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Again, there are certain mathematical theorems that are universal,
extending beyond these substances. Here then we shall have another
intermediate substance separate both from the Ideas and from the
intermediates,-a substance which is neither number nor points nor
spatial magnitude nor time. And if this is impossible, plainly it is
also impossible that the former entities should exist separate from
sensible things.
And, in general, conclusion contrary alike to the truth and to the
usual views follow, if one is to suppose the objects of mathematics to
exist thus as separate entities. For because they exist thus they must
be prior to sensible spatial magnitudes, but in truth they must be
posterior; for the incomplete spatial magnitude is in the order of
generation prior, but in the order of substance posterior, as the
lifeless is to the living.
Again, by virtue of what, and when, will mathematical magnitudes
be one? For things in our perceptible world are one in virtue of soul,
or of a part of soul, or of something else that is reasonable
enough; when these are not present, the thing is a plurality, and
splits up into parts. But in the case of the subjects of
mathematics, which are divisible and are quantities, what is the cause
of their being one and holding together?
Again, the modes of generation of the objects of mathematics
show that we are right. For the dimension first generated is length,
then comes breadth, lastly depth, and the process is complete
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