolute

terms

relative motion

as

is

to.

If the law

stated

these

angular velocities:

and angularvelocity,

adhered

no

were

which, instead of containingseveral differentials,

acceleration

contain

if there

could be demonstrated

and

follows.

as

to what

be regarded the pointmay be as universal, gravitation The laws of motion requireto be stated by reference

of

have

been called kinetic

axes

these

:

the

absolute

are

in

axes reality

no

component accelerations of the

We

here

cannot

introduce

that dynamical principle

data,

the

masses,

facts must

if it is to

obey the

subjectto

not

material

take

any

axes

spatialaxes,

laws

not

due particles be true if only themselves of mass,

centre

therefore

the acceleration, and

from

no

and

the

be, or

this force

to

the

have

no

are

accelerations acceleration.

for, accordingto the

be derived

the

of mass, centre Hence vice versa. any

of motion, must

forces.

It is

two

ratio mt : m^ But this will measured which to axes relatively

in the are

rotation.

having

asserted,for example,when the third law is combined with the notion of mass, that, of two particles between which there is a force, if w, m' be the masses

no

absolute acceleration and

from,observable be obtained

must

dynamical motion,

be referred to

axes

which

are

be accepted, But, if the law of gravitation

will

this satisfy

and

motions

condition.

Hence

relative to these

are

shall have

we

of

course

to

absolute

motions.

465.

In order to avoid this

essential part of

the

laws of

C. Neumann* conclusion,

motion

the

assumes

as

an

somewhere, of existence,

an

rigid Body Alpha? by reference to which all motions are absolutely l"e estimated. This suggestion misses the essence of the discussion, which is (or should be) as to the logical of dynamical promeaning "

to

*

Die ("fafifei-yewtoHxrhe

Theorie,Leipzig, 1870,

p. lo.

Absolute

464-467]

and

Relative

Motion

491

not to the as It seem.s positions, discovered. way in which they are evident that,if it is sufficiently invent fixed to a body,purely necessary and servingno hypothetical is the to be reason fixed, purpose except that what is really relevant is a fixed place, and that the body occupying it is irrelevant. It is true that Neumann does not incur "the vicious

circle which

would

all motions

while

rightlyavoids theory, would the

that

be involved in relative to

are

any be

statement

sayingthat it ; he

the

asserts

motion, which, in his evident wholly unmeaning. Nevertheless,it seems

questionwhether

as

to

its rest

Body Alpha is fixed, that*it is rigid,but

or

body is at rest or in motion must have as good a meaning as the same questionconcerningany other body ; and this seems sufficientto condemn Neumann's suggestedescape from one

absolute motion. 466. who

A

of development

refers motions axes."

and

Neumann's

what

to

These

are

views is undertaken

he calls "fundamental

defined

as

bodies

or

axes

by Streintz*,

bodies" and do not

which

damental "funrotate

independentof all outside influences. Streintz follows Kant's to admit absolute rotation Anfangsgrunde in regardingit as possible while denyingabsolute translation. This is a view which I shall discuss and which, as we shall see, though fatal to what is desired of shortly, the relational theory, is yet logically tenable,though Streintz does not show that it is so. But apart from this question, two objections may be made to his theory. (1) If motion damental relative to funmotion means bodies (and if not, their introduction is no gain from a logical becomes strictly pointof view),then the law of gravitation meaningless The if taken to be universal defend. view which seems to a impossible to any forces, that there should be matter not subject theoryrequires this is The and denied by the law of gravitation. pointis not so much that universal gravitation be tnie^ as that it must be significant must whether have already irrelevant question.(") We true or false is an that absolute acceleration* are required as even seen regardstranslations, the fact that this is due to overlooking and that the failure to perceive of the centre of mass is not a piece matter, but a spatial pointwhich is of accelerations. onlydetermined by means article similar remarks applyto Mr W, H. Macaulay's 467. Somewhat Newton's Theory of Kineticsf-"Mr Macaulay asserts that the true 011 theory(omittingpointsirrelevant to the present way to state Newton's are

"

"

*

"

issue) is

as

follows

assignmentof the rates of of the *

t of Mr Krit.

masses

:

"

Axes so

of reference

arranged,that

is

namely one possible,

a

be

so

certain

the axes, the in which

change of momenta, relative to

universe

can

chosen, and

decompositionof of all the particles components

1883 ; see Leipzig, Gruudfagen der Mertutitik, phipikalwcJuni Math. Bulletin of the American Soc.,Vol.' in. (1890-7).For a Macaula/s vieu*, see Art. Motion, htw*" of,in the new volumes (Vol.xxxi). Me

the

occur

esp. pp. 2-*,2o. later statement

Motion

imd

Matter

492

[CHAR

LVIII

of each pairbelongingto two differentparticles, ; the members pairs and being oppositein direction,in the line joiningthe particles, equal in magnitude" (p. 368). Here again,a purelylogicalpoint but it does remains. The above statement appears unobjectionable, in

and

show

not

material,for all unsuitable

for

is

matter

our

For

absolute space, the absolute space, any axes

would

nothing. The axes can, in a relation ; and not by a constant geometrical is changed by motion relative to such axes, has changed.Thus that the absolute position avoided

not

are

If absolute

468.

by

by abandoningall that and

is,to

at

these

is

absolute space and absolute

theoryto philosophers What

intact.

of Dynamics in terms principles of space, properties

find the metrical

we

set of collinear material

a

property

answer only possible

the

and plane*. Collinearity

lines and

ask what

we

is aimed

of sensible entities.

the

state

Among

when

essence science,to keep its logical

of

men

by

or

relation to matter, but

the relational

mends

i*ecom

have

exist.

not

to l)e material

of Newton's la\\s. Macaulay'sstatement it would be possible, in question, alone were

Mr

rotation

could

axes suggested

sense, be defined

motion

by any fixed be spatial really ;

will

axes

our

therefore

be defined

even

no

were

from

apart

Thus

l)e

cannot

axes

forces,and

subjectto

they cannot

purpose ; r elation to matter. geometrical and if there

be

may

or

The

is unnecessary.

motion

that absolute

but not

straight

be

included,but coplanarity may line,there is pointschange their straight

if no

change. Hence all advocates of the relational theory, thorough,endeavour,like Leibnix*,to deduce the straight

sensible intrinsic when

they are

line from

distance.

this there is also the

For

that the field of

reason

given distance of

the field of the is all space, whereas line is onlythat straightline,whence straight

a

the

former,makes

which

the

straightlines would

then

which

is

should

have

theory seeks to

relations between

as

appear

property of

the

latter,but

not

change

as

the pointsof space, among avoid. Still,we might regard

material

in

a

admit, however,!that between particles

points,and absolute

rotation

relation l"etween material

a

logically compatiblewith to

generatingrelation

intrinsic distinction

an

relational

a

points,

relational

We theory of space. the straight line was not a wimble

which

it

relation ; and

in any for absolute translational accelerations remains the necessity fatal

case,

to any

two

relational

469.-

theoryof

Macht

has

was

a

motion.

very curious argument by which he attempts to refute the groundsin favour of absolute rotation. He remarks that,in

world, the earth

the actual

the universe

and

as

could

a

is not

find it.

we

be inferred contains *

t

rotates

giventwice Hence

if there the

very

any

over

to relating

the fixed stars, and that in different shapes, but only once,

argument that the

were essence

no

rotation

heavenlybodies of

of the

earth

is futile. This

empiricism,in

a

sense

See my article " Recent Work on Leibniz,"in Mind, 1!K":". IHe Merhanik in Hirer Kittuickelnng, 1st edition, p. 210.

in

gument ar-

which

Absolute

407-460]

radically

is

empiricism work*.

present

or

may

may

meaning

exists,

and

Apart

from

they

general rational

distribution

of

have

they false,

it the

of

existence

there

ground.

the the

philosophy

not

with be

only

absolute

Baldwin,

Cf.

for

Art. Vol.

important,

spite

"

H,

:

is

of

Absolute This

space. a

us

have

discussions

matter, as

to

the

be

and

to

so,

to

which

idealism what

that

as

falls in

respect

with

the

times

many

a

the

assert

argument

illustrating

yet, true

being

as

Madi's

reckoned

world. and

;

This

but

contention

the

all, whether

meaning

twice,

applied

so

actual

the

at

applied.

are

are

exist.

and

exists

can

empirically. conclude

to

is

*"

in

of

their

of

only

not

advocated

empiricism, known

Thus, involves

point here

they

of

is

such

apply

not

calculations,

calculations

propositions part

necessary

distributions

The

laws

exact

that

not

to

contain

entity,

an

is

the

all

in

do

the

of

they

which

be

can

part

no

which

to

that

that,

significance

which

be

our

laws

its

can

universes

to

assumed

they

to

as

possible

are

no

matter

given,

is

universe

if

be

can

is

it

entities

with

throughout

with

matter

evident

is

and

deny

to

the

applied it

which

significance,

then

be

can

Dynamics,

impossible

seems

world

tions proposi-

all

not

the

in

that

held

been

then

that

arguments,

matter

as

exigence,

assert

is

existents, has

dynamical

to

to

advocated

argument

actual

if,

493

philosophy

the

with

regard

therefore

of

For

whole

laws

throughout

or

the

these

of

basis

exist.

not

without

considered

if

logical

Motion

the

to

concerned

discussions,

previous

It

opposed

The

essentially

are

which

Relative

and

been

Nativism 1902.

powerful

fact,

essential

is

motion

which

is

a

of

confirmation

difficulty the

logic

and

Dynamics,

to

in

sophies, philo-

current

upon

which

our

based.

"

in

the

Dictionary

of

Philosophy

and

Pxyvhoiogy,

edited

by

LIX.

CHAPTER

DYNAMICS.

HERTZ'S

470. evidence a

which

has

that

these

laws

In

order

from examine

what

suitable

most

The

work

that that

which

are

there

only of

manner. a

considered: as

a

of

means

stage

which

motions

the

thinkable

there and

are

mass,

(These

may

a

n

has particles

result of

motions

introduces

Hertz

of

in

free

which

when

form

introduce

of

series

regarded

are

But

stage. system

the

geometrical the

of

motion a

sense

so

His in

mass.

though

out

specificationof

appears

as

are

merely one-one

At

this

far

independent: independent are all so

to

kinetics,

introducing time,

relations

connections of

instants.

coining

Without

time

here

a particle,

a

of

before

direct

and

space

Matter

and

system.

in

and

system

a

energy,

stages in the

material

time

or

coordinates, all

"n

all

the

the

points

intermediate

an

any

of

the

purpose

mathematically. in a satisfactory difficulty

kinematical.

series

to

form

a

briefly.

them

carry

relations

only the

purely

well

Dimple

so

force

as

to

three

establishing, through

between

the

is

tonian, speciallyNew-

construct

to

the

overcome

first stage,

seen

gravitation.

concepts, space, time, and

difficult

system,

are

expound

to

concept, such

have

to

stage

collection

a

fundamental

in his

are,

the

this

correlation

three

fourth

have

of

this

For

theory

writers, is

recent

however,

In

motion.

while

by theory, is

There

Hertz's

most

a

causation

We

is

self-

of Hertz*.

that

worth

seems

evidentlydemanded seems,

be

to

of

subject,we "shall do fundamental principlesin

Electricity.

as

principlesof of

elimination

law

the

the

to

the

re-state

sciences

seems

it

object,like

Hertz

to

such

fundamental

admirable

The

reallyessential

is

applicable to

more

contradict

they

elementary Dynamics,

in

what,

attempts

some

wholly lacking in

are

usuallybeen held to be indubitable. are speciallysuggestive of the law

eliminate

to

Laws

Newton's

indeed, that

so,

form

also

that

seen

much

so

"

in

have

WE

between of

the

space

system.

involving velocities, but

that independent of time in the sense they are expressed at all times by the same equations, and that these do not contain the time thinkable motions which explicitly.)Those among satisfythe equations

they

are

*

Principien der Mechanik, Leipzig,

1894.

470, 471]

Hertz's

of connection

partsof

called

are

system

a

well-defined

sense

495

powtibkmotions.

the The connections among further to be continuous in a certain

assumed

are

Dynamics

(p.89).

It then

follows that

theycan

be

expressed

by homogeneous linear the coordinates. among

But

differentialequationsof the first order among further principle now is needed to discriminate a

possiblemotions,

motion, which

and

here

Hertz

introduces

his

only law

of

is as follows :

in its state of rest or of uniform motion Every free system persists in a straightest path." This law requires In the firstplace, some there are when explanation. in a system unequalparticles, each is split into a number of particles its this to all mass. become means proportional By particles equal. there are 3n If now their n coordinates particles, are regardedas the coordinates of a pointin space of 3?i dimensions. The above law then asserts that, in a free system,the velocity of this representative pointis its from and a givenpointto another constant, path point neighbouring the possible in a given direction is that one, among pathsthroughthese which has the smallest curvature. Such a path is called a two points, natural path,and motion in it is called a natural motion. It will be seen that this system, though far simpler and more 471. in than Newton's, does not differ very greatlyin philosophical form in the preceding the discussed to problems chapter. We still regard "

have, what for three

we

found to be the

essence

in order configurations

of the law of

to obtain

a

the necessity inertia,

causal relation.

This

broad

ordinaryDynamics, reappear in every system at all resemblingfor differentialequationsof the second and is exhibited in the necessity fact must

order,which

pervadesall Physics. But

between

Hertz's

there is

system and Newton's

"

one a

very material

ference dif-

difference which,

as

the two decision between pointsout, renders an experimental laws,other than the laws at least theoretically possible.The special laws of motion, which regulate system,are for Newton any particular For itself. such Hertz, as gravitation concerningmutual accelerations, of in the connections contained all laws are these special geometrical the system, and are expressedin equationsinvolvingonly velocities and is shown by Hertz (v.p. 48). This is a considerable simplification, conformable to phenomena in all departmentsexcept where to be more to have It is also a great simplification is concerned. only gravitation for the three. But Newton's of instead law of motion, philosopher, one introduced differentials second involves are law this (which so long as that the minor matter it is a comparatively through the curvature), laws of special systemsshould be of the first order. special it should be observed, of particles, The definition of mass as number mathematical device,and is not, I think,regardedby Hertz as is a mere of (v.p. 54). Not only must we allow the possibility anythingmore it if this difficulty were but even overcome, incommensurable masses, Hertz

and

Matter

496 stillremain

would

would

that all particles would their

This

therefore stillbe

be of the

happento

would

effect any retain shall do well,therefore, to mass mass.

a

not

ultimate

particles magnitude,only magnitudeas regards

that all

to assert significant

Mass

equal.

were

[CHAP.LIX

Motion our

of variety same

theoretical

and we simplification,

quantityof which without any implicato a certain particle, a certain magnitudebelongs tion that the particle is divisible. There is,in fact,no valid ground for The whole to different particles. different masses denyingultimately and the philosopher should, in this questionis,indeed, purelyempirical, what finds the physicist matter, acceptpassively requisite. ether its relations to With and to matter, a similar remark regard in the philosophical Ether is,of course, matter to be applicable. seems as

intensive

an

present state of Science will scarcely permit where, elseIt should be observed, to go. as however, that in Electricity, us thus of the second order, that the our indicating equationsare stillholds good. in the preceding law of inertia, as chapter, interpreted sense

but

;

this the

beyond

for philosophy, of indeed,to be the chief result,

This broad fact seems, discussion

of dynamical

principles. results : have two principal up, is relation between the a (1) In any independentsystem, there which is such three at that,given the configurations given times, configurations of the times,the configuration at two at the third time is

our

472.

