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. 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
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