The Basic Works of Aristotle (Modern Library Classics) (153 page)

8
     
[1083a]
First of all it is well to determine what is the differentia of a number—and of a unit, if it has a differentia. Units must differ either in quantity or in quality; and neither of these seems to be possible. But number
qua
number differs in quantity. And if the units also did differ in quantity,
(5)
number would differ from number, though equal
in number of units. Again, are the first units greater or smaller, and do the later ones increase or diminish? All these are irrational suppositions. But neither can they differ in
quality
. For no attribute can attach to them; for even to numbers quality is said to belong
after
quantity.
(10)
Again, quality could not come to them either from the 1 or from the dyad; for the former has no quality, and the latter gives
quantity
; for this entity is what makes things to be many. If the facts are really otherwise, they should state this quite at the beginning and determine if possible,
(15)
regarding the differentia of the unit, why it must exist, and, failing this, what differentia they mean.

Evidently then, if the Ideas are numbers, the units cannot all be associable, nor can they be inassociable in either of the two ways.
⁴⁰
But neither is the way in which some others speak about numbers correct.
(20)
These are those who do not think there are Ideas, either without qualification or as identified with certain numbers, but think the objects of mathematics exist and the numbers are the first of existing things, and the 1-itself is the starting-point of them. It is paradoxical that there should be a 1 which is first of 1’s, as
they
say,
(25)
but not a 2 which is first of 2’s, nor a 3 of 3’s; for the same reasoning applies to all. If, then, the facts with regard to number are so, and one supposes mathematical number alone to exist, the 1 is not the starting-point (for this sort of 1 must differ from the other units; and if this is so,
(30)
there must also be a 2 which is first of 2’s, and similarly with the other successive numbers). But if the 1 is the starting-point, the truth about the numbers must rather be what Plato used to say, and there must be a first 2 and 3, and the numbers must not be associable with one another. But if on the other hand one supposes this, many impossible results, as we have said,
⁴¹
follow.
(35)
But either this or the other
must
be the case, so that if neither is, number cannot exist separately.

It is evident, also, from this that the third version
⁴²
is the worst—the view ideal and mathematical number is the same.
[1083b]
For two mistakes must then meet in the one opinion. (1) Mathematical number cannot be of this sort, but the holder of this view has to spin it out by making suppositions peculiar to himself.
(5)
And (2) he must also admit all the consequences that confront those who speak of number in the sense of ‘Forms’.

The Pythagorean version in one way affords fewer difficulties than those before named, but in another way has others peculiar to itself.
(10)
For not thinking of number as capable of existing separately removes many of the impossible consequences; but that bodies should be composed of numbers, and that this should be mathematical number, is impossible. For it is not true to speak of indivisible spatial magnitudes; and however much there might be magnitudes of this sort,
(15)
units at least have not magnitude; and how can a magnitude be composed of indivisibles? But arithmetical number, at least, consists of units, while these thinkers identify number with real things; at any rate they apply their propositions to bodies as if they consisted of those numbers.

If, then, it is necessary, if number is a self-subsistent real thing,
(20)
that it should exist in one of these ways which have been mentioned,
⁴³
and if it cannot exist in any of these, evidently number has no such nature as those who make it separable set up for it.

Again, does each unit come from the great and the small, equalized, or one from the small, another from the great? (
a
) If the latter,
(25)
neither does each thing contain all the elements, nor are the units without difference; for in one there is the great and in another the small, which is contrary in its nature to the great. Again, how is it with the units in the 3-itself? One of them is an odd unit. But perhaps it is for this reason that they give 1-itself the middle place in odd numbers.
(30)
(
b
) But if each of the two units consists of both the great and the small, equalized, how will the 2, which is a single thing, consist of the great and the small? Or how will it differ from the unit? Again, the unit is prior to the 2; for when it is destroyed the 2 is destroyed. It must, then, be the Idea of an Idea since it is prior to an Idea,
(35)
and it must have come into being before it. From what, then? Not from the indefinite dyad, for
its
function was to double.