Thus

to

sum

we

determinate. whole laws

is

There

(2)

material as

universe; but if two

the actual universe

distance

from

a

be found These

473.

in the two

same

When

these

are

have

the

except the

the two

within

this

at

a

great

regionwill

i.e.an

upper sets of motions. "

causal

same

regardto the matter

universes

to principles applyequally

to that of Hertz.

science

universes which

differ only in

for the difference between two

actual world

in the

the relative motions givenregion,

the approximately

be

a

independent system

no

limit

can

the

Dynamics of Newton and will give abandoned, other principles

having but littleresemblance

to received

Dynamics.

which is commonly stated as vital to generalprinciple, This is the principle Dynamics,deserves at least a passingmention. the effect and that cause are equal. Owing to pre-occupationwith and of it appears to have not been ignorance symboliclogic, quantity that this statement is equivalentto the assertion that the perceived and effectis mutual. between cause All equations, at bottom, implication i.e. mutual are logical equations, implications; quantitative equality between

One

variables,such

Thus implication.

as

cause

the

and

effect,involves

a

mutual

formal

in questioncan principle only be maintained the cause level,which, with the same logical it is no were we to to compelled interpretation give causality, longer to do. when state of the universe is given, Nevertheless, one possible others have a mutual and this is the source of implication any two ; if

and

effectl"e placedon

471-474]

Hertz's

the various laws of conservation

truth

underlyingthe We

474.

may

in the of

these, the

most

which

now

review the whole

puzzlingis the

discussed in

Part the

497

pervadeDynamics,a*nd givethe

of supposedequality

In presentwork. deduction,and of

the nature

Dynamics

and effect.

cause

of the

course

I, an

attempt

arguments

is made

logical concepts involved

notion

of

class"and

from

tained con-

analyze

to

in it.

the

Of

diction contra-

Chapter x it *), appearedthat types

(though this is

perhapssoluble by the doctrine of a tenable theoryas to the nature of classes is very hard to obtain. In subsequent Parts, it was shown that existingpure mathematics (includingGeometry and Rational be derived whollyfrom the indefinables and indemonDynamics) can strables of Part I. In this are specially important: process, two points the definitions and the existence-theorems. A definition is alwayseither the definition of

class :

this is be

a

a

or class,

the definition of the

necessary result of the

singlemember

plainfact

effected

that

of

unit

a

definition

a

can

of the

only object or objectsto by assigninga property function which i.e. defined, by statinga prepositional they are to

be

kind

of grammar controls definitions, making it impossible but possible to define the class of Euclidean e.g. to define Euclidean Space,

satisfy.A

And of abstraction is employed, wherever the principle i.e.where spaces. the objectto be defined is obtained from a transitive symmetricalrelation,

class of classes will

some

alwaysbe

the

objectrequired.When

used,the requirementsof are symbolic expressions become

grammar

entitydefined

is in

evident,and no

way existence-theorems

The

classes defined

various

The

class is

a

unit-class

that, if

n

the

that

seen

member

may

logical type

of not

are

mathematics null

"

i.e. the

"

a

of the

almost

are

of it ; the existence

of 1 from

proofsthat

all obtained

finite number,

n

+

1 is the number

the existence-theorem (both inclusive),

the from

importantof

more

fact that

the

the fact that

(forthe null-classis its only member). Hence, be

be called

optional.

be well here to collect the It may is derived from the existence of zero

Arithmetic.

them.

it is

what

of numbers

null-

zero

from from

is

a

the fact 0 to

n

follows for all finite numbers.

themselves,follows from the class of the finite cardinal numbers cardinal numbers; and existence of "0, the smallest of the infinite

Hence, the

from

the

existence of

the

series of finite cardinals in order of magnitude follows the the definition of "", the smallest of infinite ordinals. From

rational numbers

and

of their order

of

magnitudefollows series ;

the

thence, type of endless compact denumerable real the existence of the of series rationals, from the segments of the of the series. The terms numbers, and of 0, the type of continuous series of well-ordered types are proved to exist from the two facts: (1) that the number of well-ordered types from 0 to a is a -fl, (") that existence

of ??, the

*

See

Appendix

B.

if

be

u

all

prove

the

existence

the

points

of

dimensions

of

outside

we

this of

the

of

process,

logical

fundamental theorems

existence-

mathematics

are

is

established

is

spaces,

the

prove

the

some

in

class

employed

the

but

throughout.

such Thus

and

the

as

are

purely

space

of

with

LVI,

Throughout

definable

chain

class

Chapter

worlds.

the

the

metrical

a

in

explained

dynamical

of

constants.

complete,

ways

of

prove

methods

various

points

any

removing

of

the

By

the

of

we

by

existence

spaces of

XLVI,

with

u.

(Chapter

thence,

spaces. of

existence

series

and

of

every

spaces

Chapter

of

process

we

correlating

of

entities

no

the

(hyperbolic)

continuous

existence

the

prove

a

by

by

Euclidean

of

class

the

numbers

complex

LIX

series

than

greater

type

a

of

the

maximum,

no

of

definition

quadric,

the

prove

Lastly,

terms

the

we

itself

protective

descriptive

XLVIII,

properties.

of

closed

a

non-Euclidean

Chapter

all

;

having

is

of

thence,

class

the

of

the

existence

the

u

every

[CHAP.

Motion

types

6, by

of

existence

we

number

than

greater

not

the

XLIV),

well-ordered

of

class

a

types

From

and

and

Matter

498

of

logical

in

terms

definitions nature

of

LIST

Bs.

Denkens.

Gl.

der der

Begriff

FT.

Ueber

fiir

Function

BuG.

SuB.

KB.

Ueber

Ueber

BP.

griff

Ueber

und

Sinn

fiir

Archiv

die

Begriff

einiger fur

der

in

des

der

Sitzung

Yierteljahrschrift

Punkte

in

Yol.

Phil.,

syst.

Herrn

Konigl.

Peano

Sachs.

Jenaischen

9.

vom

den

Gesell-

fur

fiir

Phil,

Schroder's

E. i

\viss.

und

Januar, Jena,

Natnrwissenschaft.

und

2eitschrift

ssclirift

Classe

physischen

Medicin

Bedeutung,

Beleuchtung

Logik.

tlber

1885.

gehalten

Vortrag

der

Sitzungsberichte

Arithmetik.

Gegenstand.

und

TJntersuchung

logisch-Daathematische

1884.

Naturwissenschaft,

Gesellschaft

Kritische der

der und

Begriff.

und

Be

Eine

Theorien

Jenaischen

reinen

1879.

Breslau,

Medicin

des

Formelsprache

nachgebildete

arithmetischen

Arithmetik.

Zahi

formale

schaft

FuB.

a/S,

Halle

Grundlagen

der

Eine

Begriff stchrift.

ABBREVIATIONS.

OF

PM1.,

phil.

2

xvi

Kritik,

vol.

fiber

Vorlesungen

der

1891, 1891.

(1892).

100

die

(1892). Algebra

(1895). und

Gesellsohaft

rneine

eigene. der

Berichte

der

Wissenschaften

zu

math.-

Leipzk

(1896). Gg.

Grundgesetze Vol.

n.

der 1903.

Arithmetik.

Begriffsschriftlich

abgeleitet.

Vol.

i

Jena

1893

APPENDIX

THE

LOGICAL

475.

AND

THE

deserves,contains present work, and avoids

DOCTRINES

FREGR

OF

Frege,which

many where

the differences demand and

ARITHMETICAL

of

work

A.

than it appears to be far, less known of the doctrines set forth in Partis I and II of the it differs from

discussion.

all the usual

Frege'swork

fallacies which

symbolism, though unfortunately so

which

the views

abounds

beset

cumbrous

I have

writers

be

to

as

advocated,

in subtle

tions, distinc-

Logic.

on

His

difficult to

very

in

is based upon an practice, analysisof logicalnotions much more is and Peano's, profound philosophically very superior to its more convenient rival. In what to expound Frege's follows, I shall try briefly theories on the most important points,and to explain my grounds for I do differ. But where the points of disagreement are differing very few

employ

than

slightcompared to those of agreement. They all result from difference three points:(1)Frege does not think that there is a contradiction in the on notion of concepts which cannot be made logicalsubjects(see" 49 supra) ; the proposition can in a proposition, (2) he thinks that, if a term a occurs be about vn) assertion and into a a an analysed always (see Chapter ; and

(3)he

is not

aware

discussed

of the contradiction

in

matters, and it will be well here very fundamental in almost written since the previous discussion was

Chapter to

x.

discuss

These

them

are

afresh,

completeignorance of

Frege'swork. Frege senses

is

compelled,as

which

departmore

different from Some

mine, a

I have

been,

to

employ

common

words

in technical

As his departures are frequently usage. of his terms. translation the arises as regards difficulty

of these, to avoid

or

less from

confusion, I shall leave untranslated,

since

every

in a. think of has been alreadyemployed by me that I can Englishequivalent different sense. slightly The principal heads under which Frege'sdoctrines' may be discussed are the following: (1) meaning and indication; (2) truth-values and judgment; and symboliclogic; (3)Begriffand Gegenstand;(4)classes;(5) implication matical abstraction of ; (7) mathe(6) the definition of integersand the principle I shall deal successively induction and the theory of progressions, with

these

topics.

Appendix

502

[476-

A

Meaning and indication. The distinction between meaning (Sinn) and indication (Bedeutung)*is roughly,though not exactly,equivalentto 476.

distinction between

my

("96). Frege did under

consideration

it appears first in Before making the of

names

not

a

concept

as

what

this distinction in the

possess

and (theBegriffsschrift BuG.

and

such

objects(Bs. p. 13) :

"

A

concept denotes of the works

first two

Orundlagen der

Arithmetik) "

dealt with specially thought that identityhas to do

(cf.p. 198),and

he distinction,

the

the

is

is identical with

B"

in with

SuB. the

he says, that

means,

(Bs.p. 15) a definition signification circularity.But later he explains he Identity," explainedin " 64. identityin much the same way as it was it and which calls attach for to reflection not are owing to questions says, A relation between Is it a relation? Gegenstande? quite easy to answer. between must or or names signs of Gegenstande?"(SuB. p. 25). We of the is contained. he which the in being distinguish, says, way meaning, given,from what is indicated (fromthe Bedeutung). Thus "the evening star" but not the same and "the morning star" have the same indication, meaning. A word ordinarilystands for its indication; if we wish to speak of its such device (pp.27-8). The or some meaning, we must use inverted commas indication of a proper name is the objectwhich it indicates ; the presentation which between the two lies the meaning, goes with it is quitesubjective; which is not subjective and yet is not the object (p.30). A proper name expresses its meaning, and indicates its indication (p.31). This theory of indication is more sweeping and generalthan mine, as the two is supposed to have appears from the fact that every proper name sides. It seems from that only such proper names derived to me cepts conas are of the can be said to have meaning, and that such words as by means John merely indicate without meaning. If one allows,as I do, that concepts be objects and have can it seems fairlyevident that their names, proper as without a names, rule, will indicate them proper having any distinct meaning ; but the oppositeview, though it leads to an endless regress, does not appear to be logically impossible.The further discussion of this point be postponeduntil we must to Frege'stheory of Begriffe. come 477. Truth-values and Judgment. The problem to be discussed under this head is the same the one raised in " 52 1, concerningthe difference as between asserted and unasserted propositions.But Frege'spositionon this is subtle radical analysisof question more than mine, and involves a more judgment. His Begriffsschrift^ of the distinction owing to the absence between and has a simplertheorythan his later works. meaning indication,

the

and

sign A

which,

the

verballyat

sign B have the same least,suffers from

"

"

"

I shall therefore omit

it from

the discussions.

There

told (Gg. p. x),three elements in judgment: (1)the are are, we of truth,(2) the Gedanke, recognition truth-value the (Wahrheitswerth). (3) I

do

meaning as

not

denotingfor t This

translate

different from

Hcdeutunfjby denotation, because this word Frege's,and also because bedeuten, for him, is

has not

a

technical

quite the

same

me.

is the

logicalside of the problem of Annahmen, raised by Meinong in his able subject,Leipzig,1902. The logical, though not the psychological,part of Meinong'swork appears to have been completely anticipatedby Frege. work

on

the

The

478]

Logicaland

Here

the Gedanke

what

I called

by

togetherwith

is what

Arithmetical Doctrines

I have called an

this name

It wiJl be

distinct notions ; I shall call the Gedanke truth-value of a Gedanke I shall call an

assumption does

not

concept: whatever the true

means

(FuB.

require that

x

if x

21).

In

falsehood of #"; these

are

false

p.

only assertions of truth and

is not

has

a

be,

may

is true, and

"the

propositionor rather, "

well to have

alone

for these two

names

propositional concept;

a

assumption*.Formallyat

its content

should

truth of x"

is

if # is false or

503

alone and the Gedanke

both the Gedanke

covers

its truth- value.

unasserted

of Frege

a

be

a

the

least,an

propositional

definite notion.

This

the it means proposition accordingto Frege, there is "the assertions and negations not but of propositions, of falsity, i.e.negationbelongsto what is assei'ted, not

a

like manner,

or

the

oppositeof assertion!. Thus we have first a propositional truth or falsity its next the assertion the case concept, as may be, and finally of its truth or falsity.Thus in a hypothetical have we a relation, judgment^ of two judgments, not but of two propositional concepts (SuB.p. 43). This theory is connected the- theory of with in a very curious way meaning and indication. It is held that every assumption indicates the the false (which are it means true while or the called truth-values), indicates correspondingpropositionaiconcept. The assumption "22=4" the true, we are told,just as "22" indicates 4J (FuB. p. 13; SuB. p. 32). In a dependent clause,or where a name occurs (suchas -Odysseus)which indicates nothing,a sentence But when have no indication. a sentence may truth-value,this

is a (Behauptungssatz)

is its indication. proper

Thus

assertive sentence

indicates the true

which

name,

every

the false

or

does sign of judgment (Urtlieilstrich) indicates combine with other signs to denote an not object;a judgment^ for but asserts judgment, symbol nothing, something. Frege has a special which is something distinct from and additional to the truth-value of a propositional concept (Gg. pp. 9"10).

(SuB.

32"4;

pp.

478.

There

Gg.

are

well to discuss.

In the first

it would of

any

that what

seem

Caesar/'not

"

place,it seems

real

analysis.If

we

consider,say,

is not, I

"

Caesar

is asserted is the propositional concept "the

the truth

This latter

of the death of Caesar."

think, the

same

will^be

doubtful whether the introduction

merely another propositional concept,asserted true," which

f

it

theory which

difficultiesin the above

some

of truth-values marks

The

7).

p.

in

"

the

death

seems

death to

be

of Caesar died."

"Caesar as proposition

died,37

is

There

elements here, and it would in avoiding psychological great difficulty that Frege has allowed them to intrude in describingjudgment as seem is due to the fact that the recognition of truth (Gg. p. x). The difficulty

is

there

is

a

sense psychological

of

assertion,which

is what

as

lacking to the logical

with Meinong'sAnnahmen, and that this does not run parallel be whether true or false, may sense. proposition, Psychologically, any merely thought of, or may be actuallyasserted : but for this po"bihty, true only are asserted, would be impossible.But logically, propositions error ^

*

Frege, like Meinong, calls

this

aa

Annahine:

I^uB. p. 21.

twU^ermwMohfn'dica^isitsdf spoken of.asopposed to be

Frege

uses

inverted

commas.

Cf. " 56,

to what

ft in***..

Appendix

504

[478-

A

in an unasserted form as parts of other propositions. though they may occur In up impliesq" either or both of the propositions be true, yet p, q may not in a nd this in is unasserted a each, merely in a logical, proposition, definite place among has Thus assertion a logical psychological, sense. which assertion to of notion notions,though there is a psychological nothing of to be a constituent logicalcorresponds. But assertion does not seem in contained asserted an an sense, although it is, in some proposition, truth is which asserted proposition.Jf p is a proposition, a concept "jp's has thus "p's truth" is not the same if p is false,and as being even p asserted. is equivalentto p asserted, Thus no concept can be found which "

and

therefore assertion

is not

a

constituent

asserted.

in p

Yet

assertion

a term to which asserted,has an external relation ; for any p, when relation would need we to be itself asserted in order to yield what

is not such

arises owing to the apparent fact, which difficulty may be part of however be doubted, that an never asserted propositioncan another is made statement be a fact,where : thus, if this proposition any about p asserted,it is not reallyabout p asserted,but only about the of Frege'sone assertion of p. This difficulty becomes serious in the case and and is true only principleof inference (Bs. p. 9) : "p p implies q ; therefore q is true*." Here it is quiteessential that there should be three actual from deduced assertions, otherwise the assertion of propositions asserted premi?Qes would be impossible;yet the three assertions together form without whose unity is shown one proposition, by the word therefore, Also

want.

which

q

fresh

a

would

have

not

deduced,

but

would

been

have

asserted

as

a

premiss.

It

is also almost

at impossible,

truth, as

Frege does.

the

as

the

been

same

content

true

a

of

a

An

least to

asserted

proposition. We

me,

to

divorce

it would proposition, may

allow

proposition(Bs. p. 4), and

that

assertion from seem,

must

be

negation belongs to

regard

every

assertion

as

We shall then correlate p and assertingsomething to be true. not-p as unasserted propositions, and regard"pis false" as meaning "not-p is true." But to divorce assertion from truth seems by taking assertion only possible in a psychological sense. 479. for the true Frege'stheory that assumptions are proper names the also the untenable. Direct or case false,as be, appears to me may to show that the relation of a proposition to the true inspectionseems the false is quite different from "the that of (say), or present King of if view this correct England to Edward VII. on Moreover, were Frege's point, we should have to hold that in an asserted propositionit is the that is asserted,for otherwise, all asserted meaning, not the indication, would assert the very same propositions thing,namely the true, (forfalse Thus asserted not would not differ propositionsare asserted). propositions from one another in any way, but would and simply identical. be all strictly Asserted propositionshave no indication (FuB. p. 21),and can only differ, if at all,in some Thus to the analogous meaning. meaning of the way unasserted proposition togetherwith its truth-value must l"ewhat is asserted, "

'

*

Cf. *uj)jYr, " 1

9, (4)and " 38.

