it disposes of many of the impossibilities; but that bodies should be composed of numbers, and that these numbers should be mathematical, is impossible. See Introduction. For (a) it is not true to speak of indivisible magnitudes This is proved in Aristot. De Gen. et. Corr. 315b 24-317a 17.; (b) assuming that this view is perfectly true, still units at any rate have no magnitude; and how can a magnitude be composed of indivisible parts? Moreover arithmetical number consists of abstract units. But the Pythagoreans identify number with existing things; at least they apply mathematical propositions to bodies as though they consisted of those numbers. See Introduction. Thus if number, if it is a self-subsistent reality, must be regarded in one of the ways described above, and if it cannot be regarded in any of these ways, clearly number has no such nature as is invented for it by those who treat it as separable. Again, does each unit come from the Great and the Small, when they are equalized Cf. Aristot. Met. 13.7.5 n.
Aristotle is obviously referring to the two units in the Ideal 2.; or does one come from the Small and another from the Great? If the latter, each thing is not composed of all the elements, nor are the units undifferentiated; for one contains the Great, and the other the Small, which is by nature contrary to the Great. Again, what of the units in the Ideal 3? because there is one over. But no doubt it is for this reason that in an odd number they make the Ideal One the middle unit. Cf. DieIs, Vorsokratiker 270. 18. If on the other hand each of the units comes from both Great and Small, when they are equalized, how can the Ideal 2 be a single entity composed of the Great and Small? How will it differ from one of its units? Again, the unit is prior to the 2; because when the unit disappears the 2 disappears. Therefore the unit must be the Idea of an Idea, since it is prior to an Idea, and must have been generated before it. From what, then? for the indeterminate dyad, as we have seen, Aristot. Met. 13.7.18.
causes duality. Again, number must be either infinite or finite (for they make number separable, so that one of these alternatives must be true). The point seems to be that if number is self-subsistent it must be actually finite or infinite. Aristotle himself holds that number is infinite only potentially; i.e., however high you can count, you can always count higher. Now it is obvious that it cannot be infinite, because infinite number is neither odd nor even, and numbers are always generated either from odd or from even number. By one process, when 1 is added to an even number, we get an odd number; by another, when 1 is multiplied by 2, we get ascending powers of 2; and by another, when powers of 2 are multiplied by odd numbers, we get the remaining even numbers. Again, if every Idea is an Idea of something, and the numbers are Ideas, infinite number will also be an Idea of something, either sensible or otherwise. This, however, is impossible, both logically i.e., as implying an actual infinite. and on their own assumption, i.e., as inconsistent with the conception of an Idea as a determining principle. since they regard the Ideas as they do. If, on the other hand, number is finite, what is its limit? In reply to this we must not only assert the fact, but give the reason. Now if number only goes up to 10, as some hold, Cf. Aristot. Met. 12.8.2. The Platonists derived this view from the Pythagoreans; see Introduction. in the first place the Forms will soon run short. For example, if 3 is the Idea of Man, what number will be the Idea of Horse? Each number up to 10 is an Idea; the Idea of Horse, then, must be one of the numbers in this series, for they are substances or Ideas. But the fact remains that they will run short, because the different types of animals will outnumber them. At the same time it is clear that if in this way the Ideal 3 is the Idea of Man, so will the other 3's be also (for the 3's in the same numbers Robin is probably right in taking this to mean that the 3 which is in the ideal 4 is like the 3 which is in the 4 which is in a higher ideal number, and so on ( La Theorie platonicienne des Idees et des Nombres d'apres Aristote, p. 352). are similar), so that there will be an infinite number of men; and if each 3 is an Idea, each man will be an Idea of Man; or if not, they will still be men. And if the smaller number is part of the greater, when it is composed of the addible units contained in the same number, then if the Ideal 4 is the Idea of something, e.g. "horse" or "white," then "man" will be part of "horse," if "man" is 2. It is absurd also that there should be an Idea of 10 and not of 11, nor of the following numbers.
Again, some things exist and come into being of which there are no Forms Cf. Aristot. Met. 13.4.7, 8; Aristot. Met.
