[1087a] [29]

With regard to this kind of substance,1 then, let the foregoing account suffice. All thinkers make the first principles contraries; as in the realm of natural objects, so too in respect of the unchangeable substances.Now if nothing can be prior to the first principle of all things, that first principle cannot be first principle if it is an attribute of something else. This would be as absurd as to say that “white” is the first principle, not qua anything else but qua white, and yet that it is predicable of a subject, and is white because it is an attribute of something else; because the latter will be prior to it.Moreover, all things are generated from contraries as from a substrate, and therefore contraries must most certainly have a substrate. [1087b] [1] Therefore all contraries are predicated of a subject, and none of them exists separately. But there is no contrary to substance; not only is this apparent, but it is borne out by reasoned consideration.2 Thus none of the contraries is strictly a first principle; the first principle is something different.

But the Platonists treat one of the contraries as matter, some opposing “the unequal” to Unity (on the ground that the former is of the nature of plurality) and others plurality.For according to some,3 numbers are generated from the unequal dyad of the Great and Small; and according to another,4 from plurality; but in both cases they are generated by the essence of unity. For he who speaks of “the unequal” and Unity as elements, and describes the unequal as a dyad composed of Great and Small, speaks of the unequal, i.e. the Great and Small, as being one; and does not draw the distinction that they are one in formula but not in number.5

Again, they state the first principles, which they call elements, badly; some say that the Great and the Small, together with Unity (making 36 in all), are the elements of numbers; the two former as matter, and Unity as form. Others speak of the Many and Few, because the Great and the Small are in their nature more suited to be the principles of magnitude; and others use the more general term which covers these—“the exceeding” and “the exceeded.”But none of these variations makes any appreciable difference with respect to some of the consequences of the theory; [20] they only affect the abstract difficulties, which these thinkers escape because the proofs which they themselves employ are abstract.There is, however, this exception: if “the exceeding” and “the exceeded” are the first principles, and not the Great and the Small, on the same principle number should be derived from the elements before 2 is derived; for as “the exceeding and the exceeded” is more universal than the Great and Small, so number is more universal than 2. But in point of fact they assert the one and not the other.

Others oppose “the different” or “other” to Unity; and others contrast Plurality and Unity.Now if, as they maintain, existing things are derived from contraries, and if there is either no contrary to unity, or if there is to be any contrary it is plurality; and if the unequal is contrary to the equal, and the different to the same, and the other to the thing itself then those who oppose unity to plurality have the best claim to credibility—but even their theory is inadequate, because then unity will be few. For plurality is opposed to paucity, and many to few.

That “unity” denotes a measure7 is obvious. And in every case there is something else which underlies it; e.g., in the scale there is the quarter-tone; in spatial magnitude the inch or foot or some similar thing; and in rhythms the foot or syllable. Similarly in the case of gravity there is some definite weight. Unity is predicated of all things in the same way; [1088a] [1] of qualities as a quality, and of quantities as a quantity.(The measure is indivisible, in the former case in kind, and in the latter to our senses.) This shows that unity is not any independent substance. And this is reasonable; because unity denotes a measure of some plurality, and number denotes a measured plurality and a plurality of measures. (Hence too it stands to reason that unity is not a number; for the measure is not measures, but the measure and unity are starting-points.)The measure must always be something which applies to all alike; e.g., if the things are horses, the measure is a horse; if they are men, the measure is a man; and if they are man, horse and god, the measure will presumably be an animate being, and the number of them animate beings.If the things are “man,” “white” and “walking,” there will scarcely be a number of them, because they all belong to a subject which is one and the same in number; however, their number will be a number of genera, or some other such appellation.

Those8 who regard the unequal as a unity, and the dyad as an indeterminate compound of great and small, hold theories which are very far from being probable or possible. For these terms represent affections and attributes, rather than substrates, of numbers and magnitudes—“many” and “few” applying to number, and “great” and “small” to magnitude— [20] just as odd and even, smooth and rough, straight and crooked, are attributes.Further, in addition to this error, “great” and “small” and all other such terms must be relative. And the relative is of all the categories in the least degree a definite entity or substance; it is posterior to quality and quantity. The relative is an affection of quantity, as we have said, and not its matter; since there is something else distinct which is the matter both of the relative in general and of its parts and kinds.There is nothing great or small, many or few, or in general relative, which is many or few, great or small, or relative to something else without having a distinct nature of its own. That the relative is in the lowest degree a substance and a real thing is shown by the fact that of it alone9 there is neither generation nor destruction nor change in the sense that in respect of quantity there is increase and decrease, in respect of quality, alteration, in respect of place, locomotion, and in respect of substance, absolute generation and destruction.There is no real change in respect of the relative; for without any change in itself, one term will be now greater, now smaller or equal, as the other term undergoes quantitative change. [1088b] [1] Moreover, the matter of every thing, and therefore of substance, must be that which is potentially of that nature; but the relative is neither potentially substance nor actually.

