Aristotle, Metaphysics

Since there is no contact in numbers, but units which have nothing between them—e.g. those in 2 or 3—are successive, the question might be raised whether or not they are successive to Unity itself, and whether of the numbers which succeed it 2 or one of the units in 2 is prior.

We find similar difficulties in the case of the genera posterior to number¹—the line, plane and solid. Some derive these from the species of the Great and Small; viz. lines from the Long and Short, planes from the Broad and Narrow, and solids from the Deep and Shallow. These are species of the Great and Small.As for the geometrical first principle which corresponds to the arithmetical One, different Platonists propound different views.² In these too we can see innumerable impossibilities, fictions and contradictions of all reasonable probability. For (a) we get that the geometrical forms are unconnected with each other, unless their principles also are so associated that the Broad and Narrow is also Long and Short; and if this is so, the plane will be a line and the solid a plane. [20] Moreover, how can angles and figures, etc., be explained? And (b) the same result follows as in the case of number; for these concepts are modifications of magnitude, but magnitude is not generated from them, any more than a line is generated from the Straight and Crooked, or solids from the Smooth and Rough.

Common to all these Platonic theories is the same problem which presents itself in the case of species of a genus when we posit universals—viz. whether it is the Ideal animal that is present in the particular animal, or some other "animal" distinct from the Ideal animal. This question will cause no difficulty if the universal is not separable; but if, as the Platonists say, Unity and the numbers exist separately, then it is not easy to solve (if we should apply the phrase "not easy" to what is impossible).For when we think of the one in 2, or in number generally, are we thinking of an Idea or of something else?

These thinkers, then, generate geometrical magnitudes from this sort of material principle, but others³ generate them from the point (they regard the point not as a unity but as similar to Unity) and another material principle which is not plurality but is similar to it; yet in the case of these principles none the less we get the same difficulties.For if the matter is one, line, plane and solid will be the same; because the product of the same elements must be one and the same.