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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,
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METAPHYSICS 177
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 number, that of the Forms and that which is mathematical, neither
have said nor can say how mathematical number is to exist and of what
it is to consist. For they place it between ideal and sensible number.
If (i) it consists of the great and small, it will be the same as
the other-ideal-number (he makes spatial magnitudes out of some other
small and great). And if (ii) he names some other element, he will
be making his elements rather many. And if the principle of each of
the two kinds of number is a 1, unity will be something common to
these, and we must inquire how the one is these many things, while
at the same time number, according to him, cannot be generated except
from one and an indefinite dyad.
"All this is absurd, and conflicts both with itself and with the
probabilities,
and we seem to see in it Simonides 'long rigmarole' for the long rigmarole
comes into play, like those of slaves, when men have nothing sound
to say. And the very elements-the great and the small-seem to cry
out against the violence that is done to them; for they cannot in
any way generate numbers other than those got from 1 by doubling.
"It is strange also to attribute generation to things that are eternal,
or rather this is one of the things that are impossible. There need
be no doubt whether the Pythagoreans attribute generation to them
or not; for they say plainly that when the one had been constructed,
whether out of planes or of surface or of seed or of elements which
they cannot express, immediately the nearest part of the unlimited
began to be constrained and limited by the limit. But since they are [ Pobierz całość w formacie PDF ]