It is absurd, then, or rather impossible, to represent non-substance as an element of substance and prior to it; for all the other categories are posterior to substance. And further, the elements are not predicated of those things of which they are elements; yet “many” and “few” are predicated, both separately and together, of number; and “long” and “short” are predicated of the line, and the Plane is both broad and narrow.If, then, there is a plurality of which one term, viz. “few,” is always predicable, e.g. 2 (for if 2 is many, 1 will be few10), then there will be an absolute “many”; e.g., 10 will be many (if there is nothing more than 1011), or 10,000. How, then, in this light, can number be derived from Few and Many? Either both ought to be predicated of it, or neither; but according to this view only one or the other is predicated.

But we must inquire in general whether eternal things can be composed of elements. If so, they will have matter; for everything which consists of elements is composite.Assuming, then, that that which consists of anything, whether it has always existed or it came into being, must come into being <if at all> out of that of which it consists; and that everything comes to be that which it comes to be out of that which is it potentially (for it could not have come to be out of that which was not potentially such, nor could it have consisted of it); and that the potential can either be actualized or not; [20] then however everlasting number or anything else which has matter may be, it would be possible for it not to exist, just as that which is any number of years old is as capable of not existing as that which is one day old. And if this is so, that which has existed for so long a time that there is no limit to it may also not exist.Therefore things which contain matter cannot be eternal, that is, if that which is capable of not existing is not eternal, as we have had occasion to say elsewhere.12 Now if what we have just been saying—that no substance is eternal unless it is actuality—is true universally, and the elements are the matter of substance, an eternal substance can have no elements of which, as inherent in it, it consists.

There are some who, while making the element which acts conjointly with unity the indeterminate dyad, object to “the unequal,” quite reasonably, on the score of the difficulties which it involves. But they are rid only of those difficulties13 which necessarily attend the theory of those who make the unequal, i.e. the relative, an element; all the difficulties which are independent of this view must apply to their theories also, whether it is Ideal or mathematical number that they construct out of these elements.

There are many causes for their resorting to these explanations, [1089a] [1] the chief being that they visualized the problem in an archaic form. They supposed that all existing things would be one, absolute Being, unless they encountered and refuted Parmenides' dictum:

It will ne'er be proved that things which are not, are,14

i.e., that they must show that that which is not, is; for only so—of that which is, and of something else—could existing things be composed, if they are more than one.15

However, (i) in the first place, if “being” has several meanings (for sometimes it means substance, sometimes quality, sometimes quantity, and so on with the other categories), what sort of unity will all the things that are constitute, if not-being is not to be? Will it be the substances that are one, or the affections (and similarly with the other categories), or all the categories together? in which case the “this” and the “such” and the “so great,” and all the other categories which denote some sense of Being, will be one.But it is absurd, or rather impossible, that the introduction of one thing should account for the fact that “what is” sometimes means “so-and-so,” sometimes “such-and-such,” sometimes “of such-and-such a size,” sometimes “in such-and-such a place.”

(2) Of what sort of not-being and Being do real things consist? Not-being, too, has several senses, inasmuch as Being has; and “not-man” means “not so-and-so,” whereas “not straight” means “not such-and-such,” and “not five feet long” means “not of such-and-such a size.” What sort of Being and not-being, then, make existing things a plurality? [20] This thinker means by the not-being which together with Being makes existing things a plurality, falsity and everything of this nature16; and for this reason also it was said17 that we must assume something which is false, just as geometricians assume that a line is a foot long when it is not.But this cannot be so; for (a) the geometricians do not assume anything that is false (since the proposition is not part of the logical inference18), and (b) existing things are not generated from or resolved into not-being in this sense. But not only has “not-being” in its various cases as many meanings as there are categories, but moreover the false and the potential are called “not-being”; and it is from the latter that generation takes place—man comes to be from that which is not man but is potentially man, and white from that which is not white but is potentially white; no matter whether one thing is generated or many.

Clearly the point at issue is how “being” in the sense of the substances is many; for the things that are generated are numbers and lines and bodies. It is absurd to inquire how Being as substance is many, and not how qualities or quantities are many.Surely the indeterminate dyad or the Great and Small is no reason why there should be two whites or many colors or flavors or shapes; [1089b] [1] for then these too would be numbers and units. But if the Platonists had pursued this inquiry, they would have perceived the cause of plurality in substances as well; for the cause19 is the same, or analogous.

