The Complete Aristotle (eng.)

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.
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3
    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;
for if number did not exist, none the less spatial magnitudes would
exist for those who maintain the existence of the objects of
mathematics only, and if spatial magnitudes did not exist, soul and
sensible bodies would exist. But the observed facts show that
nature is not a series of episodes, like a bad tragedy. As for the
believers in the Ideas, this difficulty misses them; for they
construct spatial magnitudes out of matter and number, lines out of
the number planes doubtless out of solids out of or they use other
numbers, which makes no difference. But will these magnitudes be
Ideas, or what is their manner of existence, and what do they
contribute to things? These contribute nothing, as the objects of
mathematics contribute nothing. But not even is any theorem true of
them, unless we want to change the objects of mathematics and
invent doctrines of our own. But it is not hard to assume any
random hypotheses and spin out a long string of conclusions. These
thinkers, then, are wrong in this way, in wanting to unite the
objects of mathematics with the Ideas. And those who first posited
two kinds of