The Pythagorean Numbers

Some people believe that the Ideas exist as numbers. They do this by setting out each term apart from its instances. Out of the unity of each general term, they try to explain why number must exist.

Their reasons, however, are neither conclusive nor possible in themselves. This is why we should not assert the existence of number.

The Pythagoreans saw many attributes of numbers belonging to sensible bodies. They supposed real things to be numbers themselves.

This is because the attributes of numbers are present:

• in a musical scale
• in the heavens
• in many other things.

Those who say that mathematical number alone exists also say that these sensible things could not be the subject of the sciences.

But I believe that they are subject to the sciences. The objects of mathematics do not exist apart. If they did, then their attributes would not have been present in bodies.

The Pythagoreans do not object this. Instead, they construct natural bodies out of numbers. They say that things that have lightness and weight come out of things that do not weight or lightness. These come from another dimension which is beyond the senses.

But they think that those numbers exist in that other dimension. These then greet the soul in mathematics.

My rival theory will say the opposite.

The point is the limit and end of a line. The line of the plane is the end of the plane. The plane of the solid is the end of the solud.

But such ends are not substances, merely limits. Even the movement of walking has an end. To the Pythagoreans, the end of walking will be the end and a sustance. But that is absurd.

If the end of walking is a substance, then all the ends of walking will be things in this world.

“Again, if we are not too easily satisfied, we may, regarding all number and the objects of mathematics, press this difficulty, that they contribute nothing to one another, the prior to the posterior; for

If physical number did not exist, then physical distances would not exist.

If distances did not exist, then if spatial magnitudes did not exist, soul and sensible bodies would exist. But the observed facts show that nature is not a series of episodes, like a bad tragedy.

The believers in the Ideas miss this difficulty. They construct distances out of matter and physical number.

But will these distances be Ideas? These contribute nothing, as the objects of mathematics contribute nothing.

But not even is any theorem true of them, unless we want to change the objects of mathematics and invent doctrines of our own. But it is not hard to assume any random hypotheses and spin out a long string of conclusions.

The Pythagoreans thus are wrong in wanting to unite the objects of mathematics with the Ideas. They proposed two kinds of numbers:

• the number of the metaphysical Forms and
• the physical mathematical numbers.

But they never said how mathematical numbers were to exist or what they were made of. Instead, they place it between the idea-number and the physical-number.

If (i) number is made up of big and small, then number will be the same as the idea-number.

If (ii) he names some other element, he will be making his elements rather many.

If the principle of each of the two kinds of number is a 1, unity will be something common to these, and we must inquire how the one is these many things, while at the same time number, according to him, cannot be generated except from one and an indefinite dyad.

All this is absurd. It conflicts both with itself and with the probabilities.

and we seem to see in it Simonides ’long rigmarole’ for the long rigmarole comes into play, like those of slaves, when men have nothing sound to say.

The very elements-the great and the small-seem to cry out against the violence that is done to them. for they cannot in any way generate numbers other than those got from 1 by doubling.

It is strange also to attribute generation to things that are eternal, or rather this is one of the things that are impossible.

There need be no doubt whether the Pythagoreans attribute generation to them or not; for they say plainly that when the one had been constructed, whether out of planes or of surface or of seed or of elements which they cannot express, immediately the nearest part of the unlimited began to be constrained and limited by the limit.

But since they are constructing a world and wish to speak the language of natural science, it is fair to make some examination of their physical theorics, but to let them off from the present inquiry; for we are investigating the principles at work in unchangeable things, so that it is numbers of this kind whose genesis we must study.