This deviation of theirs was the reason why in seeking the opposite of Being and unity, from which in combination with Being and unity existing things are derived, they posited the relative (i.e. the unequal), which is neither the contrary nor the negation of Being and unity, but is a single characteristic of existing things, just like substance or quality. They should have investigated this question also; how it is that relations are many, and not one.As it is, they inquire how it is that there are many units besides the primary unity, but not how there are many unequal things besides the Unequal. Yet they employ in their arguments and speak of Great and Small, Many and Few (of which numbers are composed), Long and Short (of which the line is composed), Broad and Narrow (of which the plane is composed), Deep and Shallow (of which solids are composed); and they mention still further kinds of relation.20 Now what is the cause of plurality in these relations?

We must, then, as I say, presuppose in the case of each thing that which is it potentially. The author21 of this theory further explained what it is that is potentially a particular thing or substance, but is not per se existent—that it is the relative (he might as well have said “quality”); which is neither potentially unity or Being, nor a negation of unity or Being, [20] but just a particular kind of Being. And it was still more necessary, as we have said,22 that, if he was inquiring how it is that things are many, he should not confine his inquiry to things in the same category, and ask how it is that substances or qualities are many, but that he should ask how it is that things in general are many; for some things are substances, some affections, and some relations.Now in the case of the other categories there is an additional difficulty in discovering how they are many. For it may be said that since they are not separable, it is because the substrate becomes or is many that qualities and quantities are many; yet there must be some matter for each class of entities, only it cannot be separable from substances.In the case of particular substances, however, it is explicable how the particular thing can be many, if we do not regard a thing both as a particular substance and as a certain characteristic.23 The real difficulty which arises from these considerations is how substances are actually many and not one.

Again, even if a particular thing and a quantity are not the same, it is not explained how and why existing things are many, but only how quantities are many;for all number denotes quantity, and the unit, if it does not mean a measure, means that which is quantitatively indivisible. If, then, quantity and substance are different, it is not explained whence or how substance is many; [1090a] [1] but if they are the same, he who holds this has to face many logical contradictions.

One might fasten also upon the question with respect to numbers, whence we should derive the belief that they exist.For one24 who posits Ideas, numbers supply a kind of cause for existing things; that is if each of the numbers is a kind of Idea, and the Idea is, in some way or other, the cause of existence for other things; for let us grant them this assumption.But as for him25 who does not hold this belief, because he can see the difficulties inherent in the Ideal theory (and so has not this reason for positing numbers), and yet posits mathematical number, what grounds have we for believing his statement that there is a number of this kind, and what good is this number to other things? He who maintains its existence does not claim that it is the cause of anything, but regards it as an independent entity; nor can we observe it to be the cause of anything; for the theorems of the arithmeticians will all apply equally well to sensible things, as we have said.26

Those, then, who posit the Ideas and identify them with numbers, by their assumption (in accordance with their method of abstracting each general term from its several concrete examples) that every general term is a unity, make some attempt to explain why number exists.27 Since, however, their arguments are neither necessarily true nor indeed possible, [20] there is no justification on this ground for maintaining the existence of number.The Pythagoreans, on the other hand, observing that many attributes of numbers apply to sensible bodies, assumed that real things are numbers; not that numbers exist separately, but that real things are composed of numbers.28 But why? Because the attributes of numbers are to be found in a musical scale, in the heavens, and in many other connections.29

As for those who hold that mathematical number alone exists,30 they cannot allege anything of this kind31 consistently with their hypotheses; what they did say was that the sciences could not have sensible things as their objects. But we maintain that they can; as we have said before. And clearly the objects of mathematics do not exist in separation; for if they did their attributes would not be present in corporeal things.Thus in this respect the Pythagoreans are immune from criticism; but in so far as they construct natural bodies, which have lightness and weight, out of numbers which have no weight or lightness, they appear to be treating of another universe and other bodies, not of sensible ones.32 But those who treat number as separable assume that it exists and is separable because the axioms will not apply to sensible objects; whereas the statements of mathematics are true and appeal to the soul.33 [1090b] [1] The same applies to mathematical extended magnitudes.

It is clear, then, both that the contrary theory34 can make out a case for the contrary view, and that those who hold this theory must find a solution for the difficulty which was recently raised35—why it is that while numbers are in no way present in sensible things, their attributes are present in sensible things.

There are some36 who think that, because the point is the limit and extreme of the line, and the line of the plane, and the plane of the solid, there must be entities of this kind.We must, then, examine this argument also, and see whether it is not exceptionally weak. For (1.) extremes are not substances; rather all such things are merely limits. Even walking, and motion in general, has some limit; so on the view which we are criticizing this will be an individual thing, and a kind of substance. But this is absurd. And moreover (2.) even if they are substances, they will all be substances of particular sensible things, since it was to these that the argument applied. Why, then, should they be separable?

