Book 13
5
These are they who do not believe in Ideas, either absolutely or as being a kind of numbers, but believe that the objects of mathematics exist, and that the numbers are the first of existing things, and that their principle is Unity itself. For it is absurd that if, as they say, there is a 1 which is first of the 1’s,i.e., Speusippus recognized unity or the One as a formal principle, but admitted no other ideal numbers. Aristotle argues that this is inconsistent. there should not be a 2 first of the 2’s, nor a 3 of the 3’s; for the same principle applies to all cases.
6
Now if this is the truth with regard to number, and we posit only mathematical number as existing, Unity is not a principle. For the Unity which is of this nature must differ from the other units; and if so, then there must be some 2 which is first of the 2’s; and similarly with the other numbers in succession.
7
But if Unity is a principle, then the truth about numbers must rather be as Plato used to maintain; there must be a first 2 and first 3, and the numbers cannot be addible to each other. But then again, if we assume this, many impossibilities result, as has been already stated.Aristot. Met. 13.7.1-8.3. Moreover, the truth must lie one way or the other; so that if neither view is sound, number cannot have a separate abstract existence.
8
From these considerations it is also clear that the third alternativeCf. Aristot. Met. 13.6.7.—that Ideal number and mathematical number are the same—is the worst; for two errors have to be combined to make one theory. (1.) Mathematical number cannot be of this nature, but the propounder of this view has to spin it out by making peculiar assumptions; (2.) his theory must admit all the difficulties which confront those who speak of Ideal number.
