Those who treat the unequal as one thing, and the dyad as an indefinite compound of great and small, (15) say what is very far from being probable or possible. For (a) these are modifications and accidents, rather than substrata, of numbers and magnitudes—the many and few of number, and the great and small of magnitude—like even and odd, (20) smooth and rough, straight and curved. Again, (b) apart from this mistake, the great and the small, and so on, must be relative to something; but what is relative is least of all things a kind of entity or substance, and is posterior to quality and quantity; and the relative is an accident of quantity, (25) as was said, not its matter, since something with a distinct nature of its own must serve as matter both to the relative in general and to its parts and kinds. For there is nothing either great or small, many or few, or, in general, relative to something else, which without having a nature of its own is many or few, great or small, or relative to something else. A sign that the relative is least of all a substance and a real thing is the fact that it alone has no proper generation or destruction or movement, (30) as in respect of quantity there is increase and diminution, in respect of quality alteration, in respect of place locomotion, in respect of substance simple generation and destruction. In respect of relation there is no proper change; for, without changing, a thing will be now greater and now less or equal, if that with which it is compared has changed in quantity. (35) And (c) the matter of each thing, and therefore of substance, must be that which is potentially of the nature in question; but the relative is neither potentially nor actually substance. [1088b] It is strange, then, or rather impossible, to make not-substance an element in, and prior to, substance; for all the categories are posterior to substance. Again, (d) elements are not predicated of the things of which they are elements, (5) but many and few are predicated both apart and together of number, and long and short of the line, and both broad and narrow apply to the plane. If there is a plurality, then, of which the one term, viz. few, is always predicated, e. g. 2 (which cannot be many, for if it were many, (10) 1 would be few), there must be also one which is absolutely many, e. g. 10 is many (if there is no number which is greater than 10), or 10,000. How then, in view of this, can number consist of few and many? Either both ought to be predicated of it, or neither; but in fact only the one or the other is predicated.
2 We must inquire generally, whether eternal things can consist of elements. (15) If they do, they will have matter; for everything that consists of elements is composite. Since, then, even if a thing exists for ever, out of that of which it consists it would necessarily also, if it had come into being, have come into being, and since everything comes to be what it comes to be out of that which is it potentially (for it could not have come to be out of that which had not this capacity, nor could it consist of such elements), and since the potential can be either actual or not—this being so, (20) however everlasting number or anything else that has matter is, it must be capable of not existing, just as that which is any number of years old is as capable of not existing as that which is a day old; if this is capable of not existing, so is that which has lasted for a time so long that it has no limit. They cannot, then, be eternal, since that which is capable of not existing is not eternal, as we had occasion to show in another context.8 If that which we are now saying is true universally—that no substance is eternal unless it is actuality—and if the elements are matter that underlies substance, (25) no eternal substance can have elements present in it, of which it consists.
There are some9 who describe the element which acts with the One as an indefinite dyad, and object to ‘the unequal’, reasonably enough, because of the ensuing difficulties; but they have got rid only of those objections which inevitably arise from the treatment of the unequal, (30) i. e. the relative, as an element; those which arise apart from this opinion must confront even these thinkers, whether it is ideal number, or mathematical, that they construct out of those elements.
There are many causes which led them off into these explanations, (35) and especially the fact that they framed the difficulty in an obsolete form. [1089a] For they thought that all things that are would be one (viz. Being itself), if one did not join issue with and refute the saying of Parmenides:
‘For never will this be proved, that things that are not are.’
They thought it necessary to prove that that which is not is; for only thus—of that which is and something else—could the things that are be composed, (5) if they are many.
But, first, if ‘being’ has many senses (for it means sometimes substance, sometimes that it is of a certain quality, sometimes that it is of a certain quantity, and at other times the other categories), what sort of ‘one’, then, are all the things that are, if non-being is to be supposed not to be? Is it the substances that are one, (10) or the affections and similarly the other categories as well, or all together—so that the ‘this’ and the ‘such’ and the ‘so much’ and the other categories that indicate each some one class of being will all be one? But it is strange, or rather impossible, that the coming into play of a single thing10 should bring it about that part of that which is is a ‘this’, part a ‘such’, part a ‘so much’, part a ‘here’.
