For those things are prior in substantiality which
when separated from other things surpass them in the power of
independent existence, but things are prior in definition to those
whose definitions are compounded out of their definitions; and these
two properties are not coextensive. For if attributes do not exist
apart from the substances (e.g. a 'mobile' or a pale'), pale is
prior to the pale man in definition, but not in substantiality. For it
cannot exist separately, but is always along with the concrete
thing; and by the concrete thing I mean the pale man. Therefore it
is plain that neither is the result of abstraction prior nor that
which is produced by adding determinants posterior; for it is by
adding a determinant to pale that we speak of the pale man.
It has, then, been sufficiently pointed out that the objects of
mathematics are not substances in a higher degree than bodies are, and
that they are not prior to sensibles in being, but only in definition,
and that they cannot exist somewhere apart. But since it was not
possible for them to exist in sensibles either, it is plain that
they either do not exist at all or exist in a special sense and
therefore do not 'exist' without qualification. For 'exist' has many
senses.
3
For just as the universal propositions of mathematics deal not
with objects which exist separately, apart from extended magnitudes
and from numbers, but with magnitudes and numbers, not however qua
such as to have magnitude or to be divisible, clearly it is possible
that there should also be both propositions and demonstrations about
sensible magnitudes, not however qua sensible but qua possessed of
certain definite qualities.
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