Metaphysics

​Now those who posit that the Ideas exist, and that they are numbers, by taking each single thing for each case apart from the many according to the exposition, try at least to explain in some way why they exist; nevertheless, since these things are neither necessary nor possible, one must not say that number exists for these reasons. On the other hand, the Pythagoreans, because they saw many properties of numbers belonging to sensible bodies, made existing things to be numbers, not indeed as separable, but made existing things out of numbers. And why? Because the properties of numbers exist in harmony, both in the heaven and in many other things. But for those who say that only mathematical number exists, it is not possible to say any such thing according to their hypotheses, but it was said that their sciences would not exist. But we say that they do exist, just as we said before. And it is clear that mathematical objects are not separated; for if they were separated, their properties would not exist in bodies. Therefore, while the Pythagoreans are not liable to any charge in this respect, yet in respect of making physical bodies out of numbers, making things that have lightness and weight out of things that have neither weight nor lightness, they seem to be speaking about another heaven and other bodies, and not about the sensible ones. But those who make them separate, because the axioms will not be true of sensible things, while the statements made are true and soothe the soul, assume that they both exist and are separate; and similarly with mathematical magnitudes. It is clear, then, that the opposing argument also will say the contrary, and the difficulty which was just raised must be solved by those who speak in this way: why, if they do not exist in sensible things at all, do their properties exist in sensible things? And there are some who, from the fact that points are the limits and extremities of a line, the line of a plane, and the plane of a solid, think that there must be such natures. We must indeed examine this argument too, lest it be too weak. For extremities are not substances, but rather all these are limits (since there is also some limit of walking and of motion in general; will this then be a ‘this’ and some substance? But that is absurd). Nevertheless, even if they do exist, they will all be limits of these sensible things (for the argument was spoken of these); why then will they be separate? — And further, one who is not too credulous might inquire concerning all number and mathematical objects, why the prior contribute nothing to the posterior (for if number does not exist, magnitudes will exist no less for those who say that only mathematical objects exist, and if these do not exist, the soul and sensible bodies will still exist;