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For when we apprehend the unity in 2, or in general in a number, do
we apprehend a thing-itself or something else?).
"Some, then, generate spatial magnitudes from matter of this sort,
others from the point -and the point is thought by them to be not
1 but something like 1-and from other matter like plurality, but not
identical with it; about which principles none the less the same difficulties
occur. For if the matter is one, line and plane-and soli will be the
same; for from the same elements will come one and the same thing.
But if the matters are more than one, and there is one for the line
and a second for the plane and another for the solid, they either
are implied in one another or not, so that the same 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
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METAPHYSICS 169
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 [ Pobierz całość w formacie PDF ]