Book 13
13
Again, what of the units in the Ideal 3? because there is one over. But no doubt it is for this reason that in an odd number they make the Ideal One the middle unit.Cf. DieIs, Vorsokratiker 270. 18. If on the other hand each of the units comes from both Great and Small, when they are equalized, how can the Ideal 2 be a single entity composed of the Great and Small? How will it differ from one of its units? Again, the unit is prior to the 2; because when the unit disappears the 2 disappears.
14
Therefore the unit must be the Idea of an Idea, since it is prior to an Idea, and must have been generated before it. From what, then? for the indeterminate dyad, as we have seen,Aristot. Met. 13.7.18. causes duality.
14
Again, number must be either infinite or finite (for they make number separable, so that one of these alternatives must be true).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.
15
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.
16
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 logicallyi.e., as implying an actual infinite. and on their own assumption,i.e., as inconsistent with the conception of an Idea as a determining principle. since they regard the Ideas as they do.
16
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.
