The Complete Aristotle (eng.)

explained?). And (ii) the
same happens as in regard to number; for ‘long and short’, &c.,
are attributes of magnitude, but magnitude does not consist of
these, any more than the line consists of ‘straight and curved’, or
solids of ‘smooth and rough’.
    (All these views share a difficulty which occurs with regard to
species-of-a-genus, when one posits the universals, viz. whether it
is animal-itself or something other than animal-itself that is in
the particular animal. True, if the universal is not separable from
sensible things, this will present no difficulty; but if the 1 and
the numbers are separable, as those who express these views say, it
is not easy to solve the difficulty, if one may apply the words
‘not easy’ to the impossible. 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 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