by Aristotle
There are many causes which led them off into these explanations, and [1089a1] especially the fact that they framed the difficulty in an old-fashioned way. 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:2
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 thus—of that [5] which is and something else—could the things that are be composed, if they are many.
But firstly, if ‘being’ has many senses (for it means sometimes substance, sometimes quality, sometimes quantity, and at other times the other categories), what sort of one are all the things that are, if non-being is to be supposed not to be? Is it the substances that are one, or the affections and the other categories as well, or [10] everything—so that the ‘this’ and the ‘such’ and the ‘so much’ and the other categories that indicate each some one thing will all be one? But it is strange, or rather impossible, that a single nature should bring it about that part of that which is is a ‘this’, part a ‘such’, part a ‘so much’, part somewhere.
Secondly, of what sort of non-being and being do the things that are consist? [15] For ‘non-being’ also has many senses, since ‘being’ has; and not being a man means not being a certain ‘this’, not straight not being of a certain quality, not three cubits long not being of a certain quantity. From what sort of being and non-being, then, do the things that are come to be many? He means by the non-being, the union of [20] which with being makes the things that are many, the false and the character of falsity. This is also why it was 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 proposition in [25] question is extraneous to the inference), nor are the things that are, generated from or resolved into non-being in this sense. But since non-being in the various cases has as many senses as there are categories, and besides this the false is said not to be and so is the potential, generation proceeds from the latter, man from that which is not man but potentially man, and white from that which is not white but potentially [30] white, and this whether it is one thing that is generated or many.
The question evidently is, how being in the sense of substances is many; for the things that are generated are numbers and lines and bodies. It is strange to inquire how being in the sense of essence is many, and not how either qualities or quantities are many. For surely the indefinite dyad or the great and the small are not a cause of there being two kinds of white or many colours or flavors or shapes; for then these [1089b1] also would be numbers and units. But if they had attacked this point, they would have seen the cause of the plurality in substances also; for the cause is the same or analogous. This aberration is the reason also why in seeking the opposite of being and the one, from which and being and the one the things that are proceed, they [5] posited the relative (i.e. the unequal), which is neither the contrary nor the contradictory of these, but is one kind of being as substance and quality are.
They should have inquired also how relatives 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 [10] how there are many unequals besides the unequal. 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 relatives. What is the reason, then, why there is a plurality of these? [15]
It is necessary, as we say, to presuppose for each thing that which is it potentially; and the holder of these views further declared what that is which is potentially a ‘this’ and a substance but is not in itself being—viz. that it is the relative (as if he had said the qualitative), which is neither potentially the one or [20] being, nor the contradictory of the one nor of being, but one among beings. And it was much more necessary, as we said, 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 [25] another matter to give us pause, viz. how can there be many? For since they are not separable, qualities and quantities are many only because their substrate becomes and is many. Yet there ought to be a matter for each category; only it cannot be separable from substances. But in the case of a ‘this’, it is possible to explain how the [30] ‘this’ is many things, unless a thing is to be treated as both a ‘this’ and a general character. The difficulty arising from these facts is rather this, 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 [35] number means a quantity, and so does the unit, unless it means merely a measure or the indivisible in quantity. If then the quantitative and essence are different, we are [1090a1] not told whence or how essence is many; but if any one says they are the same, he has to face many inconsistencies.
One might fix one’s attention also on the question, regarding the numbers,—what justifies the belief that they exist. To the believer in the Ideas they provide a [5] cause for existing things, 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 [10] who posits mathematical number, why must we believe his statement that such number exists, 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 in itself), nor is it observed to be the cause of anything; for the theorems of [15] arithmeticians will all be found true even of sensible things, as was said.
3 · Those who suppose the Ideas to exist and to be numbers, take each to be one thing by setting each out apart from the many—so that they try at least to explain somehow why numbers exist. Since their reasons, however, are neither conclusive nor in themselves possible, one must not, on this account at least, assert [20] the existence of number. But 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 [25] and in many other things. But those who say that mathematical number alone exists cannot according to their hypotheses say anything of this sort; indeed, they used to say that those numbers could not be objects of the sciences. But we maintain that they are, as we said before. 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 [30] bodies. 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. But those who make number separable assume [35] that it exists and is separable because the axioms would not be true of sensible things, while the statements of mathematics are true and delight the soul; and similarly with the magnitudes of mathematics. It is evident, then, both that our [1090b1] contrary theory will say the contrary of this, and that the difficulty we raised just now, why if numbers are in no way present in sensible things their attributes are present in sensible things, is solved for those who hold our views.
There are some who, because the point is the limit and extreme of the line, the [5] line of the plane, and the plane of the solid, think there must be real things of this sort. We must therefore examin
e this argument too, and see whether it is not remarkably weak. For extremes are not substances, but rather all these things are mere limits. For even walking, and movement in general, has a limit, so that on their [10] theory this will be a ‘this’ and a substance. But that is absurd. Even if they are substances, they will all be the substances of particular sensible things; 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 [15] another, the prior to the posterior; for if number did not exist, none the less magnitudes would exist for those who maintain the existence of the objects of mathematics only, and if magnitudes did not exist, soul and sensible bodies would exist. But the phenomena show that nature is not a series of episodes, like a bad [20] tragedy. The believers in the Ideas escape this difficulty; for they construct magnitudes out of matter and number, lines out of 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 [25] contribute to things? These contribute nothing, as the objects of mathematics contribute nothing. But not even is any theorem true of them, unless we want to change 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. These thinkers, [30] 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 the other which is mathematical, neither have said nor can say in the least how mathematical number is to exist and of what it is to consist. For they place it [35] between ideal and sensible number. If it consists of the great and small, it will be the same as the other—ideal number. (And from what other3 great and small can he [1091a1] produce magnitudes?) And if 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 [5] except from one and the indefinite dyad.
