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Metaphysics   


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 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 constructing a world and wish to speak the language
of natural science, it is fair to make some examination of their
physical theorics, but to let them off from the present inquiry; for
we are investigating the principles at work in unchangeable things, so
that it is numbers of this kind whose genesis we must study.
4
These thinkers say there is no generation of the odd number, which
evidently implies that there is generation of the even; and some
present the even as produced first from unequals-the great and the
small-when these are equalized. The inequality, then, must belong to
them before they are equalized. If they had always been equalized,
they would not have been unequal before; for there is nothing before
that which is always. Therefore evidently they are not giving their
account of the generation of numbers merely to assist contemplation of
their nature.
A difficulty, and a reproach to any one who finds it no
difficulty, are contained in the question how the elements and the
principles are related to the good and the beautiful; the difficulty
is this, whether any of the elements is such a thing as we mean by the
good itself and the best, or this is not so, but these are later in
origin than the elements. The theologians seem to agree with some
thinkers of the present day, who answer the question in the

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