Again, we may, if we are not unduly acquiescent, further object with regard to all number and mathematical objects that they contribute nothing to each other, the prior to the posterior. For if number does not exist, none the less spatial magnitudes will exist for those who maintain that only the objects of mathematics exist; and if the latter do not exist, the soul and sensible bodies will exist.37 But it does not appear, to judge from the observed facts, that the natural system lacks cohesion, [20] like a poorly constructed drama. Those38 who posit the Ideas escape this difficulty, because they construct spatial magnitudes out of matter and a number—2 in the case of lines, and 3, presumably, in that of planes, and 4 in that of solids; or out of other numbers, for it makes no difference.But are we to regard these magnitudes as Ideas, or what is their mode of existence? and what contribution do they make to reality? They contribute nothing; just as the objects of mathematics contribute nothing. Moreover, no mathematical theorem applies to them, unless one chooses to interfere with the principles of mathematics and invent peculiar theories39 of one's own. But it is not difficult to take any chance hypotheses and enlarge upon them and draw out a long string of conclusions.

These thinkers, then, are quite wrong in thus striving to connect the objects of mathematics with the Ideas. But those who first recognized two kinds of number, the Ideal and the mathematical as well, neither have explained nor can explain in any way how mathematical number will exist and of what it will be composed; for they make it intermediate between Ideal and sensible number.For if it is composed of the Great and Small, it will be the same as the former, i.e. Ideal, number. But of what other Great and Small can it be composed? for Plato makes spatial magnitudes out of a Great and Small.40 [1091a] [1] And if he speaks of some other component, he will be maintaining too many elements; while if some one thing is the first principle of each kind of number, unity will be something common to these several kinds.We must inquire how it is that unity is these many things, when at the same time number, according to him, cannot be derived otherwise than from unity and an indeterminate dyad.41

All these views are irrational; they conflict both with one another and with sound logic, and it seems that in them we have a case of Simonides' “long story42”; for men have recourse to the “long story,” such as slaves tell, when they have nothing satisfactory to say.The very elements too, the Great and Small, seem to protest at being dragged in; for they cannot possibly generate numbers except rising powers of 2.43

It is absurd also, or rather it is one of the impossibilities of this theory, to introduce generation of things which are eternal.There is no reason to doubt whether the Pythagoreans do or do not introduce it; for they clearly state that when the One had been constituted—whether out of planes or superficies or seed or out of something that they cannot explain—immediately the nearest part of the Infinite began to be drawn in and limited by the Limit.44 However, since they are here explaining the construction of the universe and meaning to speak in terms of physics, although we may somewhat criticize their physical theories, [20] it is only fair to exempt them from the present inquiry; for it is the first principles in unchangeable things that we are investigating, and therefore we have to consider the generation of this kind of numbers.

They45 say that there is no generation of odd numbers,46 which clearly implies that there is generation of even ones; and some hold that the even is constructed first out of unequals—the Great and Small—when they are equalized.47 Therefore the inequality must apply to them before they are equalized. If they had always been equalized they would not have been unequal before; for there is nothing prior to that which has always been.Hence evidently it is not for the sake of a logical theory that they introduce the generation of numbers

A difficulty, and a discredit to those who make light of the difficulty, arises out of the question how the elements and first principles are related to the the Good and the Beautiful. The difficulty is this: whether any of the elements is such as we mean when we48 speak of the Good or the Supreme Good, or whether on the contrary these are later in generation than the elements.It would seem that there is an agreement between the mythologists and some present-day thinkers,49 who deny that there is such an element, and say that it was only after some evolution in the natural order of things that both the Good and the Beautiful appeared. They do this to avoid a real difficulty which confronts those who hold, as some do, that unity is a first principle. [1091b] [1] This difficulty arises not from ascribing goodness to the first principle as an attribute, but from treating unity as a principle, and a principle in the sense of an element, and then deriving number from unity. The early poets agree with this view in so far as they assert that it was not the original forces—such as Night, Heaven, Chaos or Ocean—but Zeus who was king and ruler.It was, however, on the ground of the changing of the rulers of the world that the poets were led to state these theories; because those of them who compromise by not describing everything in mythological language—e.g. Pherecydes50 and certain others—make the primary generator the Supreme Good; and so do the Magi,51 and some of the later philosophers such as Empedocles and Anaxagoras: the one making Love an element,52 and the other making Mind a first principle.53 And of those who hold that unchangeable substances exist, some54 identify absolute unity with absolute goodness; but they considered that the essence of goodness was primarily unity.

