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Works by Aristotle
Pages of Metaphysics

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results will follow even so; for either the plane will not contain a
line or it will he a line.
Again, how number can consist of the one and plurality, they
make no attempt to explain; but however they express themselves, the
same objections arise as confront those who construct number out of
the one and the indefinite dyad. For the one view generates number
from the universally predicated plurality, and not from a particular
plurality; and the other generates it from a particular plurality, but
the first; for 2 is said to be a 'first plurality'. Therefore there is
practically no difference, but the same difficulties will follow,-is
it intermixture or position or blending or generation? and so on.
Above all one might press the question 'if each unit is one, what does
it come from?' Certainly each is not the one-itself. It must, then,
come from the one itself and plurality, or a part of plurality. To say
that the unit is a plurality is impossible, for it is indivisible; and
to generate it from a part of plurality involves many other
objections; for (a) each of the parts must be indivisible (or it
will be a plurality and the unit will be divisible) and the elements
will not be the one and plurality; for the single units do not come
from plurality and the one. Again, (,the holder of this view does
nothing but presuppose another number; for his plurality of
indivisibles is a number. Again, we must inquire, in view of this
theory also, whether the number is infinite or finite. For there was
at first, as it seems, a plurality that was itself finite, from
which and from the one comes the finite number of units. And there
is another plurality that is plurality-itself and infinite
plurality; which sort of plurality, then, is the element which
co-operates with the one? One might inquire similarly about the point,
i.e. the element out of which they make spatial magnitudes. For surely
this is not the one and only point; at any rate, then, let them say
out of what each of the points is formed. Certainly not of some
distance + the point-itself. Nor again can there be indivisible
parts of a distance, as the elements out of which the units are said
to be made are indivisible parts of plurality; for number consists
of indivisibles, but spatial magnitudes do not.
All these objections, then, and others of the sort make it evident
that number and spatial magnitudes cannot exist apart from things.
Again, the discord about numbers between the various versions is a
sign that it is the incorrectness of the alleged facts themselves that
brings confusion into the theories. For those who make the objects
of mathematics alone exist apart from sensible things, seeing the
difficulty about the Forms and their fictitiousness, abandoned ideal
number and posited mathematical. But those who wished to make the
Forms at the same time also numbers, but did not see, if one assumed
these principles, how mathematical number was to exist apart from
ideal, made ideal and mathematical number the same-in words, since
in fact mathematical number has been destroyed; for they state
hypotheses peculiar to themselves and not those of mathematics. And he
who first supposed that the Forms exist and that the Forms are numbers
and that the objects of mathematics exist, naturally separated the
two. Therefore it turns out that all of them are right in some
respect, but on the whole not right. And they themselves confirm this,
for their statements do not agree but conflict. The cause is that
their hypotheses and their principles are false. And it is hard to
make a good case out of bad materials, according to Epicharmus: 'as
soon as 'tis said, 'tis seen to be wrong.'
But regarding numbers the questions we have raised and the
conclusions we have reached are sufficient (for while he who is
already convinced might be further convinced by a longer discussion,
one not yet convinced would not come any nearer to conviction);
regarding the first principles and the first causes and elements,
the views expressed by those who discuss only sensible substance
have been partly stated in our works on nature, and partly do not
belong to the present inquiry; but the views of those who assert

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