Some, (30) then, generate spatial magnitudes from matter of this sort, others54 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.
10 Let us now mention a point which presents a certain difficulty both to those who believe in the Ideas and to those who do not, (15) and which was stated before, at the beginning, among the problems.65 If we do not suppose substances to be separate, and in the way in which individual things are said to be separate, we shall destroy substance in the sense in which we understand ‘substance’; but if we conceive substances to be separable, how are we to conceive their elements and their principles?
If they are individual and not universal, (20) (a) real things will be just of the same number as the elements, and (b) the elements will not be knowable. For (a) let the syllables in speech be substances, and their elements elements of substances; then there must be only one ba and one of each of the syllables, (25) since they are not universal and the same in form but each is one in number and a ‘this’ and not a kind possessed of a common name (and again they suppose that the ‘just what a thing is’66 is in each case one). And if the syllables are unique, so too are the parts of which they consist; there will not, then, be more a’s than one, nor more than one of any of the other elements, on the same principle on which an identical syllable cannot exist in the plural number. (30) But if this is so, there will not be other things existing besides the elements, but only the elements. (b) Again, the elements will not be even knowable; for they are not universal, and knowledge is of universals. This is clear from demonstrations and from definitions; for we do not conclude that this triangle has its angles equal to two right angles, unless every triangle has its angles equal to two right angles, (35) nor that this man is an animal, unless every man is an animal.
But if the principles are universal, either the substances composed of them are also universal, or non-substance will be prior to substance; for the universal is not a substance, but the element
or principle is universal, and the element or principle is prior to the things of which it is the principle or element. [1087a]
All these difficulties follow naturally, when they make the Ideas out of elements and at the same time claim that apart from the substances which have the same form there are Ideas, (5) a single separate entity. But if, e. g., in the case of the elements of speech, the a’s and the b’s may quite well be many and there need be no a-itself and b-itself besides the many, there may be, so far as this goes, an infinite number of similar syllables. The statement that all knowledge is universal, (10) so that the principles of things must also be universal and not separate substances, presents indeed, of all the points we have mentioned, the greatest difficulty, but yet the statement is in a sense true, although in a sense it is not. For knowledge, like the verb ‘to know’, (15) means two things, of which one is potential and one actual. The potency, being, as matter, universal and indefinite, deals with the universal and indefinite; but the actuality, being definite, deals with a definite object—being a ‘this’, it deals with a ‘this’. But per accidens sight sees universal colour, because this individual colour which it sees is colour; and this individual a which the grammarian investigates is an a. (20) For if the principles must be universal, what is derived from them must also be universal, as in demonstrations67; and if this is so, there will be nothing capable of separate existence—i. e. no substance. But evidently in a sense knowledge is universal, and in a sense it is not. (25)
* * *
1 Phys. i.
2 Met. vii, viii, ix.
3 Plato, Xenocrates, and the Pythagoreans and Speusippus, respectively, are meant.
4 Cf. chs. 2, 3.
5 Cf. chs. 4, 5.
6 Cf. chs. 6–9.
7 Cf. iii. 998a 7–19.
8 Which nevertheless the theory in question represents as Ideas apart from sensible things.
9 iii. 997b 12–34.
10 Cf. 1076a 38–b 11.
11 Cf. vi. 1026a 25, xiii. 1077a 9.
12 sc. indivisibility and humanity.
13 The reference is apparently to Aristippus; Cf. iii. 996a 32.
14 Apparently an unfulfilled promise.
15 Chs. 2, 3.
16 1077a 17–20, 24–b 11.
17 Cf. vii. 1039a 2, Soph. El. 178b 36–179a 10, and Plato, Parmenides, 132 AB, D-133 A.
18 i. e. the relative in general is more general than, and therefore (on Platonic principles) prior to, number. Number is similarly prior to the dyad. Therefore the relative is prior to the dyad, which vet is held to be absolute.
19 With 1078b 34–1079b 3 Cf. i. 990b 2–991a 8.
20 sc. in the essence of man.
21 100 D.
22 With 1079b 12–1080a 8 Cf. i. 991a 8–b 9.
23 ll. 15–20.
24 ll. 23–35.
25 Cf. 1076a 38–b 11.
26 Plato is meant.
27 i. e. in which the numbers differ in kind.
28 Speusippus is meant.
29 Some unknown Platonist.
30 Xenocrates is meant.
31 This refers to Plato; Cf. i. 992b 13–18.
32 Speusippus is meant.
33 Xenocrates is meant.
34 l. 19.
35 Cf. 1080a 18–20, 23–35.
36 Plato.
37 The theory of ideal number holds that 2 comes next after the original 1, which with the ‘indefinite 2’ is the source of number. But if all units are different in species, one of the units in 2 is prior to the other and to 2, and comes next after the original 1. Similarly between 2 and 3 there will be the first unit in 3, and so on.
38 i. e. if there is a difference of kind between the numbers.
39 1081a 5–17.
40 Cf. 1080a 18–20, 23–35.
41 Cf. 1080b 37–1083a 17.
42 That of Xenocrates; Cf. 1080b 22.
43 1080a 15–b 36.
44 This includes Plato (Cf. Phys. 206b 32) and probably Speusippus.
45 i. e. to account for the oddness of odd numbers they identify the odd with the 1, which is a principle present in all numbers, not with the 3, which on their theory is not present in other numbers.
