although in actual fact it is
untenable.For
assuming that there is a first unit or first 1,i.e., the Ideal One. it is reasonable that
the units should be prior and posterior; and similarly in the case of
2's, if there is a first 2. For it is reasonable and indeed necessary
that after the first there should be a second; and if a second, a
third; and so on with the rest in sequence.But the two statements, that there is after 1
a first and a second unit, and that there is a first 2, are
incompatible. These thinkers, however, recognize a first unit and
first 1, but not a second and third; and they recognize a first 2, but
not a second and third.It is also
evident that if all units are inaddible, there cannot be an Ideal 2
and 3, and similarly with the other numbers;for whether the units are indistinguishable or
each is different in kind from every other, numbers must be produced
by addition; e.g. 2 by adding 1 to another 1, and 3 by adding another
1 to the 2, and 4 similarly.This is of course not true of the natural
numbers.
This being so, numbers cannot be generated as these thinkers try to
generate them, from Unity and the dyad; because 2 becomes a part of
3,i.e., 3 is produced by
adding 1 to 2. and 3 of 4,and the same applies to the following
numbers.But
according to them 4 was generated from the first 2 and the
indeterminate dyad, thus consisting of two 2's apart from the Ideal
2.Cf. sect. 18.
Otherwise 4 will consist of the Ideal 2 and another 2 added to it, and
the Ideal 2 will consist of the Ideal 1 and another 1; and if this is
so the other element cannot be the indeterminate dyad, because it
produces one unit and not a definite 2.The general argument is: Numbers are produced by
addition; but this is incompatible with the belief in the
Indeterminate Dyad as a generative principle, because, being
duplicative, it cannot produce single units. Again, how can there be other 3's and 2's besides the Ideal numbers
3 and 2, and in what way can they be composed of prior and posterior
units? All these theories are absurd and fictitious, and there can be
no primary 2 and Ideal 3. Yet there must be, if we are to regard Unity
and the indeterminate dyad as elements.i.e., if numbers are not generated by addition,
there must be Ideal (or natural) numbers. But if the consequences are
impossible, the principles cannot be of this nature.If, then, any one unit differs in kind from any
other, these and other similar consequences necessarily follow. If, on
the other hand, while the units in different numbers are different,
those which are in the same number are alone indistinguishable from
one another, even so the consequences which follow are no less
difficult.