Secondly, (15) of what sort of non-being and being do the things that are consist? For ‘non-being’ also has many senses, since ‘being’ has; and ‘not being a man’ means not being a certain substance, ‘not being straight’ not being of a certain quality, ‘not being three cubits long’ not being of a certain quantity. What sort of being and non-being, then, by their union pluralize the things that are? This thinker11 means by the non-being, (20) the union of which with being pluralizes the things that are, the false and the character of falsity. This is also why it used to be said that we must assume something that is false, as geometers assume the line which is not a foot long to be a foot long. But this cannot be so. For neither do geometers assume anything false (for the enunciation is extraneous to the inference), (25) nor is it non-being in this sense that the things that are are generated from or resolved into. But since ‘non-being’ taken in its various cases12 has as many senses as there are categories, and besides this the false is said not to be, and so is the potential, it is from this that generation proceeds, man from that which is not man but potentially man, (30) and white from that which is not white but potentially white, and this whether it is some one thing that is generated or many.
The question evidently is, how being, in the sense of ‘the substances’, is many; for the things that are generated are numbers and lines and bodies. Now it is strange to inquire how being in the sense of the ‘what’ is many, (35) and not how either qualities or quantities are many. For surely the indefinite dyad or ‘the great and the small’ is not a reason why there should be two kinds of white or many colours or flavours or shapes; for then these also would be numbers and units. [1089b] But if they had attacked these other categories, they would have seen the cause of the plurality in substances also; for the same thing or something analogous is the cause. (5) This aberration is the reason also why in seeking the opposite of being and the one, from which with being and the one the things that are proceed, they posited the relative term (i. e. the unequal), which is neither the contrary nor the contradictory of these, and is one kind of being as ‘what’ and quality also are.
They should have asked this question also, how relative terms are many and not one. But as it is, they inquire how there are many units besides the first 1, but do not go on to inquire how there are many unequals besides the unequal. (10) Yet they use them and speak of great and small, many and few (from which proceed numbers), long and short (from which proceeds the line), broad and narrow (from which proceeds the plane), deep and shallow (from which proceed solids); and they speak of yet more kinds of relative term. What is the reason, then, why there is a plurality of these?
It is necessary, then, as we say, to presuppose for each thing that which is it potentially; and the holder of these views further declared what that i
s which is potentially a ‘this’ and a substance but is not in itself being—viz. (15) that it is the relative (as if he had said ‘the qualitative’), which is neither potentially the one or being, nor the negation of the one nor of being, but one among beings. (20) And it was much more necessary, as we said,13 if he was inquiring how beings are many, not to inquire about those in the same category—how there are many substances or many qualities—but how beings as a whole are many; for some are substances, some modifications, some relations. In the categories other than substance there is yet another problem involved in the existence of plurality. Since they are not separable from substances, qualities and quantities are many just because their substratum becomes and is many; yet there ought to be a matter for each category; only it cannot be separable from substances. (25) But in the case of ‘thises’, it is possible to explain how the ‘this’ is many things, unless a thing is to be treated as both a ‘this’ and a general character.14 The difficulty arising from the facts about substances is rather this, (30) how there are actually many substances and not one.