All this is absurd, and conflicts both with itself and with the probabilities, and we seem to see in it Simonides’ ‘long story’; for the long story comes into play, like those which slaves tell, when men have nothing sound to say. And the very [10] 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.
It is strange also to attribute generation to eternal things, or rather this is one of the things that are impossible. There need be no doubt whether the Pythagoreans [15] attribute generation to them or not; for they obviously say 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 drawn in 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 explanation of their [20] account of nature, 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 say the even is [25] produced first from unequals—the great and the small—when these are equalized. 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 as a theoretical account.
[30] A difficulty, and a reproach to any one who finds it no difficulty, 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. The mythologists seem to agree with some thinkers of the present day, who answer the question in the negative, and say that both the good and the beautiful appear [35] only when nature has made some progress. This they do to avoid a real objection which confronts those who say, as some do, that the one is a first principle. [1091b1] (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.) And the old poets agree with [5] this inasmuch as they say that not those who are first in time, e.g. Night and Heaven or Chaos or Ocean, reign and rule, but Zeus. These poets, however, speak thus only because they think of the rulers of the world as changing; for those of them who combine 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 [10] so do the Magi, and some of the later sages also, e.g. Empedocles and Anaxagoras, of whom one made friendship an element, and the other made thought 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.
This, then, is the problem,—which of the two ways of speaking is right. It [15] would be strange if to that which is primary and eternal and most self-sufficient this very quality—self-sufficiency and self-maintenance—belongs primarily in some other way than as a good. But indeed it can be for no other reason indestructible or self-sufficient than because its nature is good. Therefore to say that the first principle is good is probably correct; but that this principle should be the one or, if [20] not that, an element, and an element of numbers, is impossible. Powerful objections arise, to avoid which some have given up the theory (viz. those who agree that the one is a first principle and element, but only of mathematical number). For all the units become identical with species of good, and there is a great profusion of goods. [25] Again, if the Forms are numbers, all the Forms are identical with species of good. But let a man assume Ideas of anything he pleases. If these are Ideas only of goods, the Ideas will not be substances; but if the Ideas are also Ideas of substances, all animals and plants and all things that share in Ideas will be good. [30]
These absurdities follow, and it also follows that the contrary element, whether it is plurality or the unequal, i.e. the great and small, is the bad-itself. (Hence one thinker avoided attaching the good to the one, because it would necessarily follow, since generation is from contraries, that badness is the fundamental nature of plurality; others say inequality is the nature of the bad.) It follows, then, that all [35] things partake of the bad except one—the one itself, and that numbers partake of it in a more undiluted form than magnitudes, and that the bad is the space in which [1092a1] the good is realized, and that it partakes in and desires that which tends to destroy it; for contrary tends to destroy contrary. And if, as we said, the matter is that which is potentially each thing, e.g. that of actual fire is that which is potentially fire, the bad will be just the potentially good. [5]
All these objections, then, follow, partly because they make every principle an element, partly because they make contraries principles, partly because they make the one a principle, partly because they treat the numbers as the first substances, and as capable of existing apart, and as Forms.
5 · If, then, it is equally impossible not to put the good among first principles and to put it among them in this way, evidently the principles are not being correctly [10] described, nor are the first substances. Nor do we conceive the matter correctly if we compare the principles of the universe to that of animals and plants, on the ground that the more complete always comes from the indefinite and incomplete—which is what leads this thinker to say that t
his is also true of the first principles of reality, so that the one itself is not even an existing thing. For here too the principles [15] from which these come are complete; for it is a man that produces a man, and the seed is not first.
It is strange, also, to generate place simultaneously with the mathematical solids (for place is peculiar to the individual things, and hence they are separable in [20] place, but mathematical objects are nowhere), and to say that they must be somewhere, but not say what the place is.
Those who say that the things that are come from elements and that the first of things that are are the numbers, should have first distinguished the senses in which one thing comes from another, and then said in which sense number comes from its elements.
By intermixture? But not everything is capable of intermixture, and that [25] which is produced by it is different, and on this view the one will not be separate or a distinct entity; but they want it to be so.
By juxtaposition, like a syllable? But then the elements must have position; and he who thinks of the one and plurality must think of them apart; number then will be this—a unit and plurality, or the one and the unequal.
Coming from certain things means in one sense that these are still to be found [30] in the product and in another that they are not; in which sense does number come from these elements? Only things that are generated can come from elements which are present in them. Does number come from its elements as from seed? But nothing can come from that which is indivisible. Does it come from its contrary, its contrary not persisting? But all things that come in this way come also from something else which does persist. Since, then, one thinker places the 1 as contrary [1092b1] to plurality, and another places it as contrary to the unequal, treating the 1 as equal, number is treated as coming from contraries. There will then be something else that persists, from which and from one contrary the compound is or has come to be. Again, why in the world do the other things that come from contraries, or that have contraries, perish (even when all of the contrary is used to produce them), while [5] number does not? Nothing is said about this. Yet whether present or not present in the compound the contrary destroys it, e.g. strife destroys the mixture (yet it should not; for it is not to that that it is contrary).