the former. The same is also true of mechanics. Thus if we regard objects independently of their attributes and investigate any aspect of them as so regarded, we shall not be guilty of any error on this account, any more than when we draw a diagram on the ground and say that a line is a foot long when it is not; because the error is not in the premisses. Cf. Aristot. Met. 14.2.9, 10. The best way to conduct an investigation in every case is to take that which does not exist in separation and consider it separately; which is just what the arithmetician or the geometrician does. For man, qua man, is one indivisible thing; and the arithmetician assumes man to be one indivisible thing, and then considers whether there is any attribute of man qua indivisible. And the geometrician considers man neither qua man nor qua indivisible, but qua something solid. For clearly the attributes which would have belonged to "man" even if man were somehow not indivisible can belong to man irrespectively of his humanity or indivisibility. Hence for this reason the geometricians are right in what they maintain, and treat of what really exists; i.e., the objects of geometry really exist. For things can exist in two ways, either in complete reality or as matter. i.e., potentially. And since goodness is distinct from beauty (for it is always in actions that goodness is present, whereas beauty is also in immovable things), they Cf. Aristot. Met. 3.2.4.
are in error who assert that the mathematical sciences tell us nothing about beauty or goodness; for they describe and manifest these qualities in the highest degree, since it does not follow, because they manifest the effects and principles of beauty and goodness without naming them, that they do not treat of these qualities. The main species of beauty are orderly arrangement, proportion, and definiteness; and these are especially manifested by the mathematical sciences. And inasmuch as it is evident that these (I mean, e.g., orderly arrangement and definiteness) are causes of many things, obviously they must also to some extent treat of the cause in this sense, i.e. the cause in the sense of the Beautiful. But we shall deal with this subject more explicitly elsewhere. There is no obvious fulfilment of this promise. As regards the objects of mathematics, then, the foregoing account may be taken as sufficient to show that they exist, and in what sense they exist, and in what sense they are prior and in what they are not. But as regards the Ideas we must first consider the actual theory in relation to the Idea, without connecting it in any way with the nature of numbers, but approaching it in the form in which it was originally propounded by the first exponents It seems quite obvious that Aristotle intends this vague phrase to refer to Plato. Cf. Aristot. Met. 1.6.1-3, with which the following sections 2-5 should be compared. On the whole subject see Introduction. of the Ideas. The theory of Forms occurred to those who enunciated it because they were convinced as to the true nature of reality by the doctrine of Heraclitus, that all sensible things are always in a state of flux; so that if there is to be any knowledge or thought about anything, there must be certain other entities, besides sensible ones, which persist.
For there can be no knowledge of that which is in flux. Now Socrates devoted his attention to the moral virtues, and was the first to seek a general definition of these (for of the Physicists Democritus gained only a superficial grasp of the subject Cf. Aristot. Phys. 194a 20, Aristot. De Part. Anim. 642a 24.
and defined, after a fashion, "the hot" and "the cold"; while the Pythagoreans Cf. Aristot. Met. 1.5.2, 16. at an earlier date had arrived at definitions of some few things—whose formulae they connected with numbers—e.g., what "opportunity" is, or "justice" or "marriage"); and he naturally inquired into the essence of things; for he was trying to reason logically, and the starting-point of all logical reasoning is the essence. At that time there was as yet no such proficiency in Dialectic that men could study contraries independently of the essence, and consider whether both contraries come under the same science. There are two innovations This is perhaps too strong a word. What Aristotle means is that Socrates was the first thinker who attached importance to general definitions and systematically used arguments from analogy in order to arrive at them. The Greeks as a whole were only too readily impressed by analogy; Socrates merely developed an already prevalent tendency. For an example of his method see the reference at Aristot. Met. 5.29.5 n.
which, may fairly be ascribed to Socrates: inductive reasoning and general definition.
Both of these are associated with the starting-point of scientific knowledge. But whereas Socrates regarded neither universals nor definitions as existing in separation, the Idealists gave them a separate existence, and to these universals and definitions of existing things they gave the name of Ideas. Cf.