But further, if the ‘this’ and the quantitative are not the same, we are not told how and why the things that are are many, but how quantities are many. For all ‘number’ means a quantity, (35) and so does the ‘unit’, unless it means a measure or the quantitatively indivisible. If, then, the quantitative and the ‘what’ are different, we are not told whence or how the ‘what’ is many; but if any one says they are the same, he has to face many inconsistencies. [1090a]
One might fix one’s attention also on the question, regarding the numbers, what justifies the belief that they exist. To the believer in Ideas they provide some sort of cause for existing things, (5) since each number is an Idea, and the Idea is to other things somehow or other the cause of their being; for let this supposition be granted them. But as for him who does not hold this view because he sees the inherent objections to the Ideas (so that it is not for this reason that he posits numbers), but who posits mathematical number,15 why must we believe his statement that such number exists, (10) and of what use is such number to other things? Neither does he who says it exists maintain that it is the cause of anything (he rather says it is a thing existing by itself), nor is it observed to be the cause of anything; for the theorems of arithmeticians will all be found true even of sensible things, (15) as was said before.16
3 As for those, then, who suppose the Ideas to exist and to be numbers, by their assumption—in virtue of the method of setting out each term apart from its instances—of the unity of each general term they try at least to explain somehow why number must exist. Since their reasons, however, are neither conclusive nor in themselves possible, one must not, for these reasons at least, (20) assert the existence of number. Again, the Pythagoreans, because they saw many attributes of numbers belonging to sensible bodies, supposed real things to be numbers—not separable numbers, however, but numbers of which real things consist. But why? Because the attributes of numbers are present in a musical scale and in the heavens and in many other things.17 (25) Those, however, who say that mathematical number alone exists18 cannot according to their hypotheses say anything of this sort, but it used to be urged that these sensible things could not be the subject of the sciences. But we maintain that they are, as we said before.19 And it is evident that the objects of mathematics do not exist apart; for if they existed apart their attributes would not have been present in bodies. (30) Now the Pythagoreans in this point are open to no objection; but in that they construct natural bodies out of numbers, things that have lightness and weight out of things that have not weight or lightness, they seem to speak of another heaven and other bodies, not of the sensible. (35) But those who make number separable20 assume that it both exists and is separable because the axioms would not be true of sensible things, while the statements of mathematics are true and ‘greet the soul’; and similarly with the spatial magnitudes of mathematics. [1090b] It is evident, then, both that the rival theory21 will say the contrary of this, and that the difficulty we raised just now,22 why if numbers are in no way present in sensible things their attributes are present in sensible things, has to be solved by those who hold these views.
There are some who, because the point is the limit and extreme of the line, (5) the line of the plane, and the plane of the solid, think there must be real things of this sort. We must therefore examine this argument too, and see whether it is not remarkably weak. For (i) extremes are not substances, but rather all these things are limits. (10) For even walking, and movement in general, has a limit, so that on their theory this will be a ‘this’ and a substance. But that is absurd. Not but what (ii) even if they are substances, they will all be the substances of the sensible things in this world; for it is to these that the argument applied. Why then should they be capable of existing apart?
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 number did not exist, (15) none the less spatial magnitudes would exist for those who maintain the existence of the objects of mathematics only,23 and 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, (20) like a bad tragedy. As for the believers in the Ideas, this difficulty misses them; for they construct spatial magnitudes out of matter and number, lines out of the number 2, planes doubtless out of 3, solids out of 4—or they use other numbers, which makes no difference. But will these magnitudes be Ideas, or what is their manner of existence, and what do they contribute to things? These contribute nothing, (25) 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. (30) These thinkers,24 then, are wrong in this way, in wanting to unite the objects of mathematics with the Ideas. And those who first posited two kinds of number, that of the Forms and that which is mathematical, neither have said nor can say how mathematical number is to exist and of what it is to consist. For they place it between ideal and sensible number. (35) If (i) it consists of the great and small, it will be the same as the other—ideal—number (he25 makes spatial magnitudes out of some other small and great26). [1091a] And if (ii) he names some other element, he will be making his elements rather many. And 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, (5) and 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. And 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. (10)
It is strange also to attribute generation to things that are eternal, or rather this is one of the things that are impossible. (15) 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 theories, (20) 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.
4 These thinkers say there is no generation of the odd number, which evidently implies that there is generation of the even; and some present the eve
n as produced first from unequals—the great and the small—when these are equalized. (25) 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 evidently they are not giving their account of the generation of numbers merely to assist contemplation of their nature.27
A difficulty, and a reproach to any one who finds it no difficulty, (30) 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 seem to agree with some thinkers of the present day,28 who answer the question in the negative, and say that both the good and the beautiful appear in the nature of things only when that nature has made some progress. (35) (This they do to avoid a real objection which confronts those who say, as some do, that the one is a first principle. [1091b] 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, (5) 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, (10) 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.
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