Aristotle's Philosophy of Mathematics

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ARISTOTLE'S PHILOSOPHY OF MATHEMATICS

ARISTOTLE'S PHILOSOPHY � \I

OF MATHEMATICS By

Hippocrat�s George Apostle /:;

THE

UNIVERSITY OF

CHICAGO

PRESS

II ,:f.,,.) 1" --Iif ":t 52..-137 go THE UNIVERSITY OF CHICAGO PRESS, CHICAGO

37

Cambridge University Press, London, N.W. 1, England Copyright 1952 by The University of Chicago. All rights reserved Published 1952. Composed and printed by THE UNIVERSITY OF CHICAGO PRESS, Chicago, Illinois, U.S.A.

To

RICHARD P. McKEON

O

PREFACE

F ARISTOTLE'S extant works no one treats of math­ ematics systematically, and in none of them is there a reference to any such work. A book on mathematics by Aris­ totle is listed in the catalogues of Diogenes Laertius and of Hesychius, but some thinkers have questioned the reliability of those catalogues. However, numerous passages on mathe­ matics are distributed throughout the 'Yorks we possess and indicate a definite philosophy of mathematics, so that an at­ tempt to construct or reconstruct that philosophy with a fairly high degree of accuracy is possible. Perhaps such an attempt is desirable at this time, for the present interest and activity in the philosophy of mathe­ matics can find their parallel only in the days of Plato and Aristotle. Unfortunately, modern mathematical philoso­ phers made no attempt to trace the early history of the phi­ losophy of mathematics, and some of them who ventured to state or criticize early views on the subject were more anx­ ious to dismiss than to discover them. The method chosen to present Aristotle's philosophy of mathematics is his own. It is discussed in the Posterior Analytics and carried out in the Physics, Metaphysics, De Anima, and the rest of the sciences. Deviations from that method are introduced for the sake of the reader. Thus, Aristotle states and criticizes the views of his predecessors and contemporaries before proceeding to expound his own, but I reversed this order. My main purpose is to present Aristotle's view; and clarity and understanding demand that the principles of criticism, which are part of or implied by the critic's position, should precede criticism itself. The general order of presentation is what Aristotle calls "sci­ entific" or "order of nature," the order of proceeding from the universal to the particular or from causes to effects; but vu

PREFACE

Vlll

the manner of introducing principles by means of induction and example is what Aristotle calls "the order clearer to us," the order of proceeding from the particular to the uni­ versal. Both procedures are used by Aristotle to suit the occasion. No attempt has been made to go beyond what is found in Aristotle's works and what i� implied by them. Some impor­ tant definitions which I have introduced, although not to be found in the works we possess, are implied by or are in ac­ cord with Aristotle's general principles of definition. Such is the definition of ratio. The accuracy or reliability of Aris­ totle's account of the mathematical philosophies of his pred­ ecessors and contemporaries has often been questioned. I am of the opinion that the account is fairly accurate. To retain the definiteness and clarity of Aristotle's terms and meaning, it was found necessary to include and adhere strictly to an English-Greek dictionary of the important philosophical terms. The notes refer to the pages and lines in Bekker's edition ofAristotle's works (Berlin, 1831). Through­ out the work I found it convenient to drop the third-person singular in favor of the first plural.

I wish to express my deep obligation to Professor Richard P. McKean, who has introduced me to philosophy and has guided me toward the understanding of philosophy and the completion of this work. Special thanks are due to Miss Natalie Lincoln for her generous help, her delightful philo­ sophical discussions on this work, and her spiritual encour­ agement. I am indebted to Professors John D. Wild, Raphael Demos, and Werner Jaeger for their helpful criti­ cisms and valuable suggestions and to my friend George Tsucalas, who has been a constant source of inspiration. To Mrs. Rowland Chase, who has gone over the manuscript and made many valuable corrections, I owe many thanks. GRINNELL COLLEGE

March 1951

H.G. A.

TABLE OF CONTENTS I.

UNIVERSAL MATHEMATICS

.: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. II.

1

Mathematics Is a Theoretical Science Mathematics Investigates the Properties of Quantities Nature of Quantity . Accidental Quantities Manner of Existence of Quantities Motion, Generation, Place, Position, Not Essential Attributes of Quantities How Quantities Are Known Method in Mathematics Causes in Mathematics . Mathematical Definition Common Attributes of Quantities Nature of Mathematical Axioms The Infinite . First Division of the Mathematical Sciences

ARITHMETIC

1. Arithmetic Is More Accurate than Geometry 2. The Unit as Principle of Number 3. Number III.

GEOMETRY

1. 2. 3. 4.

IV.

1

3 4

8 11

17 23 32 49 54 55 66 67 80 81 81 83 89 96 96 97 100

Only Three Dimensions in Geometry The Limit Is Prior in Definition to the Limited The Limited Is Prior in Substance to the Limit Whether the Kinds of Magnitudes Are Simultaneous by Nature or Not . 5. Order of Scientific Knowledge in Magnitudes . 6. Material Cause and Definition in Magnitudes 7. Magnitudes Are Not Composed of Indivisible Elements

102 104 106 120

COMPOSITE SCIENCES

131

.

1. Composite Sciences Are Possible Because Quantity Is Present in Physical Bodies 131 IX

TABLE OF CONTENTS

X

2. Kinds of Subordination in Composite Sciences 3. Principles of Composite Sciences 4. Demonstration in Composite Sciences V. CRITICISM OF VARIOUS VIEWS ON MATHEMATICS

1. 2. 3. 4.

Statement of Various Views Criticism of First Principles of Things Criticism of Generation from Principles Criticism of Nature and 'Existence of Things

132 134 136

140 140 153 166

180

NOTES

207

DICTION ARy

219

INDEX

225

Chapter I UNIVERSAL MATHEMATICS

O

1. Mathematics Is a Theoretical Science

F THE three theoretical sciences-first philosophy, mathematics, and physics 1 -mathematics seems to have made the earliest advance2 with.respect to both ac­ curacy and truth. The priests of Egypt who had leisure for theoretical inquiry began with the mathematical arts; and Thales, the Pythagoreans, and other thinkers, who investi­ gated the objects of both mathematics and physics, were more successful with the former than with the latter. This happened to be so not without reason; for the principles and causes of first philosophy are furthest removed from the senses3 and so, being hardest to attain, are reached last, while those of physics are more numerous than those of mathematics 4 and require longer experience. Indeed, the familiarity and success with the objects of mathematics and the observation of many mathematical attributes in physical things led many thinkers so far as to reduce the principles and elements of all things to those of the objects of mathe­ matics. According to some Pythagoreans, physical objects and their attributes are made out of numbers; 5 according to Plato, the Ideas which are the truest of all existing things and are the causes of sensible things are themselves numbers. 6 That mathematics is a theoretical science is evident from the nature of a theoretical science and the aim of those who have pursued mathematics. Sciences are divided into theo­ retical, practical, and productive. 7 The aim of a productive science is the production of something, for example, a bridge or a house. 8 Consequently, the artist, as an artist, seeks 1

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knowledge only in so far as it is useful to that end. He must know only the fact, that such and such is the case, but he need not inquire into its causes, because they are of no use to him in the making of the product. For example, in the pro­ duction of a house he must know that, if bricks are placed vertically one on top of the other, they will not fall or are least likely to fall; but he need not bother with the causes of it, as they are of no use to him in the building of the house. 9 The aim of a practical science, on the other hand, is action, and no product is sought besides that action. 10 For an action to take place there must be a principle of action and that is the will, that is, choice preceded by deliber­ ation. 11 Hence, here, too, knowledge is necessary, but a prac­ tical science does not end in knowledge. This is seen in the case of ethics, which is a practical science. We study ethics not merely for the sake of knowing what virtue is, what hap­ piness is, and how they are attained but in order to become virtuous and attain happiness; and this end cannot be at­ tained unless we deliberate and act in accordance with de­ liberation. If this end is not achieved, then our study of ethics has been in vain. 12 Finally, the aim of a theoretical science is neither action nor the production of something but the acquisition of scientific truth. 13 Now knowledge is of three kinds, namely, of things which exist necessarily, of things which exist simply (but not neces­ sarily), and of things which exist for the most part. Ex­ amples of the first kind are the necessary existence of motion and the fact that the sum of the angles of a triangle is equal to two right angles; that is, it is impossible that all things be at rest or that the sum of the angles of a triangle be unequal to two right angles. If it is a fact that all geometers can read, then we have an example of things which exist simply, for it is not necessary that all geometers be able to read. An example of things existing for the most part is the fact that most men prefer acquiring wealth to philosophiz-

UNIVERSAL MA THEMATICS

3

ing. 14 Since the first truths or principles of things on which all other knowledge rests are necessary (such are the prin­ ciples that the same attribute cannot both belong and not belong to the same subject at the same time and in the same respect and that either the affirmation of something or its denial is true), 15 the aim of a theoretical science is truth, primarily of that which exists necessarily, and only second­ arily of that which exists simply or for the most part. 16 That mathematics is a theoretical science is evident from those who have pursued that science; they have been lovers of wisdom and have sought to discover eternal truths. 17 Thus, it is not without reason that some thinkers have even separated the objects of mathematics from sensible objects which are subject to change and have regarded them as a class by themselves. Others have not separated them but made them principles of all things. 18 Clearly, then, those who have investigated the objects of mathematics have done so not for gain or use but for the sake of truth. As for astronomy, mechanics, optics, mensuration, the art of building, and other such mathematical sciences and arts, some of them are concerned with eternal objects; others with the useful and the temporary. But their objects, al­ though treated mathematically, are not quantities properly speaking; and all these sciences and arts borrow truths from theoretical mathematics, and so their discussion is posterior and need not concern us here. 19