1.9.2, 3.; why, then, are there not Forms of these too? It follows that the Forms are not the causes of things. Again, it is absurd that number up to 10 should be more really existent, and a Form, than 10 itself; although the former is not generated as a unity, whereas the latter is. However, they try to make out that the series up to 10 is a complete number; at least they generate the derivatives, e.g. the void, proportion, the odd, etc., from within the decad. Some, such as motion, rest, good and evil, they assign to the first principles; the rest to numbers. From the Dyad were derived void ( Theophrastus, Met. 312.18-313.3 ) and motion (cf. Aristot. Met. 1.9.29, Aristot. Met. 11.9.8 ). Rest would naturally be derived from unity. For good and evil see Aristot. Met. 1.6.10. Proportion alone of the "derivatives" here mentioned appears to be derived from number. As Syrianus says, the three types of proportion can be illustrated by numbers from within the harmonic 2. 3. 6. Hence they identify the odd with Unity; because if oddness depended on 3, how could 5 be odd? sc. because (on their theory) 3 is not contained in 5. Thus oddness had to be referred to not a number but a principle—unity. Again, they hold that spatial magnitudes and the like have a certain limit; e.g. the first or indivisible line, then the 2, and so on; these too extending up to 10. The "indivisible line" or point was connected with 1, the line with 2, the plane with 3 and the solid with 4 ( Aristot. Met. 14.3.9 ); and 1+2+3+4=10. Again, if number is separable, the question might be raised whether Unity is prior, or 3 or 2. Now if we regard number as composite, Unity is prior; but if we regard the universal or form as prior, number is prior, because each unit is a material part of number, while number is the form of the units. And there is a sense in which the right angle is prior to the acute angle—since it is definite and is involved in the definition of the acute angle—and another sense in which the acute angle is prior, because it is a part of the other, i.e., the right angle is divided into acute angles. Thus regarded as matter the acute angle and element and unit are prior; but with respect to form and substance in the sense of formula, the right angle, and the whole composed of matter and form, is prior. For the concrete whole is nearer to the form or subject of the definition, although in generation it is posterior. Cf. Aristot. Met. 7.10, 11. In what sense, then, is the One a first principle?
Because, they say, it is indivisible. But the universal and the part or element are also indivisible. Yes, but they are prior in a different sense; the one in formula and the other in time. In which sense, then, is the One a first principle? for, as we have just said, both the right angle seems to be prior to the acute angle, and the latter prior to the former; and each of them is one. Accordingly the Platonists make the One a first principle in both senses. But this is impossible; for in one sense it is the One qua form or essence, and in the other the One qua part or matter, that is primary. There is a sense in which both number and unit are one; they are so in truth potentially—that is, if a number is not an aggregate but a unity consisting of units distinct from those of other numbers, as the Platonists hold— but each of the two Aristotle takes the number two as an example, but the principle is of course universal. In a sense both number and unit are one; but if the number exists as an actual unity, the unit can only exist potentially. units is not one in complete reality. The cause of the error which befell the Platonists was that they were pursuing their inquiry from two points of view—that of mathematics and that of general definition—at the same time. Hence as a result of the former they conceived of the One or first principle as a point, for the unit is a point without position. (Thus they too, just like certain others, represented existing things as composed of that which is smallest.) Perhaps the Atomists; but cf.
Aristot. Met. 1.8.3, 4. We get, then, that the unit is the material element of numbers, and at the same time is prior to the number 2; and again we get that it is posterior to 2 regarded as a whole or unity or form. On the other hand, through looking for the universal, they were led to speak of the unity predicated of a given number as a part in the formal sense also.
But these two characteristics cannot belong simultaneously to the same thing. And if Unity itself must only be without position If the text is sound (and no convincing emendation has been suggested), it seems best to understand A)/QETON in a rather wider sense than the semi-technical one put forward by Ross. "Without position"=not localized, i.e.
abstract. Unity as a principle has no concrete instance. (for it differs only in that it is a principle) and 2 is divisible whereas the unit is not, the unit will be more nearly akin to Unity itself; and if this is so, Unity itself will also be more nearly akin to the unit than to 2. Hence each of the units in 2 will be prior to 2. But this they deny; at least they make out that 2 is generated first. Cf. Aristot. Met. 13.7.5.
