^{1}but in the sense that sensible things are composed of numbers which are present in them

^{2}）—either some of them and not others, or all of them.

^{3}These are of necessity the only ways in which the numbers can exist. Now of those who say that unity is the beginning and substance and element of all things, and that number is derived from it and something else, almost everyone has described number in one of these ways (except that no one has maintained that all units are inaddible

^{4});and this is natural enough, because there can be no other way apart from those which we have mentioned. Some hold that both kinds of number exist, that which involves priority and posteriority being identical with the Ideas, and mathematical number being distinct from Ideas and sensible things, and both kinds being separable from sensible things

^{5}; others hold that mathematical number alone exists,

^{6}being the primary reality and separate from sensible things.

The Pythagoreans also believe in one
kind of number—the mathematical; only they maintain that it
is not separate, but that sensible substances are composed of it. For
they construct the whole universe of numbers, but not of numbers
consisting of abstract units;
[20]
they suppose the units to be extended—but as for how the
first extended unit was formed they appear to be at a loss.^{7}

Another thinker holds that
primary or Ideal number alone exists; and some^{8} identify this with mathematical
number.

The same applies in the
case of lines, planes and solids.Some^{9} distinguish mathematical objects from those which
"come after the Ideas"^{10}; and of those who treat the subject
in a different manner some^{11} speak of the mathematical
objects and in a mathematical way—viz. those who do not
regard the Ideas as numbers, nor indeed hold that the Ideas
exist—and others^{12} speak of the mathematical
objects, but not in a mathematical way; for they deny that every
spatial magnitude is divisible into extended magnitudes, or that any
two given units make 2.But all who hold that Unity is an element and principle of existing
things regard numbers as consisting of abstract units, except the
Pythagoreans; and they regard number as having spatial magnitude, as
has been previously stated.^{13}

It is clear from
the foregoing account (1.) in how many ways it is possible to speak of
numbers, and that all the ways have been described. They are all
impossible, but doubtless some^{14} are more so than others.

First, then, we must inquire whether the limits are addible or inaddible;