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'two-footed', and also 'man-himself', will be Forms of man. Again, the
Forms are patterns not only of sensible things, but of Forms
themselves also; i.e. the genus is the pattern of the various
forms-of-a-genus; therefore the same thing will be pattern and copy.
Again, it would seem impossible that substance and that whose
substance it is should exist apart; how, therefore, could the Ideas,
being the substances of things, exist apart?
In the Phaedo the case is stated in this way-that the Forms are
causes both of being and of becoming. Yet though the Forms exist,
still things do not come into being, unless there is something to
originate movement; and many other things come into being (e.g. a
house or a ring) of which they say there are no Forms. Clearly
therefore even the things of which they say there are Ideas can both
be and come into being owing to such causes as produce the things just
mentioned, and not owing to the Forms. But regarding the Ideas it is
possible, both in this way and by more abstract and accurate
arguments, to collect many objections like those we have considered.

Since we have discussed these points, it is well to consider again
the results regarding numbers which confront those who say that
numbers are separable substances and first causes of things. If number
is an entity and its substance is nothing other than just number, as
some say, it follows that either (1) there is a first in it and a
second, each being different in species,-and either (a) this is true
of the units without exception, and any unit is inassociable with
any unit, or (b) they are all without exception successive, and any of
them are associable with any, as they say is the case with
mathematical number; for in mathematical number no one unit is in
any way different from another. Or (c) some units must be associable
and some not; e.g. suppose that 2 is first after 1, and then comes 3
and then the rest of the number series, and the units in each number
are associable, e.g. those in the first 2 are associable with one
another, and those in the first 3 with one another, and so with the
other numbers; but the units in the '2-itself' are inassociable with
those in the '3-itself'; and similarly in the case of the other
successive numbers. 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. 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

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