2. Mathematics Investigates the Properties of Quantities Let us now turn our attention to mathematics in general, beginning with its objects. Since mathematics is a science, and since each science, marking off a genus of things, in­ vestigates the properties of those things (only first philosophy as a science is concerned with all things in so far as they are just things), evidently, mathematics deals with some one

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genus of things and investigates their properties. 20 If we at­ tend to the inquiries of those who have been called "mathe­ maticians," perhaps we shall be in a better position to iden­ tify that genus. In fact, they were concerned with numbers and such attributes of numbers as oddness, evenness, equal­ ity, excess, symmetry, and proportion; and they were also concerned with lines, planes, and solids, which are magni­ tudes, and investigated such of their attributes as symmetry, asymmetry, intersection, equality, parallelism, and relative position. Yet, although they called the special science which is concerned with numbers "arithmetic" and that which is concerned with magnitudes "geometry," most of them re­ garded these two sciences as parts of one science or as re­ ducible to one, evidently on account of some common na­ ture in their objects. Moreover, equality, inequality, excess, defect, ratio, and proportion are attributes which belong to both numbers and magnitudes. 21 Further, such axioms as "If equals are subtracted from equals, the remainders are equal" and "If two quantities are equal to the same quan­ tity, they are equal to each other" are common to both numbers and magnitudes; 22 and theorems which were for­ merly demonstrated separately for numbers, lines, solids, and time-for example, the theorem that proportionals are also proportional by alternation-are nowadays given by a single demonstration because of something common to all these objects. 23 From what we have said it is evident that mathematics investigates the properties of things commg under one genus, and that genus is "quantity." 24

3. Nature of Quantity It is desirable, however, to speak a little more clearly about quantity. A solid is a quantity, and it is divisible into solids which constitute it as parts. Each part has a common boundary, a surface, with other parts and also a relative position; 25 for it is possible to show where each part lies rela-

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5

tive to the other parts and to the whole. Further, after divi­ sion, the parts themselves become solids and can be exhib­ ited by themselves apart from the others. Surfaces and lines are like solids in this respect, except that surfaces have lines, and lines have points as boundaries. All three quantities are called "magnitudes" or "continuous" by virtue of the fact that the parts of each have common boundaries. 26 An army, which is a number and so a quantity, can likewise be divided into parts which constitute it, and each of them, after division, can be exhibited by itself apart from the others. The parts themselves, if they are composed of more than one soldier, may be divided further; but such division must finally come to an end when each part is one soldier. Here, no part of the army has a common boundary with any other part. It is true that a man, in so far as he is a body and continuous, can touch another man as a body; but a man as a man is a substance and indivisible and not a body with a surface. This is also evident from the fact that a body is divisible into bodies, but a man is not divisible into men and is divisible into bodies only qua27 body. It is in this way that the mathematician considers a number. Three, which is a part of ten, has no common boundary with the other part, seven; and no mention whatsoever is made of boundaries or of relative position of the parts or of the units when the at­ tributes of ten are investigated. 28 Numbers are called "dis­ crete quantities," that is, they are quantities divisible into units as ultimate parts, each of which is indivisible. 29 The essential quantities having been considered,30 we are in a position to state what is common to all quantities. Quantity is said to be that which is divisible into constitu­ ents, either of which, or each of which, after division, is by nature some one thing and can be exhibited apart from the other parts. 31 This formula, however, is not to be taken as a definition of quantity. A definition must have a genus and a differentia, and "quantity" is an ultimate genus; hence,

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quantity cannot be defined.32 "Being" (or "thing") is not a genus but is immediately subdivided into "substance," "quantity," "quality," "relation," and the rest of the cate­ gories. 33 Being, primarily speaking, is substance, for sub­ stance alone is separate, but the rest which come under the remaining categories are attributes of and present in sub­ stances; for example, color is a quality present in a body. 34 If "being" were the genus of "quantity," and "Q" its clif­ ferentia, then Q would itself be a being, and being would be an attribute of Q. But this is impossible, for the genus is not a predicate of the di:fferentia. For example, three-sidedness is not a triangle, and "triangle" is not a predicate of three­ sidedness. 35 Since, therefore, the formula given for quantity is neither a definition of, nor accidental to, quantity, nor a property of a specific quantity as it is equally a predicate of all, nor yet a predicate of all things, it must be a property of quantity. 36 Quantities do not admit of contraries. 37 There is no con­ trary to the number five, or to a line three feet long, or to a triangle or a cube. Straight lines seem to be contrary to cir­ cular lines, but it is not in so far as they are lines or quan­ tities that they are so opposed but rather in so far as they are straight. 38 But the di:fferentiae "straightness" and "circular­ ity" signify qualities within quantity, and it is qua qualities that lines admit of contraries. 39 The terms "much" and "little," "great" and "small," seem to be contraries, but these terms signify relations and not quantities. A mountain is called "small" not in virtue of itself but when compared or related to other mountains, and a house is called ''great" for similar reasons; otherwise, the mountain, which is greater than the house, would be called tsmall" and the house "great." Moreover, ten is little compared to twenty and much compared to five, so it would be little and much at the same time. But it is impossible for the same thing to admit of contraries at the same time. Hence, the terms in

UNIVERSAL MATHEMATICS

7

question signify relations and not quantities. Perhaps it is better to use the terms "greater" and q'smaller" instead of the terms ''great'' and ''small.'' Quantities do not admit of being more or less. 40 One quantity is not more or less of a quantity than another in the sense in which one body is more white or less white than another body. One instance of three is not more three at one time than it is at another time, and it is not more three than another instance of three. Similarly, one line is not more of a line than another line, although it may be greater, and one surface is not more of a surface than another surface. One line may be more curved than another line, but this com­ parison is made with respect to curvature, which is a qual­ ity, and qualities do admit of the more and the less. It is proper to all quantities that equality and inequality are attributes of them. 41 One line is either equal or unequal to another line, one number is either equal or unequal to another number; but outside of the genus of quantity no such affirmation can be truly made. A color, for example, which is a quality, is neither equal nor unequal to another color but is like or unlike another color. Quantity is first subdivided into magnitude and number. Number is defined as discrete quantity and magnitude as continuous quantity. 42 The objects of arithmetic are num­ bers and their properties, and those of geometry are magni­ tudes and their properties. 43 (By quantities we mean finite quantities; discussion of the infinite will be taken up later.) A number is numerable, that is, finitely exhausted by count­ ing its units one by one, 44 and each of its units is indivisible according to quantity and without position. 45 A magnitude is measurable, and each of its parts has relative position and is again a magnitude. 46 A straight line, for example, is meas­ urable, that is, finitely exhausted by taking some unit of length as a measure and going over its entire length, and its parts have relative position and are straight lines. Since the

8

ARISTOTLE'S PHILOSOPHY OF MATHEMATICS

parts of a magnitude are themselves magnitudes, and since the parts of the parts themselves are also magnitudes, etc., it follows that a magnitude is infinitely divisible, that is, di­ visible without an end or without the possibility of ever reaching ultimate indivisible parts. Magnitudes are sub­ divided into lines, surfaces, and solids. A line is a quantity continuous in one dimension, a surface is a quantity con­ tinuous in two dimensions, and a solid is a quantity con­ tinuous in three dimensions. 47

4. Accidental Quantities In attributing length or width to a substance, we some­ times substitute a derivative term to signify that substance. We say, for example, that the white is wide or that the heavy is long, and the terms "white" and "heavy'' signify the sub­ stance but are derived from the terms "whiteness" and "weight," respectively, which signify attributes present in that substance. But just as one white body is equal to or greater than another white body in virtue of its length or area or volume but not in virtue of its whiteness, so the white (substance) is wide not in virtue of its whiteness but in virtue of its width. Thus, we say that the white or the heavy is accidentally a quantity and that naming the white "wide" or the wide "white" is accidental naming. But if the wide surface is named "a surface" or "wide" or "a quantity" or "continuous," then it is said to be essentially so named, that is, these terms appear implicitly or explicitly in the defini­ tion of a wide surface, and a definition signifies the essence of a thing; or else the terms signify necessary or demon­ strable attributes of the wide surface. Whiteness, on the other hand, in no way appears in the definition of a wide surface, nor is a wide surface necessarily white. We use such accidental predication because both the white and the sur­ face are together. The two form a sort of unity or a part of another unity, that is, of the body in which they are, the

UNIVERSAL MATHEMATICS

9

body being one thing. Accordingly, the body is named "a body," or "white," or "wide," and the surface of that body is accidentally named "white." 48 In the case of motion and time it is more difficult to see how they are accidental quantities, since the quantitative terms "continuous" and "divisible" seem to be predicates of each of them. The three kinds of motion are locomotion, or motion with respect to place; alteration, or motion with re­ spect to quality; and motion with respect to quantity, of which the two kinds are increase and decrease. 49 Examples of these in the order given are the motion of a stone from one place to another, the change of a body from being black to being white, and the increase in volume of a growing tree. The body which moves has a magnitude, and that through which it moves may be a magnitude, as the distance in loco­ motion; and it is because the body and the distance are con­ tinuous and divisible that we attribute continuity and divisi­ bility to motion. Just as division of the white body will di­ vide its whiteness accidentally, so division of the moving body will divide its motion accidentally. That division of the whiteness of a body is accidental is evident from the fact that the whiteness of one part is not equal or unequal to the whiteness of another part, but the magnitude of one part is either equal or unequal to that of another part. Similarly, the body B, which has moved from point P to point S (or from state P to state S in the case of alteration or of the other motions), is divisible into parts, let us say, Bi, B2, and B a. Consequently, we may say that the motion of B is di­ visible into the motion of Bi, that of B2 , and that of B a. Again, the interval PS is divisible into parts, say, PQ, QR, and RS. The motion of B may then be said to be divisible into the motion from P to Q, that from Q to R, and that from R to S. But such divisions of motion are accidental. 5 ° Fur­ ther, the body B, being one and the same, can actually be divided according to quantity into its parts, and so can the

