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differentia they mean.
Evidently then, if the Ideas are numbers, the units cannot all
be associable, nor can they be inassociable in either of the two ways.
But neither is the way in which some others speak about numbers
correct. These are those who do not think there are Ideas, either
without qualification or as identified with certain numbers, but think
the objects of mathematics exist and the numbers are the first of
existing things, and the 1-itself is the starting-point of them. It is
paradoxical that there should be a 1 which is first of 1's, as they
say, but not a 2 which is first of 2's, nor a 3 of 3's; for the same
reasoning applies to all. If, then, the facts with regard to number
are so, and one supposes mathematical number alone to exist, the 1
is not the starting-point (for this sort of 1 must differ from
the-other units; and if this is so, there must also be a 2 which is
first of 2's, and similarly with the other successive numbers). But if
the 1 is the starting-point, the truth about the numbers must rather
be what Plato used to say, and there must be a first 2 and 3 and
numbers must not be associable with one another. But if on the other
hand one supposes this, many impossible results, as we have said,
follow. But either this or the other must be the case, so that if
neither is, number cannot exist separately.
It is evident, also, from this that the third version is the
worst,-the view ideal and mathematical number is the same. For two
mistakes must then meet in the one opinion. (1) Mathematical number
cannot be of this sort, but the holder of this view has to spin it out
by making suppositions peculiar to himself. And (2) he must also admit
all the consequences that confront those who speak of number in the
sense of 'Forms'.
The Pythagorean version in one way affords fewer difficulties than
those before named, but in another way has others peculiar to
itself. For not thinking of number as capable of existing separately
removes many of the impossible consequences; but that bodies should be
composed of numbers, and that this 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

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