Now that we have dealt with the problems concerning the Ideas, we had better re-investigate the problems connected with numbers that follow from the theory that numbers are separate substances and primary causes of existing things. Now if number is a kind of entity, and has nothing else as its substance, but only number itself, as some maintain; then either (a) there must be some one part of number which is primary, and some other part next in succession, and so on, each part being specifically different Footnote 1This statement bears two meanings, which Aristotle confuses: (i) There must be more than one number-series, each series being different in kind from every other series; (2) All numbers are different in kind, and inaddible. Confusion (or textual inaccuracy) is further suggested by the fact that Aristotle offers no alternative statement of the nature of number in general, such as we should expect from his language. In any case the classification is arbitrary and incomplete. —
and this applies directly to units, and any given unit is inaddible to any other given unit; or (b) they Footnote 2The units. are all directly successive, and any units can be added to any other units, as is held of mathematical number; for in mathematical number no one unit differs in any way from another.
Or (c) some units must be addible and others not. E.g., 2 is first after 1, and then 3, and so on with the other numbers; and the units in each number are addible, e.g. the units in the first Footnote 3i.e., Ideal or natural. 2 are addible to one another, and those in the first 3 to one another, and so on in the case of the other numbers; but the units in the Ideal 2 are inaddible to those in the Ideal 3;
and similarly in the case of the other successive numbers. Hence whereas mathematical number is counted thus: after 1, 2 (which consists of another 1 added to the former) and 3 (which consists of another 1 added to these two) and the other numbers in the same way, Ideal number is counted like this: after 1, a distinct 2 not including the original 1; and a 3 not including the 2, and the rest of the numbers similarly.
Or (d) one kind of number must be such as we first described, and another or such as the mathematicians maintain, and that which we have last described must be a third kind.
Again, these numbers must exist either in separation from things, or not in separation, but in sensible things (not, however, in the way which we first considered, Footnote 4In Aristot. Met. 13.2.1-3. but in the sense that sensible things are composed of numbers which are present in them Footnote 5The Pythagorean number-atomist view; See Introduction. )—either some of them and not others, or all of them. Footnote 6i.e., either all numbers are material elements of things, or some are and others are not.
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 Footnote 7Cf. sect. 2. );
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 Footnote 8Cf. Aristot. Met. 1.6.4. ; others hold that mathematical number alone exists, Footnote 9Cf. Aristot. Met. 12.10.14. 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; they suppose the units to be extended—but as for how the first extended unit was formed they appear to be at a loss. Footnote 10Cf. Aristot. Met. 13.8.9, 10, Aristot. Met. 14.3.15, Aristot. Met. 14.5.7, and see Introduction.
Another thinker holds that primary or Ideal number alone exists; and some Footnote 11Cf. 10ff., Aristot. Met. 13.1.4. identify this with mathematical number.
The same applies in the case of lines, planes and solids.
Some Footnote 12Plato. distinguish mathematical objects from those which come after the Ideas
Footnote 13i.e., the (semi-)Ideal lines, planes, etc. Cf. Aristot. Met. 1.9.30. ; and of those who treat the subject in a different manner some Footnote 14Speusippus; cf. sect. 7 above. 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 Footnote 15Xenocrates. For his belief in indivisible lines see Ritter and Preller 362. Aristotle ascribes the doctrine to Plato in Aristot. Met. 1.9.25. 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. Footnote 16sect. 8.
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 Footnote 17sc. the view of Xenocrates (cf. Aristot. Met. 13.8.8). are more so than others.