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interval PS in the case of locomotion; but the motion of B from P to S, once completed, cannot be actually divided, for as one part comes another goes. 5 1 Consequently, each of the partial motions cannot be properly said to be a motion because it cannot be one, but in a way it may be called "part of a motion." That motion is not an essential quantity is also clear from its definition arrived at in physics. When a body is in motion, there is an underlying subject which remains the same during the motion and also an interval between con­ traries (or intermediate states) within which the subject moves. 52 For example, Socrates, while remaining Socrates, moves from being sick to being well. The subject, then, has the potentiality of receiving contraries and various states be­ tween contraries. When at rest, the subject has a definite actuality which is one of the contraries or some state be­ tween them. For example, Socrates stays sick. While in mo­ tion, the subject has no such actuality but is in a state of in­ completeness with respect to such actuality, and the actual­ ity of such a state is motion. Hence, motion is the actuality of that which is potential, not in so far as the potential has some definite actuality in the interval between contraries, for then there would be rest and no motion, but in so far as the potential still persists in the state of being potentially something or other. For example, the actuality of the curable in so far as it is curable is curing, and the actuality of the alterable as such is alteration. Motion, then, is defined as the actuality of the potential qua potential, and the terms in this definition are not quantitative. 5 3 Since the parts of time are not constituents which can be simultaneously exhibited, for as one part comes another goes, time itself does not seem to have the property of quan­ tity given earlier. Moreover, it was discussed elsewhere that time is a number of motion with respect to before and after, and so it is an attribute of motion; and motion is an acci-

•

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dental quantity.5 4 In fact, as the magnitude through which a body moves is continuous and divisible, its motion is thought to be continuous and divisible but accidentally; and, as the motion is continuous and divisible, the corresponding time is in a similar way thought to be continuous and divisible accidentally. And just as there is succession and priority and posteriority in the parts of the magnitude, so is there in the corresponding parts of motion and of time, except secondar­ ily, and the order is similar in all of them. 55 Place, too, seems to be a quantity. Place is said to be the first inner motionless boundary of a containing body; for example, the inner surface of a bottle. 56 The differentiae of place are up, down, left, right, forward, and backward, and these have absolute existence in the universe. 5 7 Thus, the inner boundaries of two containing bodies may have the same shape, but one of them may be up near the extremity of the universe while the other is down near the Earth. Fur­ ther, some things are in place by nature and others by force; for example, earth's place is by nature down at the center of the universe but by force up near the extremity. 5 8 Hence, qua motionless boundary, place is a surface and a quantity; but qua having in it certain bodies by nature and others by force, it is not a quantity.

5. Manner of Existence of Quantities It was stated earlier that the aim of a theoretical science is truth and that mathematics is such a science. Since truth is of that which exists, 5 9 if quantities did not exist, no propo­ sition affirming a positive attribute of a quantity would be true, and mathematics would be concerned with fictions or not-being. But sciences are concerned with what exists and not with what does not exist. 60 Since, however, there is agreement as to the existence of quantities but disagreement as to the manner of their existence, it is proper to discuss the manner of their existence. 61 This will also help us later in our

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criticism of the various views put forward which differ from ours. Perhaps it is better to begin with what is clearer to us and proceed little by little to that which is clearer by nature, for this is the way of attaining knowledge of first principles. 6 2 In our discussion of the nature of quantities we used sen­ sible objects as examples and so indicated at the same time that quantities exist and are somehow related to sensible ob­ jects. That quantities are not the sensible bodies themselves may be gathered from what the mathematician says about them. His solids have no color, no weight, no principles of motion, no hardness or softness. Nor do they have place, for they are neither up nor down and have no tendency to move in one direction rather than in another; at most, they have relative place or position. One magnitude may be within another, or relative to us one triangle may be con­ ceived as to the right of another. 63 As for the units in a num­ ber, they have no relative position and none of the other qualities to be found in sensible things in so far as they are sensible; they are just indivisible. One unit qua unit does not differ from another with respect to quantity or quality but only in so far as it is another unit. 6 4 But if quantities are not the sensible bodies; if they are not with them, since otherwise two bodies would occupy the same place; 65 if they are not in the intellect (how can a solid a mile long be in the intellect?); 66 and if we have no evidence that they exist apart by themselves; then, perhaps, they must be somehow related to sensible bodies or be present in them. 67 This is also evident from the fact that we attribute numerical and other mathe­ matical attributes to sensible bodies. Sensible bodies are sometimes at rest and sometimes in motion. They are colored, have weight, and have other sensible qualities such as heat, cold, and hardness. It is evi­ dent that these attributes cannot be separated from the sensible bodies and be exhibited apart by themselves. Mo­ tion is present in the moving body and cannot exist apart

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13

from that body; color is likewise present in the surface of a body and cannot exist apart from it; and the same may be said of the other sensible attributes. Similarly, sensible bod­ ies have length, width, and depth-for do we not say that they are so long or so deep? They are also numbered. And it is impossible to separate the length, or the surface, or the number from the sensible body to which it belongs and present it by itself as something separate. In general, sub­ stances alone-and men, . animals, trees, stones, and other sensible bodies are substances-have the property of being neither attributes of nor present in something else, for they are separate. All others are attributes and present in these substances. 68 By "being present in a subject" we mean, not being in place as a man is in a room or being a part of a whole in the sense in which the hand is a part of the body, but being incapable of existing apart from that subject. 69 For example, the color or weight or length of a body is said to be present in that body. Now, we predicate the term "being" or "thing" not only of substances but of these others as well. For example, we say that a color, or a surface, or a relation is a being or a thing. As it was stated elsewhere, the kinds of being are as many as there are categories. Being, in the primary and unqualified sense, is substance, which is separate. Quality, quantity, relation, and the rest, in so far as they are incapable of existing apart by themselves, are called "being" only in a secondary and a qualified way. 70 In fact, the treatment of qualified being is analogous to the universal treatment of mathematical objects. In mathematics there are axioms and theorems regarding quantities in general, as, for example, the axioms "The whole is greater than a part of it" and "Proportional quantities are also proportional by composi­ tion"; and such axioms and theorems do not postulate the existence of quantity apart from numbers and lines and solids and the rest of the quantities but treat of these latter

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universally. They treat of numbers not qua numbers or qua discrete but qua quantities only; and they treat of lines and solids not qua continuous in one and three dimensions, re­ spectively, nor yet qua continuous, but qua quantities only. By the expression "they treat of lines qua quantities" we mean that they investigate those attributes of lines which belong to the lines only in so far as they are quantities and not in so far as they are · one-dimensional or continuous. 7 1 Similarly, a triangle has its angles equal to two right angles, although a triangle does not exist apart from isosceles, scalene, and equilateral triangles, since that which is a tri­ angle must be one of these. In the same way there are propo­ sitions and demonstrations concerning moving bodies, not qua hard or soft or colored, but qua moving; and it is not necessary, for that reason, that motion exist apart from the moving body. So, too, there is a science which deals with these same sensible bodies, not qua moving or hard or soft or colored, and, in general, not qua sensible; but it deals with them qua solids which are continuous in three dimen­ sions, and, again, qua having surfaces and lines and points, and, finally, qua indivisible only. In this way the objects of mathematics are arrived at by abstraction. We take a substance, let us say a man, and in thought we disregard all that belongs to him qua living and rational; what remains is a physical body, still capable of being moved from one place to another, and in this it has much in common with other bodies, such as stones, chairs, and metals. Further, we remove in thought its principles of motion and along with them the sensible qualities, such as whiteness, heat, hardness, weight, and their contraries. What remains now is an immovable solid, continuous in three dimensions. This solid has length, width, depth, and also surfaces, lines, and points. We may remove in thought continuity itself, thus arriving at something which is in­ divisible, the unit. 72

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Evidently, the more we remove in thought, the less is left there, and so the less possibility of error and the greater the accuracy. There is greater accuracy in the science of things which are simpler, prior in definition, and more universal than in the science of things which are composite or less simple, posterior in definition, and less universal or nearer to particulars. Consequently, there is greater accuracy in the treatment of bodies if they are treated without motion than if they are treated with motion; and, if with motion, there is greater accuracy if the simple motions are attributed to them (i.e., the circular or the straight) than if the composite ones are attributed to them. We have greater accuracy still if we remove magnitude or continuity and regard them as indivisible with respect to quantity. 73 For this reason, arith­ metic, which is concerned with numbers whose units are indivisible with respect to quantity, is regarded as one of the most accurate sciences. 7 4 The removal in thought of attributes does not falsify our statements about things. We do not say that the one is sepa­ rated or can be separated from the other (i.e., the motion from the body or the color from the surface or the surface from the sensible body), but we pay attention now to one aspect of a man, for example, and now to another. 75 Certain attributes belong to a man qua solid, certain other attributes belong to him qua having weight, and still others belong to him qua being indivisible. Man, then, separates in thought the objects of mathematics from sensible bodies; that is, he treats sensible bodies without reference to their principles of motion and investigates the attributes which belong to them in so far as they are quantities of some kind or another or in so far as they are just quantities. Indeed, if to know a thing scientifically is to know it through its principles and causes and elements, then such separation or abstraction is both necessary and desirable. If a white stone is thrown upward, it falls. Is its whiteness

