then how can the Ideas, if they are the
substances of things, exist in separation from them?In thePhaedoPlat. Phaedo 100d. this
statement is made: that the Forms are causes both of being and of
generation. Yet assuming that the Forms exist, still there is no
generation unless there is something to impart motion; and many other
things are generated (e.g. house and ring) of which the Idealists say
that there are no Forms.Thus it is clearly possible that those things of which they say that
there are Ideas may also exist and be generated through the same kind
of causes as those of the things which we have just mentioned, and not
because of the Forms. Indeed, as regards the Ideas, we can collect
against them plenty of evidence similar to that which we have now
considered; not only by the foregoing methods, but by means of more
abstract and exact reasoning. Now that we have dealt with
the problems concerning the Ideas, we had better re-investigate the
problems connected with numbers that follow from the theory that
numbers are separate substances and primary causes of existing things.
Now if number is a kind of entity, and has nothing else as its
substance, but only number itself, as some maintain; then either (a)
there must be some one part of number which is primary, and some other
part next in succession, and so on, each part being specifically
differentThis statement
bears two meanings, which Aristotle confuses: (i) There must be
more than one number-series, each series being different in kind
from every other series; (2) All numbers are different in kind,
and inaddible. Confusion (or textual inaccuracy) is further
suggested by the fact that Aristotle offers no alternative
statement of the nature of number in general, such as we should
expect from his language. In any case the classification is
arbitrary and incomplete.— and this applies directly to units,
and any given unit is inaddible to any other given unit;or (b) theyThe units. are all
directly successive, and any units can be added to any other units, as
is held of mathematical number; for in mathematical number no one unit
differs in any way from another.Or (c) some units must be addible and others
not. E.g., 2 is first after 1, and then 3, and so on with the other
numbers; and the units in each number are addible, e.g. the units in
the firsti.e., Ideal or
natural.2 are addible to one another, and those in the
first 3 to one another, and so on in the case of the other numbers;
but the units in the Ideal 2 are inaddible to those in the Ideal
3;and similarly in
the case of the other successive numbers. Hence whereas mathematical
number is counted thus: after 1, 2 (which consists of another 1 added
to the former) and 3 (which consists of another 1 added to these two)
and the other numbers in the same way, Ideal number is counted like
this: after 1, a distinct 2 not including the original 1; and a 3 not
including the 2, and the rest of the numbers similarly.Or (d) one kind of number must
be such as we first described, and another or such as the
mathematicians maintain, and that which we have last described must be
a third kind.Again, these numbers must
exist either in separation from things,