Introduction.
Hence on their view it followed by virtually the same argument that there are Ideas of all terms which are predicated universally With sect. 6-13 cf. Aristot. Met. 1.9.1-8, which are almost verbally the same. See Introduction.; and the result was very nearly the same as if a man who wishes to count a number of things were to suppose that he could not do so when they are few, and yet were to try to count them when he has added to them. For it is hardly an exaggeration to say that there are more Forms than there are particular sensible things (in seeking for whose causes these thinkers were led on from particulars to Ideas); because corresponding to each thing there is a synonymous entity, apart from the substances (and in the case of non-substantial things there is a One over the Many) both in our everyday world and in the realm of eternal entities. Again, not one of the ways in which it is attempted to prove that the Forms exist demonstrates their point; from some of them no necessary conclusion follows, and from others it follows that there are Form of things of which they hold that there are no Forms. For according to the arguments from the sciences, there will be Forms of all things of which there are sciences; and according to the "One-over-Many" argument, of negations too; and according to the argument that "we have some conception of what has perished" there will be Forms of perishable things, because we have a mental picture of these things. Further, of the most exact arguments some establish Ideas of relations, of which the Idealists deny that there is a separate genus, and others state the "Third Man." And in general the arguments for the Forms do away with things which are more important to the exponents of the Forms than the existence of the Ideas; for they imply that it is not the Dyad that is primary, but Number; and that the relative is prior to number, and therefore to the absolute; and all the other conclusions in respect of which certain persons by following up the views held about the Forms have gone against the principles of the theory. Again, according to the assumption by which they hold that the Ideas exist, there will be Forms not only of substances but of many other things (since the concept is one not only in the case of substances but in the case of non-substantial things as well; and there can be sciences not only of substances but also of other things; and there are a thousand other similar consequences); but it follows necessarily from the views generally held about them that if the Forms are participated in, there can only be Ideas of substances, because they are not participated in accidentally; things can only participate in a Form in so far as it is not predicated of a subject. I mean, e.g., that if a thing participates in absolute doubleness, it participates also in something eternal, but only accidentally; because it is an accident of "doubleness" to be eternal. Thus the Ideas will be substance. But the same terms denote substance in the sensible as in the Ideal world; otherwise what meaning will there be in saying that something exists besides the particulars, i.e. the unity comprising their multiplicity? If the form of the Ideas and of the things which participate in them is the same, they will have something in common (for why should duality mean one and the same thing in the case of perishable 2's and the 2's which are many but eternal, and not in the case of absolute duality and a particular 2?). But if the form is not the same, they will simply be homonyms; just as though one were to call both Callias and a piece of wood "man," without remarking any property common to them. sect.
14, 15 have no counterpart in Book 1. And if we profess that in all other respects the common definitions apply to the Forms, e.g. that "plane figure" and the other parts of the definition apply to the Ideal circle, only that we must also state of what the Form is a Form, we must beware lest this is a quite meaningless statement. The suggestion is that the definition of an Ideal circle is the same as that of a particular circle, except that it must have added to it the statement of what particular the Idea is an Idea. For to what element of the definition must the addition be made? to "center," or "plane" or all of them? For all the elements in the essence of an Idea are Ideas; e.g. "animal" and "two-footed." sc. in the definition or essence of "Ideal man." Further, it is obvious that "being an Idea," just like "plane," must be a definite characteristic which belongs as genus to all its species. i.e., "being an idea" will be a characteristic common to all ideas, and so must be itself an Idea. This chapter corresponds almost verbally to Aristot. Met. 1.9.9-15. Cf. note on Aristot. Met. 13.4.6. Above all we might examine the question what on earth the Ideas contribute to sensible things, whether eternal or subject to generation and decay; for they are not the cause of any motion or change in them. Moreover they are no help towards the knowledge of other things (for they are not the substance of particulars, otherwise they would be in particulars) or to their existence (since they are not present in the things which participate in them. If they were, they might perhaps seem to be causes, in the sense in which the admixture of white causes a thing to be white. But this theory, which was stated first by Anaxagoras and later by Eudoxus in his discussion of difficulties, and by others also, is very readily refuted; for it is easy to adduce plenty of impossibilities against such a view). Again, other things are not in any accepted sense derived from the Forms. To say that the Forms are patterns, and that other things participate in them, is to use empty phrases and poetical metaphors; for what is it that fashions things on the model of the Ideas? Besides, anything may both be and come to be without being imitated from something else; thus a man may become like Socrates whether Socrates exists or not, and even if Socrates were eternal, clearly the case would be the same. Also there will be several "patterns" (and therefore Forms) of the same thing; e.g., "animal" and "two-footed" will be patterns of "man," and so too will the Idea of man. Further, the Forms will be patterns not only of sensible things but of Ideas; e.g.
