The Complete Aristotle (eng.)

should be
mathematical number, is impossible. For it is not true to speak of
indivisible spatial magnitudes; and however much there might be
magnitudes of this sort, units at least have not magnitude; and how
can a magnitude be composed of indivisibles? But arithmetical
number, at least, consists of units, while these thinkers identify
number with real things; at any rate they apply their propositions
to bodies as if they consisted of those numbers.
    If, then, it is necessary, if number is a self-subsistent real
thing, that it should exist in one of these ways which have been
mentioned, and if it cannot exist in any of these, evidently number
has no such nature as those who make it separable set up for
it.
    Again, does each unit come from the great and the small,
equalized, or one from the small, another from the great? (a) If
the latter, neither does each thing contain all the elements, nor
are the units without difference; for in one there is the great and
in another the small, which is contrary in its nature to the great.
Again, how is it with the units in the 3-itself? One of them is an
odd unit. But perhaps it is for this reason that they give 1-itself
the middle place in odd numbers. (b) But if each of the two units
consists of both the great and the small, equalized, how will the 2
which is a single thing, consist of the great and the small? Or how
will it differ from the unit? Again, the unit is prior to the 2;
for when it is destroyed the 2 is destroyed. It must, then, be the
Idea of an Idea since it is prior to an Idea, and it must have come
into being before it. From what, then? Not from the indefinite
dyad, for its function was to double.
    Again, number must be either infinite or finite; for these
thinkers think of number as capable of existing separately, so that
it is not possible that neither of those alternatives should be
true. Clearly it cannot be infinite; for infinite number is neither
odd nor even, but the generation of numbers is always the
generation either of an odd or of an even number; in one way, when
1 operates on an even number, an odd number is produced; in another
way, when 2 operates, the numbers got from 1 by doubling are
produced; in another way, when the odd numbers operate, the other
even numbers are produced. Again, if every Idea is an Idea of
something, and the numbers are Ideas, infinite number itself will
be an Idea of something, either of some sensible thing or of
something else. Yet this is not possible in view of their thesis
any more than it is reasonable in itself, at least if they arrange
the Ideas as they do.
    But if number is finite, how far does it go? With regard to this
not only the fact but the reason should be stated. But if number
goes only up to 10 as some say, firstly the Forms will soon run
short; e.g. if 3 is man-himself, what number will be the
horse-itself? The series of the numbers which are the several
things-themselves goes up to 10. It must, then, be one of the
numbers within these limits; for it is these that are substances
and Ideas. Yet they will run short; for the various forms of animal
will outnumber them. At the same time it is clear that if in this
way the 3 is man-himself, the other 3’s are so also (for those in
identical numbers are similar), so that there will be an infinite
number of men; if each 3 is an Idea, each of the numbers will be
man-himself, and if not, they will at least be men. And if the
smaller number is part of the greater (being number of such a sort
that the units in the same number are associable), then if the
4-itself is an Idea of something, e.g. of ‘horse’ or of ‘white’,
man will be a part of horse, if man is It is paradoxical also that
there should be an Idea of 10 but not of 11, nor of the succeeding
numbers. Again, there both are and come to be certain things of
which there are no Forms; why, then, are there not Forms of them
also? We infer that the Forms are not causes. Again, it is
paradoxical-if the number series up to 10 is more of a real thing
and a Form than 10 itself. There is no generation of the former as
one thing, and there is of the latter. But they try to work on the
assumption that the series of numbers up to 10 is a complete
series. At least they generate the derivatives-e.g. the void,
proportion, the odd, and the others of this kind-within the decade.
For some things, e.g. movement and rest, good and bad, they assign
to the originative principles, and the others to the numbers. This
is why they