This, then, is the problem: which of these two views we should hold.Now it is remarkable if that which is primary and eternal and supremely self-sufficient does not possess this very quality, viz. self-sufficiency and immunity, in a primary degree and as something good. Moreover, it is imperishable and self-sufficient for no other reason than because it is good. [20] Hence it is probably true to say that the first principle is of this nature. But to say that this principle is unity, or if not that, that it is an element, and an element of numbers, is impossible; for this involves a serious difficulty, to avoid which some thinkers55 have abandoned the theory (viz. those who agree that unity is a first principle and element, but of mathematical number). For on this view all units become identical with some good, and we get a great abundance of goods.56 Further, if the Forms are numbers, all Forms become identical with some good. Again, let us assume that there are Ideas of anything that we choose. If there are Ideas only of goods, the Ideas will not be substances57; and if there are Ideas of substances also, all animals and plants, and all things that participate in the Ideas, will be goods.58

Not only do these absurdities follow, but it also follows that the contrary element, whether it is plurality or the unequal, i.e. the Great and Small, is absolute badness. Hence one thinker59 avoided associating the Good with unity, on the ground that since generation proceeds from contraries, the nature of plurality would then necessarily be bad.Others60 hold that inequality is the nature of the bad. It follows, then, that all things partake of the Bad except one—absolute unity; and that numbers partake of it in a more unmitigated form than do spatial magnitudes61; [1092a] [1] and that the Bad is the province for the activity of the Good, and partakes of and tends towards that which is destructive of the Good; for a contrary is destructive of its contrary.And if, as we said,62 the matter of each thing is that which is it potentially—e.g., the matter of actual fire is that which is potentially fire—then the Bad will be simply the potentially Good.

Thus all these objections follow because (1.) they make every principle an element; (2.) they make contraries principles; (3.) they make unity a principle; and (4.) they make numbers the primary substances, and separable, and Forms.

If, then, it is impossible both not to include the Good among the first principles, and to include it in this way, it is clear that the first principles are not being rightly represented, nor are the primary substances. Nor is a certain thinker63 right in his assumption when he likens the principles of the universe to that of animals and plants, on the ground that the more perfect forms are always produced from those which are indeterminate and imperfect, and is led by this to assert that this is true also of the ultimate principles; so that not even unity itself is a real thing.64 He is wrong; for even in the natural world the principles from which these things are derived are perfect and complete—for it is man that begets man; the seed does not come first.65 It is absurd also to generate space simultaneously with the mathematical solids (for space is peculiar to particular things, which is why they are separable in space, whereas the objects of mathematics have no position) [20] and to say that they must be somewhere, and yet not explain what their spatial position is.

Those who assert that reality is derived from elements, and that numbers are the primary realities, ought to have first distinguished the senses in which one thing is derived from another, and then explained in what way number is derived from the first principles. Is it by mixture? But (a) not everything admits of mixture66; (b) the result of mixture is something different; and unity will not be separable,67 nor will it be a distinct entity, as they intend it to be.Is it by composition, as we hold of the syllable? But (a) this necessarily implies position; (b) in thinking of unity and plurality we shall think of them separately. This, then, is what number will be—a unit plus plurality, or unity plus the Unequal.

And since a thing is derived from elements either as inherent or as not inherent in it, in which way is number so derived? Derivation from inherent elements is only possible for things which admit of generation.68 Is it derived as from seed?But nothing can be emitted from that which is indivisible.69 Is it derived from a contrary which does not persist? But all things which derive their being in this way derive it also from something else which does persist. Since, therefore, one thinker70 regards unity as contrary to plurality, [1092b] [1] and another (treating it as the Equal) as contrary to the Unequal, number must be derived as from contraries.Hence there is something else which persists from which, together with one contrary, number is or has been derived.71

Further, why on earth is it that whereas all other things which are derived from contraries or have contraries perish, even if the contrary is exhausted in producing them,72 number does not perish? Of this no explanation is given; yet whether it is inherent or not, a contrary is destructive; e.g., Strife destroys the mixture.73 It should not, however, do this; because the mixture is not its contrary.

Nor is it in any way defined in which sense numbers are the causes of substances and of Being; whether as bounds,74 e.g. as points are the bounds of spatial magnitudes,75 and as Eurytus76 determined which number belongs to which thing—e.g. this number to man, and this to horse—by using pebbles to copy the shape of natural objects, like those who arrange numbers in the form of geometrical figures, the triangle and the square.77 Or is it because harmony is a ratio of numbers, and so too is man and everything else? But in what sense are attributes—white, and sweet, and hot—numbers?78 And clearly numbers are not the essence of things, nor are they causes of the form; for the ratio79 is the essence, and number80 is matter.E.g. the essence of flesh or bone is number only in the sense that it is three parts of fire and two of earth.81 And the number, [20] whatever it is, is always a number of something; of particles of fire or earth, or of units. But the essence is the proportion of one quantity to another in the mixture; i.e. no longer a number, but a ratio of the mixture of numbers, either of corporeal particles or of any other kind. Thus number is not an efficient cause—neither number in general, nor that which consists of abstract units—nor is it the matter, nor the formula or form of things. Nor again is it a final cause.

