
Metaphysics
construct the whole universe out of numbersonly not numbers
consisting of abstract units; they suppose the units to have spatial
magnitude. But how the first 1 was constructed so as to have
magnitude, they seem unable to say.
Another thinker says the first kind of number, that of the
Forms, alone exists, and some say mathematical number is identical
with this.
The case of lines, planes, and solids is similar. For some think
that those which are the objects of mathematics are different from
those which come after the Ideas; and of those who express
themselves otherwise some speak of the objects of mathematics and in a
mathematical wayviz. those who do not make the Ideas numbers nor
say that Ideas exist; and others speak of the objects of
mathematics, but not mathematically; for they say that neither is
every spatial magnitude divisible into magnitudes, nor do any two
units taken at random make 2. All who say the 1 is an element and
principle of things suppose numbers to consist of abstract units,
except the Pythagoreans; but they suppose the numbers to have
magnitude, as has been said before. It is clear from this statement,
then, in how many ways numbers may be described, and that all the ways
have been mentioned; and all these views are impossible, but some
perhaps more than others.
7
First, then, let us inquire if the units are associable or
inassociable, and if inassociable, in which of the two ways we
distinguished. For it is possible that any unity is inassociable
with any, and it is possible that those in the 'itself' are
inassociable with those in the 'itself', and, generally, that those in
each ideal number are inassociable with those in other ideal
numbers. Now (1) all units are associable and without difference, we
get mathematical numberonly one kind of number, and the Ideas
cannot be the numbers. For what sort of number will manhimself or
animalitself or any other Form be? There is one Idea of each thing
e.g. one of manhimself and another one of animalitself; but the
similar and undifferentiated numbers are infinitely many, so that
any particular 3 is no more manhimself than any other 3. But if the
Ideas are not numbers, neither can they exist at all. For from what
principles will the Ideas come? It is number that comes from the 1 and
the indefinite dyad, and the principles or elements are said to be
principles and elements of number, and the Ideas cannot be ranked as
either prior or posterior to the numbers.
But (2) if the units are inassociable, and inassociable in the
sense that any is inassociable with any other, number of this sort
cannot be mathematical number; for mathematical number consists of
undifferentiated units, and the truths proved of it suit this
character. Nor can it be ideal number. For 2 will not proceed
immediately from 1 and the indefinite dyad, and be followed by the
successive numbers, as they say '2,3,4' for the units in the ideal are
generated at the same time, whether, as the first holder of the theory
said, from unequals (coming into being when these were equalized) or
in some other waysince, if one unit is to be prior to the other, it
will be prior also to 2 the composed of these; for when there is one
thing prior and another posterior, the resultant of these will be
prior to one and posterior to the other. Again, since the 1itself is
first, and then there is a particular 1 which is first among the
others and next after the 1itself, and again a third which is next
after the second and next but one after the first 1,so the units must
be prior to the numbers after which they are named when we count them;
e.g. there will be a third unit in 2 before 3 exists, and a fourth and
a fifth in 3 before the numbers 4 and 5 exist.Now none of these
thinkers has said the units are inassociable in this way, but
according to their principles it is reasonable that they should be
so even in this way, though in truth it is impossible. For it is