the genus will be the pattern of its species; hence the same thing will be pattern and copy. Further, it would seem impossible for the substance and that of which it is the substance to exist in separation; then how can the Ideas, if they are the substances of things, exist in separation from them? In the Phaedo Plat. Phaedo 100d. this statement is made: that the Forms are causes both of being and of generation. Yet assuming that the Forms exist, still there is no generation unless there is something to impart motion; and many other things are generated (e.g. house and ring) of which the Idealists say that there are no Forms. Thus it is clearly possible that those things of which they say that there are Ideas may also exist and be generated through the same kind of causes as those of the things which we have just mentioned, and not because of the Forms. Indeed, as regards the Ideas, we can collect against them plenty of evidence similar to that which we have now considered; not only by the foregoing methods, but by means of more abstract and exact reasoning. Now that we have dealt with the problems concerning the Ideas, we had better re-investigate the problems connected with numbers that follow from the theory that numbers are separate substances and primary causes of existing things.
Now if number is a kind of entity, and has nothing else as its substance, but only number itself, as some maintain; then either (a) there must be some one part of number which is primary, and some other part next in succession, and so on, each part being specifically different This statement bears two meanings, which Aristotle confuses: (i) There must be more than one number-series, each series being different in kind from every other series; (2) All numbers are different in kind, and inaddible. Confusion (or textual inaccuracy) is further suggested by the fact that Aristotle offers no alternative statement of the nature of number in general, such as we should expect from his language. In any case the classification is arbitrary and incomplete. — and this applies directly to units, and any given unit is inaddible to any other given unit; or (b) they The units. are all directly successive, and any units can be added to any other units, as is held of mathematical number; for in mathematical number no one unit differs in any way from another. Or (c) some units must be addible and others not. E.g., 2 is first after 1, and then 3, and so on with the other numbers; and the units in each number are addible, e.g. the units in the first i.e., Ideal or natural. 2 are addible to one another, and those in the first 3 to one another, and so on in the case of the other numbers; but the units in the Ideal 2 are inaddible to those in the Ideal 3; and similarly in the case of the other successive numbers. Hence whereas mathematical number is counted thus: after 1, 2 (which consists of another 1 added to the former) and 3 (which consists of another 1 added to these two) and the other numbers in the same way, Ideal number is counted like this: after 1, a distinct 2 not including the original 1; and a 3 not including the 2, and the rest of the numbers similarly. Or (d) one kind of number must be such as we first described, and another or such as the mathematicians maintain, and that which we have last described must be a third kind. Again, these numbers must exist either in separation from things, or not in separation, but in sensible things (not, however, in the way which we first considered, In Aristot. Met. 13.2.1-3. but in the sense that sensible things are composed of numbers which are present in them The Pythagorean number-atomist view; See Introduction. ）—either some of them and not others, or all of them. i.e., either all numbers are material elements of things, or some are and others are not. These are of necessity the only ways in which the numbers can exist. Now of those who say that unity is the beginning and substance and element of all things, and that number is derived from it and something else, almost everyone has described number in one of these ways (except that no one has maintained that all units are inaddible Cf.
sect. 2. ); and this is natural enough, because there can be no other way apart from those which we have mentioned. Some hold that both kinds of number exist, that which involves priority and posteriority being identical with the Ideas, and mathematical number being distinct from Ideas and sensible things, and both kinds being separable from sensible things Cf. Aristot. Met. 1.6.4.; others hold that mathematical number alone exists, Cf. Aristot. Met.