Further, if 2 itself and 3 itself are each one thing, both together make 2. From what, then, does this 2 come? Since there is no contact in numbers, but units which have nothing between them—e.g. those in 2 or 3—are successive, the question might be raised whether or not they are successive to Unity itself, and whether of the numbers which succeed it 2 or one of the units in 2 is prior. We find similar difficulties in the case of the genera posterior to number Cf. Aristot. Met. 13.6.10. —the line, plane and solid. Some derive these from the species of the Great and Small; viz. lines from the Long and Short, planes from the Broad and Narrow, and solids from the Deep and Shallow. These are species of the Great and Small. As for the geometrical first principle which corresponds to the arithmetical One, different Platonists propound different views. Cf.
Aristot. Met. 3.4.34, Aristot. Met. 14.3.9. In these too we can see innumerable impossibilities, fictions and contradictions of all reasonable probability. For (a) we get that the geometrical forms are unconnected with each other, unless their principles also are so associated that the Broad and Narrow is also Long and Short; and if this is so, the plane will be a line and the solid a plane. Moreover, how can angles and figures, etc., be explained? And (b) the same result follows as in the case of number; for these concepts are modifications of magnitude, but magnitude is not generated from them, any more than a line is generated from the Straight and Crooked, or solids from the Smooth and Rough. Common to all these Platonic theories is the same problem which presents itself in the case of species of a genus when we posit universals—viz. whether it is the Ideal animal that is present in the particular animal, or some other "animal" distinct from the Ideal animal. This question will cause no difficulty if the universal is not separable; but if, as the Platonists say, Unity and the numbers exist separately, then it is not easy to solve (if we should apply the phrase "not easy" to what is impossible). For when we think of the one in 2, or in number generally, are we thinking of an Idea or of something else? These thinkers, then, generate geometrical magnitudes from this sort of material principle, but others The reference is probably to Speusippus; Plato and Xenocrates did not believe in points ( Aristot. Met.
1.9.25, Aristot. Met. 13.5.10 n ). generate them from the point (they regard the point not as a unity but as similar to Unity) and another material principle which is not plurality but is similar to it; yet in the case of these principles none the less we get the same difficulties. For if the matter is one, line, plane and solid will be the same; because the product of the same elements must be one and the same. If on the other hand there is more than one kind of matter—one of the line, another of the plane, and another of the solid—either the kinds are associated with each other, or they are not. Thus the same result will follow in this case also; for either the plane will not contain a line, or it will be a line. Further, no attempt is made to explain how number can be generated from unity and plurality; but howsoever they account for this, they have to meet the same difficulties as those who generate number from unity and the indeterminate dyad. The one school generates number not from a particular plurality but from that which is universally predicated; the other from a particular plurality, but the first; for they hold that the dyad is the first plurality. Aristotle again identifies the indeterminate dyad with the number 2. Thus there is practically no difference between the two views; the same difficulties will be involved with regard to mixture, position, blending, generation and the other similar modes of combination. sc. of the elements of number. We might very well ask the further question: if each unit is one, of what it is composed; for clearly each unit is not absolute unity. It must be generated from absolute unity and either plurality or a part of plurality. Now we cannot hold that the unit is a plurality, because the unit is indivisible; but the view that it is derived from a part of plurality involves many further difficulties, because (a) each part must be indivisible; otherwise it will be a plurality and the unit will be divisible, and unity and plurality will not be its elements, because each unit will not be generated from plurality sc. but from an indivisible part of plurality—which is not a plurality but a unity. and unity. (b) The exponent of this theory merely introduces another number; because plurality is a number of indivisible parts. i.e., to say that number is derived from plurality is to say that number is derived from number—which explains nothing. Again, we must inquire from the exponent of this theory whether the number sc. which plurality has been shown to be. is infinite or finite. There was, it appears, a finite plurality from which, in combination with Unity, the finite units were generated; and absolute plurality is different from finite plurality.