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necessary for its falling? No, for it can, be painted red and still fall, and other nonwhite stones fall. Also, it falls regard­ less of its shape or size. If, then, these do not contribute to its falling, they are not the causes of it. In the same way the stone is hard neither because it is white nor because it is of this or that shape. Thus, the physical body, which is at first sensed confusedly as a whole and as one, is later found to be somehow a composite arid is analyzed into its principles and causes and elements. 76 The abstraction of the objects of mathematics from sen­ sible bodies, however, is not the same as that of the triangle from specific triangles or of animal from man, dog, and the rest of the spec_ific animals. In the latter case "animal" is a part in the definition of man, and a man is named both "a man'' and ''an animal,'' each of these signifying the man as a whole. The terms "man" and "animal" are in the same line of predication, and each signifies a substance; only if one calls a man "a man," he gives information which is greater and more proper to the thing named than if he calls him by the more universal name "animal." 77 In the former case the thing abstracted is not in the same line of predication, for it is signified neither by the genus nor by the differentia of that from which it is abstracted. This is also evident when we name things. We do not name a man or a physical body "a magnitude" or "a quantity," and quantitative terms are predicated of them only derivatively. The body is long, we say, or white, and both "long" and "white" are derivative terms; the one is derived from the term "length," which sig­ nifies a quantity, and the other from the term "whiteness," which signifies a quality. 78 A difficulty arises with respect to the elements of geome­ try. Sensible lines do not seem to be straight or circular as defined by the mathematician, and sensible surfaces do not seem to be plane, yet the straight and the circular and the plane are elements and principles in geometry. 79 And if the

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elements cannot exist, nothing which contains them as parts or is defined in terms of them can exist, with the result that most of the propositions in geometry will be false. However, being is divided into actual and potential, and the objects of geometry exist potentially in sensible objects. Matter in ge­ ometry is the continuous, and this cannot exist apart by it­ self but only in sensible objects. Further, just as the form of a statue is potentially in the marble, so are circles and spheres and cubes and any other figures, but only potentially; for either all of them are there or none. 80

6. Motion, Generation, Place, Position, Not Essential Attributes of Quantities We shall next consider whether or not motion, genera­ tion, place, and position are attributes of the objects of mathematics in any way. As we said earlier, there are three principles in motion, that which moves, that from which it moves, and that to which it moves, the first being the subject and the other two contraries8 1 (under the term "contraries" here we may also include states between contraries, for in motion we often oppose one contrary or state against another contrary or state, as in the case of motion from white to gray); 82 and the three kinds of motion are locomotion, alteration, and motion with respect to quantity. 83 For example, Socrates walks from the house to the Agora, water becomes warm from being cold, and iron expands from volume A to volume B. It is evi­ dent that the subject remains one and the same throughout the motion, for Socrates is still one and the same Socrates whether in the house or in the Agora; and the same may be said concerning iron and water. When we speak of the sub­ ject qua subject, then, we do not include in it any of the con­ traries, so that Socrates qua subject is not to be taken as being in this or that place, although qua body he must be in place, universally considered. 8 4 The two contraries are with-

18

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in the same genus or are themselves under contrary genera. 86 A and B are volumes, cold and heat are sensible qualities, being in the house or in the Agora is being in place, and "goodness" and "badness" are contrary genera. Moreover, the contraries are not within the same genus as the subject, for a contrary is an attribute of a subject not in the sense in which animal and man are attributes of Socrates but in the sense in which whiteness i-S an attribute of a body; that is, a contrary is present in a subject. 86 Socrates would not be Socrates or remain one and the same subject if he changed from man or animal to something else, since man and ani­ mal are essential attributes of Socrates. S_ince motion belongs to that which has a principle of being moved, this principle being sensible matter, 87 and since quantities are arrived at by the removal in thought of sensible matter from physical bodies, 88 quantities cannot be moved, that is, essentially moved. 89 Moreover, what kind of motion would it be? Quantities cannot move with respect to place, for, not being heavy or light or a mixture of the two, they cannot move up or down or in some other direction. Quantities cannot move with respect to sensible qualities. They are not warm or cold or heavy or light, and none of the other sensible qualities is an attribute of them. Perhaps they move with respect to shape, as in the case of solids or sur­ faces. But they are not hard or soft or elastic; and such mo­ tion presupposes motion in place. 90 A solid cannot move with respect to quantity, for if solid A and solid B were the two end terms of the motion, A being less than B, what would the subject underlying the motion be? If the end terms are warm water and cold water, the subject is water. In the case of A and B, that which is gained is C, the excess of B over A; and this is itself a solid and a quantity. But con­ traries are not in the same genus as the subject. Perhaps the matter underlying A and B has moved, this matter being depth or the continuous in three dimensions. But mathemat-

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ical matter is neither dense nor rare, for these are attributes of physical bodies. This is true also for a number, for its units possess no such attributes as weight, color, shape, and the rest; they are merely indivisible. 9 1 Quantities, then, do not move essentially in any way. But since quantities are in physical bodies and physical bodies do move, quantities may be said to move accidentally. 92 This is also the sense in which qualities, relations, and the rest of the accidents of physical bodies move. 93 A physical body moves from one place to another ; the quantity in that body also moves from the first place to the second, but accidentally. The body may undergo alteration only; in this case there is no motion of its quantity, ex�ept accidentally, as when we say that the wide changed from cold to warm. If the body increases from volume A to volume B, we do not have a quantity remaining one and the same of which we can at­ tribute now one contrary and now another. Now, genera­ tion and destruction differ from motion in that motion is a change with respect to an attribute of a body or a substance, but generation or destruction is a change with respect to the essence or nature of that body or substance. 9 4 Since, how­ ever, a physical substance is a composite of matter and form, 95 in generation or in destruction there is matter which as subject underlies the change. But quantities do not pos­ sess such matter; hence, they are accidentally generated or destroyed, that is, at one moment they exist and at another they do not exist, but they themselves are not in the process of being generated or destroyed. A body, for example, is di­ vided in two, and that division gives rise to two new con­ gruent surfaces; later, the two parts may be placed in their original position. Thus, before division, the two congruent surfaces do not exist except potentially in the body, but after division they exist actually; later, they again exist only potentially. There is no subject to underlie the change from the actual existence to the nonexistence and back again of

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the two surfaces, 96 and existence and nonexistence are not contraries. 97 Similarly, if the rest of the attributes change, it is only accidentally. Are the objects of mathematics in place? Only physical bodies are in place, either naturally or by force. 98 The heavier bodies are naturally down and nearer the center (of the Earth); the lighter ones are naturally up and farther from the center. 9 9 If a body is higher or lower than its nat­ ural place, then it is in place by force, since it is kept there by force. For example, rocks are held up by force, and, if re­ leased, they naturally go down. Since quantities, being ab­ stracted from all that is sensible, are not heavy or light, then they are not up or down or in some other place. 100 They cannot be heavenly bodies, like the sun and the stars, be­ cause they are neither substances nor in motion. However, quantities are in place secondarily, that is, acci­ dentally or relatively. They are accidentally in place be­ cause they are present in physical bodies which are in place; 101 and, relatively to us and to one another, they may be thought of as being up, or down, or within one another, or to the right of one another, or intersecting one another. 102 Just as being is spoken of either primarily as a substance or secondarily as an attribute, 103 so a thing is said to be in place in the same way. Only physical bodies are in place pri­ marily, as we said. One quantity may be within another, as one sphere within another sphere or a circle within a tri­ angle or a line within a line, and, in general, as one figure within another, but only secondarily. 10 4 It is because a body with a spherical shape may be within another body with the same shape that we transfer the expression to quantities and say that one sphere is within another sphere. Neither the sphere which is said to contain nor the one which is said to be contained can be truly said to be in place either naturally or by force, since neither the one nor the other is either heavy or light.

