The Pythagoreans, then, may be dismissed for the present, for it is enough to touch upon them thus briefly. 990bAs for those who posit the Forms as causes,¹ in the first place in their attempt to find the causes of things in our sensible world, they introduced an equal number of other entities—as though a man who wishes to count things should suppose that it would be impossible when they are few, and should attempt to count them when he has added to them. For the Forms are as many as, or not fewer than, the things in search of whose causes these thinkers were led to the Forms; because corresponding to each thing there is a synonymous entity apart from the substances (and in the case of non-substantial things there is a One over the Many²), both in our everyday world and in the realm of eternal entities.³ Again, not one of the arguments by which we⁴ try to prove that the Forms exist demonstrates our point: from some of them no necessary conclusion follows, and from others it follows that there are Forms of things of which we hold that there are no Forms. For according to the arguments from the sciences⁵ there will be Forms of all things of which there are sciences⁶; and according to the “One-over-Many” argument,⁷ of negations too; and according to the argument that “we have some conception of what has perished,” of perishable things; because we have a mental picture of these things.⁸ Again, of Plato’s more exact arguments some establish Ideas of relations,⁹ which we do not hold to form a separate genus; and others state the “Third Man.” ¹⁰ And in general the arguments for the Forms do away with things which are more important to us exponents of the Forms than the existence of the Ideas; for they imply that it is not the Dyad that is primary, but Number¹¹; and that the relative is prior to the absolute¹²; and all the other conclusions in respect of which certain persons, by following up the views held about the Ideas, have gone against the principles of the theory. Again, according to the assumption by which we hold that the Ideas exist, there will be Forms not only of substances but of many other things (since the concept is one not only in the case of substances, but also in the case of all other things; and there are sciences not only of substances but of other things as well; and there are a thousand other similar consequences); but according to logical necessity, and from the views generally held about them, it follows that if the Forms are participated in, then there can only be Ideas of substances. For they are not participated in qua accidents; each Form can only be participated in in so far as it is not predicated of a subject. I mean, e.g., that if anything participates in “absolute Doubleness” it participates also in “eternal,” but only accidentally; because it is an accident of Doubleness to be eternal.¹³ Thus the Forms must be substance. But the same names denote substance in the sensible as in the Ideal world; 991aotherwise what meaning will there be in saying that something exists beside the particulars, i.e. the unity comprising their multiplicity? If the form of the Ideas and of the things which participate in them is the same, they will have something in common (for why should Duality mean one and the same thing in the case of perishable “twos”¹⁴ and the “twos” which are many but eternal,¹⁵ and not in the case of the Idea of Duality and a particular “two”?); but if the form is not the same, they will simply be homonyms; just as though one were to call both Callias and a piece of wood “man,” without remarking any property common to them.¹⁶ Above all we might examine the question what on earth the Forms contribute to sensible things, whether eternal or subject to generation and decay; for they are not the cause of any motion or change in them. Again, they are no help towards the knowledge of other things¹⁷(for they are not the substance of things, otherwise they would be in things), nor to their existence, since they are not present in the things which partake of them. If they were, it might perhaps seem that they are causes, in the sense in which the admixture of white causes a thing to be white; but this theory, which was first stated by Anaxagoras¹⁸ and later by Eudoxus¹⁹ and others, is very readily refutable, for it is easy to adduce plenty of impossibilities against such a view. Again, other things are not in any accepted sense derived from the Forms. To say that the Forms are patterns, and that other things participate in them, is to use empty phrases and poetical metaphors; for what is it that fashions things on the model of the Ideas²⁰ Besides, anything may both be and become like something else without being imitated from it; thus a man may become just like Socrates whether Socrates exists or not, and even if Socrates were eternal, clearly the case would be the same. Also there will be several “patterns,” and hence Forms, of the same thing; e.g. “animal” and “two-footed” will be patterns of “man,” and so too will the Idea of Man.²¹ Further, the Forms will be patterns not only of sensible things but of themselves (e.g. genus in the sense of genus of species), and thus the same thing will be both pattern and copy.²² 991bFurther, it would seem impossible that the substance and the thing of which it is the substance exist in separation; hence how can the Ideas, if they are the substances of things, exist in separation from them?²³ It is stated in the Phaedo²⁴ that the Forms are the causes both of existence and of generation. Yet, assuming that the Forms exist, still the things which participate in them are not generated unless there is something to impart motion; while many other things are generated (e.g. house, ring) of which we hold that there are no Forms. Thus it is clearly possible that all other things may both exist and be generated for the same causes as the things just mentioned. Further, if the Forms are numbers, in what sense will they be causes? Is it because things are other numbers, e.g. such and such a number Man, such and such another Socrates, such and such another Callias? then why are those numbers the causes of these? Even if the one class is eternal and the other not, it will make no difference. And if it is because the things of our world are ratios of numbers (e.g. a musical concord), clearly there is some one class of things of which they are ratios. Now if there is this something, i.e. their matter , clearly the numbers themselves will be ratios of one thing to another. I mean, e.g., that if Callias is a numerical ratio of fire, earth, water and air, the corresponding Idea too will be a number of certain other things which are its substrate. The Idea of Man, too, whether it is in a sense a number or not, will yet be an arithmetical ratio of certain things, and not a mere number; nor, on these grounds, will any Idea be a number.²⁵ Again, one number can be composed of several numbers, but how can one Form be composed of several Forms? And if the one number is not composed of the other numbers themselves, but of their constituents (e.g. those of the number 10,000), what is the relation of the units? If they are specifically alike, many absurdities will result, and also if they are not (whether (a) the units in a given number are unlike, or (b) the units in each number are unlike those in every other number).