The question might also be raised as to what the good is which things derive from numbers because their mixture can be expressed by a number, either one which is easily calculable,82 or an odd number.83 For in point of fact honey-water is no more wholesome if it is mixed in the proportion “three times three”84; it would be more beneficial mixed in no particular proportion, provided that it be diluted, than mixed in an arithmetical proportion, but strong.Again, the ratios of mixtures are expressed by the relation of numbers, and not simply by numbers; e.g., it is 3 : 2, not 3 X 285; for in products of multiplication the units must belong to the same genus. Thus the product of 1 x 2 x 3 must be measurable by 1, and the product of 4 X 5 x 7 by 4. Therefore all products which contain the same factor must be measurable by that factor. Hence the number of fire cannot be 2 X 5 X 3 X 7 if the number of water is 2 x 3.86 [1093a] [1]

If all things must share in number, it must follow that many things are the same; i.e., that the same number belongs both to this thing and to something else. Is number, then, a cause; i.e., is it because of number that the object exists? Or is this not conclusive? E.g., there is a certain number of the sun's motions, and again of the moon's,87 and indeed of the life and maturity of every animate thing. What reason, then, is there why some of these numbers should not be squares and others cubes, some equal and others double?There is no reason; all things must fall within this range of numbers if, as was assumed, all things share in number, and different things may fall under the same number. Hence if certain things happened to have the same number, on the Pythagorean view they would be the same as one another, because they would have the same form of number; e.g., sun and moon would be the same.88 But why are these numbers causes? There are seven vowels,89 seven strings to the scale,90 seven Pleiads; most animals (though not all91) lose their teeth in the seventh year; and there were seven heroes who attacked Thebes. Is it, then, because the number 7 is such as it is that there were seven heroes, or that the Pleiads consist of seven stars? Surely there were seven heroes because of the seven gates, or for some other reason, and the Pleiads are seven because we count them so; just as we count the Bear as 12, whereas others count more stars in both. [20] Indeed, they assert also that Ξ, Ψ and Ζ are concords,92 and that because there are three concords, there are three double consonants. They ignore the fact that there might be thousands of double consonants—because there might be one symbol for ΓΡ. But if they say that each of these letters is double any of the others, whereas no other is,93 and that the reason is that there are three regions94 of the mouth, and that one consonant is combined with ς in each region, it is for this reason that there are only three double consonants, and not because there are three concords—because there are really more than three; but there cannot be more than three double consonants.

Thus these thinkers are like the ancient Homeric scholars, who see minor similarities but overlook important ones.

Some say that there are many correspondences of this kind; e.g., the middle notes95 of the octave are respectively 8 and 9, and the epic hexameter has seventeen syllables, which equals the sum of these two; [1093b] [1] and the line scans in the first half with nine syllables, and in the second with eight.96 And they point out that the interval from α to ω in the alphabet is equal to that from the lowest note of a flute to the highest, whose number is equal to that of the whole system of the universe.97 We must realize that no one would find any difficulty either in discovering or in stating such correspondences as these in the realm of eternal things, since they occur even among perishable things.

As for the celebrated characteristics of number, and their contraries, and in general the mathematical properties, in the sense that some describe them and make them out to be causes of the natural world, it would seem that if we examine them along these lines, they disappear; for not one of them is a cause in any of the senses which we distinguished with until respect to the first Principles.98 There is a sense, however, in which these thinkers make it clear that goodness is predicable of numbers, and that the odd, the straight, the equal-by-equal,99 and the powers100 of certain numbers, belong to the series of the Beautiful.101 For the seasons are connected with a certain kind of number102; and the other examples which they adduce from mathematical theorems all have the same force.Hence they would seem to be mere coincidences, for they are accidental; but all the examples are appropriate to each other, and they are one by analogy. For there is analogy between all the categories of Being—as “straight” is in length, [20] so is “level” in breadth, perhaps “odd” in number, and “white” in color.

Again, it is not the Ideal numbers that are the causes of harmonic relations, etc. (for Ideal numbers, even when they are equal, differ in kind, since their units also differ in kind)103; so on this ground at least we need not posit Forms.

Such, then, are the consequences of the theory, and even more might be adduced. But the mere fact that the Platonists find so much trouble with regard to the generation of Ideal numbers, and can in no way build up a system, would seem to be a proof that the objects of mathematics are not separable from sensible things, as some maintain, and that the first principles are not those which these thinkers assume.

