[1085b] [1] If on the other hand there is more than one kind of matter—one of the line, another of the plane, and another of the solid—either the kinds are associated with each other, or they are not. Thus the same result will follow in this case also; for either the plane will not contain a line, or it will be a line.

Further, no attempt is made to explain how number can be generated from unity and plurality; but howsoever they account for this, they have to meet the same difficulties as those who generate number from unity and the indeterminate dyad. The one school generates number not from a particular plurality but from that which is universally predicated; the other from a particular plurality, but the first; for they hold that the dyad is the first plurality.1 Thus there is practically no difference between the two views; the same difficulties will be involved with regard to mixture, position, blending, generation and the other similar modes of combination.2

We might very well ask the further question: if each unit is one, of what it is composed; for clearly each unit is not absolute unity. It must be generated from absolute unity and either plurality or a part of plurality.Now we cannot hold that the unit is a plurality, because the unit is indivisible; but the view that it is derived from a part of plurality involves many further difficulties, because (a) each part must be indivisible; otherwise it will be a plurality and the unit will be divisible, [20] and unity and plurality will not be its elements, because each unit will not be generated from plurality3 and unity.(b) The exponent of this theory merely introduces another number; because plurality is a number of indivisible parts.4

Again, we must inquire from the exponent of this theory whether the number5 is infinite or finite.There was, it appears, a finite plurality from which, in combination with Unity, the finite units were generated; and absolute plurality is different from finite plurality. What sort of plurality is it, then, that is, in combination with unity, an element of number?

We might ask a similar question with regard to the point, i.e. the element out of which they create spatial magnitudes.This is surely not the one and only point. At least we may ask from what each of the other points comes; it is not, certainly, from some interval and the Ideal point. Moreover, the parts of the interval cannot be indivisible parts, any more than the parts of the plurality of which the units are composed; because although number is composed of indivisible parts, spatial magnitudes are not.

All these and other similar considerations make it clear that number and spatial magnitudes cannot exist separately.

1 Aristotle again identifies the indeterminate dyad with the number 2.

2 sc. of the elements of number.

3 sc. but from an indivisible part of plurality—which is not a plurality but a unity.

4 i.e., to say that number is derived from plurality is to say that number is derived from number—which explains nothing.

5 sc. which plurality has been shown to be.