12.10.14. being the primary reality and separate from sensible things. The Pythagoreans also believe in one kind of number—the mathematical; only they maintain that it is not separate, but that sensible substances are composed of it. For they construct the whole universe of numbers, but not of numbers consisting of abstract units; they suppose the units to be extended—but as for how the first extended unit was formed they appear to be at a loss. Cf. Aristot. Met.
13.8.9, 10, Aristot. Met.
14.3.15, Aristot. Met. 14.5.7, and see Introduction. Another thinker holds that primary or Ideal number alone exists; and some Cf. 10ff., Aristot. Met.
13.1.4. identify this with mathematical number. The same applies in the case of lines, planes and solids. Some Plato. distinguish mathematical objects from those which "come after the Ideas" i.e., the (semi-)Ideal lines, planes, etc. Cf. Aristot. Met.
1.9.30.; and of those who treat the subject in a different manner some Speusippus; cf. sect. 7 above. speak of the mathematical objects and in a mathematical way—viz. those who do not regard the Ideas as numbers, nor indeed hold that the Ideas exist—and others Xenocrates. For his belief in indivisible lines see Ritter and Preller 362.
Aristotle ascribes the doctrine to Plato in Aristot.
Met. 1.9.25. speak of the mathematical objects, but not in a mathematical way; for they deny that every spatial magnitude is divisible into extended magnitudes, or that any two given units make 2. But all who hold that Unity is an element and principle of existing things regard numbers as consisting of abstract units, except the Pythagoreans; and they regard number as having spatial magnitude, as has been previously stated. sect. 8. It is clear from the foregoing account (1.) in how many ways it is possible to speak of numbers, and that all the ways have been described. They are all impossible, but doubtless some sc. the view of Xenocrates (cf. Aristot. Met.
13.8.8 ). are more so than others.
First, then, we must inquire whether the limits are addible or inaddible; and if inaddible, in which of the two ways which we have distinguished. Aristot. Met. 13.6.2, 3. For it is possible either (a) that any one unit is inaddible to any other, or (b) that the units in the Ideal 2 are inaddible to those in the Ideal 3, and thus that the units in each Ideal number are inaddible to those in the other Ideal numbers. Now if all units are addible and do not differ in kind, we get one type of number only, the mathematical, and the Ideas cannot be the numbers thus produced; for how can we regard the Idea of Man or Animal, or any other Form, as a number? There is one Idea of each kind of thing: e.g. one of Humanity and another one of Animality; but the numbers which are similar and do not differ in kind are infinitely many, so that this is no more the Idea of Man than any other 3 is. But if the Ideas are not numbers, they cannot exist at all; for from what principles can the Ideas be derived? Number is derived from Unity and the indeterminate dyad, and the principles and elements are said to be the principles and elements of number, and the Ideas cannot be placed either as prior or as posterior to numbers. Since the only principles which Plato recognizes are Unity and the Dyad, which are numerical (Aristotle insists on regarding them as a kind of 1 and 2), and therefore clearly principles of number; and the Ideas can only be derived from these principles if they (the Ideas) are (a) numbers (which has been proved impossible) or (b) prior or posterior to numbers (i.e., causes or effects of numbers, which they cannot be if they are composed of a different kind of units); then the Ideas are not derived from any principle at all, and therefore do not exist. But if the units are inaddible in the sense that any one unit is inaddible to any other, the number so composed can be neither mathematical number (since mathematical