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Just as the term "place" is used primarily for physical bodies and secondarily for their surfaces and lines, so is it in the case of geometrical figures.105 A colored figure on the sur­ face of a physical body is in place, but only secondarily. In geometrical figures place is nearer to primary place when it is attributed to geometrical solids than when it is attributed to their surfaces or lines which are limits of solids. What is true of place in geometry is true of the kinds of place and of position and of the attributes that follow. Up and down, right and left, in front and behind, are differentiae of place and are absolute directions in the universe, and they are at­ tributes primarily of physical bodies. 106 In geometry one fig­ ure cannot be to the right of, or in front of, or higher than another figure absolutely, but only secondarily. Relative to us, one of two figures imagined may be to the right of an­ other, and we may keep its relative position unchanged no matter which direction we take; or we may imagine one straight line erected perpendicular to a horizontal line at one of its extremities, but by an appropriate rotation of the con­ figuration in thought we may interchange their positions. Similarly, the terms "horizontal," "vertical," etc., are pri­ marily used for physical bodies and secondarily for geo­ metrical objects. Since we arrive at numbers by abstracting not only the sensible qualities but continuity itself, and since place and things which are in place are continuous, it follows that place, position, intersection, and the like are not attributes of numbers. 10 7 Evidence of this is the fact that no such terms are used by the arithmetician in his study of numbers. A number is in place accidentally, for the subject to which it is attributed may be in place. For example, a body is in place, and the number of its volume is in place accidentally. If motion is not an essential attribute of mathematical ob­ jects, how is it that we find the mathematician moving his geometrical figures from one place to another in his attempt

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to demonstrate theorems concerning them? Further, do the geometrical figures change in shape or size when they are so moved? Thus, to demonstrate the equality of two triangles, one having two of its sides and the included angle equal to two corresponding sides and the included angle of the other, the mathematician so moves one triangle that it is made to coincide with the other triangle; and, to demonstrate the equality of the opposite sides of a parallelogram, he divides the latter and then demonstrates the congruence of the re­ sulting triangles. A physical body, say, T, may be moved and be placed upon another body so as to have part of their surfaces coin­ cide; consequently, the surface of T is also moved, but acci­ dentally. Similarly, if T is divided, its surface is also di­ vided, and so the shape of that surface is divided and de­ stroyed, but accidentally. If T changes in shape or size when moved from one place to another (i.e., from cold to warm surroundings), then it changes because it is being acted upon by the surrounding body and because it has a principle of being acted upon. Since, however, quantities have no such principle and hence neither move nor are in place essen­ tially, any change attributed to them by the mathematician is accidental. Consequently, geometrical figures neither gain nor lose attributes arising out of their own nature by being made to move accidentally by the mathematician from one place to another; and, if they change accidentally in size or shape by such motion, the new attributes are no longer at­ tributes of the original figures. In short, the mathematician is concerned with the attri­ butes of motionless objects, and in this he differs from the physicist, who is concerned with the attributes of physical bodies in so far as they have a principle of motion. 10 8 If so, one need not fear that the triangle ABC might change its size or shape by being moved by the mathematician, nor is such motion essential to the demonstration that triangles· ABC

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and A' B'C' are congruent. The distance between the two triangles and their relative position do not affect their con­ gruence or the equality of their areas, for these arise only out of the natures of the triangles. Likewise, the division of paral­ lelogram PQRS into triangles PQS and QSR is accidental and does not change the length of its sides or the angles P and R. Since quantities do not change essentially, perhaps it is better to change the mode of expression and to regard one geometrical figure now next to another figure, now upon it, and now related to it in some other way. For the sake of the learner, however, such accuracy need not be present in the demonstration. In introducing him to mathematics, per­ haps it is better to present the science in terms familiar to him. 109

7. How Quantities Are Known Since quantities are arrived at by the removal in thought of all that is sensible in physical bodies, they cannot be known directly by sensation. This is also evident by enumer­ ating the faculties of sensation. They are five : vision, hear­ ing, taste, smell, and touch. 110 Their corresponding objects are colors, sounds, flavors, odors, and in the case of touch such differentiae of bodies as hardness and softness, heat and cold, and wetness and dryness. 11 1 Each of the faculties is lim­ ited to its objects and cannot sense directly the objects which come under another faculty.112 Hearing is limited to the genus of sounds, and it alone can sense directly sounds and their differences. Vision, taste, or any of the other faculties cannot be properly said to sense a sound. In the same thing, however, one accident may always, or for the most part, fol­ low another, or both may follow each other; and, on sensing the one, one may conclude that the thing has also the other. On seeing something white, one may conclude that it is sweet, taking it to be sugar; and, on smelling something, one may conclude that it is yellow and transparent, taking

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it to be oil. Hence, substances, too, are somehow known through sensation. In the first case, vision is said to sense ac­ cidentally the sugar or its sweetness, and, in the second, smell is said to sense accidentally the oil or its yellowness. Neither the sugar as a substance nor its sweetness directly affects vision, and likewise with the second case. 113 The term "proper sensibles" is applicable to objects proper to some faculty of sensation in so far as they are sensible by that faculty (i.e., colors in the case of vision); and the term "ac­ cidental sensibles" is applicable to substances or to objects proper to some faculty of sensation in so far as they are sensible by another faculty (i.e., colors and substances in the case of hearing). 11 4 In sensing the proper sensibles, there is no error (or there is very Iittle error, as in the case of a defective organ),115 but in the case of accidental sensibles there is error. 116 The white may be bitter and not sweet, salt and not sugar. Shape, magnitude in general, number, motion, and rest are somehow sensed through more than one of the five facul­ ties, and, in so far as they are so sensed, they are called "common sensibles. 117 The shape of a coin or of a ball or of something triangular, the area of a sheet of paper, the vol­ ume of a book, the corners of a cube, can each be sensed by touch as well as by sight. Motion is sensed by sight, as in the case of seeing a horse running; by touch, as in the case of touching a rotating wheel; by hearing, when we are listening to a rising pitch; and similarly by the other two faculties. We come to know number through a succession of different colors or different tones or spoken syllables, and the length or distance between two points by going over it or measur­ ing it. Thus, we come to know the common sensibles by means of motion, either in us or in the objects. 11 8 Rest itself is likewise known, for it is the privation of motion in that which naturally admits of motion, and motion is prior in

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definition to the privation of motion; 1 1 9 and there is motion somehow in the soul while the thing is resting.120 Like the accidenta] sensibles, the common sensibles do not directly affect the faculties of sensation. 121 Vision is not af­ fected by the boundary line within a sheet of paper painted red on the upper half and green on the lower half but only by the red and green colors. Yet there is a difference be­ tween the common and the accidental sensibles. The bound­ ary line is judged by virtue of the differences in the two colors. Vision being affected now by the red coming from the upper part and now by the green coming from the lower part, one judges that there is a boundary· line between the two at which the one ends and the other begins. Likewise, we come to know number through a succession of different colors or different tones, etc. Further, vision may be en­ tirely mistaken as to that which is white, for it may be bitter and not sweet, salt and not sugar, but it cannot be so mis­ taken as to the boundary line within the paper. However, there is error in judgment in the case of the common sen­ sibles, too, especially with respect to the lower differentiae. One truly judges the sun to have a magnitude, but he may judge it to be smaller than it is; and, similarly, he may judge the rod to be longer than it is, the circular shape to be el­ liptical, and ten horses to be nine. 122 In general, vision is superior to the other faculties in judging accurately the com­ mon sensibles. 123 The common sensibles, then, seem to oc­ cupy a place between the proper and the accidental sen­ sibles. 124 It was stated that the common sensibles as such do not affect the faculties of sensation and are known by them acci­ dentally but not purely accidentally, as in the case of the accidental sensibles, because motion and the differentiae of the proper sensibles contribute to the knowledge of the com­ mon sensibles. This is also evident from the fact that no

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previous knowledge is required to judge that a given object seen for the first time has a magnitude or is moving, but pre­ vious knowledge is required to know how it tastes. More­ over, whether a common sensible comes to be known through one faculty of sensation or through another, our knowledge of it is the same. Our knowledge of the number of three men qua three is the same, whether we see them, hear them, or sense them in some other way. If so, there must be some faculty which senses or judges the common sensibles, does so by way of the faculties of sensation, and to which the faculties of sensation terminate and thus contrib­ ute to the knowledge of the common sensibles. Let this fac­ ulty be called "common sensation. " 125 The number of the faculties of sensation and the extent of the discriminating capacities within each of them help a great deal and reduce the error in receiving knowledge of the common sensibles. 126 Lack of vision, the most accurate of the faculties and most used in acquiring knowledge,127 would retard such knowledge considerably. If we had only one faculty, vision, for example, and if this were limited to white colors without any differences as to brilliancy, shade, etc., all things would be seen as white, as one expanse of whiteness, and it would be difficult if not impossible to dis­ tinguish the surface from the color in the things. 128 As it is, there are white balls and green balls and red balls, and the roundness of a white ball is easily distinguished from its whiteness; and touch gives us the same sensations of all, be­ cause color is different from surface and does not affect touch. It was discussed elsewhere that the faculties of sensation are only five. 129 Since nature always tends for the best,130 it is reasonable that man, the most complete of animals, 1a 1 should have all the faculties of sensation and that he should use vision most for the sake of contemplation, his proper activity . 132

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From what has been said it is evident that knowledge of the common sensibles and therefore of quantities cannot be received unless we possess the faculties of sensation. 133 As a theoretical science, mathematics is concerned with quantities universally and not individually, and hence its principles, which are axioms and hypotheses and defini­ tions, are universal and predicable of many individuals. 13 4 For example, if A, B, and C are individuals, triangle A may be present in body B and be greater than triangle C. Mathe­ matics, however, is not concerned with such individual at­ tributes of A; it is concerned only with such of its attributes which belong to it in so far as it is a triangle, or a triangle of some kind, or a polygon, or a quantity of some kind, or just a quantity, or a triangle in some relation to some other spe­ cific or generic quantity universally considered. The terms "triangle" and "isosceles triangle" and the corresponding definitions are universal, and the axiom "Equals from equals, the results are equal" is universally true for all kinds of quantities. A question arises whether the principles are inborn in us or are acquired and, if they are acquired, how they are ac­ quired. It is absurd to admit that they are inborn, for that would be to admit that we fail to notice our most accurate knowledge. 135 Plato says that they are inborn but that knowledge is recollection. 136 If this were so, why should eminent men hold contradictory views about the same thing? Moreover, recollection of colors and their properties is useless to men who are born blind, for such men are in­ capable of ever forming ideas of colors, much less of under­ standing propositions concerning them. 137 Hence, since the principles are acquired and are not things that can be sensed, but since they cannot be acquired without the sensa­ tion of particulars, there must be a faculty in us, other than that of sensation, which enables us to arrive at them after the sensation of particulars.