Again, number must be either infinite or finite; for these thinkers think of number as capable of existing separately, so that it is not possible that neither of those alternatives should be true.
[1084a]
Clearly it cannot be
infinite
; for infinite number is neither odd nor even, but the generation of numbers is always the generation either of an odd or of an even number; in one way, when 1 operates on an even number,
(5)
an odd number is produced; in another way, when 2 operates, the numbers got from 1 by doubling are produced; in another way, when the odd numbers operate, the other even numbers are produced. Again, if every Idea is an Idea of something, and the numbers are Ideas, infinite number itself will be an Idea of something, either of some sensible thing or of something else. Yet this is not possible in
view of their thesis any more than it is reasonable in itself, at least if they arrange the Ideas as they do.

But if number is
finite
, how far does it go? With regard to this not only the fact but the reason should be stated.
(10)
But if number goes only up to 10, as some say,
⁴⁴
firstly the Forms will soon run short; e. g. if 3 is man-himself, what number will be the horse-itself? The series of the numbers which are the several things-themselves goes up to 10.
(15)
It must, then, be one of the numbers within these limits; for it is these that are substances and Ideas. Yet they will run short; for the various forms of animal will outnumber them. At the same time it is clear that if in this way
the
3 is man-himself, the other 3’s are so also (for those in identical numbers are similar),
(20)
so that there will be an infinite number of men; if each 3 is an Idea, each of the numbers will be man-himself, and if not, they will at least be men. And if the smaller number is part of the greater (being number of such a sort that the units in the same number are associable), then if the 4-itself is an Idea of something, e. g. of ‘horse’ or of ‘white’, man will be a part of horse, if man is 2. It is paradoxical,
(25)
also that there should be an Idea of 10, but not of 11, nor of the succeeding numbers. Again, there both are and come to be certain things of which there are no Forms; why, then, are there not Forms of them also? We infer that the Forms are not causes. Again, it is paradoxical if the number-series up to 10 is more of a real thing and a Form than 10 itself.
(30)
There is no generation of the former as one thing, and there is of the latter. But they try to work on the assumption that the series of numbers up to 10 is a complete series. At least they generate the derivatives—e. g. the void, proportion, the odd, and the others of this kind—within the decade. For some things, e. g. movement and rest, good and bad, they assign to the originative principles, and the others to the numbers.
(35)
This is why they identify the odd with 1; for if the odd implied 3, how would 5 be odd?
⁴⁵
Again, spatial magnitudes and all such things are explained without going beyond a definite number; e. g. the first, the indivisible, line,
⁴⁶
then the 2, &c.
[1084b]
; these entities also extend only up to 10.
⁴⁷

Again, if number can exist separately, one might ask which is
prior—1, or 3 or 2? Inasmuch as the number is composite, 1 is prior, but inasmuch as the universal and the form is prior,
(5)
the number is prior; for each of the units is part of the number as its matter, and the number acts as form. And in a sense the right angle is prior to the acute, because it is determinate and in virtue of its definition; but in a sense the acute is prior, because it is a part and the right angle is divided into acute angles. As matter, then, the acute angle and the element and the unit are prior,
(10)
but in respect of the form and of the substance as expressed in the definition, the right angle, and the whole consisting of the matter and the form, are prior; for the concrete thing is nearer to the form and to what is expressed in the definition, though in generation it is later. How then is 1 the starting-point? Because it is not divisible, they say; but both the universal, and the particular or the element,
(15)
are indivisible. But they are starting-points in different ways, one in definition and the other in time. In which way, then, is 1 the starting-point? As has been said, the right angle is thought to be prior to the acute, and the acute to the right, and each is one. Accordingly they make 1 the starting-point in both ways. But this is impossible. For the universal is one as form or substance,
(20)
while the element is one as a part or as matter. For each of the two is in a sense one—in
truth
each of the two units exists potentially (at least if the number is a unity and not like a heap, i. e. if different numbers consist of differentiated units, as they say), but not in complete reality; and the cause of the
error
they fell into is that they were conducting their inquiry at the same time from the standpoint of mathematics and from that of universal definitions,
(25)
so that (1) from the former standpoint they treated unity, their first principle, as a point; for the unit is a point without position. They put things together out of the smallest parts, as some others
⁴⁸
also have done. Therefore the unit becomes the matter of numbers and at the same time prior to 2; and again posterior,
(30)
2 being treated as a whole, a unity, and a form. But (2) because they were seeking the universal they treated the unity which can be predicated of a number, as in this sense also
⁴⁹
a part of the number. But these characteristics cannot belong at the same time to the same thing.

