Book 13
6
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 inaddibleCf. sect. 2.);
7
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 thingsCf. Aristot. Met. 1.6.4.; others hold that mathematical number alone exists,Cf. Aristot. Met. 12.10.14. being the primary reality and separate from sensible things.
8
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; they suppose the units to be extended—but as for how the first extended unit was formed they appear to be at a loss.Cf. Aristot. Met. 13.8.9, 10, Aristot. Met. 14.3.15, Aristot. Met. 14.5.7, and see Introduction.
9
Another thinker holds that primary or Ideal number alone exists; and someCf. 10ff., Aristot. Met. 13.1.4. identify this with mathematical number.
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The same applies in the case of lines, planes and solids.
