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, 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, Parmenides Fr. 7 (Diels). 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. Cf. Plat. Soph. 237a, 241d, 256e. 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? This thinker means by the not-being which together with Being makes existing things a plurality, falsity and everything of this nature Plat. Soph. 237a, 240; but Aristotle's statement assumes too much.; and for this reason also it was said Presumably by some Platonist. 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 inference i.e., the validity of a geometrical proof does not depend upon the accuracy of the figure. ), 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; 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 cause Matter, according to Aristotle; and there is matter, or something analogous to it, in every category. Cf. Aristot. Met.
12.5. 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. Cf. Aristot. Met. 14.1.6, 18, Aristot.
Met. 1.9.23. 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 author Plato. 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, but just a particular kind of Being. And it was still more necessary, as we have said, sect. 11.
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. This, according to Aristotle, is how the Platonists regard the Ideas.
See Introduction. 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; 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 one Plato and his orthodox followers. 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 him Speusippus. 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. Aristot. Met. 13.3.1. 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. I have followed Ross's text and interpretation of this sentence. For the meaning cf. Aristot. Met. 14.2.20. Since, however, their arguments are neither necessarily true nor indeed possible, 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. See Introduction. But why? Because the attributes of numbers are to be found in a musical scale, in the heavens, and in many other connections. Cf. Aristot. Met. 14.6.5. As for those who hold that mathematical number alone exists, Cf. Aristot. Met.
14.2.21. they cannot allege anything of this kind i.e., that things are composed of numbers. 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. See Introduction.
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. 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.
The same applies to mathematical extended magnitudes. It is clear, then, both that the contrary theory 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. 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 raised sect. 3. —why it is that while numbers are in no way present in sensible things, their attributes are present in sensible things. There are some Probably Pythagoreans. Cf. Aristot.
Met. 7.2.2, Aristot. Met.
3.5.3. 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. That the criticism is directed against Speusippus is clear from Aristot. Met. 7.2.4. Cf. Aristot.
Met. 12.10.14. But it does not appear, to judge from the observed facts, that the natural system lacks cohesion, like a poorly constructed drama.
Those Xenocrates (that the reference is not to Plato is clear from sect. 11). 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 theories e.g. that of "indivisible lines." 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. This interpretation (Ross's second alternative, reading TI/NOS for TINOS ) seems to be the most satisfactory. For the objection cf. Aristot.
Met. 3.4.34.
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. 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. 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 story The exact reference is uncertain, but Aristotle probably means Simonides of Ceos. Cf. Simonides Fr. 189 (Bergk). "; 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. Assuming that the Great-and-Small, or indeterminate dyad, is duplicative ( Aristot. Met.
13.7.18 ). 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. Cf. Aristot. Physics 3.4, Aristot. Physics 4.6, and Burnet, E.G.P. sect. 53. 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, 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. They The Platonists. say that there is no generation of odd numbers, 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.). 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. Cf. Aristot. Met. 13.7.5. 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 we Aristotle speaks as a Platonist. See Introduction. 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, The Pythagoreans and Speusippus; cf. Aristot. Met. 12.7.10.
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.
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. Pherecydes Of Syros (circa 600 -525 B.C.). He made Zeus one of the three primary beings (Diels, Vorsokratiker 201, 202). and certain others—make the primary generator the Supreme Good; and so do the Magi, The Zoroastrian priestly caste. and some of the later philosophers such as Empedocles and Anaxagoras: the one making Love an element, Cf. Aristot. Met. 3.1.13. and the other making Mind a first principle. Cf.
Aristot. Met. 1.3.16. And of those who hold that unchangeable substances exist, some Plato; cf. Aristot. Met.
