by Aristotle
It has, then, been sufficiently pointed out that the objects of mathematics are not substances in a higher sense than bodies are, and that they are not prior to sensibles in being, but only in formula, and that they cannot in any way exist separately. But since they could not exist in sensibles either, it is plain that they [15] either do not exist at all or exist in a special way and therefore do not exist without qualification. For ‘exist’ has many senses.
3 · Just as the universal part of mathematics deals not with objects which exist separately, apart from magnitudes and from numbers, but with magnitudes [20] and numbers, not however qua such as to have magnitude or to be divisible, clearly it is possible that there should also be both formulae and demonstrations about sensible magnitudes, not however qua sensible but qua possessed of certain definite qualities. For as there are many formulae about things merely considered as in motion, apart from the essence of each such thing and from their accidents, and as it [25] is not therefore necessary that there should be either something in motion separate from sensibles, or a separate substance in the sensibles, so too in the case of moving things there will be formulae and sciences which treat them not qua moving but only qua bodies, or again only qua planes, or only qua lines, or qua divisibles, or qua [30] indivisibles having position, or only qua indivisibles.
Thus since it is true to say without qualification that not only things which are separable but also things which are inseparable exist—for instance, that moving things exist,—it is true also to say, without qualification, that the objects of mathematics exist, and with the character ascribed to them by mathematicians. And it is true to say of the other sciences too, without qualification, that they deal [35] with such and such a subject—not with what is accidental to it (e.g. not with the white, if the white thing is healthy, and the science has the healthy as its subject), [1078a1] but with that which is the subject of each science—with the healthy if it treats things qua healthy, with man if qua man. So too is it with geometry; if its subjects happen to be sensible, though it does not treat them qua sensible, the mathematical sciences will not for that reason be sciences of sensibles—nor, on the other hand, of other things separate from sensibles.
[5] Many properties attach to things in virtue of their own nature as possessed of some such property; e.g. there are attributes peculiar to the animal qua female or qua male, yet there is no female nor male separate from animals. And so also there are attributes which belong to things merely as lengths or as planes. And in proportion as we are dealing with things which are prior in formula and simpler, our [10] knowledge will have more accuracy, i.e. simplicity. Thus a science which abstracts from the magnitude of things is more precise than one which takes it into account; and a science is most precise if it abstracts from movement, but if it takes account of movement, it is most precise if it deals with the primary movement, for this is the simplest; and of this again uniform movement is the simplest form. The same account may be given of harmonics and optics; for neither considers its objects qua [15] light-ray or qua voice, but qua lines and numbers; but the latter are attributes proper to the former. And mechanics too proceeds in the same way. Thus if we suppose things separated from their attributes and make any inquiry concerning them as such, we shall not for this reason be in error, any more than when one draws [20] a line on the ground and calls it a foot long when it is not; for the error is not included in the propositions.
Each question will be best investigated in this way—by supposing separate what is not separate, as the arithmetician and the geometer do. For a man qua man is one indivisible thing; and the arithmetician supposes one indivisible thing, and [25] then considers whether any attribute belongs to a man qua indivisible. But the geometer treats him neither qua man nor qua indivisible, but as a solid. For evidently the attributes which would have belonged to him even if he had not been indivisible, can belong to him apart from these attributes. Thus, then, geometers speak correctly—they talk about existing things, and their subjects do exist; for being has two forms—it exists not only in fulfillment but also as matter. [30]
Now since the good and the beautiful are different (for the former always implies conduct as its subject, while the beautiful is found also in motionless things), those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a very great deal about them; for if they do not expressly mention them, but prove attributes which are their results or [35] their formulae, it is not true to say that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical [1078b1] sciences demonstrate in a special degree. And since these (e.g. order and definiteness) are obviously causes of many things, evidently these sciences must treat this sort of cause also (i.e. the beautiful) as in some sense a cause. But we shall [5] speak more plainly elsewhere about these matters.
4 · So much then for the objects of mathematics; we have said that they exist and in what sense they exist, and in what sense they are prior and in what sense not prior. Now, regarding the Ideas, we must first examine the ideal theory by itself, not connecting it in any way with the nature of numbers, but treating it in the form in [10] which it was originally understood by those who first maintained the existence of Ideas. The supporters of the ideal theory were led to it because they were persuaded of the truth of the Heraclitean doctrine that all sensible things are ever passing away, so that if knowledge or thought is to have an object, there must be some other [15] and permanent entities, apart from those which are sensible; for there can be no knowledge of things which are in a state of flux. Socrates occupied himself with the excellences of character, and in connection with them became the first to raise the problem of universal definitions—for of the natural scientists, only Democritus touched on the matter and defined, after a fashion, the hot and the cold; while the [20] Pythagoreans had before this treated of a few things, whose formulae they connected with numbers—e.g. opportunity, justice, or marriage. But it was natural that Socrates should seek the essence. For he was seeking to deduce, and the essence is the starting-point of deductions. For there was as yet none of the dialectical power [25] which enables people even without knowledge of the essence to speculate about contraries and inquire whether the same science deals with contraries. For two things may be fairly ascribed by Socrates—inductive arguments and universal definition, both of which are concerned with the starting-point of science. But Socrates did not make the universals or the definitions exist apart; his successors, [30] however, gave them separate existence, and this was the kind of thing they called Ideas.
