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. And so while mathematical number is counted
thus-after 1, 2 (which consists of another 1 besides the former 1),
and 3 which consists of another 1 besides these two), and the other
numbers similarly, ideal number is counted thus-after 1, a distinct
2 which does not include the first 1, and a 3 which does not include
the 2 and the rest of the number series similarly. Or (2) one kind
of number must be like the first that was named, one like that which
the mathematicians speak of, and that which we have named last must be
a third kind.
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
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