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Metaphysics   


other senses as well, and the other objects of sense, will exist
apart; for why should one set of them do so and another not? And if
this is so, there will also be animals existing apart, since there
will be senses.
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. If,
then, that which is posterior in the order of generation is prior in
the order of substantiality, the solid will be prior to the plane
and the line. And in this way also it is both more complete and more
whole, because it can become animate. How, on the other hand, could
a line or a plane be animate? The supposition passes the power of
our senses.
Again, the solid is a sort of substance; for it already has in a
sense completeness. But how can lines be substances? Neither as a form
or shape, as the soul perhaps is, nor as matter, like the solid; for
we have no experience of anything that can be put together out of
lines or planes or points, while if these had been a sort of
material substance, we should have observed things which could be
put together out of them.
Grant, then, that they are prior in definition. Still not all
things that are prior in definition are also prior in
substantiality. 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.
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