number consists of units which do not differ, and the facts demonstrated of it fit in with this character) nor Ideal number. For on this view 2 will not be the first number generated from Unity and the indeterminate dyad, and then the other numbers in succession, as they The Platonists. say 2, 3, because the units in the primary 2 are generated at the same time, This was the orthodox Platonist view of the generation of ideal numbers; or at least Aristotle is intending to describe the orthodox view. Plato should not have regarded the Ideal numbers as composed of units at all, and there is no real reason to suppose that he did (see Introduction). But Aristotle infers from the fact that the Ideal 2 is the first number generated (and then the other Ideal numbers in the natural order) that the units of the Ideal 2 are generated simultaneously, and then goes on to show that this is incompatible with the theory of inaddible units. whether, as the originator of the theory held, from unequals i.e., the Great-and-Small, which Aristotle wrongly understands as two unequal things. It is practically certain that Plato used the term (as he did that of "Indeterminate Dyad") to describe indeterminate quantity. See Introduction. (coming into being when these were equalized), or otherwise— since if we regard the one unit as prior to the other, This is a necessary implication of the theory of inaddible units (cf. Aristot. Met. 13.6.1, 2 ). it will be prior also to the 2 which is composed of them; because whenever one thing is prior and another posterior, their compound will be prior to the latter and posterior to the former. So the order of generation will be: (i) Unity (ungenerated); (2) first unit in 2; (3) second unit in 2; and the Ideal 2 will come between (2) and (3).
Further, since the Ideal 1 is first, and then comes a particular 1 which is first of the other 1's but second after the Ideal 1, and then a third 1 which is next after the second but third after the first 1, it follows that the units will be prior to the numbers after which they are called; e.g., there will be a third unit in 2 before 3 exists, and a fourth and fifth in 3 before these numbers exist. This is a corollary to the previous argument, and depends upon an identification of "ones" (including the Ideal One or Unity) with units. It is true that nobody has represented the units of numbers as inaddible in this way; but according to the principles held by these thinkers even this view is quite reasonable, although in actual fact it is untenable. For assuming that there is a first unit or first 1, i.e., the Ideal One. it is reasonable that the units should be prior and posterior; and similarly in the case of 2's, if there is a first 2. For it is reasonable and indeed necessary that after the first there should be a second; and if a second, a third; and so on with the rest in sequence. But the two statements, that there is after 1 a first and a second unit, and that there is a first 2, are incompatible. These thinkers, however, recognize a first unit and first 1, but not a second and third; and they recognize a first 2, but not a second and third. It is also evident that if all units are inaddible, there cannot be an Ideal 2 and 3, and similarly with the other numbers; for whether the units are indistinguishable or each is different in kind from every other, numbers must be produced by addition; e.g. 2 by adding 1 to another 1, and 3 by adding another 1 to the 2, and 4 similarly. This is of course not true of the natural numbers.
This being so, numbers cannot be generated as these thinkers try to generate them, from Unity and the dyad; because 2 becomes a part of 3, i.e., 3 is produced by adding 1 to 2. and 3 of 4, and the same applies to the following numbers. But according to them 4 was generated from the first 2 and the indeterminate dyad, thus consisting of two 2's apart from the Ideal 2. Cf. sect. 18.
