The Principles of Mathematics
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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.