Book 14
12
Again, it is not the Ideal numbers that are the causes of harmonic relations, etc. (for Ideal numbers, even when they are equal, differ in kind, since their units also differ in kind)Aristotle has argued (Aristot. Met. 13.6-8.) that if the Ideal numbers differ in kind, their units must differ in kind. Hence even equal numbers, being composed of different units, must be different in kind. In point of fact, since each ideal number is unique, no two of them could be equal.; so on this ground at least we need not posit Forms.
13
Such, then, are the consequences of the theory, and even more might be adduced. But the mere fact that the Platonists find so much trouble with regard to the generation of Ideal numbers, and can in no way build up a system, would seem to be a proof that the objects of mathematics are not separable from sensible things, as some maintain, and that the first principles are not those which these thinkers assume.