1.6.10. 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. 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 thinkers Speusippus and his followers; cf. sect. 3. 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. If unity is goodness, and every unit is a kind of unity, every unit must be a kind of goodness—which is absurd. 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 substances Because they are Ideas not of substances but of qualities.; and if there are Ideas of substances also, all animals and plants, and all things that participate in the Ideas, will be goods. Because the Ideas are goods. 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 thinker Speusippus. avoided associating the Good with unity, on the ground that since generation proceeds from contraries, the nature of plurality would then necessarily be bad. Others Plato and Xenocrates. 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 magnitudes As being more directly derived from the first principles. Cf. Aristot. Met. 1.9.23 n.; 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, Aristot. Met. 14.1.17.
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 thinker Evidently Speusippus; cf. Aristot. Met. 14.4.3.
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. 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. 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. Cf. Aristot. Met.
9.8.5. 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) 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 mixture 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.; (b) the result of mixture is something different; and unity will not be separable, sc. when it has once been mixed. Cf. Aristot. De Gen. et Corr. 327b 21-26. 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. And numbers are supposed to be eternal. Cf. Aristot. Met. 14.2.1-3. Is it derived as from seed? But nothing can be emitted from that which is indivisible. i.e., unity, being indivisible, cannot contribute the formal principle of generation in the way that the male parent contributes it. 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 thinker Speusippus: Plato. Cf. Aristot. Met. 14.1.5. regards unity as contrary to plurality, 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. 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. 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, Because it may be regarded as still potentially present. 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. According to Empedocles Fr.
17 (Diels). 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, The theories criticized from this point onwards to Aristot. Met. 14.6.11 are primarily Pythagorean.
See Introduction. e.g. as points are the bounds of spatial magnitudes, e.g. the line by 2 points, the triangle (the simplest plane figure) by 3, the tetrahedron (the simplest solid figure) by 4. and as Eurytus Disciple of Philolaus; he "flourished" in the early fourth century B.C. 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. cf.
Burnet, E.G.P. sect. 47. 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? This is an objection to the view that numbers are causes as bounds. And clearly numbers are not the essence of things, nor are they causes of the form; for the ratio Or "formula." is the essence, and number In the sense of a number of material particles. 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. Cf. Empedocles Fr. 96 (Diels).
And the number, 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, i.e., a simple ratio. or an odd number. 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 ). For in point of fact honey-water is no more wholesome if it is mixed in the proportion "three times three" Apparently the Pythagoreans meant by this "three parts of water to three of honey." Aristotle goes on to criticize this way of expressing ratios.; 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 2 Cf. previous note.; 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. sc. because if so, a particle of fire would simply equal 35 particles of water.
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, 5 in each case, according to Aristotle; cf. Aristot. Met. 12.7.9, 11. 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. Cf. previous note. But why are these numbers causes? There are seven vowels, In the Greek alphabet.
seven strings to the scale, In the old heptachord; cf. note on Aristot. Met.
5.11.4. seven Pleiads; most animals (though not all Cf. Aristot. Hist. An. 576a 6. ) 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. Indeed, they assert also that *C, *Y and *Z are concords, According to Alexander Z was connected with the fourth, C with the fifth, and Y with the octave. 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 *G*R. But if they say that each of these letters is double any of the others, whereas no other is, Q, F, and X are aspirated, not double, consonants. and that the reason is that there are three regions Palate, lips, and teeth. of the mouth, and that one consonant is combined with S 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 notes i.e., the ME/SH (fourth) and PARAME/SH (fifth), whose ratios can be expressed as 8: 6, 9: 6. of the octave are respectively 8 and 9, and the epic hexameter has seventeen syllables, which equals the sum of these two; and the line scans in the first half with nine syllables, and in the second with eight. 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 TO\ DECIO/N meaning "the first part" of a metrical system see Bassett, Journal of Classical Philology 11.458-460. And they point out that the interval from A to W 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. 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. 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. Cf. Aristot. Met. 1.3.1, Aristot.
Met. 5.1, 2.
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 square. and the powers Probably their "power" of being represented as regular figures; e.g. the triangularity of 3 or 6. of certain numbers, belong to the series of the Beautiful. Cf. Aristot. Met.
1.5.6. For the seasons are connected with a certain kind of number i.e., 4.; 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, 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) 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.; 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