Book 13
2
and this applies directly to units, and any given unit is inaddible to any other given unit; or (b) theyThe 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.
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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 firsti.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;
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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.
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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.
5
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,In Aristot. Met. 13.2.1-3. but in the sense that sensible things are composed of numbers which are present in themThe Pythagorean number-atomist view; See Introduction.)—either some of them and not others, or all of them.i.e., either all numbers are material elements of things, or some are and others are not.
