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 Footnote 1It 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 Footnote 2Cf. Aristot. Phys. 194a 20, Aristot. De Part. Anim. 642a 24. and defined, after a fashion, the hot
and the cold
; while the Pythagoreans Footnote 3Cf. 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 Footnote 4This 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. Footnote 5Cf. Introduction.
Hence on their view it followed by virtually the same argument that there are Ideas of all terms which are predicated universally Footnote 6With 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.
Footnote 7sect. 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. Footnote 8The 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.
Footnote 9sc. 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. Footnote 10i.e., being an idea
will be a characteristic common to all ideas, and so must be itself an Idea.