The general propositions in mathematics are not concerned with objects which exist separately apart from magnitudes and numbers; they are concerned with magnitudes and numbers, but not with them as possessing magnitude or being divisible. It is clearly possible that in the same way propositions and logical proofs may apply to sensible magnitudes; not qua sensible, but qua having certain characteristics. For just as there can be many propositions about things merely qua movable, without any reference to the essential nature of each one or to their attributes, and it does not necessarily follow from this either that there is something movable which exists in separation from sensible things or that there is a distinct movable nature in sensible things; so too there will be propositions and sciences which apply to movable things, not qua movable but qua corporeal only; and again qua planes only and qua lines only, and qua divisible, and qua indivisible but having position, and qua indivisible only. Therefore since it is true to say in a general sense not only that things which are separable but that things which are inseparable exist, e.g., that movable things exist, it is also true to say in a general sense that mathematical objects exist, and in such a form as mathematicians describe them. And just as it is true to say generally of the other sciences that they deal with a particular subject—not with that which is accidental to it (e.g. not with “white” if “the healthy” is white, and the subject of the science is “the healthy”), but with that which is the subject of the particular science; 1078awith the healthy if it treats of things qua healthy, and with man if qua man—so this is also true of geometry. If the things of which it treats are accidentally sensible although it does not treat of them qua sensible, it does not follow that the mathematical sciences treat of sensible things—nor, on the other hand, that they treat of other things which exist independently apart from these. Many attributes are essential properties of things as possessing a particular characteristic; e.g., there are attributes peculiar to an animal qua female or qua male, although there is no such thing as female or male in separation from animals. Hence there are also attributes which are peculiar to things merely qua lines or planes. And in proportion as the things which we are considering are prior in formula and simpler, they admit of greater exactness; for simplicity implies exactness. Hence we find greater exactness where there is no magnitude, and the greatest exactness where there is no motion; or if motion is involved, where it is primary, because this is the simplest kind; and the simplest kind of primary motion is uniform motion.¹ The same principle applies to both harmonics and optics, for neither of these sciences studies objects qua sight or qua sound, but qua lines and numbers²; yet the latter are affections peculiar to 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.³ 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.⁴ And since goodness is distinct from beauty (for it is always in actions that goodness is present, whereas beauty is also in immovable things), they⁵ 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; 1078band 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.⁶

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 exponents⁷ 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 subject⁸ and defined, after a fashion, “the hot” and “the cold”; while the Pythagoreans⁹ 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 innovations¹⁰ 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.¹¹ Hence on their view it followed by virtually the same argument that there are Ideas of all terms which are predicated universally¹²; 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 1079a(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, 1079band 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. ¹³And 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.¹⁴ 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.” ¹⁵ Further, it is obvious that “being an Idea,” just like “plane,” must be a definite characteristic which belongs as genus to all its species.¹⁶