If the 1-itself must be unitary (for it differs in nothing from other 1’s except that it is the starting-point), and the 2 is divisible but the unit is not, the unit must be liker the 1-itself than the 2 is.
But if the unit is liker it,
it
must be liker to the unit than to the 2; therefore each of the units in 2 must be prior to the 2.
(35)
But they deny this; at least they generate the 2 first.
[1085a]
Again, if the 2-itself is a unity and the 3-itself is one also, both form a 2. From what, then, is this 2 produced?

9
     Since there is not contact in numbers, but succession, viz. between the units between which there is nothing,
(5)
e. g. 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
⁵⁰
construct these out of the species of the ‘great and small’; e. g. lines from the ‘long and short’,
(10)
planes from the ‘broad and narrow’, masses from the ‘deep and shallow’; which are species of the ‘great and small’. And the originative principle of such things which answers to the 1
⁵¹
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.
(15)
For (i) the geometrical classes are severed from one another, 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 a line and the solid a plane;
⁵²
again, how will angles and figures and such things be explained?). And (ii) the same happens as in regard to number; for ‘long and short’,
(20)
&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’.
⁵³

(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.
(25)
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,
(30)
then, generate spatial magnitudes from matter of this sort, others
⁵⁴
from the point—and the point is thought by them to be not 1 but something like 1—and from other matter like plurality,
(35)
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 solid will be the same; for from the same elements will come one and the same thing.
[1085b]
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
be
a line.

Again, how number can consist of the one and plurality,
(5)
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.
⁵⁵
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,
(10)
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.
(15)
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.
(20)
Again, (
b
) 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,
⁵⁶
whether the number is infinite or finite. For there was at first, as it seems, a plurality that was itself finite,
(25)
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 other points is formed. Certainly
not of some
distance
+ the point-itself. Nor again can there be indivisible parts of a distance,
(30)
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.
⁵⁷

All these objections, then, and others of the sort make it evident that number and spatial magnitudes cannot exist apart from things.
(35)
Again, 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.
[1086a]
For those who make the objects of mathematics alone exist apart from sensible things,
⁵⁸
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,
(5)
but did not see, if one assumed these principles, how mathematical number was to exist apart from ideal,
⁵⁹
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.
(10)
And he who first supposed that the Forms exist and that the Forms are numbers and that the objects of mathematics exist,
⁶⁰
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.
(15)
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,
(20)
the views expressed by those who discuss only sensible substance have been partly stated in our works on nature,
⁶¹
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.
(25)
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,
(30)
must be considered later;
⁶²
but as regards those who believe in the Ideas one might survey at the same time their way of thinking and the difficulty into which they fall. For they at the same time make the Ideas universal and again treat them as separable and as individuals.
(35)
That this is not possible has been argued before.
⁶³
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.
[1086b]
They thought that the particulars in the sensible world were in a state of flux and none of them remained, but that the universal was apart from these and something different. And Socrates gave the impulse to this theory, as we said in our earlier discussion,
⁶⁴
by reason of his definitions, but he did not
separate
universals from individuals; and in this he thought rightly,
(5)
in not separating them. This is plain from the results; for without the universal it is not possible 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,
(10)
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.