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All animals have the faculty of sensation. In some of them, after the sensation of a particular, there is something left behind, an image or a memory of that particular. Man has additional faculties. Many memories of the same par­ ticular generate one experience of that particular, and from many experiences of particulars of the same kind one universal can be generated. From the sensation of Socrates and Callias and other indiyidual men arises the idea of man, from the ideas of man and horse and lion arise the ideas of animal and of biped and of reason, and from the ideas of animal and of plant arise the ideas of substance and of sen­ sation and of life; and this process can go on until the ulti­ mate elements of the definition of man are reached. Again, from the particular fact that seven men remain in each case when each of two committees of ten men is reduced by three and from equal reductions in other particular cases of equal numbers arises the knowledge that equal numbers re­ main when equal numbers are taken away from equal num­ bers; and from this truth and similar other truths about lines, surfaces, solids, time, and the rest of the quantities arises the axiom that equal quantities remain when equal quantities are subtracted from equal quantities. Thus, start­ ing from sensation of particulars, we arrive at universals by abstraction and induction. If these universals deal with gen­ eration, they are principles of art, as in the art of war or of carpentry or of healing; if they are necessary truths about things, they are principles of theoretical science, as in mathe­ matics and physics; and if they are truths which do not hold of necessity but hold for the most part, they are principles of the approximate sciences or of sciences spoken of in a secondary sense. The faculty which corresponds to the prin­ ciples of a theoretical science is called the "intellect." 13 8 The intellect differs from opinion or scientific knowledge, and it is not clear whether truth and falsity are attributed to all three; so let us first define truth and falsity.

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As it was stated earlier, some things or facts exist of neces­ sity and others exist but not of necessity. 13 9 In the first case, the attribute A belongs to the subject B of necessity, or else A of necessity does not belong to B; for examp]e, sixteen is of necessity a square number, and the diagonal of a square of necessity is not commensurable with the side. In the second case, A is an attribute of B or is not an attribute of B but not of necessity; for example, it may be true that all five-year­ olds have two feet, but this is not a necessary fact, since it is possible for some of them not to have both feet. Now truth and falsity are in the soul (in thought) and not in things. 1 4 0 If A is an attribute of B (or is not an attribute of B) in a cer­ tain manner (of necessity or not of necessity), and if we think that A is an attribute of B (or is not an attribute of B) in the same manner, then we are said to think truly concern­ ing this relation of A to B; and let this be the definition of truth. Similarly, we are said to think false]y concerning the relation of A to B if A is not related to B in the way we think. 1 41 We are said to have opinion concerning the relation of A to B if we think that A is an attribute of B (or is not an at­ tribute of B) but not of necessity; consequently, we have true opinion concerning the relation of A to B if we think truly concerning that relation and false opinion if we think falsely. But if A of necessity is an attribute of B or is not an attribute of B, to have opinion concerning the relation of A to B is to think that A is an attribute of B or is not an attri­ bute of B but not of necessity; hence, opinion of necessary relations must be false. 1 42 If A of necessity is an attribute of B (or is not an attribute of B), we are said to have scientific knowledge of it if we think that A of necessity is an attribute of B (or is not an attribute of B) and if we have a demonstration of this fact from its first principles and causes. 1 43 This scientific knowl­ edge is sometimes called "absolute"; for we also use the

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term "scientific knowledge" in a qualified way, and this is when we have knowledge without demonstration of a fact which is demonstrable. 1 4 4 If we think, however, that A of necessity is an attribute of B when this is not the case (either because A of necessity is not an attribute of B or because A is or is not an attribute of B but not of necessity), or that A of necessity is not an attribute of B when this is not the case (for similar reasons), theq we are said to be ignorantly dis­ posed. We are also said to be ignorantly disposed when we have false opinion. 1 45 We must distinguish ignorance due to disposition from what we may call "negative ignorance." We are said to be ignorant in this latter way when we are in no way disposed toward a fact, and in this way we do not differ from inani­ mate objects or plants. 1 46 The axioms, hypotheses, and definitions, of which there is no demonstration, are known by the intellect. 1 47 Since the axioms and hypotheses are necessarily true, for in them something is either affirmed or denied of something else and first principles cannot be false, the intellect which knows them is true. But if we affirm that which should be denied or deny that which should be affirmed or are in error in some other way concerning them, then we have false opinion of or are ignorantly disposed toward them; and it is negative igno­ rance if we are in no way disposed toward them. In the case of essence or definition, it is no longer a matter of whether an attribute belongs or does not belong to a subject, for no such relation exists. Of them there is either knowledge or negative ignorance, and it is the intellect which conceives them. If the definition of a triangle is "a three-sided plane figure," for example, it is either conceived by the intellect or it is not. The intellect here may be compared to vision, and the disposition of the intellect when it conceives the essence of a triangle may be compared to the disposition of

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vision when it sees white. Just as sight cannot be mistaken about its proper objects, which are colors, so the intellect cannot be mistaken about its proper objects. However, if there is a problem of whether the definition ' ' a three-sided plane figure" belongs or does not belong to an object before us, then it is a question of an attribute belonging or not belonging to a subject; and it is opinion, which is true or false, and not the intellect that affirms or denies the at­ tribution or else it is ignorant disposition. Hence, the intel­ lect cannot be false; and, when it is said that the intellect is always true, by "truth" we are to understand that the intel­ lect cannot be mistaken about the principles it conceives or knows. 1 4 8 From what has been said it is evident that universals do not exist as things separate by themselves, 1 49 nor do they exist in the individuals of which they are attributes or predi­ cates, except accidentally . 150 The universal "animal" does not exist in a particular horse, for if it did, being one and the same, it could not exist in any other animal; nor can it exist in Socrates except accidentally, that is, not in the na­ ture of Socrates but only in his intellect as knowledge. Uni­ versals exist in the forms of sensation which are in the fac­ ulty of sensation, and hence, without the possibility of ever sensing an object or its principles, we can neither acquire nor have the corresponding universal. If one is color-blin1 with respect to red, one can neither acquire nor understand the universal "redness" or any universal in which "redness" is a principle. When we think or theorize about redness, we must have an image of red before us, and it is in this image that the universal "redness" exists and is thought by the thinking faculty. Quantities, too, which are without sensible qualities, such as color, hardness, and weight, are thought of and conceived as universals in the images of sensible bodies, not as existing

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apart from the sensible bodies or the sensible qualities in those bodies, but qua different from these and considered by themselves. 151 The relation of images to universals is in some way analogous to that of particular triangles used in demonstra­ tion to the universal term ''triangle'' or to universal proposi­ tions about triangles. The triangle drawn has three sides but also a definite area and ,. other such attributes; yet we may mention only three-sidedness in the demonstration, or just sidedness, or even less than that. Similarly, we have an im­ age of an isosceles triangle before us, yet we may think of that triangle as a three-sided figure, or as a surface, or as continuous; but the continuity, or the surface, or the three­ sidedness cannot exist apart from the triangle of which we have the image. 152

8. Method in Mathematics As a theoretical science, mathematics proceeds according to a method. This is especially true of the first parts of mathematics, arithmetic and geometry, whose elements have been abstracted from all that is sensible. Each of these sciences, having few and definite principles and elements, lends itself easily to demonstration. In this respect, mathe­ matics differs from physics. Physics investigates sensible ob­ jects in so far as they are in motion, and moving bodies qua moving require, in order to be known scientifically, all four causes: a formal cause, a moving cause or mover, an under­ lying matter or material cause, and a final cause or that to­ ward which they move. 153 The objects of mathematics, on the other hand, require only a formal cause in their defini­ tion,16 4 and a material cause only in a qualified way, as will be discussed later. Consquently, the principles, causes, and elements of physics are more numerous and harder to arrive at than those of mathematics and require more experience and careful observation. Evidence of this is what happens to

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young people who study mathematics and physics. They often become geometricians and mathematicians but not physicists, and they learn and give mathematical demon­ strations more from memory and less from conviction. 165 Being thus an accurate science, 156 mathematics hardly needs dialectical discussion for the introduction of its principles and elements; but definitions and the other principles are laid down at the start, and properties of numbers or of mag­ nitudes are demonstrated. Indeed, mathematicians who have been little exposed to a method other than their own find it difficult to accept anything but a demonstration in ap­ proaching other subjects; their own subject being without sensible matter and hence hardly receptive of the more and the less, they demand accuracy and demonstration from a subject which may not admit of them to the same extent as mathematics. 15 7 Since the mathematician is concerned with a genus of things and, starting with principles which are true of those things, proceeds to investigate the attributes belonging to them, let us say a few things about that genus, its principles, and the investigation of its attributes. It was stated elsewhere that first philosophy is concerned with things in so far as they are just things (with being qua being) and the attributes which belong to them as such. 158 Being in the primary sense is substance, for substance alone is separate. Quantities, qualities, relations, and the rest are also called "being" because they are present in substances. 159 The term "being" is analogous to the term "health." Some things are called "healthy" because they preserve health, others because they bring about health, still others because they are a sign of health. All, howev:er, come under one sci­ ence, medicine, which is the science of health. In the same way, some things are qualities of primary being which is substance, others are quantities of it, still others are motions or relations or actions, but all of them are attributes of sub-