The

480] if the

Logicaland

Arithmetical Doctrines

meaning simplyis rejected. truth- value

the

is proposition

here

it

:

But

there

seems

no

ofFrege

ing in introduc-

purpose

quite sufficient to say that is true, and that to say the

seems

whose

505

an

asserted

meaning meaning is conclude We asserted. then is meaning might that true propositions, even when as they occur parts of others,are always and essentially asserted,while false propositionsare always unasserted,thus about ttierefore escapingthe difficulty discussed above. It may also be objected the true to Frege that and the false," as opposedto truth and falsehood, do not denote single definite things,but rather the classes of true and false respectively.This objection,however, would be met by his propositions theory of ranges, which correspond approximatelyto my classes; these, he says, are things,and the true and the false are ranges (t?.inf.). 480. to a point in I come Begriffand Gegenstand. Functions. now which Frege'swork is very important, and requirescareful examination. of the word His use Begriffdoes not correspondexactly to any notion in assertion as to the notion of an though it comes my vocabulary, very near defined in "43, and discussed in Chapter vn. On the other hand, his to correspond exactlyto what I have called a iking("48). Gegenstandseems I shall therefore translate Gegenstand by thing. The meaning of proper for him name seems to be the same for me, but he regardsthe range of as confined can to things,because they alone,in his opinion, as proper names be logical subjects. and is set forth simplyin FuB. Frege'stheoryof functions and Begriffe defended against the criticisms of Kerry* in BuG. He regardsfunctions and and in this I agree with him than predicates fundamental as more relations;but he adopts concerning functions the theory of subjectand assertion which we The acceptance of discussed and rejected in Chapter vii. this view gives a simplicity been unable to his I which have to exposition of the to persuade me attain; but I do not find anything in his work legitimacyof his analysis. An arithmetical function,e.g. 2x3 + x, does not denote, Frege says, the result of an arithmetical operation, for that is merely a number, which would is left when be nothing new The of a function is what essence (FuB. p. 5). the x is taken away, i.e., in the above instance,2 ( )*+ ( ). The argument whole does the not to a x function, but the two togethermake belong (ib.p. 6). A function may be a propositionfor every value of the variable ; divided its value is then always a truth-value (p.13). A proposition may be true

is the

one

same

the

to say

as

"

"

"

"

"

The former Frege calls "conquered Gaul" is a possible the argument, the latter the function. Any thing whatever for division of function a corresponds propositions (p.17). (This argument and assertion as explainedin " 43, but Frege does not exactlyto my subject

into two

"Caesar"

parts, as

restrict this method which two

is not

a

of

function,i.e. whose

(1) "If

one

in

*

of the an

of

the

latest

Chaptervii.) A thingis anything expressionleaves no empty place. The

I do in

analysisas

followingaccounts

earliest and

and

nature

of

of

a

function

Frege'sworks

expression,whose

content

fur wiss. Phil.,vol. Vierteljahrschrift

are

quoted from

the

respectively. need

not

be

xi, pp. 249-307.

propositional

Appendix

506

[480,

A

places,and of these places, by something else, more which call the part of the expression part we function,and the replaceable in

a (faurtheilbar), simple or compositesign occurs

in one regardit as replaceable, by the same everywhere,then

we

but

invariable

remains call its

"If

from

the

a

a

proper in of the first,

the whole

such

we

this process

in

a

argument" (Bs. p. 16).

(2) or

or

more

or

one

way

which is part proper name, but in it where all of the places occurs, exclude

we

some

or

a

and to be filledby one as recognizable (as argument positionsof the first kind),

placesremain

these

that

name

arbitraryproper name of a function of the first order with we thereby obtain the name fills the which Such a one togetherwith a proper name argument. name, forms a (Gg. p. 44). argument-places, proper name" The latter definition may become examples. plainerby the help of some and The present king of England is,accordingto Frege, a proper name, which is part of it. Thus here we regard "England is a proper name may Thus we England as the argument, and "the present king of" as function. have will a This led are to "the always expression present king of x" meaning, but it will not have an indication except for those values of x which at present are monarchies. The above function is not propositional. But "Caesar conqueredGaul" leads to "a; conqueredGaul" ; here we have a function. There is here a minor pointto be noticed : the asserted propositional but only the assumptionis a proper name is not a proposition proper name, for the true or the false (v.supra) ; thus it is not Caesar conquered Gaul the as asserted,but only correspondingassumption,that is involved in the This is indeed sufficiently function. obvious, since genesisof a propositional there whereas wish a; to be able to be any thing in we x conquered Gaul," is no such asserted proposition perform this feat. except when x did actually Again consider "Socrates is a man impliesSocrates is a mortal. " This for the true. is,accordingto Frege,a proper name By varying (unasserted) the proper name obtain three propositional Socrates, we can functions, Socrates is is Socrates is man man a a a mortal," namely x implies implies Of these the first and a; is a mortal/7"a: is a man impliesx is a mortal." third are true for all values of xt the second is true when and only when is x same

I call what

"

"

"

"

"

"

'

"

"

a

"

mortal.

By suppressingin

like

of the first order with first order with we

one

get

from 1 " 2," arguments (Gg. p. 44). Thus e.g. starting of a function of the first order with 2," which is the name

"

argument, and with

Frege says, Thus

"

thence

two

e.g. the assertion

function of tc, but be

"

:

of

a

of existence

in the mathematical

may

thing, but

regarded as be

one a

value of

function

a

function

order

(Gg.p. 44). is

sense

function

a "

#

"f"x satisfying

of "t". Here

function.

of the

function in like manner,

function of the second

There is at least be

of

name

By suppressinga

arguments. name

is the

which

"#"y/'

obtain the

we

of the second order

account

a

two

first "or

first order

one

of a function in the name proper name of a function of the argument, we obtain the name manner

Thus

must "t"

this

is not on

a no

proposition, any quitedifferent from functions of the first order,by the fact that the possible arguments are different. Thus given any consider either f(x),the function of the first proposition, say /(a),we may considered

a

as

a

function

may

of "",is

The

481] order

Logicaland

Arithmetical

from varyinga resulting

Doctrines

of Frege

507

and

the function keeping/ constant, or ""("), by varying/ aud keepinga fixed ; or, finally, we may consider ""(a?), in which hoth / and a are varied. be is to (It separately observed that such notions as "f" (#),in which we consider any proposition involved in of indiscernibles as stated in " 43.) the identity concerninga, are Functions of the first order with two variables,Frege pointsout, express relations (Bs. both subjectsIn a are p. 17); the referent and the relatum relational proposition (GL p. 82). Belations,just as much as predicates, belong,Frege rightlysays, to pure logic(ib.p. 83). 481. The word JBegriff is used by Frege to mean thing nearlythe same as prepositional function (e.g.FuB. p. 28)*; when there are two variables, the Begriffis a relation. A thing is anything not a function,i.e. anything whose expression leaves no empfcy place(ib. p. 18). To Frege'stheory of the essential cleavage between thingsand Begriffe, cit. p. 272 ff.) Kerry objects(lac, that Begriffe also can occur as subjects. To this Frege makes two replies. In the firstplace, it is,he says, an importantdistinction that some terms can while others can if also onlyoccur as subjects, as even occur Begriffe concepts, also occur can as subjects(BuG. p. 195), In this I agree with him entirely ; the distinction is the one second he 49. But in to a employed g" 48, goes on of the second

order got

point which

appears

fallingunder

to

We

mistaken.

me

he

can,

falls under

a says, have he means, man,

concept

as not higher one (as but its Greek falls under man) ; but in such cases, it is not the concept itself, he that is in question(BuG. p. 195). "The name, concept horse," says, is is indicated by inverted commas not a concept, but a thing ; the peculiar use

(ib.p. involve when

a

196). a

Socrates

But

few

a

somethingis

and

when

A

asserted

concept does not fit an

involved,though similar,are to reconcile

these

remarks

to

When

a

fall under

the

not

with those

which

statements

even

which

made

be

can

thing is said to fall under a

higher concept, the

(ib.p. 201).

same

but

195;

of p.

to

seem

predicative essentially

concept,he says, is of it : an assertion

object.

concept is said

a

later he makes

pages

different view.

a

two

of

a

concept, relations

It is difficult to

I shall return

to

me

this

point shortly. the unity of a proposition : of the parts of a propositional Frege recognizes but be incomplete at least must one concept, he says, not all can be complete, the parts would otherwise not cohere (ib. or predicative, (ungesattigi) p. 205). the from oddities resulting He also,though he does not discuss, recognizes that every positive thus he remarks integer any and every and such words : of four squares, but "every positive is the sum integer"is not a possible of four squares/' The meaning of "every of x in "a is the sum value the context he says, dependsupon (Bs,p. 17) a remark positiveinteger," "

is doubtless

which

notions

concept

*

with

"We one

pp. 8*2"3.

is

admitted

concepts:

are

as

F

the

"

is

a

whatever thing a may proposition of a predicate the indication ; a thing is

concept J?" is

the

does

not

dictory subject. Self-contraconcept if a fallsunder

exhaust

correct, but

have

here

argument

we

a

a

function have

whose

value is always

called Begriffe; with

two,

we

a

be

(GL

what

truth- value. call them

p.

can

Such

87). never

A

be

functions

relations."

Cf. GL

[481,

Appendix A

508 whole

the

indication

of

though predicate,

a

it may

be that

of

a

subject

(BuG. p. 198). theory,in spiteof close resemblance,differs in some Before important points from the theory set forth in Part I above. I shall briefly own theory. recapitulate my examining the differences, Given any prepositional concept,or any unity (see" 136),which may of two sorts : (1)those in the limit be simple,its constituents are in general without whatever be which destroyingthe replacedby anything else may unityof the whole ; (2)those which have not this property. Thus in the death of Caesar," anything else may be substituted for Caesar,but a proper be be substituted for death, and must not name hardly anything can class constituents of the former substituted for of. Of the unityin question, have then, in regard to any will be called terras^ the latter concepts. We objects: unity,to consider the following (1) What remains of the said unity when one of its terms is simply from it is removed several times, when removed, or, if the term occurs than has more it occurs, or, if the unity of the placesin which or more one 482.

The

above

"

term, when

one

placeswhere

two

or

of its terms

more

they occur.

This

are

removed

Frege calls

is what

a

from

some

or

all of the

function.

all,only by of the places in one or more replaced, of its other terms, or by the fact that two or more where it occurs, by some have been thus replaced terms by other terms. of the class (2). (3) Any member The of the class (2)is true. assertion that every member (4) (5) The assertion that some member of the class (2)is true. of the class (2)to the value which the (6) The relation of a member

(2)

The

the fact that

variable

The

class of unities

one

of its terms

the

said

unity,if

at

has been

has in that member.

fundamental

case

is that where

this is derived the usual

From

from differing

our

mathematical

unity is a propositional concept. notion of function,which might function,its value propositional

simpler. If f(x) i* not a is the term y x given (f(x)being assumed to be one- valued) function the propositional for the given satisfying y =/(#), i.e. satisfying, value of X, some this relational proposition relational is involved proposition ; in the definition of f(x),and some such propositional function is required in the definition of any function which is not propositional. As maintained regards(1),confiningourselves to one variable,it was in Chapter vn where from which the proposition start that, except we is predicative else asserts a fixed relation to a fixed term, there is no or such entity: the analysisinto argument and assertion cannot be performed Thus in the manner calls what if conclusion a function, our Frege required. was sound, is in generala non-entity.Another point of difference from he in in to be the which, however, Frege, right,lies in the fact appears that I place no restriction upon the variation of the variable,whereas of to the function,confines the variable to the nature Frege, according functions of the first things,functions of the first order with one variable, order with two of functions the second order with one variables, variable, at

first

for

and

sightseem

value

a

so

on.

There

of

are

thus

for him

an

infinite number

of different kinds

482] of

The

Logicaland

This variability.

have in

identified. functions

Instead

of

of the

from

arises such

concept occurringas

Arithmetical

and

For

me,

the

first of

rump

(3) analysisinto argument

the

Doctrines

fact that

ojFrege

regardsas distinct the concept occurringas term, which I ("49)

the

he

the

functions,which cannot be values and non-entities false order, are

propositionconsidered

a

(4) accordingto

509

circumstances.

in

(1),I

The

of variables abstractions.

(2)

substitute

regarding the

ground always possibleis that, when is removed term from is apt one a propositional concept, the remainder of unity,but to fall apart into a set of disjointed to have no sort terms. Thus what is fundamental in such a case is (2). Frege'sgeneraldefinition of a function,which is intended also functions which to cover not are be shown to be inadequateby considering what propositional, may may i.e. a; as a function be called the identical function, of x. If we follow in hopesof having the function left, find x Frege'sadvice,and remove we that nothing is left at all ; yet nothing is not the meaning of the identical function. Frege wishes to have the empty placeswhere the argument is

or

to

or

be

and

indicated

inserted

in

is

function

fact

be

the

singleentity,is the

a

relatum

of this

relation

any "Socrates formal

be

can

is

apparatus

it presupposes is a function

want

the

is

of the

the function of

involved

any

stand empty placesreally

for.

The

the

notion

(6) above; of all

assertion

of the

terms

expressedin

of

terms

of relations

calculus

we or

empty is

no

in 2a?

+ y" certain

a

function, relata,and

correspondingreferent,as Socrates.

cannot

be

We

functions. propositional

of the

some

the

x

consider any

then

can

+

there

:

of

expressedin

man"

a

mean

from

relation

or relation,

be indicated

letter cannot

2a? two

term

we our

not

as

thus he says that in way; here his requirementthat the

same

we

that

that this is what

class,and as

to

seems

some

But

2( )3-t-( ). places are to be filled by what of distinguishing way The

function

for

But

the

usual

employed,because a propositional

that

may say for the class of its has all terms relation which many-one contained its relata and has propositions*:or, if we referents, among prefer,we may call the class of relata of such a relation a propositional these statements is the air of formal definition about But function.

functions fallacious,since propositional class

of referents and

relata

of

a

are

presupposed in definingthe

relation.

collected Into are functions,propositions propositional But we may also collect (Theseclasses are not mutuallyexclusive.)

by

Thus classes.

means

of

which

in them

all

containing propositions giventerm a will form a class. In this way we obtain propositions functions. the "" is In the notation ""(a?)? concerningvariable propositional take some variable ; if we wish it not to be so, we must particular essentially class a." Thus is "a; a or about cc, such as "as implies ""(#) proposition decided that is if have variables. not a we two contains But, "" essentially cannot we regard"" itself as the second variable. It will separableentity, either the relation of x to tf" variable to take as our (x),or else be necessary but for constant ""(y) for different values of y "". the class of propositions for but; be it is clear to to as logic important This does not matter formally,

them

into

classes

by

the

terms

occur

:

a

"

*

Not

all relations

having this property

are

functions propositional

; v.

inf.

the

meaning

of what

obtain in this way

"j".We

but again these into classes, propositions

division of

another

the variation of

as

appears

[482-

A

Appendix

510

classes

not

are

mutually exclusive. In

the above

functions

it would

manner,

we

seem,

having to introduce

without

the

can

make

be observed, however, that the kind tional functions are defined is less generalthan

prepositional

Frege calls functions.

of relation

It is to

of

use

objectswhich

by

which

having their domain coextensive with terms in propositions.For in this way any proposition would, for be relatum relation, for

to

any

the term

term, whereas

function, be a constituent propositional relatum*. This point illustrates again that a

is fundamental

the

which class of relations,

above

Otherwise,

""(2)is not

dealingwith

are

of 2 with

relatum

of

a

regardto

discussed in

soiue

Chapter functions propositional x

we

and proposition,

which indefinable,

an

seems

which proposition

it would

;

have

to have

demands

that

but for the

subjectswithout logical

into

The

concept

change of

meaning.

name

to seem

a

is named

by

this name?"

of the

concept that

the first place,the

If 'there

be

concepts can

Frege'stheory,that

concept,but its name, is involved,"has subject. In the second place,it seems always

is it that

"what

the

tion mystery lurks in the variatheoryof different present,Frege's

the

not

be

contradiction

some

the name to be done it is really this appears bear In investigation. involved,will not, I think, "

its range

""(2)should

that

when

assertion

considering as

call propositional relations. may is indeed meaningless,for we

kinds of variables must, I think,be accepted. whether 483. It remains to discusC*afresh the question made

also to show

seem

relation. propositional show

to

is its

class of relations involved

are variability Frege'sdifferent kinds would t he variable where is variable, (say)"f" "" (2),

the

suitable

a

is referent must,

unavoidable,for in

of

that

which

of the

But

definition.

