Number and Unit

There is no contact in numbers, but succession. Between the units, there is nothing. Between those in 2 or in 3 one might ask whether these succeed the 1-itself or not, and whether, of the terms that succeed it, 2 or either of the units in 2 is prior.

Similar difficulties occur with regard to the classes of things posterior to number,-the line, the plane, and the solid. For some construct these out of the species of the ‘great and small’; e.g. lines from the ’long and short’, planes from the ‘broad and narrow’, masses from the ‘deep and shallow’; which are species of the ‘great and small’.

The originative principle of such things which answers to the 1 different thinkers describe in different ways, And in these also the impossibilities, the fictions, and the contradictions of all probability are seen to be innumerable.

1. Geometrical classes are severed from one another.

This is unless the principles of these are implied in one another in such a way that the ‘broad and narrow’ is also ’long and short’ (but if this is so, the plane will be line and the solid a plane; again, how will angles and figures and such things be explained?).

2. The same happens as in regard to number

long and short’, &c., are attributes of magnitude, but magnitude does not consist of these, any more than the line consists of ‘straight and curved’, or solids of ‘smooth and rough’.

(All these views share a difficulty which occurs with regard to species-of-a-genus, when one posits the universals, viz. whether it is animal-itself or something other than animal-itself that is in the particular animal. True, if the universal is not separable from sensible things, this will present no difficulty; but if the 1 and the numbers are separable, as those who express these views say, it is not easy to solve the difficulty, if one may apply the words ’not easy’ to the impossible. For when we apprehend the unity in 2, or in general in a number, do we apprehend a thing-itself or something else?).

“Some, then, generate spatial magnitudes from matter of this sort, others from the point -and the point is thought by them to be not 1 but something like 1-and from other matter like plurality, but not identical with it; about which principles none the less the same difficulties occur. For if the matter is one, line and plane-and soli will be the same; for from the same elements will come one and the same thing. But if the matters are more than one, and there is one for the line and a second for the plane and another for the solid, they either are implied in one another or not, so that the same results will follow even so; for either the plane will not contain a line or it will he a line.

“Again, how number can consist of the one and plurality, they make no attempt to explain; but however they express themselves, the same objections arise as confront those who construct number out of the one and the indefinite dyad. For the one view generates number from the universally predicated plurality, and not from a particular plurality; and the other generates it from a particular plurality, but the first; for 2 is said to be a ‘first plurality’. Therefore there is practically no difference, but the same difficulties will follow,-is it intermixture or position or blending or generation? and so on.

Above all one might press the question ‘if each unit is one, what does it come from?’ Certainly each is not the one-itself. It must, then, come from the one itself and plurality, or a part of plurality. To say that the unit is a plurality is impossible, for it is indivisible; and to generate it from a part of plurality involves many other objections; for (a) each of the parts must be indivisible (or it will be a plurality and the unit will be divisible) and the elements will not be the one and plurality; for the single units do not come from plurality and the one.

Again, (,the holder of this view does nothing but presuppose another number; for his plurality of indivisibles is a number. Again, we must inquire, in view of this theory also, whether the number is infinite or finite. For there was at first, as it seems, a plurality that was itself finite, from which and from the one comes the finite number of units. And there is another plurality that is plurality-itself and infinite plurality; which sort of plurality, then, is the element which co-operates with the one? One might inquire similarly about the point, i.e. the element out of which they make spatial magnitudes. For surely this is not the one and only point; at any rate, then, let them say out of what each of the points is formed. Certainly not of some distance + the point-itself. Nor again can there be indivisible parts of a distance, as the elements out of which the units are said to be made are indivisible parts of plurality; for number consists of indivisibles, but spatial magnitudes do not.

“All these objections, then, and others of the sort make it evident that number and spatial magnitudes cannot exist apart from things.

The discord about numbers between the various versions is a sign that it is the incorrectness of the alleged facts themselves that brings confusion into the theories.

For those who make the objects of mathematics alone exist apart from sensible things, seeing the difficulty about the Forms and their fictitiousness, abandoned ideal number and posited mathematical. But those who wished to make the Forms at the same time also numbers, but did not see, if one assumed these principles, how mathematical number was to exist apart from ideal, made ideal and mathematical number the same-in words, since in fact mathematical number has been destroyed; for they state hypotheses peculiar to themselves and not those of mathematics.

He who first supposed that the Forms exist and that the Forms are numbers and that the objects of mathematics exist, naturally separated the two.

Therefore it turns out that all of them are right in some respect, but on the whole not right. And they themselves confirm this, for their statements do not agree but conflict.

The cause is that their hypotheses and their principles are false. And it is hard to make a good case out of bad materials, according to Epicharmus: ‘as soon as ’tis said, ’tis seen to be wrong.’

But regarding numbers the questions we have raised and the conclusions we have reached are sufficient (for while he who is already convinced might be further convinced by a longer discussion, one not yet convinced would not come any nearer to conviction); regarding the first principles and the first causes and elements, the views expressed by those who discuss only sensible substance have been partly stated in our works on nature, and partly do not belong to the present inquiry; but the views of those who assert that there are other substances besides the sensible must be considered next after those we have been mentioning. Since, then, some say that the Ideas and the numbers are such substances, and that the elements of these are elements and principles of real things, we must inquire regarding these what they say and in what sense they say it.

Those who posit numbers only, and these mathematical, must be considered later.

Those who believe in the Ideas have a certain way of thinking which causes their difficulties.

They believe that the Ideas are universal. Yet they treat them as separable and as individuals.

I think this is not possible.

The reason why those who described their substances as universal combined these two characteristics in one thing, is that they did not make substances identical with sensible things.

They thought that the particulars in the sensible world were a state of flux and none of them remained, but that the universal was apart from these and something different.

Socrates gave the impulse to this theory by reason of his definitions.

But he did not separate universals from individuals.

In this he thought rightly, in not separating them.

This is plain from the results.

Without the universal, it is impossible to get knowledge.

But the separation is the cause of the objections that arise with regard to the Ideas.

His successors, however, treating it as necessary, if there are to be any substances besides the sensible and transient substances, that they must be separable, had no others, but gave separate existence to these universally predicated substances, so that it followed that universals and individuals were almost the same sort of thing. This in itself, then, would be one difficulty in the view we have mentioned.