Otherwise 4 will consist of the Ideal 2 and another 2 added to it, and the Ideal 2 will consist of the Ideal 1 and another 1; and if this is so the other element cannot be the indeterminate dyad, because it produces one unit and not a definite 2. The general argument is: Numbers are produced by addition; but this is incompatible with the belief in the Indeterminate Dyad as a generative principle, because, being duplicative, it cannot produce single units. Again, how can there be other 3's and 2's besides the Ideal numbers 3 and 2, and in what way can they be composed of prior and posterior units? All these theories are absurd and fictitious, and there can be no primary 2 and Ideal 3. Yet there must be, if we are to regard Unity and the indeterminate dyad as elements. i.e., if numbers are not generated by addition, there must be Ideal (or natural) numbers. But if the consequences are impossible, the principles cannot be of this nature. If, then, any one unit differs in kind from any other, these and other similar consequences necessarily follow. If, on the other hand, while the units in different numbers are different, those which are in the same number are alone indistinguishable from one another, even so the consequences which follow are no less difficult. For example, in the Ideal number 10 there are ten units, and 10 is composed both of these and of two 5's. Now since the Ideal 10 is not a chance number, I think Ross's interpretation of this passage must be right. The Ideal 10 is a unique number, and the numbers contained in it must be ideal and unique; therefore the two 5's must be specifically different, and so must their units—which contradicts the view under discussion. and is not composed of chance 5's, any more than of chance units, the units in this number 10 must be different; for if they are not different, the 5's of which the 10 is composed will not be different; but since these are different, the units must be different too. Now if the units are different, will there or will there not be other 5's in this 10, and not only the two? If there are not, the thing is absurd i.e., it is only reasonable to suppose that other 5's might be made up out of different combinations of the units.; whereas if there are, what sort of 10 will be composed of them? for there is no other 10 in 10 besides the 10 itself: Again, it must also be true that 4 is not composed of chance 2's. For according to them the indeterminate dyad, receiving the determinate dyad, made two dyads; for it was capable of duplicating that which it received. Cf.
Introduction. Again, how is it possible that 2 can be a definite entity existing besides the two units, and 3 besides the three units? Either by participation of the one in the other, as "white man" exists besides "white" and "man," because it partakes of these concepts; or when the one is a differentia of the other, as "man" exists besides "animal" and "two-footed." Again, some things are one by contact, others by mixture, and others by position; but none of these alternatives can possibly apply to the units of which 2 and 3 consist. Just as two men do not constitute any one thing distinct from both of them, so it must be with the units. The fact that the units are indivisible will make no difference; because points are indivisible also, but nevertheless a pair of points is not anything distinct from the two single points. Moreover we must not fail to realize this: that on this theory it follows that 2's are prior and posterior, and the other numbers similarly. Let it be granted that the 2's in 4 are contemporaneous; yet they are prior to those in 8, and just as the <determinate> 2 produced the 2's in 4, so In each case the other factor is the indeterminate dyad (cf. sect. 18). they produced the 4's in 8. Hence if the original 2 is an Idea, these 2's will also be Ideas of a sort. And the same argument applies to the units, because the units in the original 2 produce the four units in 4; and so all the units become Ideas, and an Idea will be composed of Ideas. Hence clearly those things also of which these things are Ideas will be composite; e.g., one might say that animals are composed of animals, if there are Ideas of animals. In general, to regard units as different in any way whatsoever is absurd and fictitious (by "fictitious" I mean "dragged in to support a hypothesis"). For we can see that one unit differs from another neither in quantity nor in quality; and a number must be either equal or unequal—this applies to all numbers, but especially to numbers consisting of abstract units. Thus if a number is neither more nor less, it is equal; and things which are equal and entirely without difference we assume, in the sphere of number, to be identical. Otherwise even the 2's in the Ideal 10 will be different, although they are equal; for if anyone maintains that they are not different, what reason will he be able to allege? Again, if every unit plus another unit makes 2, a unit from the Ideal 2 plus one from the Ideal 3 will make 2—a 2 composed of different units Which conflicts with the view under discussion.; will this be prior or posterior to 3?
It rather seems that it must be prior, because one of the units is contemporaneous with 3, and the other with 2. The implication seems to be, as Ross says, that the Platonists will refuse to admit that there is a number between 2 and 3. We assume that in general 1 and 1, whether the things are equal or unequal, make 2; e.g. good and bad, or man and horse; but the supporters of this theory say that not even two units make 2. If the number of the Ideal 3 is not greater than that of the Ideal 2, it is strange; and if it is greater, then clearly there is a number in it equal to the 2, so that this number is not different from the Ideal 2. But this is impossible, if there is a first and second number. i.e., if numbers are specifically different. Cf. Aristot. Met.
