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. 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
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