of incapable

and

proposi-

the class of many-one relations domain and their converse tained con-

were

is

mere

alreadymade

the

to -ask : legitimate no

answer,

the

could not be a name ; but if there is an answer, the concept,as opposed made be its name, a can subject.(Frege,it may be observed,does not the logicaland to have elements of clearlydisentangled linguistic former

depend upon denoting,and have, I think,a much more Frege allows them.) It is true that we found difficulties everythingcan be a logical subject: as regards"any a," for example,and also as regardsplurals.But in the case of "any a," there is ambiguity,which introduces a new class of problems; and as regards there which i n the many behave like a logical are plurals, propositions subject in every respect except that they are many subjectsand not one only (see"" 127, 128). In the case of concepts,however, no such escapes The case of asserted propositions but is met, I think, is difficult, are possible. that asserted proposition is merely a true proposition, and is an by holding

naming

:

the

restricted range than in the doctrine that

therefore the

asserted wherever

it occurs, even when would lead to grammar Thus, on the whole, the doctrine of concepts which

oppositeconclusion. be made subjects untenable. seems 484. Classes, Frege'stheoryof classes

cannot

*

The

notion

of

a

constituent of

a

is very

propositionappears

and difficult,

to be

a

I

am

logicalindefinable.

not

The

484]

Logicaland

Arithmetical

Doctrines

of Frege

that I have

sure

lauf* to class

as

thoroughlyunderstood it. He gives the as entitywhich appears to be nearlythe same

an

The

one*

concept of the

class,and

the class

as

Werthver-

name

I call the

what

do not

many,

511

appear

in his

exposition.He differs from the theoryset forth in Chapter vi chiefly the fact that he adopts a more intensional view of classes than hy I have done, beingled thereto mainly by the desirability of admittingthe null-class and of distinguishing term from a class whose a it is. I agree only member that these two cannot be attained entirely extensional objects by an theory, though I have tried to show how to satisfythe requirementsof formalism (""69, 73). The of a Begriff, extension Frege says, is the range of a function whose value for every argument is a truth-value (FuB. things, p. 16). Banges are whereas functions are not ($. p. 19). There would be no null-class, if classes taken in extension ; for the null-class is only possible were if a class is not collection of terms (KB. pp. 436-7). If x be a term, we a cannot identify view requires, with the class whose #, as the extensional only member is x ; for suppose to be 2 different members two

class

a

of

; then

x

than

having more if

member,

one

let y,

the class whose

is identical with

x

and

z

be

only

member

is #, y and z will both be members of this class,and will therefore be identical with x and with each other,contrary to the hypothesis!. The extension of a Begriff has its beingin the Begriff not in the individuals itself,

under the Begriff(-tb. I say something about all men, falling p. 451). When I say nothing about some wretch in the centre of Africa,who is in no way indicated, and does not belong to the indication of man (p.454). Begriffe are priorto their extension,and it is a mistake to attempt, as Schroder does, extension to base on individuals; this leads to the calculus of regions not to Logic (p.455). (Gebiete), What Frege understands by a range, and in what way it is to be he endeavours to explainin his conceived without reference to objects, He Grundgesetzeder Aritkmetik. beginsby decidingthat two prepositional value for have the same functions to are they have the same range when of

value

every

This

(pp. 7, 14)

the

determines X

(")be

x,

i.e. for every is laid down

a

as

equalityof

of

x

both

true

are

what

not

same

they value

the

same

of themselves

range

decide what notion

the

since

Thus

value.

of

of the function

ranges

is not

range

a

is true

"x

are

M

to be

(p.16).

yet fixed

(asan

"

only If

for different values

of "

("') =

Let

that

X

"fccand

of equality

the conditions for the

false

this

themselves.

in

are

and if we denote by ""'the range of "fa we shall have X and only when equal,i.e.when and only when ""'and i//are have

both

or

primitiveproposition.But

ranges, has the never

function which

a

value

ranges

decide

us

the

true

assumption,not

an

(^')when \ffxalways do not

arbitrarily

is to

be

the

asserted proposition),

every term the false is to be the range of the function a"=not the true when is of that the follows It "fcr is identical with itself." range the Begriff""#; the the true and nothing else falls under and only when

and

range

under

*

of

fa

the

is the false when

Begrifffa',

I shall translate thia

in

as

and other

rdnge.

only when cases, the

the false and range

nothing else

is neither

t I". p. 444.

the

Cf. *wpr*f

true

" 74.

falls nor

A

Appendix

512 the false

(pp. 17

If

18).

"

thing is distinct from

the

the

as

is the

reason

same

thing falls under a cojacept,this one the concept in question(p. 18, note)

range of that mentioned

"

above.

of the range argument (p.49) to prove that the name that i.e. the has an symbol employed for indication, always

There

is

function

an

meaningless*.In

never

only one

484,

of the

view

contradiction

discussed in

of

a

it is

Chapter

x,

tion meaning to a range when we have a proposiand tf" where /is constant of the form ""[/("")]" variable,or of the form function which is determinat fx(x),where x is variable and fx is a propositional value of from varies but to when is a? another one x given, and the into is things dependent whenj^ part analyzed concepts, provided, but contains also at least one does not consist only of things, x on concept. This is a very complicatedcase, in which, I should say, there is no class thus escape for saying so being that we the can as only reason one, my be inclined

I should

deny

to

a

"

contradiction. 485.

By

relation which

definition of the to

a

of variable

means

class of which

it is

a

functions,Frege obtains a propositional Peano calls c, namely the relation of a term

member*.

definition

The

is

follows

as

:

"

"

atu

or many) x such (orthe range of terms if there be none function that is the range of such which is there is a propositional u "" "" and "f"ais identical with x (p. 53). It is observed that this defines atu whatever things a and n may be. In the first place, suppose u to be a range.

is to

the term

mean

that

range is u" and any two whose range is u are regarded by Frege as identical. Thus we may speak of the function "" In this case, a*u is the proposition is u. whose is true """,which range Then

there

when

a

u

is

is not

is

a

range, is the

aeu,

one

there

then range

null-range.Thus of

member

member

of

u

whose "t"

of u, and

member

a

a

therefore i.e. the

is at least

atu

of

is false otherwise. is

a

aeu

no

such

If,in

which

function propositional indicates

second

the

function propositional

the

when

true

indicates the false when

u

is

u

is

It is to

only infer

"", and

as

is

always false,

a

range

and

; range indicates the other in aeu cases, null-range. ; that from the equivalenceof x*u be observed and

u

a

place,

a

and

a

is not

a

for all

xcv

identityof u and v when tc- and v are they are not ranges, the equivalencewill always hold, since ranges. the null-range and xev for all values of $c; thus if we are xtu allowed the inference in this case, any two objectswhich are be not ranges would absurd. One is which be doubt to whether and identical, u ^* might tempted when, they are with be identical even must intensional of an view : ranges this becomes classes, open to question. Frege proceeds(p.55) to an analogousdefinition of the propositional function of three variables which I have symbolisedas x JK y, and here again does not placeany restrictions on the he givesa definition which variability of E. This is done by introducing a double range, defined by a propositional function of two variables;we may regard this as a class of coupleswith and if (x ; y) is a member f. If then E is such a class of couples, sense of this values of

we

x

can

the

When

*

t

Cf. "" 21,

76, snpra. Neglecting,for the present, our

with sense,

cf .

"

98.

doubts

as

to there

being any such entityas

a

couple

The

487]

Logicaland

Arithmetical Doctrines

class,x R y is to hold; in other cases this basis,Frege successfully erects

requiredfor his of variability

R

the present work 486.

The

to the kind

Arithmetic;

which

it is to be false as

and

he

of the

much

ofFrege null

or

the

arise from the intensional view

On

before.

as

logicof relations

is free from

513

restrictions

of relations

is

as

the

on

adopted in

(cf." 83).

chief

which difficulty

arises in the above

theoryof

classes is

as

of

which led me, against entitythat a range is to be. The reason to inclination, of an exteusional view of the necessity adopt was my classes, discoveringsome entitydeterminate for a given propositional function,and the same for any equivalent is Thus function. "x is a man" propositional equivalent(we will suppose)to "# is a featherless biped,"and we wish to discover some one entitywhich is determined in the same way by both these functions. The have I able to discover been propositional only singleentity is the class as one except the derivative class (alsoas one)of propositional functions This functions. equivalentto either of the given propositional latter class is plainlya more complex notion, which will not enable us to with the notion of class, but this more dispense general complex notion (so we agreedin " 73) must be substituted for the class of terms in the symbolictreatment, if there is to be any null-classand if the class whose only is a given term member is to be distinguished from that term. It would to admit, as certainlybe a very great simplification Frege does, a range the terms which is .somethingother than the whole of composed satisfying the propositional reveals tofunction in question; but for my part,inspection me such this also of the account on contradiction, no entity.On ground,and of I feel compelledto adhere to the extensional theory classes, though not quiteas set forth in Chapter vi. That modification in that doctrine is necessary, is proved by 487. some This argument appears the argument of KB. capableof proving p. 444. "

that

class,even

a

only member.

as

In

between

distinction

cannot

one,

the class

as

s argument Frege'

and of

if

whence

as

a

same

is the

thing as only term

of one

one

term

of

be

term

a

This

a.

suppose a to be is identical with that For

collection,the above

If

follows.

to

the class of which

argument as

a

a

one

is

many,

class of

a

it is the

by

met

was

the

but this contention

this reason, it is necessary

to re-examine

than

more

term,

one

only term of the

is a, then to be a term class whose only term is #,

appears to prove not merely but rather that it is wholly inadequate,

argument

of classes is

that .the extensional view

inadmissible.

one

the class whose

is identical with

a

is the

a

is

the

and the class

For

appears the whole doctrine of classes.

now

that

I contended

" 74,

mistaken.

to me

be identified with

and suppose collection, term. Then, if a can

argument proves

that

a

is the

that

be

only term

a

collection

regardedas of

a.

We

relation to the

is to be a the or class-concept escape by sayingthat e if there is such for class the class as entityas or any many, concept of the call which we may terms ", between the class as one, there will be a relation, leads the conclusion above to the Thus argument and their classes as one.

cannot

that either

(a)a

only term

collection whose all in the

case

collection of

of

a

it

more

is,or

collection of many

than

one

there ()S)

identical with the

term

is not

is

collection

no

as

terms, but the collection is

one

term

at

and strictly

One

only many. argument. 488. former this many,

of

it must

but

objections.The grave of nature realize the paradoxical

there

Frege and Peano. To be clearlygrasped that

is the view

view,

views

either of these

To

(a)

the collection

are

only the

it is not

the

that is distinct from

one,

as

of the above

in virtue

be admitted

of these must

other

or

[487

A

Appendix

514

collection

as

collection whose

because it is importantto examine (I speak of collections, of an extensional the possibility the bearing of Frege's argument upon the one which standpoint.)This view, in spiteof its paradox,is certainly essential that is should It we seems to be requiredby the symbolism. quite that there should be a null-class, be able to regarda class as a singleobject, the and at any rate)be identical with that a term should not (in general, It is subjectto these conditions that "class of which it is the only member. the symbolicmeaning of class has to be interpreted. Frege's notion of a

only

it is.

term

be identified with the collection as one, and all will then go well. range may But it is very hard to see any entitysuch as Frege'srange, and the argument that there must be such an entitygivesus little help. Moreover, in virtue

of the

there certainly are cases contradiction, collection as one but no (" 104). Let

many,

whether

this offers

(ft) Let that

a

there

us

is not

a

question.

In

and shows

that

of many

singleterm this view

it has

to its class ; if

term

is that

us

collection

a

examine

and (/3),

as

see

collection of

one

term, and

one

are)those many terms, so that in is the collection of the many at all which terms there is, at first sight at any rate, nothing paradoxical, of

is

(orrather

what the Contradiction admittinguniversally

In this case, unless we will have to be a relation of a term the

dogmas, c

a

terms

the merit

to be sometimes

fundamental

have

we

then

better solution.

a

suppose

collection

where

case.

abandon to

its

one

of

our

class-concept,

the class what appears symbolically as class-concept, will whose is be term the under a only (one might suppose) class-concept which falls only the concept a, which is of course if not always) (ingeneral, We of the contradiction, different from a. shall maintain, on account that there is not always a class-concept for a given propositional function ""#, i.e+ that there is not always,for every "",some such that x*a a class-concept is equivalentto "fccfor all values of x ; and the cases where there is no such will in be which is form. cases class-concept "f" a quadratic So far, all goes well. But now no we longer have one definite entity which is determined equallyby any one of a set of equivalentpropositional i.e. there is,it might be urged,no meaning of class left which is functions, determined by the extension alone. Thus, to take a case where this leads to

not

confusion,if

and

a

else.

Thus

cannot

we

can

find

to u"

to

of

u

such that X"a class-concepts the class-concept under which

be identical with

get is

any

one. a

is the

combinations

that under

same

which

of

denoting what way Or again,if u and v be

different concept from

extensional

some

that the number

problemsas

of #,

the class as

classes,"similar we

b be different

will not

correspondto

a

all values

equivalentfor nothing else

is

a

"

as

that of

as to (i.e.

v.

the number

And

a

#c"

are

falls and

falls b and

nothing symbolically

should

similar but

similar to

we meaning for class,

and

different

v"-,thus,unless

shall not be able to say all the usual

of classes of

elementary kinds specified

The

489]

Logicaland

contained in

] t would

seem

waiting

to

and Frege's maintained; range therefore

necessary

whether

see

there

are

to

acceptranges by

such

by

and

which

form

in my

in the form

or

can

act of

an

faith,without

things. class \vith the

class

one,

as

of

Frege'srange, appears unavoidable, class as many is left as the only object

process of exclusion the play the part of a class.

a

required.

fulfilsthe conditions

Nevertheless,the non-identificationof the whether

515

a

be

must

one

ofFrege

will have become impossible givenclass) and even meaningless. various reasons, an objector the class like might contend,something

For these as

Arithmetical Doctrines

By

logichitherto

modification of the

a

advocated in the present work, we shall, I think,be able at once to satisfy of the Contradiction and to keep in harmony with common the requirements sense*. Let

489.

begin by recapitulating the possible theories

us

themselves. A class may presented the concept,(y) concept of

have

the class ()8) numerical

conjunctionof the

the terms

of the

Of

they

that

from

a

doubt

to

as

that,if ranges

are

but unobjectionable,

class determinate

a

there

being such

an

have intensional, when

its terms

also

entity,and

is not

defect

the

given.

are

they have others.

terms, the contradiction is inevitable,

(")cannot appliesagainst(8);also of

are

have this defect,but

member.

one

(a)the predicate, the class, (S)Frege'srange, (e)the the class, (")the whole composed of

first three,which

render

not

three do not

other

The

do

be identified with

class.

the theories,

these

of

terms

of classes which

(")

from

suffers

the

fact

(c)is

logically singleentity,except when the class has only reason as always exist as a term, for the same a

be identified with

it cannot

the

class

on

account

Frege'sargument t. Nevertheless,without

a

Mathematics

crumbles.

for all values

of the variable

there

should be

some

Two

single object J to represent functions which propositional may

not

objectdetermined

but identical,

be

by both.

