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 thinkers^{2} 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,