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. 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.
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
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