[1084a] [1] so that one of these alternatives must be true).1 Now it is obvious that it cannot be infinite, because infinite number is neither odd nor even, and numbers are always generated either from odd or from even number. By one process, when 1 is added to an even number, we get an odd number; by another, when 1 is multiplied by 2, we get ascending powers of 2; and by another, when powers of 2 are multiplied by odd numbers, we get the remaining even numbers.

Again, if every Idea is an Idea of something, and the numbers are Ideas, infinite number will also be an Idea of something, either sensible or otherwise. This, however, is impossible, both logically2 and on their own assumption,3 since they regard the Ideas as they do.

If, on the other hand, number is finite, what is its limit? In reply to this we must not only assert the fact, but give the reason.Now if number only goes up to 10, as some hold,4 in the first place the Forms will soon run short. For example, if 3 is the Idea of Man, what number will be the Idea of Horse? Each number up to 10 is an Idea; the Idea of Horse, then, must be one of the numbers in this series, for they are substances or Ideas.But the fact remains that they will run short, because the different types of animals will outnumber them. At the same time it is clear that if in this way the Ideal 3 is the Idea of Man, so will the other 3's be also (for the 3's in the same numbers5 are similar), [20] so that there will be an infinite number of men; and if each 3 is an Idea, each man will be an Idea of Man; or if not, they will still be men.And if the smaller number is part of the greater, when it is composed of the addible units contained in the same number, then if the Ideal 4 is the Idea of something, e.g. "horse" or "white," then "man" will be part of "horse," if "man" is 2. It is absurd also that there should be an Idea of 10 and not of 11, nor of the following numbers.

Again, some things exist and come into being of which there are no Forms6; why, then, are there not Forms of these too? It follows that the Forms are not the causes of things.

Again, it is absurd that number up to 10 should be more really existent, and a Form, than 10 itself; although the former is not generated as a unity, whereas the latter is. However, they try to make out that the series up to 10 is a complete number;at least they generate the derivatives, e.g. the void, proportion, the odd, etc., from within the decad. Some, such as motion, rest, good and evil, they assign to the first principles; the rest to numbers.7 Hence they identify the odd with Unity; because if oddness depended on 3, how could 5 be odd?8

Again, they hold that spatial magnitudes and the like have a certain limit;

1 The point seems to be that if number is self-subsistent it must be actually finite or infinite. Aristotle himself holds that number is infinite only potentially; i.e., however high you can count, you can always count higher.

2 i.e., as implying an actual infinite.

3 i.e., as inconsistent with the conception of an Idea as a determining principle.

4 Cf. Aristot. Met. 12.8.2. The Platonists derived this view from the Pythagoreans; see Introduction.

5 Robin is probably right in taking this to mean that the 3 which is in the ideal 4 is like the 3 which is in the 4 which is in a higher ideal number, and so on (La Theorie platonicienne des Idees et des Nombres d'apres Aristote, p. 352).

7 From the Dyad were derived void (Theophrastus, Met. 312.18-313.3) and motion (cf. Aristot. Met. 1.9.29, Aristot. Met. 11.9.8). Rest would naturally be derived from unity. For good and evil see Aristot. Met. 1.6.10. Proportion alone of the "derivatives" here mentioned appears to be derived from number. As Syrianus says, the three types of proportion can be illustrated by numbers from within the decadâ€”arithmetical 1. 2. 3, geometrical 1. 2. 4, harmonic 2. 3. 6.

8 sc. because (on their theory) 3 is not contained in 5. Thus oddness had to be referred to not a number but a principleâ€”unity.