Any

extension,

an

equivalent

are

it is necessary that objectthat may be

We define class proposed,however, presupposes may by a optativelyas follows: A class is an objectuniquely determined determined and function, propositional equally propositional by any equivalent Now function. cannot take as this object we (asin other cases of symmetrical functions equivalentto a given the class of propositional transitive relations) Again, unless we alreadyhave the notion of class. propositional function, want be we distinct : considered intensionally, equivalentrelations, may therefore to find some one objectdetermined equallyby any one of a set of equivalentrelations. But the onlyobjectsthat suggest themselves are the class of relations or the class of couplesforming their common range ; and of the notion these both presuppose without class,elementary class. And of formed be w combinations such "how objectsn at can problems, as many time?" /become meaningless.Moreover, it appears immediately evident a the same have that there is some in saying that two class-concepts sense the

*

in

"

The

doctrine to be advocated

in what

notion

of

class.

follows is the direct denial of the

dogma

70, note.

t Archiv

" For the

i.

p. 444. of the word

use

objectin

the

" 58, see followingdiscussion,

note.

stated

Appendix

516

[489

A

objectwhich can be called the extension of a class-concept.But it is exceedinglydifficult to discover any such object, and the contradiction proves conclusively that,even functions there for if there be such an are propositional sometimes, object extension,and

which

this

requiresthat

extension

the

is not

should

there

be

some

term,

one

numbered we (")in the above enumeration,is many, We choose,represent but is many and not one. unobjectionable, may, if we of the u's." is one This "# will mean this by a singlesymbol: thus actu class

The

which

xtu

as

relation

a

conjunctionis

numerical for

as

be taken

not

must

which

be

would

not

terms, singleterm, and

a

the

of two

if for

same

u

and

x

we

u, because

substituted

we

u

the

as

wish to have

a meaning equal class vt

an

u intensionally.Thus we may regard interpreting of x to many relation terms, among one as expressinga to this view, if onlysingleterms which x is included. The main objection can if for is a symbol standing essentially many be subjects, is that, u terms, we We risk without of make cannot can error. a no u longer logicalsubject speak, one might suppose, of a class of classes; for what should be the class are of such terms*. terms not a singleterms, but are each many We would assert cannot one a predicateof many, suppose, except in is required of assertingit of each but what the sense of the many; of a predicateconcerning the many here is the assertion not as many, whole which all (ifany) concerningeach nor yet concerningthe compose. constituents its will each be Thus class of classes will be many a many's ; and therefore in any cannot one might suppose, be only many, sense, I forced constituents. Now find to maintain, in spiteof the myself single that this is precisely what is required for the apparent logicaldifficulty,

which "x

prevents

requiresthat

arises difficulty

should

can

be

we

not

should

we

yet it has

sense

have

we

terms, it is necessary

two

and should two-fold, and

If

of number.

assertion has

from

us

of the M'S"

is

the as

form

of two

one.

combine of

to the number

speak

class of classes,each

that the

be each not

a

a

members

Or

should

again,"Brown

Brown

and

Jones

members

of whose be

each

genuinely

and Jones into

a

are

singlewhole,

subject-predicate proposition.But of members

couples1

This

of seems

class of classes.

a

to

two"

now

In

a

what

require that each couple have two units,not two

singleentity; yet if it were, we should couples. requirea sense for diversityof collections, meaning thereby, if u and v are the collections in question, that xcu and xtv are apparently, for all of values not x. equivalent The 490. logicaldoctrine which is thus forced upon us is this: The subjectof a proposition may be not a singleterm, but essentially terms; many this is the case with all propositions and 1. numbers other than 0 asserting But the predicates or in propositions can or relations which class-concepts occur different those that are f rom some having pluralsubjects (with exceptions) in propositions can occur having singleterms as subjects.Although a class is many and not and diversity and one," yet there is identity classes, among thus classes can be counted as though each were a genuineunity; and in this class of of and we thfe classes which are members of a one sense can speak a

We

* Wherever the context requiresit,the reader is to add "provided the class in question do not consist of a singleterm." (or all the classes in question)

4=91 ]

The

class of

classes.

asserted of

which

One

of

which

one

LogicalDoctrines of Frege

what

it is when

is

in applicable

asserted of

a

speakingof

517

different when

be held,however, to be somewhat

must

class from

a

meaning

a

Arithmetical and

term; that is,there is term, and

one

another

is

applicablein speaking of one class,but there is also a general which The fundamental doctrine upon meaning applicableto both cases. all rests is the doctrine that the subject and of a proposition be plural, may that such pluralsubjects what is meant than are by classes which have more term*.

one

It will of

be necessary

distinguish (1) terms, (2)classes,(3) classes shall have to hold that no member of so on infinitum; we member be that should of any other set, and that x*u x requires degreelower by one than the set to which u belongs. Thus xcx the contradiction is a meaningless proposition ; and in this way

now

and classes,

one

of

a

set

is

set

of

a a

will become

to

ad

avoided. 491.

But

problem of classes which have one member The case of the null-classmight be met by a bare denial or none. this is only inconvenient, But in the case of classes not self-contradictory. them from their sole having only one term, it is stillnecessary to distinguish members. This results from Frege'sargument, which we may repeat as than one follows. Let u be a class having more term ; let tu be the class of classes whose Then has one iu member, u has many ; only member is u. must

we

consider the

now

"

hence

and

u

this argument

of a

a

is,so

doubted, at

identical.

to many

of many

terms

The

It may be relation of x to

terms

u

; the relation

(assubject)to

the argument breaks down. of u and that u is a member

spiteof

many

terms

t(as predicate) the

from

It is in different

of

iu

sight,whether

; thus

u

and

previousone

senses

iu

This

may

that

is

x

; "

be identical

the

argument. attempt, however,

This

first

expressedby X"U is a relation of u to iu expressedby utm, is

objectormight contend,a different relation

an

thus

member in

not

is valid.

singleterm

relation

and

are

at,

escape from Frege'sargument, is all the purposes of Arithmetic,to begin with, and to

capableof

for many is have i t for to which is a " meaning logic, necessary purposes of a class to a class of to the relation of a term to a class, equally applicable refutation.

For

of the

of

classes,and

so

on.

But

the chief

pointis that,if every singleterm

is

a

class,

be admissible. must propositionx*x, which givesrise to the Contradiction, that in the u must and and It is only by distinguishing xeu x insisting tx, always be of a type higherby one than a;, that the contradiction can be avoided; Thus, although we identifythe class with the numerical may conjunction of its terms, wherever there are many terms, yet where there is object distinct range as an only one term we shall have to accept iTrege's done And also admit a of from its only term. course this,we may having We function. shall differ from of a null prepositional in the case range in but in no a an case term, regardinga range as objectof a Frege only that a prepositional in function ""(#), different logicaltype, in the sense if be any term, is in generalmeaningless for x we substitute a which x may

the

*

Cf.

t

The

"" 128, 132 supra. word

predicateis here

used

loosely,not

in the

precisesense

defined in " 48.

Appendix A

518 range; and if if for x

will in generalbe meaningless range of terms, ""(x) a or substitute either a term range of ranges of terms.

x

may

we

are Banges,finally,

cardinal

that

[491

be any

what

properlyto

are

numbers

and be called classes,

it is of them

asserted.

are

According to the view here advocated, it will be necessary, with is terms, classes, its field of significance to indicate whether every variable, will be variable not A able,except in special classes of classes, on*. so or 492.

of these sets into another ; and cases, to extend from one the u must always belongto different types ; c will not be

objectsof the

type, but

same

"e

tR* t will

or

a

be, providedR

in xw, the x and relation between is

We

so.

shall

accordingto the types to which domains and their domains belong; also variables whose fields converse classes of couples, will not as a as these being understood include relations, relations will be different rule include anything else,and relations between This seems to givethe truth in type from relations between terms. though in a thoroughlyextensional form underlyingFrege'sdistinction between Moreover the opinion here kinds of functions. and the various terms indeed to common advocated seems to adhere sense. very closely of correct classes is even Thus the final conclusion theory is,that the the class that is the than that of as extensional more Chapter vi; many function,and that this is only objectalways defined by a propositional class the that for formal as composed one, or the whole adequate purposes; of the terms of .the class,is probablya genuineentityexcept where the class is defined by a quadratic function (see" 103),but that in these cases, and in is the onlyobjectuniquelydefined. the class as many other cases possibly, The theorythat there are different kinds of variables demands a reform in the doctrine of formal implication.In a formal implication, the variable does not, in general, take all the values of which variables are susceptible, but all function in those that make the propositional only questiona proposition. For other values of the variable,it must be held that any givenpropositional function becomes meaningless. Thus in #eM, u must be a class,or a class of

have

to

also distinguish

relations

among

"

"

or classes, etc.,and and so on; classes,

variables

require

a

remains

which

certain

be

work.

this

a

It remains

to

deal

class,a

class if

u

is

have

*

class of

asserted.

are

the

in the

end of the

more

philosophical part of Frege's

with his Symbolic Logic and Arithmetic; briefly completeagreement with him that it is hardly which, acknowledgehis discoveryof propositions been

new.

Implicationand I

a

myselfin such

employs as fundamental what

a

This fact will given case. of Symbolic Logic; but it principles all propositionsbelongingto a implication,

than necessary to do more when I wrote, I believed to have

as

is

u

of the

formal

to

come

we

but here I find

493.

if

admissible

not

are

function given prepositional With

term

a

modification

that,in

true

must

in every propositional function there will be some range the variable,but in general there will be possiblevalues for

to permissible

other

x

See

called

Symbolic Logic. The relation which Frege in the logicof propositions is not exactlythe same implication:it is a relation which holds between

Appendix B.

f On

this

notation,see

"" 28, 97.

494] p I

and

The

519

the relation which q is true or p is not true, whereas holds whenever p and q are propositions, and q is true or p is false.

is to

whatever

Frege'srelation holds

say,

be;

q may

definition has

His

LogicalDoctrines of Frege

and

q whenever

employ

That

Arithmetical

mine

the

does not

when

hold

is not

p

unless p

and

propositionat all, propositions. q are

a

advantagethat it avoids the necessityfor and but it has the propositions"; q are

formal

hypotheses of the form "p disadvantagethat it does not lead to a definition of propositionand of negation. In fact,negationis taken by Frege as indefinable;propositionis introduced by means Whatever of the indefinable notion of a truth-value. be, " the truth-value of x " is to indicate the true if x is true, and the may false in all other cases, Frege'snotation has certain advantagesover Peano's,

x

He spiteof the fact that it is exceedinglycumbrous and difficult to use. defines Peano's all of for values the whereas invariably variable, expressions definitions are often precededby a hypothesis. He has a special symbol for a nd he is values of able for function to all assert a prepositional assertion, x Peano's which not statingan implication, symbolism will not do. He also the of and between German Latin letters respectively, use distinguishes, by of a certain propositional function and all such propositions. any proposition By always using implications, Frege avoids the logicalproduct of two and therefore has axioms to Importationand no propositions, corresponding Exportation*. Thus the jointassertion of p and q is the denial of "p implies not^." 494. Arithmetic. definition of cardinal Frege gives exactlythe same numbers I have given,at least if we as identifyhis range with my class f. But followinghis intensional theory of classes,he regardsthe number as a If u be a range, not of the class in extension. property of the class-concept, in

the

number

of

Grundlayen

In the range similar to u." der Arithmetik,other possible discusssed theories of number are range

often

very

concept

"

that

In is

emendation it

can

cannot

be

asserted

of

difficultto discover the difference between

demonstration. some

Numbers

of the

objects,because the same have different numbers set of objectsmay assignedto them (Gl.p. 29); for of example, one army is so many regiments and such another number soldiers. This view seems to me : to involve too physicala view of objects I do not consider the army to be the same object as the regiments. A stronger argument for the same view is that 0 will not apply to objects, but only to concepts (p.59). This argument is,I think, conclusive up to a certain point; but it is satisfied by the viexv of the symbolic meaning of Numbers classes set forth in " 73, like other ranges, are things themselves, as general (p. 67). For definingnumbers ranges, Frege gives the same abstraction I of have what call the 1 as J. ground given,namely principle of In the Grundgesetzeder Arithmetic,various theorems in the foundations cardinal Arithmetic are proved with great elaboration,so great that it is and

dismissed.

is the

u

do

more

view

of the contradiction

successive

of Chapter x, it is

in Frege'sprinciples; but required than

introduce

some

it is hard

generallimitation

which

steps in to

believe

leaves the

details unaffected.

*

See " 18, (7),(8). t See GL

pp.

79, 85; Gg. p. 57, Df. Z.

a

plainthat

" Gl. p. 79; cf. $ 111

Appendix

520 495.

In addition

to his work

[495

A

cardinal numbers,

on

Frege has, already

rather or a JBegri/sschrift, theory of progressions, very admirable does relations. not be Frege can generatedby many-one in onlyone direction, relations: as long as we move confine himself to one-one relation also will generatea series. In some a many-one parts of his theory, for any he even deals with generalrelations. He begins by considering, relation /(a?, y) holds,then F (x) y\ functions F which are such that,if /(as, the property F is If that this condition holds,Frege says impliesF(y). inherited in the /-series (Bs,pp. 55 58). From this he goes on to define, of the without to " some numbers, a relation which is equivalent use positive in follows. defined The relation of This is the relation." as given power

in

the

of all series that

"

and y if every property F9 which is inherited in the that /(#, z)impliesF(z) for all values of ", belongsto y

question holds between

x

and is such /"series a non-numerical (Bs.p. 60). On this basis, theoryof series is veiry successfully and is in the erected, concerning the proof of propositions applied Gg. to of Unite numbers and kindred topics. This is,so far as I know, the number best method such and Frege'sdefinition just quoted of treating questions, But induction. as no gives,apparently,the best form of mathematical controversy is involved, I shall not pursue this subjectany further. admirable criticism of the psychological Frege'sworks contain much and also of the formalist theoryof mathematics, which standpointin logic, believes that the actual symbols are the subject-matter dealt with, and that their propertiescan be arbitrarily assigned by definition. In both these I find in points, myself complete agreement with him. and 496. has criticized Frege very severely, professes Kerry (loc. cit.) have to proved that a purely logicaltheory of Arithmetic is impossible (p.304). On the questionwhether concepts can be made logicalsubjects, I find myself in agreement with his criticisms ; on other points, they seem to rest on would these such As as mere are misunderstandings. naturally to any one unfamiliar with symboliclogic, I shall briefly discuss them. occur The definition of numbers classes is,Kerry asserts, a vvrtpov Trporcpov. as We know that every concept has only one must extension,and we must what know one objectis; Frege'snumbers, in fact,are merely convenient be symbols for what are commonly called numbers (p. 277). It must admitted, I think,that the notion of a term is indefinable (cf. "132 supra), and is presupposedin the definition of the number 1. But Frege argues and his argument at least deserves discussion that one is not a predicate, to but has less generalmeaning, and a attaching every imaginableterm, attaches to concepts (Gl.p. 40). Thus a term is not to be analyzedinto one and term, and does not presuppose the notion of one (cf " 72 mpra). As to the assumptionthat every concept has only one extension, it is not necessary able be to to state this in language which 1 : all we employs the number need is,that if "j"xand \fa are all for values of x, equivalentpropositions then they have the same extension a primitivepropositionwhose symbolic in the no number 1. From this it follows that expression way presupposes if a and b are both extensions of "f"x, and b are identical, which again does a not involve the number 1. In like manner, formally other objections to Frege'sdefinition can be met. "

"

.

"

496]

Arithmetical

The

LogicalDoctrines of Frege

and

521

Kerry is misled by a certain passage (GL p. 80, note)into the belief that The Frege identifies a concept with its extension. passage in question of u might be defined as the concept appears to assert that the number "similar

to

that

two

the

There reveals

u"

and

not

definitions

is

fundamental

errors

as

that

0 is

existential

the

to

it does not

of this concept; but

range

equivalent. of Frege'sproof

are

criticism

long

a

the

as

say

number, which

a

universal

of

import

The pointis to prove that,if u and t? are null-classes, propositions. they*are E similar. relation is that there to mean one-one a Frege defines similarity such that "x is aw" implies there is a v to which x stands in the relation jft" and vice versa. into conformitywith my (I have altered the expressions usual language.) This,he says, is equivalent relation there is a one-one to "

"

E

such

that 'x is '

relation E

u1 and

a

'there

is

of

terra

no

to which

v

value

be true, whatever

both

cannot

may

x

x

stands

have, and

vice

in the

";

versa

is true if "a; is a u" and are always false. proposition "y is a v This strikes Kerry as absurd (pp.287 of 9). Similarity classes,he thinks, is affirms that Frege'sassertion above He impliesthat they have terms. is contradicted by a later one is a a v, u, and nothing (GL p. 89): "If a and

this

"

"

then

'

is

a

for

true

'

a

all values

contradiction all so

and

u

; but

all

"

"

is b

a

is

term

no

of he

and propositions that

'

#."

a

do

I

which

v

universal

that

and

"

no

is b

a

where

quite know

not

evidentlydoes

has the relation

realize that false

not

have propositions

no

will both be true

if

"

a

E

'

to

both

are

a

the Kerry propositions imply finds

existential import, is the null-class.

of Frege's notion of relation. Kerry objects(p.290, note)to the generality affirms a relation and b that Frege asserts any propositioncontaining a between concludes that it is and b (GL p. 83); hence a Kerry (rightly) So related. b are to deny that a and self-contradictory generala notion, he

neither

have

can

says,

should both be constituents sense;

for

as

nor

sense

of

the

purpose, be adduced

first sightto be function not

"

E

a

formal

and

8

hold between that

suppose

E

it is

T for both E not

have

This is

might

the a

functions

equivalentto

has

E

"

are

is

this instance,on

the

equivalent there

with

containingtwo

is

Contradiction: one

variable of

variables

a

are

But the refutation

are

not

there

was

not

fixed

he is

shown

equivalentto

class,while same

and

in the

Then

as

the case

T to T."

