The case of lines, planes, and solids is similar. For some think
that those which are the objects of mathematics are different from
those which come after the Ideas; and of those who express
themselves otherwise some speak of the objects of mathematics and in a
mathematical way-viz. those who do not make the Ideas numbers nor
say that Ideas exist; and others speak of the objects of
mathematics, but not mathematically; for they say that neither is
every spatial magnitude divisible into magnitudes, nor do any two
units taken at random make 2. All who say the 1 is an element and
principle of things suppose numbers to consist of abstract units,
except the Pythagoreans; but they suppose the numbers to have
magnitude, as has been said before. It is clear from this statement,
then, in how many ways numbers may be described, and that all the ways
have been mentioned; and all these views are impossible, but some
perhaps more than others.
7
First, then, let us inquire if the units are associable or
inassociable, and if inassociable, in which of the two ways we
distinguished. For it is possible that any unity is inassociable
with any, and it is possible that those in the 'itself' are
inassociable with those in the 'itself', and, generally, that those in
each ideal number are inassociable with those in other ideal
numbers. Now (1) all units are associable and without difference, we
get mathematical number-only one kind of number, and the Ideas
cannot be the numbers.
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