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. 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. For as there are many propositions about
things merely considered as in motion, apart from what each such thing
is and from their accidents, and as it is not therefore necessary that
there should be either a mobile separate from sensibles, or a distinct
mobile entity in the sensibles, so too in the case of mobiles there
will be propositions and sciences, which treat them however not qua
mobile but only qua bodies, or again only qua planes, or only qua
lines, or qua divisibles, or qua indivisibles having position, or only
qua indivisibles
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