Let it
be granted that they are prior in formula; yet not everything which is
prior in formula is also prior in substantiality. Things are prior in
substantiality which when separated have a superior power of
existence; things are prior in formula from whose formulae the
formulae of other things are compounded. And these characteristics are
not indissociable.For if
attributes, such as "moving" or "white," do not exist apart from their
substances, "white" will be prior in formula to "white man," but not
in substantiality; for it cannot exist in separation, but always
exists conjointly with the concrete whole—by which I mean
"white man."Thus it is
obvious that neither is the result of abstraction prior, nor the
result of adding a determinant posterior—for the expression
"white man" is the result of adding a determinant to
"white."Thus we have sufficiently
shown (a) that the objects of mathematics are not more substantial
than corporeal objects; (b) that they are not prior in point of
existence to sensible things, but only in formula; and (c) that they
cannot in any way exist in separation.And since we have seensect. 1-3 above. that they cannot exist in
sensible things, it is clear that either they do not exist at all, or
they exist only in a certain way, and therefore not absolutely; for
"exist" has several senses. The general propositions in
mathematics are not concerned with objects which exist separately
apart from magnitudes and numbers; they are concerned with magnitudes
and numbers,but not with
them as possessing magnitude or being divisible. It is clearly
possible that in the same way propositions and logical proofs may
apply to sensible magnitudes; not qua sensible,
but qua having certain
characteristics.For
just as there can be many propositions about things merely qua movable, without any reference to the
essential nature of each one or to their attributes, and it does not
necessarily follow from this either that there is something movable
which exists in separation from sensible things or that there is a
distinct movable nature in sensible things; so too there will be
propositions and sciences which apply to movable things, not qua movable but qua
corporeal only; and again qua planes only and
qua lines only, and qua divisible, and qua indivisible but
having position, and qua indivisible
only.Therefore since
it is true to say in a general sense not only that things which are
separable but that things which are inseparable exist, e.g., that
movable things exist, it is also true to say in a general sense that
mathematical objects exist, and in such a form as mathematicians
describe them.And just as
it is true to say generally of the other sciences that they deal with
a particular subject—not with that which is accidental to it
(e.g. not with "white" if "the healthy" is white, and the subject of
the science is "the healthy"), but with that which is the subject of
the particular science;