1 i.e., the Platonic Ideas or numbers, which they regarded as unchangeable substances. There is, however, no definite transition to a fresh subject at this point. The criticisms of the Ideas or numbers as substances, and of the Platonic first principles, have not been grouped systematically in Books 13 and 14. Indeed there is so little distinction in subject matter between the two books that in some Mss. 14 was made to begin at 13.9.10. (Syrianus ad loc.). See Introduction.

2 Cf. Aristot. Categories 3b 24-27

3 Plato; cf. Aristot. Met. 13.7.5.

4 Probably Speusippus.

5 This shows clearly that by the Great-and Small Plato meant a single principle, i.e., indeterminate quantity. Aristotle admits this here because he is contrasting the Great-and Small with the One; but elsewhere he prefers to regard the Platonic material principle as a duality. See Introduction.

6 Cf. previous note.

7 Cf. Aristot. Met. 5.6.17, 18, Aristot. Met. 10.1.8, 21.

8 Cf. sect. 5.

9 Cf. Aristot. Met. 11.12.1. There Aristotle refers to seven categories, but here he omits “activity” and “passivity” as being virtually identical with motion.

10 Cf. Aristot. Met. 10.6.1-3.

11 Cf. Aristot. Met. 13.8.17.

12 Aristot. Met. 9.8.15-17, Aristot. De Caelo 1.12.

13 Cf. Aristot. Met. 14.1.14-17.

14 Parmenides Fr. 7 (Diels).

15 Cf. Plat. Soph. 237a, 241d, 256e.

16 Plat. Soph. 237a, 240; but Aristotle's statement assumes too much.

17 Presumably by some Platonist.

18 i.e., the validity of a geometrical proof does not depend upon the accuracy of the figure.

19 Matter, according to Aristotle; and there is matter, or something analogous to it, in every category. Cf. Aristot. Met. 12.5.

20 Cf. Aristot. Met. 14.1.6, 18, Aristot. Met. 1.9.23.

21 Plato.

22 sect. 11.

23 This, according to Aristotle, is how the Platonists regard the Ideas. See Introduction.

24 Plato and his orthodox followers.

25 Speusippus.

26 Aristot. Met. 13.3.1.

27 I have followed Ross's text and interpretation of this sentence. For the meaning cf. Aristot. Met. 14.2.20.

28 See Introduction.

29 Cf. Aristot. Met. 14.6.5.

30 Cf. Aristot. Met. 14.2.21.

31 i.e., that things are composed of numbers.

32 See Introduction.

33 The statements of mathematics appeal so strongly to our intelligence that they must be true; therefore if they are not true of sensible things, there must be some class of objects of which they are true.

34 The Pythagorean theory, which maintains that numbers not only are present in sensible things but actually compose them, is in itself an argument against the Speusippean view, which in separating numbers from sensible things has to face the question why sensible things exhibit numerical attributes.

35 sect. 3.

36 Probably Pythagoreans. Cf. Aristot. Met. 7.2.2, Aristot. Met. 3.5.3.

37 That the criticism is directed against Speusippus is clear from Aristot. Met. 7.2.4. Cf. Aristot. Met. 12.10.14.

38 Xenocrates (that the reference is not to Plato is clear from sect. 11).

39 e.g. that of “indivisible lines.”

40 This interpretation (Ross's second alternative, reading τίνος for τινος) seems to be the most satisfactory. For the objection cf. Aristot. Met. 3.4.34.

41 The argument may be summarized thus. If mathematical number cannot be derived from the Great-and-Small or a species of the Great-and-Small, either it has a different material principle (which is not economical) or its formal principle is in some sense distinct from that of the Ideal numbers. But this implies that unity is a kind of plurality, and number or plurality can only be referred to the dyad or material principle.

42 The exact reference is uncertain, but Aristotle probably means Simonides of Ceos. Cf. Simonides Fr. 189 (Bergk).

43 Assuming that the Great-and-Small, or indeterminate dyad, is duplicative (Aristot. Met. 13.7.18).

44 Cf. Aristot. Physics 3.4, Aristot. Physics 4.6, and Burnet, E.G.P. sect. 53.

45 The Platonists.

46 This statement was probably symbolical. “They described the odd numbers as ungenerated because they likened them to the One, the principle of pure form” (Ross ad loc.).

