called the individuals. Further, if unity is not a substance, evidently number also will not exist as an entity separate from the individual things; for number is units, and the unit is something whose essence it is to be one.—But (2) if there is a unity-itself and a being-itself, their substance must be unity and being; for it is not something else that is predicated universally of them, but just unity and being. But if there is to be a being-itself and a unity-itself, there is much difficulty in seeing how there will be anything else besides these,—I mean, how things will be more than one in number. For what is different from being does not exist, so that it necessarily follows, according to the argument of Parmenides, that all things that are are one and this is being.—There are objections to both views. For whether unity is not a substance or there is a unity-itself, number cannot be a substance. We have already said why this result follows if unity is not a substance: and if it is, the same difficulty arises as arose with regard to being. For whence is there to be another one besides the unity-itself? It must be not-one; but all things are either one or many, and of the many each is one.—Further, if the unity-itself is indivisible, according to Zeno's doctrine it will be nothing. For that which neither when added makes a thing greater nor when subtracted makes it less, he asserts to have no being, evidently assuming that whatever has being is a spatial magnitude. And if it is a magnitude, it is corporeal; for the corporeal has being in every dimension, while the other objects of mathematics, e.g. a plane or a line, added in one way will increase what they are added to, but in another way will not do so, and a point or a unit does so in no way. But if he argues thus, his argument is of a low order; and an indivisible thing can exist, so that in this way too the position may be defended even against him; for the indivisible when added will make the number, though not the size, greater. But how can a magnitude proceed from one such indivisible or from many? It is like saying that the line is made out
- Cf. 1001a24.
- Cf. Diels, Vorsokratiker, ed. 2, p. 130, § 21.
- e.g. a line added to another at the end makes it longer, but one which lies beside another makes it no broader.