What sort of plurality is it, then, that is, in combination with unity, an element of number? We might ask a similar question with regard to the point, i.e. the element out of which they create spatial magnitudes. This is surely not the one and only point. At least we may ask from what each of the other points comes; it is not, certainly, from some interval and the Ideal point. Moreover, the parts of the interval cannot be indivisible parts, any more than the parts of the plurality of which the units are composed; because although number is composed of indivisible parts, spatial magnitudes are not. All these and other similar considerations make it clear that number and spatial magnitudes cannot exist separately. Further, the fact that the leading authorities Alexander preferred the reading PRW/TOUS, interpreting it in this sense; and I do not see why he should not be followed. Ross objects that PRW=TOS is used in the chronological sense in 16., but this is really no argument.
For a much more serious (although different) inconsistency in the use of terms cf. Aristot. Met.
12.3.1. disagree about numbers indicates that it is the misrepresentation of the facts themselves that produces this confusion in their views. Those Speusippus and his followers. who recognize only the objects of mathematics as existing besides sensible things, abandoned Ideal number and posited mathematical number because they perceived the difficulty and artificiality of the Ideal theory. Others, Xenocrates and his followers. wishing to maintain both Forms and numbers, but not seeing how, if one posits these Unity and the indeterminate dyad; for the difficulty see Aristot.
Met. 13.7.3, 4. as first principles, mathematical number can exist besides Ideal number, identified Ideal with mathematical number,—but only in theory, since actually mathematical number is done away with, because the hypotheses which they state are peculiar to them and not mathematical. Cf. Aristot. Met.
13.6.10. And he Plato. who first assumed that there are Ideas, and that the Ideas are numbers, and that the objects of mathematics exist, naturally separated them. Thus it happens that all are right in some respect, but not altogether right; even they themselves admit as much by not agreeing but contradicting each other. The reason of this is that their assumptions and first principles are wrong; and it is difficult to propound a correct theory from faulty premisses: as Epicharmus says, "no sooner is it said than it is seen to be wrong." Epicharmus, Fr. 14, Diels. We have now examined and analyzed the questions concerning numbers to a sufficient extent; for although one who is already convinced might be still more convinced by a fuller treatment, he who is not convinced would be brought no nearer to conviction. As for the first principles and causes and elements, the views expressed by those who discuss only sensible substance either have been described in the Physics Aristot. Physics 1.4-6. or have no place in our present inquiry; but the views of those who assert that there are other substances besides sensible ones call for investigation next after those which we have just discussed. Since, then, some thinkers hold that the Ideas and numbers are such substances, and that their elements are the elements and principles of reality, we must inquire what it is that they hold, and in what sense they hold it. Those The Pythagoreans and Speusippus. who posit only numbers, and mathematical numbers at that, may be considered later Aristot. Met.
14.2.21, Aristot. Met. 14.3.2-8, 15, 16.; but as for those who speak of the Ideas, we can observe at the same time their way of thinking and the difficulties which befall them. For they not only treat the Ideas as universal substances, but also as separable and particular. (That this is impossible has been already shown Aristot. Met.
3.6.7-9. by a consideration of the difficulties involved.) The reason why those who hold substances to be universal combined these two views was that they did not identify substances with sensible things. They considered that the particulars in the sensible world are in a state of flux, and that none of them persists, but that the universal exists besides them and is something distinct from them. This theory, as we have said in an earlier passage, Aristot. Met. 13.4, and cf.
Aristot. Met.
1.6. was initiated by Socrates as a result of his definitions, but he did not separate universals from particulars; and he was right in not separating them. This is evident from the facts; for without the universal we cannot acquire knowledge, and the separation of the universal is the cause of the difficulties which we find in the Ideal theory. Others, The Platonists.
regarding it as necessary, if there are to be any substances besides those which are sensible and transitory, that they should be separable, and having no other substances, assigned separate existence to those which are universally predicated; thus it followed that universals and particulars are practically the same kind of thing.
