so that one of these alternatives must be
true).The point seems to
be that if number is self-subsistent it must be
actually finite or infinite. Aristotle himself
holds that number is infinite only potentially; i.e., however
high you can count, you can always count
higher. Now
it is obvious that it cannot be infinite, because infinite number is
neither odd nor even, and numbers are always generated either from odd
or from even number. By one process, when 1 is added to an even
number, we get an odd number; by another, when 1 is multiplied by 2,
we get ascending powers of 2; and by another, when powers of 2 are
multiplied by odd numbers, we get the remaining even
numbers. Again, if every Idea is an Idea of
something, and the numbers are Ideas, infinite number will also be an
Idea of something, either sensible or otherwise. This, however, is
impossible, both logicallyi.e.,
as implying an actual infinite. and on their own
assumption,i.e., as
inconsistent with the conception of an Idea as a determining
principle. since they regard the Ideas as they
do.If, on the other hand, number
is finite, what is its limit? In reply to this we must not only assert
the fact, but give the reason.Now if number only goes up to 10, as some
hold,Cf. Aristot. Met. 12.8.2. The Platonists derived this
view from the Pythagoreans; see Introduction. in the
first place the Forms will soon run short. For example, if 3 is the
Idea of Man, what number will be the Idea of Horse? Each number up to
10 is an Idea; the Idea of Horse, then, must be one of the numbers in
this series, for they are substances or Ideas.But the fact remains that they will run
short, because the different types of animals will outnumber them. At
the same time it is clear that if in this way the Ideal 3 is the Idea
of Man, so will the other 3's be also (for the 3's in the same
numbersRobin is probably
right in taking this to mean that the 3 which is in the ideal 4
is like the 3 which is in the 4 which is in a higher ideal
number, and so on (La Theorie platonicienne des Idees et
des Nombres d'apres Aristote, p. 352). are
similar),so that
there will be an infinite number of men; and if each 3 is an Idea,
each man will be an Idea of Man; or if not, they will still be
men.And if the
smaller number is part of the greater, when it is composed of the
addible units contained in the same number, then if the Ideal 4 is the
Idea of something, e.g. "horse" or "white," then "man" will be part of
"horse," if "man" is 2. It is absurd also that there should be an Idea
of 10 and not of 11, nor of the following numbers.
Again, some things exist and come into being of which there are no
FormsCf. Aristot. Met. 13.4.7, 8; Aristot. Met.
1.9.2, 3.; why, then, are there not Forms of
these too? It follows that the Forms are not the causes of
things.Again, it is absurd that
number up to 10 should be more really existent, and a Form, than 10
itself; although the former is not generated as a unity, whereas the
latter is. However, they try to make out that the series up to 10 is a
complete number;at least
they generate the derivatives, e.g. the void, proportion, the odd,
etc., from within the decad. Some, such as motion, rest, good and
evil, they assign to the first principles; the rest to numbers.From the Dyad were derived void
(Theophrastus, Met. 312.18-313.3) and
motion (cf. Aristot. Met. 1.9.29, Aristot. Met. 11.9.8). Rest would naturally be derived from unity. For good and
evil see Aristot. Met. 1.6.10. Proportion
alone of the "derivatives" here mentioned appears to be derived
from number. As Syrianus says, the three types of proportion can
be illustrated by numbers from within the
decadâ€”arithmetical 1. 2. 3, geometrical 1. 2. 4,
harmonic 2. 3. 6. Hence they identify the odd with Unity;
because if oddness depended on 3, how could 5 be odd?sc. because (on their theory) 3
is not contained in 5. Thus oddness had to be referred to not a
number but a principleâ€”unity.Again, they hold that spatial magnitudes and the
like have a certain limit;