34

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stances; and it is the task of first philosophy to study sub­ stances and their attributes qua natures and qua existing. Consequently, it is the task of first philosophy to inquire whether the genus of quantities exists or not and, if it exists, how it exists. 16 0 But, since it belongs to the same faculty to inquire if a thing exists and what the thing is, it belongs to first philosophy to discuss what quantities are and to give the first definitions. 161 Moreoyer, such axioms as "Equals from equals, the remainders are equal" and "If a quantity is greater than the greater of two quantities, it is greater than the lesser" are believed to be true for all quantities qua quan­ tities. Hence, it is first philosophy that investigates and dis­ cusses such axioms also. 162 The principles of science, then, are three in kind: axioms, hypotheses, and definitions. 163 An ex­ ample of a definition is : "A unit is a nature indivisible ac­ cording to quantity and without position"; an example of a hypothesis is : "Units exist"; and examples of axioms have already been given. To mathematics belongs the task of demonstrating whatever is demonstrable from these prin­ ciples, for mathematics is a demonstrative science; and the aim of such a science is scientific knowledge, that is, knowl­ edge reached by demonstration. 16 4 The aim of a theoretical science, then, is scientific knowl­ edge. We are said to have scientific knowledge of a fact when we know the first causes through which the fact is as being the causes of that fact and when we know that the fact is necessary, that is, incapable of being otherwise. 165 An ex­ ample of a necessary fact is the fact that the diagonals of a parallelogram bisect each other, and to say that this is not or may not be so is necessarily false. To know the causes of a fact, it is necessary that we have syllogistic knowledge of the fact. A syllogism is defined as discourse in which, certain things being laid down, some­ thing else follows necessarily from them. What we lay down are two premises, and what follows is the conclusion. Each

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premise has two terms, one of which, called the "middle term," is common to both premises. 166 For example, from the premises "Every straight line is a magnitude" and "Every magnitude is infinitely divisible" follows "Every straight line is infinitely divisible," "magnitude" being the middle term; and from "If one magnitude is greater than the greater of two other magnitudes, then it is greater than the lesser" and "A is greater than B, and B is greater than C" follows "A is greater than C.' ' The kinds of syllogisms, when and how they are produced, have been discussed in detail in Prior Analytics and are assumed here. Since a fact, of which there is scientific knowledge, is necessary and consequently has causes which are necessary, it follows that the proposition signifying that fact is neces­ sarily true and that that proposition, in so far as it is scien­ tifically known, is the conclusion of premises which are them­ selves necessarily true. If the premises were true but not necessarily true, and if the conclusion were necessarily true, then, although the conclusion would necessari]y follow from the premises, it would be necessarily true not because of the premises; and hence the premises would not be the causes of the conclusion in so far as the conclusion is necessarily true. 16 7 For example, at one time it may be true to say that all men wear shoes and that all those who wear shoes are mortal, and from these true premises "All men are mortal" would follow, a conclusion which is necessarily true. But the necessary truth of the conclusion does not follow from the two premises, for the premises are true but not of necessity. The premises are not causes of the conclusion, except acci­ dentally, for the middle term is not necessary; that is, men are mortal not because they wear shoes. Let a demonstration be defined as a syllogism whose premises are necessarily true and are either the first causes of the conclusion or are them­ selves the conclusions of necessarily true premises which are first causes. It is evident from what has been said that scien-

36

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tific knowledge of a fact is knowledge of it reached through demonstration. The first principles in a theoretical science are true (definitions are true in a qualified sense), necessary, in­ demonstrable, prior in knowledge to, better known by na­ ture than, more believed than, and first causes of the conclu­ sion. 16 8 The axioms and hypotheses are true, for we have framed our definition of scientific knowledge so as to deal with being and not with not-being; there is no scientific knowledge of the nonexistent. 16 9 It is easy to lay down false premises or draw conclusions in any manner, but there is little worth in such activity. 17 ° They are necessarily true, for the unchang­ ing is prior in existence to, more noble than, and the cause of the changing. 171 In a secondary sense, however, we may say that a science is of that which is true for the most part, but we are not concerned with such a science here. 172 The first principles are first causes of the conclusion, for it is from these and through these that we have scientific knowledge of the fact. 173 They are indemonstrable, for to admit that there is demonstration of them is to admit that they are not first, and an infinite regress would make the number of causes (if these be causes at all) infinite and sci­ entific knowledge impossible. But it was shown elsewhere that, since the causes and indefinable terms of a definition and the kinds of nonaccidental predication are finite, if A is a predicate of the subject B nonaccidentally, then the middle terms between A and B are finite, and the number of indemonstrable principles through which it is demon­ strated that A is a predicate of B is finite. 174 By a nonacci­ dental predicate of a thing we mean a property of it or a term in its definition, such as "the equality of the sum of the angles to two right angles" and "continuous" in the case of the triangle; for "white" is an accidental predicate of a tri­ angle, and the predication of one property of another prop-

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erty is also excluded, since the underlying subject is missing and since the properties of the triangle are posterior in knowledge to the principles and infinite in number. The first principles are prior in knowledge to the conclu­ sion; for if A is a predicate of C and if the middle term be­ tween A and C is B, understanding of the terms A and B and C is prior to knowledge of the fact that A is a predicate of C, and knowledge of the axioms used and of the facts that A is a predicate of B and B a predicate of C is prior to dem­ onstrated knowledge of the fact that A is a predicate of C. 1 75 The first principles are better known by nature than the conclusion. 1 76 What is better known by nature is not what is better known to us at first. The latter is that which is nearer to sensation and that which is a confused whole, but that which is better known by nature is that which is later reached by analysis; and this is simpler or more universal or both. For example, at first we have experience of sensible surfaces, and then we get a confused idea of surface, univer­ sally taken ; but by later analysis and abstraction we are able to define a surface as a two-dimensional continuous quan­ tity. Two-dimensionality is simpler than surface, and con­ tinuity and quantity are simpler and more universal than surface. In the same way the first premises and axioms are simpler and better known by nature than the conclusion. They are simple; that is, they are indemonstrable or without middle terms or causes. 1 7 7 And they are better known by nature, for if B is known through causes A, the causes are better known by nature than that which is caused. Similarly since it is through the principles that the conclusion is dem­ onstrated, we should have more conviction of the truth of those principles than of the truth of the conclusion ; and, if scientific knowledge is to be unshakable, our conviction of the truth of those principles must be the greatest, since any conviction of contrary or contradictory principles would lead to positive ignorance. 1 7 8 For example, we should be

38

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more convinced of the truth of such principles as "Equals added to equals, the results are equal," "The sum of the angles about a point on a straight line is equal to two right angles," and "The alternate interior angles of two parallel lines are equal" than of the truth of the theorem that the sum of the angles of a triangle is equal to two right angles (Fig. 1 ).

Fm. 1

A definition is a formula signifying the essence of a thing or what the thing is; 1 7 9 and, since the terms " essence" and "thing" are predicates of only what exists or is possible, definitions are limited only to what exists or is possible. 180 For example, if a triangle is defined as a three-sided plane figure, then "triangle" and "three-sided plane figure" have the same significance; and triangles exist or are possible. To say that "triangle" and the definition of triangle have the same significance is to say that the definition signifies the triangle with respect to the latter's essence or nature and not with respect to any of its demonstrable attributes or proper­ ties. That which is a triangle is also a sided plane figure whose angles are equal to two right angles, and vice versa; but the equality of the angles to two right angles is a demon­ strable property, and the terms "triangle" and "a sided plane figure whose angles are equal to two right angles" do not have the same significance. In the example given both "triangle" and "three-sided plane figure" signify one thing, the first signifying the thing as a whole, the second sig­ nifying it in terms of its principles, genus and differentia. 1 8 1 We say that the definition signifies one thing, for a definition is not a mere juxtaposition of terms; it is one thing in the sense that it signifies one thing. 182

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If a thing has a definition, clear knowledge of it requires that we have knowledge of its definition; for the thing is a kind of a whole with elements or parts or causes, and, if these cannot be distinguished in the thing, the result would be confused knowledge of it. 1 83 If a part in the definition is definable, its definition must also be known. Hence, since the definition of a thing cannot have an infinite number of terms, clear knowledge of the thing requires knowledge of the ultimate indefinables. 1 8 4 In the example given "figure" is the genus of triangle; but a figure is a kind of a magni­ tude, and a magnitude is a continuous quantity, and quan­ tity is indefinable. Indefinables are also found in the dif­ ferentiae and in the definitions of the rest of the attributes of quantities. For example, continuous is defined as that the parts of which are so joined that their ends ·are in the same place and one, 185 and some parts of this definition are in­ definable. Similarly, circularity, primeness, and parallelism are definable in terms of indefinables. Some definitions are immediate and first causes; 1 86 others are investigated by means of demonstration, and these there­ fore exhibit the cause. 18 7 For example, the first elements, such as points and units and magnitudes, are defined im­ mediately and are first causes. But the eclipse of the moon has a cause, namely, the interposition of the earth between the sun and the moon; for if we ask why there is an eclipse� we are seeking for the cause, and the interposition of the earth is the moving cause. If the eclipse is defined as the privation of the moon's light by the interposition of the earth, the cause is included in the definition. Such defini­ tions differ from demonstrations only in the manner of wording, for the corresponding demonstration in the ex­ ample given has as premises "The moon is so-and-so situ­ ated relative to the sun and the earth" and "All things so­ and-so situated are deprived of light." But if the eclipse is defined as the privation of the moon's light, then the cause

