this is impossible. Further, why is a number, when taken all together, one? Again, besides what has been said, if the units are diverse the Platonists should have spoken like those who say there are four, or two, elements; for each of these thinkers gives the name of element not to that which is common, e. g. to body, but to fire and earth, whether there is something common to them, viz. body, or not. But in fact the Platonists speak as if the One were homogeneous like fire or water; and if this is so, the numbers will not be substances. Evidently, if there is a One-in-itself and this is a first principle, 'one' is being used in more than one sense; for otherwise the theory is impossible.
When we Platonists wish to reduce substances to their principles, we state that lines come from the short and long (i. e. from a kind of small and great), and the plane from the broad and narrow, and the solid from the deep and shallow. Yet how then can the plane contain a line, or the solid a line or a plane? For the broad and narrow is a different class of things from the deep and shallow. Therefore, just as number is not present in these, because the many and few are different from these, evidently no other of the higher <more abstract> classes will be present in the lower <more concrete>. But again the broad is not a genus which includes the deep, for then the solid would have been a species of plane. Further, from what principle will the presence of the points in the line be derived? Plato even used to object to this class of things <sc. points> as being a geometrical fiction. What we call the point he called the principle of the line, and this is what he meant by the indivisible lines which he often posited. Yet these must have a limit; therefore the argument from which the existence of the line follows proves also the existence of the point.
In general, though philosophy seeks the cause of perceptible things, we Platonists have given this up (for we say nothing of the cause from which change takes its start), but while we fancy we are stating the substance of perceptible things, we assert the existence of a second class of substances, while
- Sc. but ordinary mathematical numbers. Cf. m. 1081a5.
- With 992a 10-19 cf. m. 1085a9-19.