13.6.1. Nor will the Ideas be numbers. For on this particular point they are right who claim that the units must be different if there are to be Ideas, as has been already stated. sect. 2-4 above. For the form is unique; but if the units are undifferentiated, the 2's and 3's will be undifferentiated. Hence they have to say that when we count like this, l, 2, we do not add to the already existing number; for if we do, (a) number will not be generated from the indeterminate dyad, and (b) a number cannot be an Idea; because one Idea will pre-exist in another, and all the Forms will be parts of one Form. i.e., the biggest number. Thus in relation to their hypothesis they are right, but absolutely they are wrong, for their view is very destructive, inasmuch as they will say that this point presents a difficulty: whether, when we count and say "1, 2, 3," we count by addition or by enumerating distinct portions. This is Apelt's interpretation of KATA\ MERI/DAS. For this sense of the word he quotes Plut. Mor. 644c. The meaning then is: If you count by addition, you regard number as exhibited only in concrete instances; if you treat each number as a "distinct portion" (i.e. generated separately), you admit another kind of number besides the mathematical. Aristotle says that number can be regarded in both ways. But we do both; and therefore it is ridiculous to refer this point to so great a difference in essence. First of all it would be well to define the differentia of a number; and of a unit, if it has a differentia. Now units must differ either in quantity or in quality; and clearly neither of these alternatives can be true. "But units may differ, as number does, in quantity." But if units also differed in quantity, number would differ from number, although equal in number of units. Again, are the first units greater or smaller, and do the later units increase in size, or the opposite? All these suggestions are absurd.
Nor can units differ in quality; for no modification can ever be applicable to them, because these thinkers hold that even in numbers quality is a later attribute than quantity. Numbers have quality as being prime or composite, "plane" or "solid" (i.e., products of two or three factors); but these qualities are clearly incidental to quantity. Cf. Aristot. Met. 5.14.2. Further, the units cannot derive quality either from unity or from the dyad; because unity has no quality, and the dyad produces quantity, because its nature causes things to be many. If, then, the units differ in some other way, they should most certainly state this at the outset, and explain, if possible, with regard to the differentia of the unit, why it must exist; or failing this, what differentia they mean.
Clearly, then, if the Ideas are numbers, the units cannot all be addible, nor can they all be inaddible in either sense. Nor again is the theory sound which certain other thinkers Cf. Aristot. Met. 13.1.4. hold concerning numbers. These are they who do not believe in Ideas, either absolutely or as being a kind of numbers, but believe that the objects of mathematics exist, and that the numbers are the first of existing things, and that their principle is Unity itself. For it is absurd that if, as they say, there is a 1 which is first of the 1's, i.e., Speusippus recognized unity or "the One" as a formal principle, but admitted no other ideal numbers.
Aristotle argues that this is inconsistent. there should not be a 2 first of the 2's, nor a 3 of the 3's; for the same principle applies to all cases. Now if this is the truth with regard to number, and we posit only mathematical number as existing, Unity is not a principle. For the Unity which is of this nature must differ from the other units; and if so, then there must be some 2 which is first of the 2's; and similarly with the other numbers in succession. But if Unity is a principle, then the truth about numbers must rather be as Plato used to maintain; there must be a first 2 and first 3, and the numbers cannot be addible to each other. But then again, if we assume this, many impossibilities result, as has been already stated. Aristot. Met.
13.7.1-8.3. Moreover, the truth must lie one way or the other; so that if neither view is sound, number cannot have a separate abstract existence. From these considerations it is also clear that the third alternative Cf. Aristot. Met.
13.6.7. —that Ideal number and mathematical number are the same—is the worst; for two errors have to be combined to make one theory. (1.) Mathematical number cannot be of this nature, but the propounder of this view has to spin it out by making peculiar assumptions; (2.) his theory must admit all the difficulties which confront those who speak of Ideal number.
The Pythagorean view in one way contains fewer difficulties than the view described above, but in another way it contains further difficulties peculiar to itself. By not regarding number as separable,