From these considerations it is also
clear that the third alternative^{1}—that Ideal number and mathematical
number are the same—is the worst; for two errors have to be
combined to make one theory. (1.) Mathematical number cannot be of
this nature, but the propounder of this view has to spin it out by
making peculiar assumptions; (2.) his theory must admit all the
difficulties which confront those who speak of Ideal number.

The Pythagorean view in one way contains fewer difficulties than the
view described above, but in another way it contains further
difficulties peculiar to itself. By not regarding number as separable,
it disposes of many of the impossibilities; but that bodies should be
composed of numbers, and that these numbers should be mathematical, is
impossible.^{2} For (a) it is not true to speak of indivisible magnitudes^{3}; (b) assuming that this view is
perfectly true, still units at any rate have no magnitude; and how can
a magnitude be composed of indivisible parts? Moreover arithmetical
number consists of abstract units. But the Pythagoreans identify
number with existing things; at least they apply mathematical
propositions to bodies as though they consisted of those numbers.^{4}

Thus if number, [20] if it is a self-subsistent reality, must be regarded in one of the ways described above, and if it cannot be regarded in any of these ways, clearly number has no such nature as is invented for it by those who treat it as separable.

Again, does each unit come from the
Great and the Small, when they are equalized^{5}; or does one come from the Small and another from the
Great? If the latter, each thing is not composed of all the elements,
nor are the units undifferentiated; for one contains the Great, and
the other the Small, which is by nature contrary to the
Great.Again, what
of the units in the Ideal 3? because there is one over. But no doubt
it is for this reason that in an odd number they make the Ideal One
the middle unit.^{6} If on the
other hand each of the units comes from both Great and Small, when
they are equalized, how can the Ideal 2 be a single entity composed of
the Great and Small? How will it differ from one of its units? Again,
the unit is prior to the 2; because when the unit disappears the 2
disappears.Therefore the unit must be the Idea of an Idea, since it is prior to
an Idea, and must have been generated before it. From what, then? for
the indeterminate dyad, as we have seen,^{7}
causes duality.

Again, number must be either infinite or finite (for they make number separable,