[1083a] [1]

First of all it would be well to define the differentia of a number; and of a unit, if it has a differentia. Now units must differ either in quantity or in quality; and clearly neither of these alternatives can be true. "But units may differ, as number does, in quantity." But if units also differed in quantity, number would differ from number, although equal in number of units.Again, are the first units greater or smaller, and do the later units increase in size, or the opposite? All these suggestions are absurd. Nor can units differ in quality; for no modification can ever be applicable to them, because these thinkers hold that even in numbers quality is a later attribute than quantity.1 Further, the units cannot derive quality either from unity or from the dyad; because unity has no quality, and the dyad produces quantity, because its nature causes things to be many. If, then, the units differ in some other way, they should most certainly state this at the outset, and explain, if possible, with regard to the differentia of the unit, why it must exist; or failing this, what differentia they mean.

Clearly, then, if the Ideas are numbers, the units cannot all be addible, [20] nor can they all be inaddible in either sense. Nor again is the theory sound which certain other thinkers2 hold concerning numbers.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,3 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.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.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.4 Moreover, the truth must lie one way or the other; so that if neither view is sound,

1 Numbers have quality as being prime or composite, "plane" or "solid" (i.e., products of two or three factors); but these qualities are clearly incidental to quantity. Cf. Aristot. Met. 5.14.2.

3 i.e., Speusippus recognized unity or "the One" as a formal principle, but admitted no other ideal numbers. Aristotle argues that this is inconsistent.

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