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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
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