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.¹ 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.² But why? Because the attributes of numbers are to be found in a musical scale, in the heavens, and in many other connections.³ As for those who hold that mathematical number alone exists,⁴ they cannot allege anything of this kind⁵ 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.⁶ 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.⁷ 1090bThe same applies to mathematical extended magnitudes. It is clear, then, both that the contrary theory⁸ 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 raised⁹—why it is that while numbers are in no way present in sensible things, their attributes are present in sensible things. There are some¹⁰ 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.¹¹ But it does not appear, to judge from the observed facts, that the natural system lacks cohesion, like a poorly constructed drama. Those¹² 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 theories¹³ 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.¹⁴ 1091aAnd 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.¹⁵ 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 story¹⁶”; 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 2.¹⁷ 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.¹⁸ 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.

They¹⁹ say that there is no generation of odd numbers,²⁰ 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.²¹ 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 we²² 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,²³ 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. 1091bThis 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. Pherecydes²⁴ and certain others—make the primary generator the Supreme Good; and so do the Magi,²⁵ and some of the later philosophers such as Empedocles and Anaxagoras: the one making Love an element,²⁶ and the other making Mind a first principle.²⁷ And of those who hold that unchangeable substances exist, some²⁸ 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 thinkers²⁹ 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.³⁰ 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 substances³¹; and if there are Ideas of substances also, all animals and plants, and all things that participate in the Ideas, will be goods.³² 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 thinker³³ avoided associating the Good with unity, on the ground that since generation proceeds from contraries, the nature of plurality would then necessarily be bad. Others³⁴ 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 magnitudes³⁵; 1092aand 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,³⁶ 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.