47 Cf. Aristot. Met. 13.7.5.

48 Aristotle speaks as a Platonist. See Introduction.

49 The Pythagoreans and Speusippus; cf. Aristot. Met. 12.7.10.

50 Of Syros (circa 600-525 B.C.). He made Zeus one of the three primary beings (Diels,Vorsokratiker201, 202).

51 The Zoroastrian priestly caste.

52 Cf. Aristot. Met. 3.1.13.

53 Cf. Aristot. Met. 1.3.16.

54 Plato; cf. Aristot. Met. 1.6.10.

55 Speusippus and his followers; cf. sect. 3.

56 If unity is goodness, and every unit is a kind of unity, every unit must be a kind of goodness—which is absurd.

57 Because they are Ideas not of substances but of qualities.

58 Because the Ideas are goods.

59 Speusippus.

60 Plato and Xenocrates.

61 As being more directly derived from the first principles. Cf. Aristot. Met. 1.9.23 n.

62 Aristot. Met. 14.1.17.

63 Evidently Speusippus; cf. Aristot. Met. 14.4.3.

64 Speusippus argued that since all things are originally imperfect, unity, which is the first principle, must be imperfect, and therefore distinct from the good. Aristotle objects that the imperfect does not really exist, and so Speusippus deprives his first principle of reality.

65 Cf. Aristot. Met. 9.8.5.

66 e.g. to admit of mixture a thing must first have a separate existence, and the Great-and-Small, which is an affection or quality of number (Aristot. Met. 14.1.14) cannot exist separately.

67 sc. when it has once been mixed. Cf. Aristot. De Gen. et Corr. 327b 21-26.

68 And numbers are supposed to be eternal. Cf. Aristot. Met. 14.2.1-3.

69 i.e., unity, being indivisible, cannot contribute the formal principle of generation in the way that the male parent contributes it.

70 Speusippus: Plato. Cf. Aristot. Met. 14.1.5.

71 The objection is directed against the Platonist treatment of the principles as contraries (cf. Aristot. Met. 14.4.12), and may be illustrated by Aristot. Met. 12.1.5-2.2. Plurality, as the contrary of unity, is privation, not matter; the Platonists should have derived numbers from unity and some other principle which is truly material.

72 Because it may be regarded as still potentially present.

73 According to Empedocles Fr. 17 (Diels).

74 The theories criticized from this point onwards to Aristot. Met. 14.6.11 are primarily Pythagorean. See Introduction.

75 e.g. the line by 2 points, the triangle (the simplest plane figure) by 3, the tetrahedron (the simplest solid figure) by 4.

76 Disciple of Philolaus; he “flourished” in the early fourth century B.C.

77 cf. Burnet, E.G.P. sect. 47.

78 This is an objection to the view that numbers are causes as bounds.

79 Or “formula.”

80 In the sense of a number of material particles.

81 Cf. Empedocles Fr. 96 (Diels).

82 i.e., a simple ratio.

83 It is hard to see exactly what this means. If the terms of a ratio are rational, one of them must be odd. Alexander says a ratio like 1 : 3 is meant. Oddness was associated with goodness (cf. Aristot. Met. 1.5.6).

84 Apparently the Pythagoreans meant by this “three parts of water to three of honey.” Aristotle goes on to criticize this way of expressing ratios.

85 Cf. previous note.

86 sc. because if so, a particle of fire would simply equal 35 particles of water.

87 5 in each case, according to Aristotle; cf. Aristot. Met. 12.7.9, 11.

88 Cf. previous note.

89 In the Greek alphabet.

90 In the old heptachord; cf. note on Aristot. Met. 5.11.4.

91 Cf. Aristot. Hist. An. 576a 6.

92 According to Alexander ζ was connected with the fourth, ξ with the fifth, and ψ with the octave.

93 θ, φ, and χ are aspirated, not double, consonants.

94 Palate, lips, and teeth.

95 i.e., the μέση(fourth) and παραμέση(fifth), whose ratios can be expressed as 8 : 6, 9 : 6.

96 i.e., a dactylic hexameter whose sixth foot is always a spondee or trochee has nine syllables in the first three feet and eight in the last three. For τὸ δεξιόν meaning “the first part” of a metrical system see Bassett,Journal of Classical Philology 11.458-460.

97 Alexander suggests that the number 24 may have been made up of the 12 signs of the zodiac, the 8 spheres (fixed stars, five planets, sun and moon) and 4 elements.

98 Cf. Aristot. Met. 1.3.1, Aristot. Met. 5.1, 2.

99 i.e., square.

100 Probably their “power” of being represented as regular figures; e.g. the triangularity of 3 or 6.

101 Cf. Aristot. Met. 1.5.6.

102 i.e., 4.

103 Aristotle has argued (Aristot. Met. 13.6-8.) that if the Ideal numbers differ in kind, their units must differ in kind. Hence even equal numbers, being composed of different units, must be different in kind. In point of fact, since each ideal number is unique, no two of them could be equal.