Part 3

The Pythagorean Numbers

by Aristotle Icon

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.

Part 4

The Pythagoreans say there is no generation of the odd number. This implies that there is generation of the even number. Some present the even as produced first from unequals-the great and the small-when these are equalized.

The inequality, then, must belong to them before they are equalized. If they had always been equalized, they would not have been unequal before; for there is nothing before that which is always.

Therefore they are not giving their account of the generation of numbers merely to assist contemplation of their nature.

A difficulty, and a reproach to any one who finds it no difficulty, are contained in the question how the elements and the principles are related to the good and the beautiful; the difficulty is this, whether any of the elements is such a thing as we mean by the good itself and the best, or this is not so, but these are later in origin than the elements.

The theologians agree with some present thinkers who answer no.

They say that both the good and the beautiful appear in the nature of things only when that nature has made some progress. (This they do to avoid a real objection which confronts those who say, as some do, that the one is a first principle.

The objection arises not from their ascribing goodness to the first principle as an attribute, but from their making the one a principle-and a principle in the sense of an element-and generating number from the one.) The old poets agree with this inasmuch as they say that not those who are first in time, e.g. Night and Heaven or Chaos or Ocean, reign and rule, but Zeus.

These poets, however, are led to speak thus only because they think of the rulers of the world as changing; for those of them who combine the two characters in that they do not use mythical language throughout, e.g. Pherecydes and some others, make the original generating agent the Best, and so do the Magi, and some of the later sages also, e.g. both Empedocles and Anaxagoras, of whom one made love an element, and the other made reason a principle. Of those who maintain the existence of the unchangeable substances some say the One itself is the good itself; but they thought its substance lay mainly in its unity.

“This, then, is the problem,-which of the two ways of speaking is right. It would be strange if to that which is primary and eternal and most self-sufficient this very quality–self-sufficiency and self-maintenance–belongs primarily in some other way than as a good. But indeed it can be for no other reason indestructible or self-sufficient than because its nature is good.

Therefore to say that the first principle is good is probably correct; but that this principle should be the One or, if not that, at least an element, and an element of numbers, is impossible. Powerful objections arise, to avoid which some have given up the theory (viz. those who agree that the One is a first principle and element, but only of mathematical number). For on this view all the units become identical with species of good, and there is a great profusion of goods. Again, if the Forms are numbers, all the Forms are identical with species of good. But let a man assume Ideas of anything he pleases. If these are Ideas only of goods, the Ideas will not be substances; but if the Ideas are also Ideas of substances, all animals and plants and all individuals that share in Ideas will be good.

These absurdities follow, and it also follows that the contrary element, whether it is plurality or the unequal, i.e. the great and small, is the bad-itself. (Hence one thinker avoided attaching the good to the One, because it would necessarily follow, since generation is from contraries, that badness is the fundamental nature of plurality; while others say inequality is the nature of the bad.)

It follows, then, that all things partake of the bad except one–the One itself, and that numbers partake of it in a more undiluted form than spatial magnitudes, and that the bad is the space in which the good is realized, and that it partakes in and desires that which tends to destroy it; for contrary tends to destroy contrary. And if, as we were saying, the matter is that which is potentially each thing, e.g. that of actual fire is that which is potentially fire, the bad will be just the potentially good.

“All these objections, then, follow, partly because they make every principle an element, partly because they make contraries principles, partly because they make the One a principle, partly because they treat the numbers as the first substances, and as capable of existing apart, and as Forms.


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