Therefore it followed for them, almost by the same argument, that there must be Ideas of all things that are spoken of universally, and it was almost as if a man wished to count certain things, and while they were few thought he would not be [35] able to count them, but made them more and then counted them; for the Forms are almost more numerous than the groups of sensible things, yet it was in seeking the [1079a1] causes of sensible things that they proceeded from these to the Forms. For to each set of substances there answers a Form which has the same name and exists apart from the substances, and so also in the other categories there is one character common to many individuals, whether these be sensible or eternal.
[5] Again, of the ways in which it is proved that the Forms exist, none is convincing; for from some no inference necessarily follows, and from some it follows that there are Forms even of things of which they think 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 argument that there is one attribute common to many things there will be Forms even of negations, and according to the argument that thought has an object when the individual object [10] has perished, there will be Forms of perishable things; for we can have an image of these. Again, of the most accurate arguments, some lead to Ideas of relations, of which they say there
is no independent class, and others involve the difficulty of the third man. And in general the arguments for the Forms destroy that for whose [15] existence the assertors of Forms are more anxious than for the existence of the Ideas; for it follows that not the dyad but number is first, and the relative is prior to that and prior to the self-dependent—and besides this there are all the other points on which certain people, by following out the opinions held about the Forms, have come into conflict with the principles of the theory.
Again, according to the assumption on which the belief in the Ideas rests, there [20] will be Forms not only of substances but also of many other things; for the concept is single, not only in the case of substances, but also in that of non-substances, and there are sciences of other things than substance; and a thousand other such conclusions also follow. But according to the necessities of the case and the opinions [25] about the Forms, if they can be shared in there must be Ideas of substances only. For they are not shared in incidentally, but each Form must be shared in as something not predicated of a subject. (E.g. if a thing shares in the double itself, it [30] shares also in eternal, but incidentally; for the double happens to be eternal.) Therefore the Forms will be substance. And the same names indicate substance in this and in the ideal world (or what will be the meaning of saying that there is something apart from the particulars—the one over many?). And if the Ideas and the things that share in them have the same Form, there will be something common: [35] for why should 2 be one and the same in all the perishable 2’s, or in the 2’s which are many but eternal, and not the same in the 2 itself as in the individual 2? But if they [1079b1] have not the same Form, they will have only the name in common, and it is as if one were to call both Callias and a piece of wood ‘man’, without observing any community between them.
But if we are to suppose that in other respects the common formulae apply to [5] the Forms, e.g. that plane figure and the other parts of the formula apply to the circle itself, but that what it is must be added, we must inquire whether this is not absolutely empty. For to what will this be added? To ‘centre’ or to ‘plane’ or to all the parts of the formula? For all the elements in the substance are Ideas, e.g. animal and two-footed. Further, the added notion must be an Idea, like plane, a definite entity which will be present as genus in all its species. [10]
5 · Above all one might discuss the question what on earth the Forms contribute to sensible things, either to those that are eternal or to those which come into being and cease to be; for they cause neither movement nor any change in them. [15] But again they help in no way towards the knowledge of other things (for they are not even the substance of these, else they would have been in them), nor towards their being, at least if they are not in the individuals which share in them—for in that case they might be thought perhaps to be causes, as white is for the white thing in which it is mixed. But this argument, which was used first by Anaxagoras, and [20] later by Eudoxus in his discussion of difficulties and by certain others, is too easily upset; for it is easy to collect many insuperable objections to such a view.
But further all other things cannot come from the Forms in any of the ways that are usually suggested. And to say that they are patterns and the other things [25] share in them is to use empty words and poetical metaphors. For what is it that works, looking to the Ideas? And any thing can both be and come into being without being copied from something else, so that, whether Socrates exists or not, a man like Socrates might come to be. And evidently this might be so even if Socrates were [30] eternal. And there will be several patterns of the same thing, and therefore several Forms, e.g. animal and two-footed, and also man-himself, will be Forms of man. Again, the Forms are patterns not only of sensible things, but of things-themselves also, e.g. the genus is the pattern of the species of the genus; therefore the same thing will be pattern and copy. [35]
Again, it might be thought impossible that substance and that whose substance it is should exist apart; how, therefore, could the Ideas, being substances of things, [1080a1] exist apart?