Frege terms

as

;

things,will closely analogousto that holds to be

prepositional function propositional

that

some

assertion

that

some

fixed

of any

of relations

There is a hierarchyof relations accordingto previouscase. terms their fields. Thus relations between objectsconstituting

the

us

T does

"

T.

relations

let

tuting substi-

T, that

relation

does

E

S;

any here it is shown

equivalentto

is the

such

no

at

prepositional

has the relation

groundthat it treats

the

it

T

"

to

ranges, which,like singleranges, result. The point involved same

in the

b

seems

relation

the relation T to S"

find,since T is identical with

and S, we T to T"

assertingmembership relation.

and the identical, E contains two variables,

This

""

Consider the

Frege'sview.

relations which

and

is,however, what

There

answer.

disproofof

contradiction, showing that

but his double

involved

in

relation

objectto

bring out

are

and

a

a seems intelligible perfectly proposition whole of the indeed of relations, logic

one

whole

mathematics, may

that

for sense,

As

purpose.

as

it

was

in

the type of are

distinct

[496

Appendix A

522 from

between

classes,and

between

those

relations.

Thus

no

these

relation

can

again are

distinct from

have itself both

as

relations

referent and

as

relatum,for if it be of the same order as the one, it must be of a higherorder function is therefore meaningless than the other; the proposed propositional for all values of the variables R

and "

It is affirmed (p. 29-1) that only the concepts of 0 and 1, not the objects themselves,are defined by Frege. But if we allow that the range of a of a concept this cannot be maintained;for the assigning is an object, Begriff will carry

it the

with

assigningof

its range.

Kerry

he proved (ib.):

uniqueness of 1 has been there might be several 1's. definition, the

I do not

and is precise supposed:the proofof uniqueness definition of immediate

The

severelycriticized (p.292

is also

of series set inherited

in

inherited."

the

perceivethat

that, with

understand

Frege's

this

how

can

be

formal.

sequence in the series of natural numbers ff.).This depends upon the generaltheory

Kerry objectsthat Frege has defined "F is "the /-series" but has not denned "/"'is nor /-series," to be not no latter essentially denned,having ought precise

forth

The

does not

thinks

in

Bs.

if necessary, as the field of the relation / easilydefined, This objection is therefore trivial. Again,there is an attack on the definition : inherited in the /series follows in x the/seriesif y has all the properties y and belongingto all terms to which x has the relation /*.'* This criterion, of such properties we are told,is of doubtful value, because no catalogue himself is itself one and further x because,as Frege exists, proves, following of these properties, whence vicious circle. This argument, to my a mind, In deduction, of deduction. misconceives the nature a proposition radically of a class, and is proved to hold concerning then be every member may asserted of a particular member: but the proposition concerningevery does of the entries in a catalogue. not necessarilyresult from enumeration Mill's of involves to Barbara,that the objection Kerry'sposition acceptance mortalityof Socrates is a necessary premiss for the mortalityof all men. The fact is,of course, that generalpropositions often be established can where no means exist of cataloguing the terms of the class for which they hold; and even, as we have abundantly seen, general propositionsfully stated hold of all terms, or, as in the above case, of all functions, of which be conceived. no can i s catalogue Kerry's argument, therefore, answered by a correct theory of deduction;and the logicaltheoryof Arithmetic is vindicated againstits critics. the former

sense;

is

"

Note. in

The

second

volume

of

which appeared too late ""?#.,

to

be noticed

the

discussion of the contradiction Appendix, contains an interesting 265),suggestingthat the solution is to be found by denyingthat (pp.253 functions which determine equalclasses must two propositional' be equivalent. "

it

As

seems

recommended

very to

*

likelythat this examine

is the true

the reader solution, Frege'sargument on the point.

is

strongly

Kerry omits the last clause, wrongly ; for not all properties inherited in the /-series belong to all its terms ; for example,the property of beinggreaterthan 100 is inherited in the number-series.

APPENDIX

DOCTRINE

THE

a

doctrine

THE

497.

possiblesolution transformed

be

to

In

well

Every its

to

of

lie if

the

first

truth,

is to "f"(x)

point

in

there the

to

is

of

range

numbers

A

second

will,

term

lowest

in

without

I

of

a

of

perhaps of

are

become

If

of

Thus

of the

same

their

occur

type

extensions

range

x,

or

follows

of

is

its

belong

and

several

and

first,

which

object-say

the

the

whole of

case

and

importance

we

members

others

are

individuals

are

classes

are

classes

simple individuals.

as

are

Chapter

whether

that

of

a

in

occur

given,but

the

Individuals

classes

of

couples.)

cannot

be

asserted. significantly

relations

different

extensional are

of

in

the

extension,

(The

relations

vi, :

space-

m

of

objects

class

the

the

(A

points,

objects

fore, there-

that

seem

of

are

are

are

what

relations when

Symbolic Logic which

all this

propositions which

of

person

material

determinate

of

as

objects

mtensional

not

the

is

substituted

concepts,

or

relational

type.

only

would

It

things actual

be

one.

as

qualities.)These

secondary

to

point

always

may

called, in

This

range.

a

certain

a

individual

its

not

is

ordinary relational propositions are are

varied;

sum

a

what

things, though

as

x

This

also

must

be

"f"may

but

in

false.

or

which

all of

precise

other

relations

e.g. the

in

of

which

of point is that ranges of "f"(x), of significance

the

existents, the

type

same

true

than

an

provided

Symbolic Logic employs, which

the

to

single type

a

objects designated by single words, type

;

object

such

reference

the

as

addition

in

within

.range

a

second

tables, chairs, apples, etc.,

some

outlines,

its main

is contended^-has,

the

type

significance. What

psychical

the

plainer.

is any

object.

individual,

an

class

with

difficulties ;

hope,

daily life,persons, is

point

proposition,any

a

loss

is

types

is less

a

all, whether

at

belongs

x

culties. diffi-

first step towards

solve.

significance,i.e.

objects,the

individual

or

type

occurs

of

all

answer

can

forth

set

to

however significance of "f"(x),

introduces

meaning

one,

a

of

of

it be

to

^"(aj)"soit

significanceis always, either

The

types.

class

found

be

proposition

a

theory

types, i.e. if

of

range

be

shape

it fails to

range

a

the

significanceform then

which

it

probability,

all

requires,in

before

Appendix

prepositionalfunction

range

must

this

in

problems

some

as

should

it

but

affording

tentatively, as

forward

put ;

subtler

however,

endeavour

here

contradiction

some

TYPES.

OF

is

types

the

into

case,

shall

I

truth,

of

of

B.

are

numbers

[497

Appendix B

524

consists of ranges or classes of individuals. " Brown to be associated with the word range.) Thus

The ideas

next

are

(No

type

ordinal

and Jones

"

sition propogeneralnot yielda significant of false if substituted for Brown in any true or proposition which Brown is a constituent. (Thisconstitutes, in a kind of way, a justification for the grammatical distinction of singular and plural ; but the analogy is is

objectof

an

this

will in

type,and

"

"

not

since close,

a

and where it has many, If u be a range in certain propositions.) singular

range

have

may

term

one

it may yet appear as determined by a propositional function

more,

or

for will consist of allobjects

not-w ""(#),

in the range of

significance of u. and contains only objects of the same ""(#), type as the members that fact the There is a difficulty two f rom in this connection,arising sitional propotruth while their of functions "f"(x), the have same u, range \p(x) may becomes of significance be different;thus not-w ambiguous. ranges may and notThere will alwaysbe a minimum which is within u contained, type u types is a may be defined as the rest of this type. (Thesum of two or more which

is false, so tf)(x)

is contained

that not-w

of

type

minimum

a

j

this Contradiction, '

is

type view

seems

falsehood of "as is

a

u," and

consequently

a

u

"

"

is

x

which

one

such

is not

view

of the

the best ; for notrw must be the range of x" must be in general "x is an meaningless ;

requirethat

must

In

sum.)

a

and

x

should be of ditferent

u

types. It is doubtful whether this result can be insured exceptby confining in this connection, ourselves, to minimum types. There is minimum

unavoidable

an

denyingthat

mixed

a

conflict with

class

type)

can

one (i.e.

be

ever

common

members

whose

of the

for example,such phrases Consider, Heine and as to be a class consisting of two individuals, the

as

"the

as

French

of the French one

as

than

more

of which

one

pointto

which

that,if be of

the as nation/'i.e.,

many,

there

is

a

different type both classes of individuals.

individuals.

from

classes

to

be understood

must we

are

of two

the other is the French

classes of individuals and

type after classesof Such

are, for

such associations, the convenient

one.

as

is a

many,

later ; for the present it is enough to remark it must, if the Contradiction is to be avoided, class,

a

next

If

If this is

speaking members,but of it is possible to form a class

as

Whether

Heine,while

}J

I shall return

there be such

The

French

not consisting

Frenchmen.

are

member

class

get a

we

class

same

of its members.

one

the French."

"

"

for necessity

all of the

not

are

type

same

in the

sense

of classes

classes of

individuals consists of classes of classes of

example,associations

clubs,are

speak of

from

of clubs ; the members of themselves classes of individuals. It will be

classesonlywhere

only where

we

have

classes of

of individuals,

have classes of classes of individuals, and For the generalnotion,I shall use the word rcwige. There is a

on.

we

so

gression pro-

of such

types,since

type, and the result is A

new

series of

a

of any range may be formed of objects of highertype than its members. range a

types beginswith

such types is what SymbolicLogictreats view of relations. We then form may

the as

couplewith a

relation of

:

given

range of this is the extensional sense.

A

relations of relations, or ranges or relationsof couples relations, in Protective (suchas separation Geometry*), *

Cf.

" 203.

498]

The

Doctrine

of Types

relations of individuals to

or

not

merely a

We

have

couples,and so but a whole singleprogression,

525

and

we in this way get, infinite series of progressions.

on

;

of triple also the types formed of trios,which the members are relations taken in extension several kinds as ranges ; but of trios there are that are reducible to previoustypes. Thus if "f"(x, y, z) be a prepositional function,it may be a product of propositions a or product fa(x) ""2(y) ""3{s) -

.

"f"i(x) ""2(#, s),or

propositionabout x and the couple (y,z),or it may be in other In such cases, a new analyzable analogousways. type does not arise. But if our proposition is not so analyzable and there seems no a priori reason it should be obtain a new then we why so always type, namely the trio. We form ranges of trios,couplesof trios, trios of trios, can couplesof a trio and an individual,and so on. All these yieldnew types. Thus we obtain immense an of and it is there difficult be how to hierarchy types, sure many be; but the method of obtainingnew may types suggests that the total number is only a0 (the number of finite integers), since the series obtained less resembles the series of rationals in the order 1, 2, or more 1/2, n, 1/3, 2/3, l/n, This, however, is only a 2/5, ...2/(2w+1), conjecture. a

-

"

"

...,

...,

...,

Each a

of the

types above enumerated

function prepositional of the above

one

...,

...

...,

which

is

minimum

a

for significant

is

one

type value

if "f"(x) be i.e.,

;

of

belongingto

x

every value of x belonging that of this I am doubtful

is significant for types,then ""(#)

the said type. But the sum of any number to

it would

seem

of minimum

"

though

types is

a

"

type, i.e. is

for certain

of

significance range this is universally

a

functions. Whether not or prepositional since f orm a certainly type, ; and every range has a number do all objects, since every objectis identical with itself. Outside the above series of types lies the type proposition ; and from this a new one hierarchy, might suppose, could be started ; but starting-point

true, all ranges so

as

there

certain

are

doubtful

whether

498.

difficultiesin the can propositions

Numbers,- also,are a type lying outside the certain difficulties, owing to the fact that

objectsout of every of the given number

certain have 0

; for every

erroneous

be

member

a

of such a view, which render way be treated like other objects.

whose

only

This renders the obvious will have

of range as

member

a

sometimes

have

and significant,

numbers, ranges in these

cases

ranges

which

definition of

which null-range,

own

that

so range is tiienull-range.Also

of types and consideration totality there may be difficulties. all ranges

its

of ranges,

of the

Since

every

senting pre-

selects

number

type of ranges, namely those ranges

members.

type

of 0 considered

0 is the range

other

and series,

above

; and

cannot

we

it

numbers

will

say that

requirea

in this consideration

is x"c range ; consequently sequently Conits denial is also significant. are

a

is false : thus the of ranges for which xtx range w of Contradiction proves that this range w does not belongto the range when be x observe that xex can We of xtx. significant only may significance is

there

is of

by

a

one

a

type of infinite order,since,in than

x

; but

the range

xcu,

u

of all ranges

must

is of

always be course

of

type higher type of infinite

of a

a

order. Since numbers

are

a

type, the

function prepositional

llx

is not

a

u,"

[498

Appendix B

526 where

is

u

a

"

of numbers, must mean this somewhat escape

range

unless,indeed, to

is

x

a

number

which

is not

u" ;

a

result,we say that, paradoxical certain propositions, they are not

a are although numbers type in regardto "J " is a x or is contained in v in such u as a regardto propositions type of which Such a view is perfectly tenable,though it leads to complications "

is hard

u" it

the end.

to see

that if it be a fact are a type results from ,the fact" propositions false. Certainly be said to be true or can significantly only propositions alone are asserted (cf true propositions appear to form a type, since they is number of But the if so, propositions as great as Appendix A. " 479). and that of all objectsabsolutely, since every objectis identical with itself, That

"

.

a?" has

is identical with

"x

however,

difficulties. First,what

two

be

to

appears

there should be individual ; consequently it seems individuals. Secondly, if it is possible, as

of

there propositions, such although propositions, ranges

ranges

These

There

first

The

499.

are,

illustrated

individuals.

Consequentlynot

result,which

is also deducible

all classes have

to

be,to

there

are

simpler ones.

somewhat

are predicates

definingpredicates.This

shows the Contradiction,

from

more

no

full discussion.

a

by

are,

objects(cf. " 343).

among

individuals ; but

classes than

more

ranges than

such

more

onlysome

are

be

point may

know,

we

be

must

very serious,and demand

difficultiesare

two

this there

propositional concept

called the

we

In

x.

always an

than propositions form

relation to

one-one

a

how

necessary

and to adhere to the extensional classes from predicates, distinguish there view of classes. more are Similarly ranges of couplesthan there are than but verbs, which there are individuals; couples,and therefore more relations individuals. are intensionally, Consequently not every express of couplesforms the extension of some verb, although every such range the extension of forms function containingtwo some propositional range variables. Although,therefore, verbs are essential in the logical genesisof such propositional the is intensional functions, standpoint inadequateto give all the objects which SymbolicLogicregardsas relations. of propositions, In the case it seems as though there were always an it is to

associated "

with

x

of

and

x

verbal

and

"

"

y

the

; and

which

noun

is

individual.

an

of x," " self-identity so

on.

The

x

We

differsfrom

verbal noun,

have "

y

which

and

is identical

"a "

is what

the difference

called the

we

to be an individual concept,appears on inspection propositional ; for "the self-identity of x" has as many impossible, values as

and objects, from

therefore

more

the fact that there

values than

there

but this is there

are

individuals. This results

are

propositions concerningevery conceivable object, shows ("26) that every objectconcerningwhich identity there are propositions, is identical with itself. The only method of evading this difficulty is to deny that propositional concepts are individuals;and

and

this

to

seems

however, that we

are

the definition of

shall have

members

are

be a

the

which

we

driven.

are

It

is

undeniable,

propositional concept and a colour are two objects ; hence to admit that it is possibleto form mixed ranges, whose

not

all of the

different type from what members of one type. The a

to

course

type

same we

may

but

such

,

call pure

ranges will be

ranges,

propositional concept

i.e. such

seems, in

as

to fact,

always of have

be

only

nothing

The

500] other

than

that

one

the

of Types

527

the difference beingmerely the psychological itself, proposition

do not

we

Doctrine

the

assert

propositionin

the

one

and

case,

do assert

it in

the other. The

500. that

there

second

point presents greater difficulties.

ranges

of

are

product of such ranges; than propositions.At solved by the fact that

for propositions,

yet

that

admit

cannot

we

often wish

we

We

there

deny logical

cannot

to assert are

the

more

ranges

first

might be thought to be sight,the difficulty there is a proposition associated with every range of propositionswhich is not null,namely the logical product of the propositions of the range*; but this does not destroy Cantor's proof that a range has Let more us sub-rangesthan members. apply the proof by assuming a w hich associates one-one relation, particular p which is not every proposition is p, while it associates a logicalproduct with the range whose only member the product of all propositions with the null-range of propositions, and with associates every other logicalproduct of propositions the range of its factors. Then the range w which, by the general principle of Cantor's own proof, is not correlated with any proposition,is the range of propositions which factors of themselves. are logicalproducts,but are not themselves But, by the definition of the correlating relation,w ought to be correlated It of will with the logical be found that the old contradiction w. product breaks out afresh ; for we can prove that the logicalproduct of w both is and is not

member

a

but the doctrine to

follow

but

that

what Let

w.

of

types does

this

state

us

I

are,

not

at

am

that there

to show

seems

Contradiction

the

these

This

of

a

why there is no such requires further subtleties loss to imagine.

fully.

more

from

there is "

w,

Consider and

a

one-one

" is not every n is true class of the whole now

having

property of

the

this class be w, and it must possess the

not

let p be the

of

as

might

productsis

be

one-one

n

be

m

doubted or

an

be different

"

is

a

Thus

w.

the contradiction

is desirable to

a

class

of

propositions,their logicalproduct

is true," which

whether

class

If we by **m. propositionscomposed

denote

of

the

the

relation of ranges

For

many-one.

