Aristotle, Metaphysics

[1078b] [1] 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.¹

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 exponents² 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 [20] (for of the Physicists Democritus gained only a superficial grasp of the subject³ and defined, after a fashion, "the hot" and "the cold"; while the Pythagoreans⁴ 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 innovations⁵ 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.⁶ Hence on their view it followed by virtually the same argument that there are Ideas of all terms which are predicated universally⁷; 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