²⁶ For in what can they differ, seeing that they have no qualities? Such a view is neither reasonable nor compatible with our conception of units. Further, it becomes necessary to set up another kind of number (with which calculation deals), and all the objects which are called “intermediate” by some thinkers.²⁷ But how or from what principles can these be derived? or on what grounds are they to be considered intermediate between things here and Ideal numbers? Further, each of the units in the number 2 comes from a prior 2; but this is impossible.²⁸ 992aFurther, why should a number 〈of units〉, taken together, be one thing? And further, in addition to the above objections, if the units are unlike, they should be treated as the thinkers who assume two or four elements treat those elements; for not one of them applies the term “element” to the common substrate, e.g. body, but to fire and earth—whether there is a common substrate (i.e. body) or not.²⁹ As it is, the One is spoken of as though it were homogeneous, like fire or water. But if this is so, the numbers will not be substances. And if there is an absolute One which is a principle, clearly the term “one” is ambiguous; otherwise this is impossible.³⁰ When we wish to refer substances to their principles we derive lines³¹ from “Long and Short,” a kind of “Great and Small”; and the plane from “Wide and Narrow,” and the solid body from “Deep and Shallow.” But in this case how can the plane contain a line, or the solid a line and a plane? for “Wide and Narrow” and “Deep and Shallow” are different genera. Nor is Number contained in these objects (because “Many and Few” is yet another class); and in the same way it is clear that none of the other higher genera will be contained in the lower. Nor, again, is the Broad the genus of which the Deep is a species; for then body would be a kind of plane. Further, how will it be possible for figures to contain points?³² Plato steadily rejected this class of objects as a geometrical fiction, but he recognized “the beginning of a line,” and he frequently assumed this latter class, i.e. the “ indivisible lines.” ³³ But these must have some limit; and so by the same argument which proves the existence of the line, the point also exists.³⁴ In general, although Wisdom is concerned with the cause of visible things, we have ignored this question (for we have no account to give of the cause from which change arises),³⁵ and in the belief that we are accounting for their substance we assert the existence of other substances; but as to how the latter are the substances of the former, our explanation is worthless—for “participation,” as we have said before,³⁶ means nothing. And as for that which we can see to be the cause in the sciences, and through which all mind and all nature works—this cause³⁷ which we hold to be one of the first principles—the Forms have not the slightest bearing upon it either. Philosophy has become mathematics for modern thinkers,³⁸ although they profess³⁹ that mathematics is only to be studied as a means to some other end. 992bFurther, one might regard the substance which they make the material substrate as too mathematical, and as being a predicate and differentia of substance or matter rather than as matter itself, I mean the “Great and Small,” which is like the “Rare and Dense” of which the physicists speak,⁴⁰ holding that they are the primary differentiae of the substrate; because these qualities are a species of excess and defect. Also with regard to motion, if the “Great and Small” is to constitute motion, obviously the Forms will be moved; if not, whence did it come? On this view the whole study of physics is abolished. And what is supposed to be easy, to prove that everything is One, does not follow; because from their exposition⁴¹ it does not follow, even if you grant them all their assumptions that everything is One, but only that there is an absolute One— and not even this, unless you grant that the universal is a class; which is impossible in some cases.⁴² Nor is there any explanation of the lines, planes and solids which “come after” the Numbers⁴³: neither as to how they exist or can exist, nor as to what their importance is. They cannot be Forms (since they are not numbers) or Intermediates (which are the objects of mathematics) or perishables; clearly they form yet another fourth class. In general, to investigate the elements of existing things without distinguishing the various senses in which things are said to exist is a hopeless task; especially when one inquires along these lines into the nature of the elements of which things are composed. For (a) we cannot surely conceive of the elements of activity or passivity or straightness; this is possible, if at all, only in the case of substances. Hence to look for, or to suppose that one has found, the elements of everything that exists, is a mistake. (b) How can one apprehend the elements of everything ? Obviously one could not have any previous knowledge of anything; because just as a man who is beginning to learn geometry can have previous knowledge of other facts, but no previous knowledge of the principles of that science or of the things about which he is to learn, so it is in the case of all other branches of knowledge. Hence if there is a science which embraces everything⁴⁴(as some say), the student of it can have no previous knowledge at all. But all learning proceeds, wholly or in part, from what is already known; whether it is through demonstration or through definition—since the parts of the definition must be already known and familiar. The same is true of induction. 993aOn the other hand, assuming that this knowledge should turn out to be innate,⁴⁵ it is astonishing that we should possess unawares the most important of the sciences. Further, how is one to know of what elements things consist? how is it to be established? Even this presents a difficulty, because the facts might be disputed, as happens in the case of certain syllables—for some say that ZA is composed of S, D and A, while others say that it is a distinct sound and not any one of those which are familiar to us.⁴⁶ Further, how can one gain knowledge of the objects of a particular sense-perception without possessing that sense? Yet it should be possible, that if the elements of which all things consist, as composite sounds consist of their peculiar⁴⁷ elements, are the same.

Thus it is obvious, from the statements of earlier thinkers also, that all inquiry is apparently directed towards the causes described in the Physics,⁴⁸ and that we cannot suggest any other cause apart from these. They were, however, only vaguely conceived; and although in one sense they have all been stated before, in another they have not been stated at all. For the earliest philosophy speaks falteringly, as it were, on all subjects; being new and in its infancy. Even Empedocles says that bone exists by virtue of its ratio,⁴⁹ which is the definition or essence of a thing. But by similar reasoning both flesh and every other thing, or else nothing at all, must be ratio; for it must be because of this, and not because of their matter—which he calls fire, earth, water and air—that flesh and bone and every other thing exists. If anyone else had stated this, he would necessarily have agreed, but his own statement was not clear. These and similar points have been explained already. We will now return to the difficulties which might be raised about these same questions, for they may throw some light upon subsequent difficulties.⁵⁰