Kitap 14
10
There is a sense, however, in which these thinkers make it clear that goodness is predicable of numbers, and that the odd, the straight, the equal-by-equal,i.e., square. and the powersProbably their power of being represented as regular figures; e.g. the triangularity of 3 or 6. of certain numbers, belong to the series of the Beautiful.Cf. Aristot. Met. 1.5.6. For the seasons are connected with a certain kind of numberi.e., 4.; and the other examples which they adduce from mathematical theorems all have the same force.
11
Hence they would seem to be mere coincidences, for they are accidental; but all the examples are appropriate to each other, and they are one by analogy. For there is analogy between all the categories of Being—as straight is in length, so is level in breadth, perhaps odd in number, and white in color.
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.