This in itself would be one difficulty in the view which we have just described. See Introduction. Let us now mention a point which presents some difficulty both to those who hold the Ideal theory and to those who do not. It has been stated already, at the beginning of our treatise, among the problems. Cf. Aristot. Met. 3.4.8-10, Aristot. Met. 3.6.7-9. If we do not suppose substances to be separate, that is in the way in which particular things are said to be separate, we shall do away with substance in the sense in which we wish to maintain it; but if we suppose substances to be separable, how are we to regard their elements and principles? If they are particular and not universal, there will be as many real things as there are elements, and the elements will not be knowable. For let us suppose that the syllables in speech are substances, and that their letters are the elements of substances. Then there must be only one BA, and only one of each of the other syllables; that is, if they are not universal and identical in form, but each is numerically one and an individual, and not a member of a class bearing a common name. (Moreover, the Platonists assume that each Ideal entity is unique.) Now if this is true of the syllables, it is also true of their letters. Hence there will not be more than one A, nor more than one of any of the other letters, This is, as a matter of fact, the assumption upon which the whole argument rests; Aristotle is arguing in a circle. on the same argument by which in the case of the syllable there cannot be more than one instance of the same syllable. But if this is so, there will be no other things besides the letters, but only the letters. Nor again will the elements be knowable; for they will not be universal, and knowledge is of the universal. This can be seen by reference to proofs and definitions; for there is no logical conclusion that a given triangle has its angles equal to two right angles unless every triangle has its angles equal to two right angles, or that a given man is an animal unless every man is an animal. On the other hand, if the first principles are universal, either the substances composed of them will be universal too, or there will be a non-substance prior to substance; because the universal is not substance, and the element or first principle is universal; and the element or first principle is prior to that of which it is an element or first principle. All this naturally follows when they compose the Ideas of elements and assert that besides the substances which have the same form there are also Ideas each of which is a separate entity. But if, as in the case of the phonetic elements, there is no reason why there should not be many A's and B's, and no "A itself" or "B itself" apart from these many, then on this basis there may be any number of similar syllables. The doctrine that all knowledge is of the universal, and hence that the principles of existing things must also be universal and not separate substances, presents the greatest difficulty of all that we have discussed; there is, however, a sense in which this statement is true, although there is another in which it is not true. Knowledge, like the verb "to know," has two senses, of which one is potential and the other actual. The potentiality being, as matter, universal and indefinite, has a universal and indefinite object; but the actuality is definite and has a definite object, because it is particular and deals with the particular. It is only accidentally that sight sees universal color, because the particular color which it sees is color; and the particular A which the grammarian studies is an A. For if the first principles must be universal, that which is derived from them must also be universal, as in the case of logical proofs "Because A)PO/DEICIS " (logical or syllogistic proof) "must be in the first figure ( Aristot. An. Post. 1.14 ), and in that figure universal premises always give a universal conclusion."
(Ross.); and if this is so there will be nothing which has a separate existence; i.e. no substance. But it is clear that although in one sense knowledge is universal, in another it is not.
With regard to this kind of substance, 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. 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. 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. Cf. Aristot.
Categories 3b 24-27 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, Plato; cf.
Aristot. Met. 13.7.5. numbers are generated from the unequal dyad of the Great and Small; and according to another, Probably Speusippus. 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. 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.
Again, they state the first principles, which they call elements, badly; some say that the Great and the Small, together with Unity (making 3 Cf. previous note. 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; 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 measure Cf. Aristot. Met. 5.6.17, 18, Aristot. Met.
10.1.8, 21. 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; 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. Those Cf. sect. 5. 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— 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 alone 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. 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. 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 few Cf. Aristot. Met.
10.6.1-3. ), then there will be an absolute "many"; e.g., 10 will be many (if there is nothing more than 10 Cf. Aristot. Met.
13.8.17. ), 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; 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. Aristot. Met. 9.8.15-17, Aristot. De Caelo 1.12. 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 difficulties Cf. Aristot. Met. 14.1.14-17. which necessarily attend the theory of those who make the unequal, i.e. the relative, an element; all the difficulties