I shall

example,

of

does

the

of

the

reopen and of

functions identityof equivalentprepositional of These two propositions. questions logicalproduct

If

of

"

contradiction,it

this

proposition"every consider the logicalproduct It

class

of "?; but this property demands that hand, if p be not a w, then p does possess

m

*

a

be itself

not

if

solution;

"

same

the

the

follows.

be

w,

seems

definingproperty

appears unavoidable. In order to deal with

questionof

for its

as

as proposition every m is true." of the form propositions every m is true," members of their Let being respectivem's. proposition every w is true." If p is a w,

the

On the other p should not be a w. therefore the definingproperty of w" and

nature

range It range.

m

or may every m is true may relation of this proposition to m:

the proposition propositions, But

If

"

"

m.

such

no

show

contradiction

new

is

the arise

is the now

of

m

propoeitionsto their logical

logicalproduct of p and

definition of the

q and

r

logicalproduct (p.21) will set this doubt at rest; for the two logical products in question,though equivalent,are relation of all ranges of propositions identical. Consequently there is a one-one by no means theorem. is directlycontradictoryto Cantor's which to some propositions, differ from

that

of pq

and

r?

A

reference

to the

together

with

is

true/

of

the

i.e.

to

is

the

of

the

the

above

it But

that

is

x

all

the

which

if

identical

with

there

close

analogous The

seem,

a

difficulty very

attention

may

of

all

the

the

proof

form

is

^tn

of

class

a

of

and

is

of

question.

hold

to

logical

in

solution,

same

course,

this

not

Thus

discussed

one

the

But

this

in

the

that

product

equivalent.

have

must

these

proposition

and

;

are

possible,

Yet

logical

true

will

identical

is

1

"Every

contradiction

factors.

as

"

"

is

"

2

The is

with

types,

type

than

two

the

is

Who

sayings

true,

self-evident

quite

identical.

*'m

the

two

It various

one

of

have

reasoning,

students

of

that

probably

is

objects,

not

special but

types,

which

logical

I

the

that

logical be,

all

m

contradiction

of

of

fundamental

foundations

class,

equivalent

the

is

it

other

and

is

m

avoiding

that

appears

doctrine

of

of

although

this

only

for not

credited.

be

to

every

solutions.

contradiction

totality

product

that

products

must

suggestion

seems

artificial it

the

is

of

are

of

up:

by

m

new

m),

and

foolish

or

composed that

suggests

highly

sum

epitaph

true,"

of

similar

propositions

To

of

member

a

prime

even

an

deeds

method

themselves

and

is

often

are

wise

class

is

analogy

very

propositions

solved

asserts

m

strongly

x

harsh

or

every

least

have

m

the

simple

no

Chapter at

an

of

"

seems

The

or

is

x

well-known

a

is

"

II.'s

Charles

of

propositions

either

of

proposition it

the

identify

we

function

impracticable,

functions

that

example,

equivalent,

are

m.

which

reality,

is, in

escape

one

if

every

logical of

member

a

Thus

every

of

class

a

propositional

for is

"

of

to

and

true

the

member.

a

an

equivalent

maintain,

of

not

such

with

both

A*"T.

prepositional

a

since

fails,

m

Thus

to

of

product

being

(*lm

product

i.e.

is

"Every

equivalent

is

logical

contradiction

logical

which

the

as

functions

to

true"

is

m

propositions

of

same

propositional

equivalent

is

"every

class

new

which

this

^"w,

[500

B

Appendix

528

or

of

difficulty. succeeded I

logic.

earnestly

all What in

contradiction there

of

is

soluble

not

the

commend

involves,

complete

as

study

it of

is

doctrine. it

solution

but

the

x

closely

one

this

by

propositions,

discovering;

Chapter

least

at

would of

affects it

the the

to

the

INDEX

The

reference

to

are

where

Absolute,

Acceleration,

490,

and

Action

519

491

tortoise, 350,

358

arithmetical,

116;

71, 133-135; ordinal, 318;

180;

relational,

118,

viduals, indi-

of

307;

logical, 17, 21, of quantities, 179, of relations, 182, 254.;

relative,

387

26,

n.

of

;

vectors,

477

67, 139,

Aggregates, and

classes

as

442;

Aliorelative,

203

105,

72,

Analysis,

far

axioms

words,

333,

332, Area,

337,

and

91 288,

indemonstrables,

no

240;

127;

relation-,

321

Assertions,

law,

Assumptions, Axioms,

in

306

200,

446,

441

449

161, 270

ff.,267, 358,

350,

363

205,

207,

Cauchy, Causal

329

a

relation

177,

ft, 282, 375,

371,

120,

119,

199,

239

".,

331, 334, 381,

390,

312-324; 335

against

;

on

nitesimal infi-

orders

on

greatest

of

number,

18

Causation,

of

422

481; 479

dynamics,

by

particulars

vii, 475, 477, Cause, equal to

particulars,

487

481,

496

effect?

n.

245,

246; element,

an

245,

246

fT.

469

347, 420

Circle,

postulate

Class,

v,

106,

486

481,

474-479,

rational

Cayley, Chain,

35

n.,

w.

laws,

ix, 510

18

of, 438, fL,

40,

view

of, 67;

132;

440

66-81,

349,

356,

ff.;

as

67, 69, 131 ff., genesis of, 67, 515;

of, 20,

intensional

526;

concept

214; is

n.

527;

ordinals,

extensional

208;

221,

n.,

ft

513,

of,

161

irrationals, 283; on continuity, 287n*.; transfinite cardinals, on 304-311;

497, n.

364

323,

157,

flf.,444,

Chasles,

185

201,

theories

373,

367

n., n.,

259

Change,

Geometry,

181

144,

n.,

437

of

503

Being, 43, 49, 71, Bernouilli, 329_n. Bernstein,

505

307

99,

90,

fL; principles of a, 376 Georg, viii, 101, 111, 112,

Cantor,

Causality,

of, 350 argument 34-36, 48, 100, 502 ff. 39, 44, 82, 83, 98, 106,

Associative

three

89, 264,

Zeno's

Assertion,

Bettazzi,

59,

252,

408

progressions,

Arrow,

Between,

55, 56,

of, 181,

;

n.

of classes, 13-18; propositional, of relations, 23-26; logical, 142; 18-23; 259, 276, infinitesimal, 304, 325-330,

in

has

Arithmetic,

47,

n.,

Carroll, Lewis, Cassirer, 287 n.

417

333,

112

infinity, 336;

351

357

n.

n.f

segments,

263, 305,

in

426;

Calculus,

on

190-193;

in

213;

393,

393

307,

n.,

367

transfinite

axiom

391,

471

448,

347,

391, 420 infinity, 188,

difference

and

210;

376

43

Burali-Forti,

466;

390,

passage*

indefinable?

201

n.,

416

of

kindred

and

terms?

24,

41,

224,

on

Kant's, 259, 458-461 Any, 45, 46, 57, 105, Archimedes,

306

Bradley,

245,

real, 466

ratio,

Antinomies,

Borel,

n.

130

of, 415,

Anharmonic

143

falsification, 141,

and

conceptual

to

cure

211;

70, 10,

305

113,

67, 69, 71, And, Angles, 205, 414;

its

sense,

121

320

type

explained.

or

Boole,

376

n.,

how

of

338

infinite,

141;

one,

universal,

Algebra, All,

42

20".,

Adjectives,

defined

Bolzano,

483

Beaction,

Addition,

;

is

protective Geometry, descriptive Geometry, Bolyai, 373

450

Activity,

321

242,

483;

the

and

of, ix, 166, 219,

497,

474,

absolute, Achilles

term

black

in

between

principle

305, 314,

285,

technical

a

448

226,

Abstraction,

References

pages*

as

many,

one,

76,

68, 76, 103, 104,

104, 106,

Index

530

De Morgan, 23, 64n.,218flM 219".,326, 376 132, 513, 523; always definable by a of Denoting, 45, 47, 53, 106, 131; predicate?98, 526; when a member and predicates, 54; and any, etc.,55, 62; itself, 102; defined by relation,97, 98; there different kinds of? 56, 61 ; and of terms not having a given relation to are finite, identity, 63 ; and infinite classes, 308 ; in72, 73, themselves,102 ; multiplicative, 72, 106, 260, 306, 356, 357; de145, 350 numerable, 309 ; and well-ordered series, Derivatives,of a series,290 ff.,323; of functions,328 322; of one term, see Individual Class-concept,19, 20, 54, 56, 58, 67, 101, Descartes, 157 Dichotomy, Zeno's argument of, 348 113; Differential coefficients, distinct from class,68, 116, 131, 514 173, 328 Clifford, 434 Dimensions, 372, 374; definable logically, of three, 388, 399 376 ; axiom Cohen, 276 n., 326, 338-345 Collections,69, 133, 140, 513, 514 Dini, 324 n., 327, 328 n., 329 n. Direction,435 Colours, 466, 467 Commutative law, 118, 240, 307, 312 Disjunction, 15 n., 17, 31; variable and constant, 22, 58 Composition, 17, 31 Concepts, 44, 211, 508; Distance, 171, 179, 182 n., 195, 252-256, as

and

such

as

terms,

45; variation of,

503, 526; can they 86; prepositional, be subjects? 46, 507, 510 Congruent figures,417 Conjunction, numerical, 57, 67, 72, 113, 131 ff.;propositional, 57; variable,57 Connection, 202, 239 Consecutive,

201

Constants, logical,3, 7, 8, 11, 106, 429; and parameters, 6 Constituent, of a proposition,356, 510; of

whole, 143, 144

a

Continuity,188, 193, 259, 2865., 368; Dedekind's axiom of,279,294; ordinal, nomies 296^03; philosophyof, 346-354; antimetry, of, 347 ff.; in projectiveGeo387, 390, 437; of Euclidean space,

438 ff.

Continuum, 440

;

in

in

of, 180, 181, 254, 408; and order, 204, 409, 419 ; and relative position, 252; not implied by order, 252,

definition of,253 ; and limits,254 ; stretch,254, 342, 352, 408 ff.,435; Arithmetic,254; axioms of, 407 ff., 413, 424; and straightline, 410; projective scriptive theory of, 422, 425, 427; detheory of, 423-5 Distributive law, 240, 307 Diversity,23; conceptual,46 460 Divisibility, infinite, Divisibility, magnitude of, 149, 151, 153, 173, 230, 333, 345, 411, 425, 428; and measurement, 178; not a property of wholes as such, 179, 412 254

;

and in

Domain,

philosophical sense,

mathematical

146,

sense, 297, 209

n.,

elements, 3447 3*7, 444 353, 440 ff.; primarilyarithmetical, Contradiction, the, vi, ix, 20, 66, 79, 97, 101-107, 305, 362, 513, 515, 517, 523, 524, 525; Frege's solution of, 522; law of, 455 385, 388, 390, Coordinates,439; projective, 422, 427 261 ; of series, Correlation, 260 ; of classes, 310; composed

of

261, 321

Counting, 114, 133, 309 Couples, are relations classes of? 24, 99, 524; with sense, 99, 512, 524 Couples, separationof, 200, 205, 214, 237; transitive asymmetrical relations, and 215, 238; in projectivegeometry, 386, 387

Coutnrat, 66, 194

w., 267 n., 291 w., 296 n., 310 n., 326 n., 410 TI., 441 n.

Cremona, 384

288, 353 ; measurement

n.,

420

Relation

see

Duality,logical, 26; geometrical,375, 392 Bois Beymond, 181 n., 254, 336 Du Dynamics, vi; as pure mathematics, 465; two principlesof, 496 Economics, mathematical,233

n.

494, Electricity,

Empiricism,

496 373, 492

Epistemology,339 Equality,219, 339

; of classes,21 ; of 24 ; of quantities, 159 Equivalence, of propositions, 15, 527

lations, re-

Ether, 485, 496 Euclid, 157, 287, 373, 404, 420, 438 ; his errors,

Euler, 329

405-407

n.

Evellin, 352

Existence,vii,449, 458, 472; of a class,21, 32 Existence-theorems,ix, 322, 431, 497; and

Euclid's

problems,404

Exponentiation, 120, 308 16 Exportation,

DedeHnd, 90, 111, 157, 199, 239 n., 245251,294,307,315, 357 n., 381, 387, 438; 278 ff. on irrationals, Deduction, 522; principles of, 4, 15, 16 Definition, 15, 27, 111, 429, 497; and the, 62; always nominal, 112; by abstraction,114, 219, 249

Extension

and

Intension, 66

Fano, 385 n. Field,see Relation Finite, 121, 192, 371 Finitude, axiom of,188, 191, 460; absolute and relative, 332

Index

531

Force, 474, 482

Heymans,

truth, 40, 105 Formalism, limits of, 16,

Hilbert,384

Formal

Idea

Fractions, 149, 150, 151 Frege, vi, viii,19, 68 n., 78 w., Ill, 124 n,, 132, 142, 451 "., 501 ff.; three points of disagreement with, 501; his

three

elements

in

405 n., 415

n.,

n.

41

267

Formula,

489

judgment, 502;

and

object,450

Identity, 20, 96, 219, 502; distinguished from equality,21; denoting,63 ; of indiscernibles,451

and

Imaginaries, 376 467, Impenetrability,

480

his

sign of judgment, 503, 519; his Implication, formal, 5, 11, 14, 36-41, 89, theory of ranges, 505, 510 ff.; his Beclass of material a 106, 518; asserts 505, 507; his Symbolic Logic,518; griff, 38; and any, etc., 91 implications, his Arithmetic, 519 ; his theory of progressions,Implication,material, 14, 26, 33-36, 106, 520 ; Kerry'scriticism of,520 203 n. ; Frege's theory of, 518 Frischauf, 410 Importation, 16 Functions, 32, 262, 263; non-serial, 263; Inclusion,of classes,19, 36, 40, 78 numerical, 265 ; complex, 266, 376 ; real, Incommensurables, 287, 438, 439 324 ; continuous, $26 ; Frege'stheory of, 233 Incompatibility, synthetic, 505 ff. Indefinables,v, 112 Functions,propositional, 13, 19, 82-88, 92, Indication,502 finable, Individual,relation to class, 18, 19, 26, 263, 356, 508 ff.; definable? 83; indethan class 88, 106; more numerous 77, 103, 512, 522; distinct from terms? 103; and the contradiction, whose it is? vi, 23, 68, 103; only member with two variables, 94, 506 ; and classes, 106, 130, 513, 514, 517 19, 88, 93, 98; variable,103, 104; cardinal Induction, 11 n., 441; mathematical, 123, number of, 367 ; range of significance 192, 240, 245, 246, 248, 260, 307, 314, of, 523 315, 357, 371, 520 Fundamental bodies,491 Inertia, law of, 482 342 Inextensive, Generalization,7; algebraical, 267, 377 10; and deduction, Inference,asyllogistic, Geometry, 199, 372 ; distance and stretch 33; two lln.; logicaland psychological, theories of, 181 ; and actual space, 372, 35 premisses unnecessary, tance, 374; three kinds of, 381; based on disInfinite,121, 259, 260, 315, 368 ; 410, 492 ; and order, 419 ; has no antinomies of, 188, 190, 355; not specially indemonstrables,429 quantitative,194; as limit of 199, 382, 393-403; Geometry, descriptive, theory of, segments, 273; mathematical indefinables of, 394, 395, 397; axioms proper, 304, 355; philosophy of, 355-368; imof, 394 ff.; their mutual independence, 331-337; orders of, 335 396; relation to protectiveGeometry, Infinitesimal, 188, 260, 276, 325, 330, 331400 ff.;and distance,423-425 337; 206, 382, 391, 399, 413; Geometry, elliptic, defined,331; instances of, 332; philosophy Euclidean, 391, 399, 442; hyperbolic, of,338-345; and continuity,344; and 255, 382, 391, 399 ; non-Euclidean, 158, change, 347 179, 255, 373, 381, 436; of position,393 Integers,infinite classes of, 299, 310 n. Geometry, metrical,382, 392, 403, 404-418; Integral,definite,329 and quantity, 407 ; and distance, 407 ; Intensity,164 and stretch,414; relation to projective Interaction,446, 453 and descriptiveGeometry, 419-428 Intuition, 260, 339, 456 199, 206, 381-392; Geometry, projective, Involution, 385, 426 and order, 385 ff., 389, 421; requires Is, 49, 64 n., 100, 106 three dimensions, 394, 399 n.; differences Isolated points,290 %

pendent indeof metrical Geometry, 419-421 ; 421,425, 427 historyof, 420 ; and distance, Gilman, 203 n. Grammar, 42, 497 Grassmann, 376 Gravitation,485, 487, 490, 491 Greater, 122, 159, 222, 306, 323, 364 from

Geometry, 419; descriptive

Groups, continuous, 436

Jevons, 376

Johnson, viii,435 Jordan, 329 n.

