Again, these kinds of numbers must either be separable from
things, or not separable but in objects of perception (not however
in the way which we first considered, in the sense that objects of
perception consists of numbers which are present in them)-either one
kind and not another, or all of them.
These are of necessity the only ways in which the numbers can
exist. And of those who say that the 1 is the beginning and
substance and element of all things, and that number is formed from
the 1 and something else, almost every one has described number in one
of these ways; only no one has said all the units are inassociable.
And this has happened reasonably enough; for there can be no way
besides those mentioned. Some say both kinds of number exist, that
which has a before and after being identical with the Ideas, and
mathematical number being different from the Ideas and from sensible
things, and both being separable from sensible things; and others
say mathematical number alone exists, as the first of realities,
separate from sensible things. And the Pythagoreans, also, believe
in one kind of number-the mathematical; only they say it is not
separate but sensible substances are formed out of it. For they
construct the whole universe out of numbers-only not numbers
consisting of abstract units; they suppose the units to have spatial
magnitude. But how the first 1 was constructed so as to have
magnitude, they seem unable to say.
Another thinker says the first kind of number, that of the
Forms, alone exists, and some say mathematical number is identical
with this.
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