Aristotle, Metaphysics

[1080b] [1] or not in separation, but in sensible things (not, however, in the way which we first considered,¹ but in the sense that sensible things are composed of numbers which are present in them²）—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 inaddible⁴);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 things⁵; 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 some⁸ identify this with mathematical number.

The same applies in the case of lines, planes and solids.Some⁹ distinguish mathematical objects from those which "come after the Ideas"¹⁰; and of those who treat the subject in a different manner some¹¹ 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 others¹² 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 some¹⁴ are more so than others.

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