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does the 'unit', unless it means a measure or the quantitatively
indivisible. If, then, the quantitative and the 'what' are
different, we are not told whence or how the 'what' is many; but if
any one says they are the same, he has to face many inconsistencies.
One might fix one's attention also on the question, regarding
the numbers, what justifies the belief that they exist. To the
believer in Ideas they provide some sort of cause for existing things,
since each number is an Idea, and the Idea is to other things
somehow or other the cause of their being; for let this supposition be
granted them. But as for him who does not hold this view because he
sees the inherent objections to the Ideas (so that it is not for
this reason that he posits numbers), but who posits mathematical
number, why must we believe his statement that such number exists, and
of what use is such number to other things? Neither does he who says
it exists maintain that it is the cause of anything (he rather says it
is a thing existing by itself), nor is it observed to be the cause
of anything; for the theorems of arithmeticians will all be found true
even of sensible things, as was said before.

As for those, then, who suppose the Ideas to exist and to be
numbers, by their assumption in virtue of the method of setting out
each term apart from its instances-of the unity of each general term
they try at least to explain somehow why number must exist. Since
their reasons, however, are neither conclusive nor in themselves
possible, one must not, for these reasons at least, assert the
existence of number. Again, the Pythagoreans, because they saw many
attributes of numbers belonging te sensible bodies, supposed real
things to be numbers-not separable numbers, however, but numbers of
which real things consist. But why? Because the attributes of
numbers are present in a musical scale and in the heavens and in
many other things. Those, however, who say that mathematical number
alone exists cannot according to their hypotheses say anything of this
sort, but it used to be urged that these sensible things could not
be the subject of the sciences. But we maintain that they are, as we
said before. And it is evident that the objects of mathematics do
not exist apart; for if they existed apart their attributes would
not have been present in bodies. Now the Pythagoreans in this point
are open to no objection; but in that they construct natural bodies
out of numbers, things that have lightness and weight out of things
that have not weight or lightness, they seem to speak of another
heaven and other bodies, not of the sensible. But those who make
number separable assume that it both exists and is separable because
the axioms would not be true of sensible things, while the
statements of mathematics are true and 'greet the soul'; and similarly
with the spatial magnitudes of mathematics. It is evident, then,
both that the rival theory will say the contrary of this, and that the
difficulty we raised just now, why if numbers are in no way present in
sensible things their attributes are present in sensible things, has
to be solved by those who hold these views.
There are some who, because the point is the limit and extreme
of the line, the line of the plane, and the plane of the solid,
think there must be real things of this sort. We must therefore
examine this argument too, and see whether it is not remarkably
weak. For (i) extremes are not substances, but rather all these things
are limits. For even walking, and movement in general, has a limit, so
that on their theory this will be a 'this' and a substance. But that
is absurd. Not but what (ii) even if they are substances, they will
all be the substances of the sensible things in this world; for it
is to these that the argument applied. Why then should they be capable
of existing apart?
Again, if we are not too easily satisfied, we may, regarding all
number and the objects of mathematics, press this difficulty, that
they contribute nothing to one another, the prior to the posterior;

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