Kant, 4, 143, 158, 168, 177, 184, 223 n.t 227, 259, 326, 339, 342, 355, 373, 442, 446, 450, 454, 456-461, 489 Kerry, 505, 520-522

Killing,400 Hamilton,

376

relation,384 Hegel, 105, 137, 287, 346, 355

Harmonic

Helmholtz,

241

Hertz, 494-496

n.

Kinetic

n., 404 n., 405 n., 415 n., 434

axes,

n.

490

Kirchoff, 474

Klein, 385, 389, 390 426 n., 434*., 43C

".,

421, 422

n., 424 n.,

Index

532

arithmetical, 119, 307, 308 Multiplication,

Kronecker, 241

;

ordinal, 318 Law, 268 Leibniz, 5, 10, 132, 143, 144, 145 "., 221, 222, 227, 228, 252, 287, 306, 325, 329 n., 338, 342, 347, 355, 410, 440 n., 445, 450, 451, 456, 461, 489, 492 Lie, 436 Likeness, 242, 261, 262, 317, 321 Limitation, principle of, 314

nth, 243, 250, 312

Necessity,454 18, Negation, of propositions,

31 ; of classes, 23, 31, 524; of relations, 25 490 Neumann, Newton, 325, 338, 469, 481, 482-492 Noe'l,348, 352 Null-class, vi, 22, 23, 32, 38, 68, 73, 106,

Limiting-point,290, 323 517, 525 Limits, 276 ff., 320, 361; and infinity, generalizationof, 267 Number, and algebraical 353; continuity, 188, 189, 260; Number, cardinal, logicaltheory of, 111 ff., conditions for existence of, 2916*.,389; 241, 519, 520-522; definable? Ill, 112, and the infinitesimal calculus,325, 339; of functions,327, 328; and magnitude,"341 130; defined, 115, 305; and classes,112, 114; 305, 306, 519; defined by abstraction, Line, see Straight 124, 432 112, 260, 304-311; finite, transfinite, Line-Gre.ometry, definition of, 247, 260, 357; Dedekind's Linearity, axiom of, 181, 252, 254, 408 249; Cantor's definition of,304; addition Lobatchewsky, 373 of, 118, 307 ; multiplication of, 119, 307, Logic, symbolic, 10-32; three parts of, 11; 308; of finite integers,122, 309, 364; and mathematics, v, 5, 8, 106, 397, 429, well-ordered,323, 364

457

310, 364

Lotze, 221, 446 ft.

; is

there

a

; of the

continuum,

greatest?101,362ff.;

362 ; numbers, 362 ; of classes, 362,526, 527 ; as a logical propositions,

of cardinal

Macaulay, 491 Mach, 474, 489, 492 Magnitude, 159, 164 ff., 194; relative theory of, 162; absolute theory of, 164; axioms of, 163, 165, 168; kinds of, 164, 334; and divisibility, 173 ; and existence, 174, 177, 342 ; extensive, 182 ; intensive,182, 326, 342; discrete and continuous, 193, negative, 229-231; 346; positiveand 332 ; limiting,341 infinitesimal, Manifold, 67 Mass, 481 n., 483, 488, 495 ; centre

of, 490

vii,3, 106, 112, 397, pure, 429, 456, 497; applied,5, 8, 112, 429; arithmetization of, 259 Matter, 465-468; as substance, 466;

Mathematics,

relation to space and definition of, 468

time, 467; logical

Maxwell, 489

McColl, 12, 13, 22 Meaning, 47, 502 Measure, Zeno's argument of, 352 Measurement,

157, 176-183, 195; Meinong, 55 n., 162 "., 168, 171 n., 173 "., 181w., 184, 187, 252, 253, 289, 419, 502n., 503

MiU, 373, 522 Mobius net, 385, 388 Monadism, 476 Monism, 44, 447 Moore, viii,24, 44 n.,

51 n., 446 w.,

448

n..

Motion, 265, 344, 405, 469-473; state of,351, 473 ; in geometry, 406, 418; logical definition of, 473; laws of, 482-488; absolute and relative, 489-493; law

of, 495

Motions, kinematical, 480; kinetic,480; thinkable, 494; possible,495; natural, 495

type, 525

Number, ordinal,240, 319; defined, 242, definition of, 248; not 317; Dedekind's prior to cardinal, 241, 249-251; trans260, 312-324 ; finite,243, ordinals,243, 313 ; second class of, 312, 315, 322 ; two principlesof formation of, 313; addition of, 317; subtraction of, 317; multiplicationof, 318; division of, 318; no greatest, 323, finite,240 260

;

n., of finite

364 ; positiveand negative,244 Number, relation-, 262, 321 Numbers, complex, 372, 376ft., 379; ordinal, series of, 323 ; positive and negative,229; real, 270 metical Numbers, irrational, 157, 270 ff., 320; ariththeories of,277 ff. dinal Numbers, rational, 149 ff.,259, 335; carnumber of, 310; ordinal type of, 296, 316, 320

Object, 55 n. Occupation (of space or time), 465, 469, 471, 472 One, 241, 356, 520; definable? 112, 130, to 135; applicable to individuals or classes?

45471.

Hertz's

of

130, 132, 517

Oppositeness,96, 205 Order, 199 ff.,207-217, 255; not psychological, 242 ; cyclic, 199 ; and 188, 189, 191, 195 ; in projective infinity, space, 385 ff.,389 ; in descriptivespace, 394, 395 Ordinal element, 200, 353 Padoa, llln,, 114 "., 125, 205 Parallelism,psychophysical,177 Parallelogram law, 477 axiom of,404 Parallels, Part,360 ; proper, 121, 246n. ; ordinal,361

;

Index three

kinds of, 138, 143; similarityto whole, 121,143,306,316,350,355, 358, 371

Pascal, 420 Pasch, 390

n.,

391 "., 393 ff.,407

n.,

Quadrilateral construction, 333, 384; metrical

in

417 relation to

geometry,

number, 157, Quantity, 169; 158, 160 ; not always divisible,160, 170 ;

417

Peano, vi, vii,4,

10 ff.,23, 26-32, 36, 62, 68, 76 ff.,Ill, 114, 115, 131, 139, 142, 152, 159n., 163"., 199, 205n., 219, 24171.,248, 270, 290, 300 n., 328 n., 334 "., 335, 341, 360, 410, 437, 443, 501, 514, 519 ; his indefinables,27, 112 ; his indemonstrable*, 29; his Arithmetic, 124-128,

238

533

sometimes

a

170-175;

and

relation, 161, 172 infinity,188;

in pure

occur

; range

does

mathematics, 158,

of, not

419

Quaternions, 432 Ranges,

511

ff.,524; extensions! 511; double, 512

or

tensions!? in-

Ratio, 149, 335

real

scriptive numbers, 274 ; on deBays, 231, 398, 414 ; order of, 415 geometry, 393 ff.; on theory of Reality,Kant's category of, 342, 344 vectors, 432 Reduction, 17 n.

; on

Pearson, 474, 489 Peirce,23, 26, 203 387 71. Pencils of

Referent, 24, 96, 99, 263 n.,

232 n., 320

n.,

376,

k

planes, 400

its function Perception,

in

philosophy,Y,

129

Regress, endless. 50, 99, 223, 348 Regression, 291, 300, 320 Relation,96, 107; peculiarto two terms, 26, 99, 268; domain of, 26, $7, 98; converse

Permutations, 316 domain of, S7, 98; field of, 97, 98; in tinguished itself and as relating, Philosophy,of Mathematics, 4, 226 ; dis49, 100; of a term from Mathematics, 128 ; and to itself,86, 96, 97, 105; definable as a Mathematics, 338 class of couples ? 99, 512 ; of a relation Pieri,199, 216 n, 382 ff.,410, 421 to its terms, 99 ; fundamental, 112 ; when Planes, projective,384 ; kinds of, 391 ; analyzable, 163; particularized by its cal, 398 ; ideal,400, 402 ; metridescriptive, terms, 51 n., 52, 211; finite,362 410 Relations, intensional view of, 24, 523, 526; Plato, 73, 355, 357, 438, 446 extensional view of, 99, 523, 526 ; monistic Pleasure, quantity of, 162, 174 ; magnitude monadistic and theories of, 221 ff. ; of, 164; and pain, 233 n. functions of two variables, 507, 521 as Pluralism, viii converse of,06, 95, 97, 201 -it.,228; reality "

Poincare, 347

444

ff.

See Number, cardinal Power, 364 n. of themselves, 45, 56 ; predicable Predicates, 96, 97, 102 Premiss, empirical,441

472.

Presentations, 446, 450 Primes, ordinal, 319 Process, endless. See Regret*. Product, logical,of propositions, 16, 519, 527 ; of classes, 21 Product, relative,06, 98 Progressions, 199, 239 ff.,247, 283, 313, 314, 520; existence of, 322, 497 Projection,390, 393

42, 44, 502 Proper names, Propositions,ix, 13, 15, 211, 502, 525; unity of,50, 51, 107, 139, 466, 507; when assertion, analyzable into subject and ly 83 ff., 106, 505-510 ; can they be infinite'number cardinal of, 145; complex? 367 ; "contradiction as to number existential theory of, viii,449,

Quadratic forms, 104, 512, Quadrics, 403

viii,99, 221, 224, 446 ff.; sense of,86, 95, 99, 107, 225, 227; difference from numbers, 95 ; with assigned domains, 26, 268 ; types of, 8, 23, 403, 436; symmetrical, 36, 96, 114, 20371.,218; asymmetrical, 95, 200, 203 n., 219-226; not-symmetrical, transitive, 25, 96, 318; transitive,114, 203, 218; inflexive, 218; not-transitive, 218; re114, 159 n., 219, 220; many-one, 114, 246 n; one-one, 113, 130, 305; nonrepeating, 232 n.; serial, 242; prepositional, 510; triangular,204, 211, 471, of?

Point-pairs,426 Points, 382, 394, 437, 443; 389 ; ideal, 400 ; rational and irrational, ginary, and improper ideal, 423 ; imaproper objections 420; logical to, 445-455; material,445*;indiscernible? 446, 451 Position,absolute and relative,220, 221,

514

of, 527 493

;

Relation

See

Verbs

-number.

See

Relatum, 24, 96, 99, of Representation,

Number,

relation-

263

system, 346

a

Resemblance, immediate,

171

Rest, 265

Reye, 403 ". Riemantf, 266 Right and left,223 Rigidity,405

".,

231, 417

Rotation, absolute, 489 ff. 13, 22, 24, 26, 142, 2H2, 306 n., 320 n., 367 n. and limits, 292; Segments, 371, 359; completed, 289, 303; of compact series, 299-302; of well-ordered series,314 n.; infinitesimal, 334, 353, 368; in projective

Schroder, 10

n.,

201 n., 221

w.,

12 n.,

geometry, 385 ff 394, 397 Semi-continuum, .

Separation. See

; in

descriptive geometry,

320

Couples

534

Index

Series,199; compact, 193 n., 203, 259, 271, 202, 204,205, 277,287,289,299-303;closed, 204, 239 ; 234-238, 297, 381, 387 ; infinite, denumerable, 296, 298 ; continuous, 205, 271, 2875.; well-ordered,310, 319, 322, 363; independent,262; by correlation, 262, 363; complete, 269, 303; perfect, 273, 288, 290, 292, 297; coherent, 274, 283, 297; cohesive, 288; fundamental, 372 ; and 283, 297 ; simpleand multiple, 204 2 04 and relations, distance, ; triangular Sheaves, 400

Sign, difference of, 227-233 Similarity,of classes,US, 249, 261, 305, 356; of null-classes, 521; of whole and part, see Part 16 Simplification, from a, 56 n., 59 Some, distinguished Space,372, 436, 442; an infinite aggregate,

Therefore,35, 504 Things,44, 106, 466,505; and change, 471 144; relational Time, an infiniteaggregate, theory of,265; Kant's theoryof,456,458 Totality,362, 368, 528 Transcendental

Aesthetic,259 ; Dialectic,259

Triangles,387, 398 Trios, 525 Tristram Shandy, paradox of, 358 Truth, 3, 35, 48, 504 Truth-values, 502, 519 Two, 135; not mental,451 103, 104, 107, 131, ]39n,, Types, logical, 367, 368, 521, 523-528; minimum, 524, of, 525; 525; mixed, 524, 526; number of infiniteorder, 525

Types, ordinal,261, 321 i

Unequal, 160 n. 143, 443, 455; absolute, 227, 445 ff.; Unit, 136, 140; material,468 finite and infinite,403; continuityof, Unities, 139, 442; infinite,144, 223 n.; 437-444; subjective?446; empty, 446, organic,466 449, 465; " priori?454; and existence, vii,458, 461 Vacuum, 468 446 n,, 456 430 Vaihinger, d efined, Spaces,projective, ; Euclidean, defined,432; Clifford's, Vailati,205, 215, 235, 393 n., 394, 395, 413 defined,434 450 Validity, Spinoza, 221, 448 Staudt, von, 199, 216,333,384,385n.,421, Variable,5, 6, 19, 89-94, 107, 264; 427 n. apparent and real, 13; range of, 36,

Stolz,90,282 n,, 283 n., 334, 336,378 rc., 379 518; as concept, 86; and generality, 90; does not vary, 90, Straightlines, elliptic, 205; projective, 90; in Arithmetic, 382 ff,, 387, 391; segments of projective, 90; conjunctiveand 344, 351 ; restricted, 92 ; individuality dependent 394-398; segments of 385; descriptive, disjunctive, of,94 ; in263 ideal, 400, descriptive,394, 397; 402; metrical,410; kinds of, 382, 391; Vectors, 432 and distance,410, 492 Velocity,473, 482 Streintz,491 Verbs,20 n., 42, 47-52, 106; and relations, Stretch,181, 182 n., 230, 254,288,342, 353, 49, 526 408 ff.,425 Vieta, 157 contained in a given Sub-classes,number Vivanti, 203 n., 288 *., 307 n., 308 Volumes, .231,333, 417, 440, 443 class,366, 527 47, 54, 77, 95, 211, Subject,and predicate, it be 221, 448, 451, 471; logical, can Ward, 474, 489 Weierstrass,111, 157, 259, 32(3,347, 473; plural?69, 76, 132, 136, 516 282 Substance, 43, 471 irrationals, on Substantives,42 Whitehead, vi, viii,119, 253 n., 299 n., Such 307 n., 308, 311 "., 322, 376 n., 377, that, 3, 11, 19, 20, 28, 79, 82 424 n., 426 Sum, logical, 26 21; relative, 161, 405 Superposition, Wholes, 77, 137; distinct from classes as Syllogism,10, 16, 21, 30, 457 and 69, 132, 134 n.; logical many, 245, 247 System, singlyinfinite, priority, 137, 147; two kinds of, 138; distinct from all their parts, 140, 141, 225; infinite, Tautology,law of, 23 143-148, 333,349; always either aggregates or unities? 146, 440, Terms, 43, 55 11., 152, 211, 448, 471, 522; of a proposition, tions 45, 95, 211; combina460; collective and distributive, 348; and enumeration, 360 of, 55, 56; simple and complex, in a 137; of a \vhole,143; principal, 297; four classes of, 465; cardinal series, Zeno, 347 ff.,355, 358 number of, 362, 366 Zero, 168, 195, 356 ; Heinong's theory of, Tetrahedra,387, 399 184, 187 ; as minimum, 185 ; of distance, Than, 100 186; as null-segment,186, 273; and The, 62 negation, 186, 187; and existence,187 Zeunelo, 306 M.