40

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is not included; and the definition differs in wording from the conclusion of the above demonstration (i.e. , from "The moon is deprived of light"). Similarly, squaring of a rec­ tangle may be defined as the finding of a square equal in area to the rectangle, and the cause is absent in this defini­ tion; but, if it is defined as the finding of a square whose side is equal to the mean between the sides of the rectangle, then the cause is included. Since quantities are immovable and abstracted from sensible matter, definitions of them and of their attributes contain formal but not final or moving or material (i.e., sensible material) causes. 188 The differentiae in a definition must be given in the proper order. 189 The first differentiae of quantity, for ex­ ample, are not the odd and the even, for these are the dif­ ferentiae of number, and continuous quantities would be left out; and they are not the circular and the straight, for these are the differentiae of lines, and surfaces and solids and numbers would be left out. But if quantities were subdivided into continuous and discrete, all of them would be consid­ ered. 19 0 Similarly, all continuous quantities come under three-dimensional, two-dimensional, and one-dimensional. The contraries are the principles according to which sub­ division into differentiae takes place, and the first contraries are privation and possession. 1 91 In colors, for example, pos­ session and privation are attributed to white and black, re­ spectively; and all other colors are intermediate between these. 192 In continuity, possession belongs to the three­ dimensional, for this is complete among continuous quan­ tities; next comes the two-dimensional, and this is deprived of one dimension; finally, we have the one-dimensional, which, among the continuous, is most deprived. 193 The genus should not be subdivided by negation, as in the subdivision of quantity into circular and noncircular or into odd and not-odd; for no positive property can be demon­ strated of, or be attributed to, quantities from negating them

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something, and there are no differentiae of not-being qua not-being. 1 94 There are many such dichotomies possible which could become differentiae of quantities, and quanti­ ties of different genera would come under a negative differ­ entia with no definite method of further subdivision. Since further subdivision by differentiae is possible, a thing may have an ultimate genus, a proximate genus, and genera between the two. 195 For example, the ultimate genus of a triangle is "quantity"; the next is "magnitude" (mag­ nitude being defined as continuous quantity contains the first differentia); the next is surface; and, continuing down the line, we come to the last or proximate genus, rectilinear plane figure or odd-numbered rectilinear plane figure or something of the sort. The ultimate genus is the most indeterminate among the genera along the same line of predication, and as such it is akin to matter. 196 As it takes on each differentia, it becomes more and more determinate, so the diflerentia must be re­ garded as akin to form. 197 Since a quantity is one thing and not many, the genus "quantity" does not signify intelligible matter. When predicated of the triangle, for example, that genus signifies one thing but indeterminately. The differen­ tia of triangle likewise signifies the triangle as one, for it is predicated of it, but it signifies it determinately. Of intel­ ligible matter, on the other hand, one cannot truly predicate unity or being, and in this respect it is unknown qua mat­ ter. 1 98 Since, however, the concept of a triangle and that of a three-sided-bounded-plane-two-dimensional-continu­ ous quantity are one and the same, the genus and the differ­ entia as concepts cannot be one unless they are somehow related, and they are related as matter to form. 199 Thus, an­ other sense of matter is the genus in a definition. It is matter in the sense of indetermination, and it becomes determinate by the differentiae which act as form. The genus, speaking analogically, acts as matter or as a subject, 200 receiving now

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one differentia and now another as form, although motion is not attributed to the genus as such. The question of what holds the parts of a triangle or of any other quantity together and so causes them to be one, whether it is the soul, as in an animal, or adhesion, or cohesion, or something else, belongs to another science and does not concern the mathemati­ cian. 2 0 1 Hypotheses are true and indemonstrable propositions laid down and signifying that something is or is not the case. 2 0 2 For example, "Units exist" and "A square is a parallelo­ gram" are hypotheses. Hypotheses are first principles and premises serving as material for scientific knowledge. Concerning the first elements in the mathematical sci­ ences, such as points and units and lines, the mathematician assumes both their definitions as laid down by the philoso­ pher and their existence. Triangles, parallelograms, pyra­ mids, and perfect numbers are composites (i.e., composed of first elements), and their existence or possibility must be demonstrated by the mathematician. 2 03 A composite whose existence is demonstrable must have a definition, for, being one thing and demonstrable and having causes, it is not in­ definable; and definitions are only of things which exist or can exist. Now, since before the demonstration of the ex­ istence of a composite it is not yet known whether the com­ posite exists or can exist or not, it cannot yet be known whether that composite can have a definition or not. Conse­ quently, a formula signifying the composite may be given at the beginning, and the demonstration of the existence of the composite would then show that the formula is a definition. For example, if the term "triangle" is introduced as signi­ fying a three-sided plane figure, then we have a formula and not a definition (except by hypothesis). That the formula is a definition has to be shown by demonstrating that a three­ sided plane figure exists or is possible. That a triangle exists or is possible can be shown by construction; hence, the for-

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mula of a triangle can be shown to be a definition . If, on the other hand, the formula for a noncompunctual triangle is given as "a triangle the angle bisectors of which do not meet at a point," then, although the terms of that formula have significance, it can be demonstrated that no such thing is possible; and so the formula cannot be a definition . Axioms are true principles without whose knowledge it is impossible to know other things.2 0 4 Axioms are not prin­ ciples in the sense of premises or hypotheses serving as ma­ terial for conclusions to be drawn; they are principles from which or according to which demonstration proceeds. 206 For example, the axiom "It is impossible for an attribute to be­ long and not to belong to the same subject at the same time and in the same respect" is true of all things; it is the most certain of all principles, and, without it, it is impossible to have any knowledge at all. 206 Since a mathematical science is concerned with some genus of things, the axioms of all things are also axioms of the things in that genus. If, for example, geometry is con­ cerned with magnitudes, then the axioms of all things are also axioms of magnitudes, for magnitudes are things. In ad­ dition, axioms proper to each genus of things are true prin­ ciples without which it is impossible to have knowledge of things in that genus. In mathematics, such are the axioms that equals from equals, the remainders are equal; that if one quantity is greater than a second, and the second is greater than a third, then the first is greater than the third; and that, if equals are added to equals, the sums are equal. 2 0 7 These axioms belong to magnitudes not qua con­ tinuous but qua quantities. Hence, they are proper to uni­ versal mathematics, which is concerned with quantities ir­ respective of their continuity or discreteness. Taking an example (Fig. 2), to demonstrate that, if O is the center of the circle and E is in the circle at a distance from 0, chords CD and AB intersecting at E do not bisect

44

ARISTOTLE'S PHILOSOPHY OF MATHEMATICS

each other, let us assume that they do bisect each other. Since AB and CD bisect each other, E is the midpoint of chords AB and CD; and, since a straight line joining the center of a circle to the midpoint of a chord is perpendicular to that chord, OE is perpendicular to chords AB and CD, and therefore angles OEB and OED are right angles, which must be equal . But angle O..ED is a part of aegle OEB, and the whole is greater than a part of it. The two angles, then, are not equal . So they are equal and not equal . But it is im­ possible for two contradictory propositions to be true (or it is

D

Fm. 2

impossible for a thing to have contradictory attributes at the same time and in the same respect) . Consequently, the first assumption is false . We have used here two axioms, each from a different science. They are : " I t is impossible for two contradictory propositions to be true" and "The whole is greater than a part of it." The first is an axiom for all propositions (or for all things) and is an axiom in the science of demonstration ; the second is an axiom in universal mathe­ matics and is true of all quantities and of quantities only. As a demonstrative science, mathematics starts from first principles and investigates demonstratively the attributes of its genus. Such attributes are oddness, evenness, commen­ surability, inclination, parallelism, and the rest. Here, as before, what the term " oddness" or " odd number" signifies is assumed by the mathematician at the start ; but that

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oddness or an odd number exists has to be demonstrated through the first principles. If its existence is demonstrated, then its formula is shown to be a definition. This is also true for the rest of the attributes. 208 Since the genus is a part of the definition of the species and the attributes of the genus are also attributes of the species, why, it may be asked, is it necessary to have a science of the genus if the science of the species would give us all the at­ tributes of the genus? If, for example, the sciences P, Q, R, and S are concerned with equilateral triangles, scalene tri­ angles, isosceles triangles, and triangles, respectively, it would seem that one studying P, or Q, or R would be also studying S, for the principles in S are also in P, and in Q, and in RJ· and all the attributes of a triangle are also attributes of an isosceles triangles, and of an equilateral triangle, and of a scalene triangle. If, for example, the angle bisectors of a tri­ angle meet at a point, so do those of a scalene triangle, for the latter is a triangle. On the other hand, since many at­ tributes are common to all the species, if the science of the genus is not necessary, then such attributes will be studied as many times as there are species. A theorem in the genus will be stated slightly differently from the corresponding theorem in each of the species. The theorem in P will be "The angle bisectors of an equilateral triangle meet at a point" (and similarly for those in Q and in R) , and that in S will be "The angle bisectors of a triangle meet at a point." If one knows a theorem in P, it does not follow that he knows the corresponding theorem in Q or in R, nor does it follow that a corresponding theorem in Q or in R exists; consequently, neither will he necessarily know the corresponding theorem in S, nor will it follow that such a theorem in S exists. To know such a theorem in S, if S be regarded as superfluous, it is necessary to have three demon­ strations, one in P, one in Q, and one in R. One might think that three demonstrations are not necessary, since one would

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have to go through the same steps in P, in Q, and in R. This would be so if there were a unique demonstration for each theorem. 20 9 Two ways will be given in which it can be demonstrated that the angle bisectors of the isosceles triangle ABC, in which AC = BC, meet at a point (Fig. 3). Since base angles of an isosceles triangle are equal, < A = < B. Let AE and BD be the angJe bisectors of