46 Cf. i. 992a 22.
47 Cf. xiv. 1090b 21–24. 1 answers to the point (the ‘indivisible line’), 2 to the line, 3 to the plane, 4 to the solid, and 1 + 2 + 3 + 4 = 10.
48 sc. the Atomists.
49 i. e. they treated the unity which is predicable of a number, as well as the unit in a number, as a part of the number.
50 This probably includes Plato himself.
51 i. e. that which is to the geometrical forms as the primary 1 is (according to the Platonic theory) to numbers.
52 With 1085a 7–19 Cf. i. 992a 10–19.
53 Cf. i. 992b 1–7, xiv. 1088a 15–21.
54 Speusippus is probably meant.
55 i. e. probably Plato and Xenocrates.
56 Cf. 1083b 36.
57 The point cannot have for an element of it (a) a distance, for this would destroy the simplicity of the point; or (b) part of a distance, for any part of a distance must be a distance.
58 Speusippus is meant.
59 Xenocrates is meant.
60 Plato.
61 Phys. i. 4–6; De Caelo, iii. 3–4; De Gen. et Corr. i. 1.
62 Speusippus is meant; Cf. N. 1090 a 7–15, 20–b 20.
63 iii. 1003a 7–17.
64 1078b 17–30.
65 iii. 999b 24–1000a 4, 1003a 5–17.
66 i. e. the Idea; Cf, 1079b 6.
67 sc. universal premisses do not give singular conclusions.
BOOK N (XIV)
1 Regarding this kind of substance, what we have said must be taken as sufficient. All philosophers make the first principles contraries: as in natural things, (30) so also in the case of unchangeable substances. But since there cannot be anything prior to the first principle of all things, the principle cannot be the principle and yet be an attribute of something else. To suggest this is like saying that the white is a first principle, not qua anything else but qua white, but yet that it is predicable of a subject, i. e. that its being white presupposes its being something else; this is absurd, (35) for then that subject will be prior. But all things which are generated from their contraries involve an underlying subject; a subject, then, must be present in the case of contraries, if anywhere. [1087b] All contraries, then, are always predicable of a subject, and none can exist apart, but just as appearances suggest that there is nothing contrary to substance, argument confirms this. No contrary, then, is the first principle of all things in the full sense; the first principle is something different.
But these thinkers make one of the contraries matter, some1 making the unequal—which they take to be the essence of plurality—matter for the One, (5) and others2 making plurality matter for the One. (The former generate numbers out of the dyad of the unequal, i. e. of the great and small, and the other thinker we have referred to generates them out of plurality, while according to both it is generated by the essence of the One.) For even the philosopher who says the unequal and the One are the elements, (10) and the unequal is a dyad composed of the great and small, treats the unequal, or the great and the small, as being one, and does not draw the distinction that they are one in definition, but not in number. But they do not describe rightly even the principles which they call elements, for some3 name the great and the small with the One and treat these three as elements of numbers, (15) two being matter, one the form; while others4 name the many and few, because the great and the small are more appropriate in their nature to magnitude than to number; and others5 name rather the universal character common to these—‘that which exceeds and that which is exceeded’. None of these varieties of opinion makes any difference to speak of, in view of some of the consequences; they affect only the abstract objections, (20) which these thinkers take care to avoid because the demonstrations they themselves offer are abstract—with this exception, that if the exceeding and the exceeded
are the principles, and not the great and the small, consistency requires that number should come from the elements before 2 does; for number is more universal than 2, as the exceeding and the exceeded are more universal than the great and the small. But as it is, (25) they say one of these things but do not say the other. Others oppose the different and the other to the One,6 and others oppose plurality to the One.7 But if, as they claim, things consist of contraries, and to the One either there is nothing contrary, or if there is to be anything it is plurality, and the unequal is contrary to the equal, and the different to the same, and the other to the thing itself, (30) those who oppose the One to plurality have most claim to plausibility, but even their view is inadequate, for the One would on their view be a few; for plurality is opposed to fewness, and the many to the few.
‘The one’ evidently means a measure. And in every case there is some underlying thing with a distinct nature of its own, (35) e. g. in the scale a quarter-tone, in spatial magnitude a finger or a foot or something of the sort, in rhythms a beat or a syllable; and similarly in gravity it is a definite weight; and in the same way in all cases, in qualities a quality, in quantities a quantity (and the measure is indivisible, in the former case in kind, and in the latter to the sense); which implies that the one is not in itself the substance of anything. [1088a] And this is reasonable; for ‘the one’ means the measure of some plurality, and ‘number’ means a measured plurality and a plurality of measures. (5) (Thus it is natural that one is not a number; for the measure is not measures, but both the measure and the one are starting-points.) The measure must always be some identical thing predicable of all the things it measures, e. g. if the things are horses, the measure is ‘horse’, and if they are men, ‘man’. If they are a man, a horse, and a god, the measure is perhaps ‘living being’, (10) and the number of them will be a number of living beings. If the things are ‘man’ and ‘pale’ and ‘walking’, these will scarcely have a number, because all belong to a subject which is one and the same in number, yet the number of these will be a number of ‘kinds’ or of some such term.
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