In the Phaedo2 it is stated in this way—that the Forms are causes both of being and of becoming. Yet though the Forms exist, still things do not come into being, unless there is something to move them; and many other things come into being (e.g. [5] a house or a ring), of which they say there are no Forms. Clearly therefore even the things of which they say there are Ideas can both be and come into being owing to such causes as produce the things just mentioned, and not owing to the Forms. But regarding the Ideas it is possible, both in this way and by more abstract and more accurate arguments, to collect many objections like those we have considered. [10]
6 · Since we have discussed these points, it is well to consider again the results regarding numbers which confront those who say that numbers are separable substances and first causes of things. If number is a real thing and its [15] substance is nothing other than just number, as some say, it follows that either there is a first in it and a second, each being different in kind, and3 this is true of the units without qualification, and any unit is non-comparable with any unit, or they are all [20] directly successive, and any of them is comparable with any, as they say is the case with mathematical number; for in mathematical number no unit is in any way different from another. Or some units must be comparable and some not, e.g. 2 is [25] first after 1, and then comes 3 and then the other numbers, and the units in each number are comparable, e.g. those in the first 2 with one another, and those in the first 3 with one another, and so with the other numbers; but the units in the 2 itself are not comparable with those in the 3 itself; and similarly in the case of the other [30] successive numbers. Therefore while mathematical number is counted thus—after 1, 2 (which consists of another 1 besides the former 1), and 3 (which consists of another 1 besides these two), and the other numbers similarly, ideal number is counted thus—after 1, a distinct 2 which does not include the first 1, and a 3 which [35] does not include the 2, and the other numbers similarly. Or one kind of number is like the first that was named, one like that which the mathematicians speak of, and that which we have named last must be a third kind.
[1080b1] Again, these numbers must either be separable from things, or not separable but in sensible things (not however in the way which we first considered, but in the sense that sensible things consist of numbers which are present in them)—either some of them and not others, or all of them.
[5] These are of necessity the only ways in which the numbers can exist. And of those who say that the 1 is the beginning and substance and element of all things, and that number is formed from the 1 and something else, almost every one has described number in one of these ways; only no one has said all the units are [10] incomparable. And this has happened reasonably enough; for there can be no way besides those mentioned. Some say both kinds of number exist, that which has a before and after being identical with the Ideas, and mathematical number being different from the Ideas and from sensible things, and both being separable from [15] sensible things; and others say mathematical number alone exists, as the first of realities, separate from sensible things.
Now the Pythagoreans, also, believe in one kind of number—the mathematical; only they say it is not separate but sensible substances are formed out of it. For they construct the whole universe out of numbers—only not numbers consisting of [20] abstract units; they suppose the units to have spatial magnitude. But how the first 1 was constructed so as to have magnitude, they seem unable to say.
Another thinker says the first kind of number, that of the Forms, alone exists, and some say mathematical number is identical with this.
The case of lines, planes, and solids is similar. For some think that those which [25] are the objects of mathematics are different from those which come after the Ideas; and of those who express themselves otherwise some speak of the objects of mathematics and in a mathematical way—viz. those who do not make the Ideas numbers nor say that Ideas exist; and others speak of the objects of
mathematics, but not mathematically; for they say that neither is every spatial magnitude divisible into magnitudes, nor do any two units make 2. All who say the 1 is an [30] element and principle of things suppose numbers to consist of abstract units, except the Pythagoreans; but they suppose the numbers to have magnitude, as has been said before. It is clear from this statement, then, in how many ways numbers may be described, and that all the ways have been mentioned; and all are impossible, but [35] some perhaps more than others.
7 · First let us inquire if the units are comparable or non-comparable, and if non-comparable, in which of the two ways we distinguished. For it is possible that [1081a1] any unit is non-comparable with any, and it is possible that those in the ideal 2 are non-comparable with those in the ideal 3, and, generally, that those in each primary number are non-comparable with one another. If all units are comparable and [5] without difference, we get mathematical number and this alone, and the Ideas cannot be the numbers. For what sort of number will the ideal man or animal or any other Form be? There is one Idea of each thing, e.g. one of ideal man and another one of ideal animal; but the similar and undifferentiated numbers are infinitely [10] many, so that this 3 is no more the ideal man than any other 3. But if the Ideas are not numbers, neither can they exist at all. For from what principles will the Ideas come? Number comes from the 1 and the indefinite dyad, and the principles and the elements are said to be principles and elements of number, and the Ideas cannot be [15] ranked as either prior or posterior to the numbers.
But if the units are non-comparable, and non-comparable in the sense that none is comparable with any other, number of this sort cannot be mathematical number; for mathematical number consists of undifferentiated units, and the truths [20] proved of it suit this character. Nor can it be ideal number. For 2 will not come first after 1 and the indefinite dyad, and be followed directly by the successive numbers, as we say ‘2, 3, 4’ (for the units in the ideal 2 are generated at the same time, whether, as the first holder of the theory said, from unequals—coming into being when these were equalized—or in some other way).4 Besides, if one unit is to be [25] prior to the other, it will be prior to the 2 composed of these; for when there is one thing prior and another posterior, the compound of these will be prior to one and posterior to the other.