The One as a Number

Why is a number, when taken all together, One?

If the units are diverse, the Platonists should have spoken like those who say there are 4 or 2 elements.

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 the Platonists speak as if the One were homogeneous like fire or water.

If so, the numbers will not be substances.

If there is a One itself and this is a first principle, ‘one’ is being used in more than one sense, otherwise the theory is impossible.

When we 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 body from the deep and shallow.

Yet how then can either the plane contain a line, or the solid a line or a plane? For the broad and narrow is a different class from the deep and shallow.

Therefore, number is not present in these because the many and few are different from these.

Evidently, no other of the higher classes will be present in the lower.

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 as being a geometrical fiction. He gave the name of principle of the line-and this he often posited-to the indivisible lines.

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 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 our account of the way in which they are the substances of perceptible things is empty talk; for ‘sharing’, as we said before, means nothing.

Nor have the Forms any connexion with what we see to be the cause in the case of the arts, that for whose sake both all mind and the whole of nature are operative,-with this cause which we assert to be one of the first principles; but mathematics has come to be identical with philosophy for modern thinkers, though they say that it should be studied for the sake of other things.

Further, one might suppose that the substance which according to them underlies as matter is too mathematical, and is a predicate and differentia of the substance, ie. of the matter, rather than matter itself; i.e. the great and the small are like the rare and the dense which the physical philosophers speak of, calling these the primary differentiae of the substratum; for these are a kind of excess and defect. And regarding movement, if the great and the small are to he movement, evidently the Forms will be moved.

But if they are not to be movement, whence did movement come? The whole study of nature has been annihilated.

What is thought to be easy-to show that all things are one-is not done; for what is proved by the method of setting out instances is not that all things are one but that there is a One itself,-if we grant all the assumptions.

And not even this follows, if we do not grant that the universal is a genus; and this in some cases it cannot be.

Nor can it be explained either how the lines and planes and solids that come after the numbers exist or can exist, or what significance they have; for these can neither be Forms (for they are not numbers), nor the intermediates (for those are the objects of mathematics), nor the perishable things. This is evidently a distinct fourth class.

If we search for the elements of existing things without distinguishing the many senses in which things are said to exist, we cannot find them, especially if the search for the elements of which things are made is conducted in this manner.

It is surely impossible to discover what ‘acting’ or ‘being acted on’, or ’the straight’, is made of, but if elements can be discovered at all, it is only the elements of substances; therefore either to seek the elements of all existing things or to think one has them is incorrect.

How could we learn the elements of all things?

We cannot start by knowing anything before. For as he who is learning geometry, though he may know other things before, knows none of the things with which the science deals and about which he is to learn, so is it in all other cases.

Therefore if there is a science of all things, such as some assert to exist, he who is learning this will know nothing before.

Yet all learning is by means of premises which are known before,-whether the learning be by demonstration or by definitions; for the elements of the definition must be known before and be familiar; and learning by induction proceeds similarly.

If the science were actually innate, it were strange that we are unaware of our possession of the greatest of sciences.

How is one to come to know what all things are made of? There might be a conflict of opinion, as there is about certain syllables. Some say za is made out of s and d and a. Others say it is a distinct sound and none of those that are familiar.

How could we know the objects of sense without having the sense in question?

Yet we ought to, if the elements of which all things consist, as complex sounds consist of the clements proper to sound, are the same.

Part 10

Everyone seeks the causes named in the Physics, and that we cannot name any beyond these; but they seek these vaguely.

In one sense, they have all been described before. In another sense, they have not been described at all.

For the earliest philosophy is, on all subjects, like one who lisps, since it is young and in its beginnings.

Even Empedocles says that bone exists by virtue of the ratio in it.

Now this is the essence and the substance of the thing.

But it is similarly necessary that flesh and each of the other tissues should be the ratio of its elements, or that not one of them should;

for it is on account of this that both flesh and bone and everything else will exist, and not on account of the matter, which he names,-fire and earth and water and air.

But while he would necessarily have agreed if another had said this, he has not said it clearly.

On these questions our views have been expressed before.

But let us return to enumerate the difficulties that might be raised on these same points; for perhaps we may get from them some help towards our later difficulties.