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Aristotle. Aristotle in 23 Volumes, Vols.17, 18, translated by Hugh Tredennick. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1933, 1989.

====== Aristotle: Metaphysics Book 3 ======


[995a] [24]

It is necessary, with a view to the science which we are investigating, that we first describe the questions which should first be discussed. These consist of all the divergent views which are held about the first principles; and also of any other view apart from these which happens to have been overlooked.Now for those who wish to get rid of perplexities it is a good plan to go into them thoroughly; for the subsequent certainty is a release from the previous perplexities, and release is impossible when we do not know the knot. The perplexity of the mind shows that there is a "knot" in the subject; for in its perplexity it is in much the same condition as men who are fettered: in both cases it is impossible to make any progress.Hence we should first have studied all the difficulties, both for the reasons given and also because those who start an inquiry without first considering the difficulties are like people who do not know where they are going; besides, one does not even know whether the thing required has been found or not. [995b] [1] To such a man the end is not clear; but it is clear to one who has already faced the difficulties.Further, one who has heard all the conflicting theories, like one who has heard both sides in a lawsuit, is necessarily more competent to judge.

The first difficulty is concerned with the subjects1 which we discussed in our prefatory remarks. (1.) Does the study of the causes belong to one science or to more than one?2(2.) Has that science only to contemplate the first principles of substance, or is it also concerned with the principles which all use for demonstration—e.g. whether it is possible at the same time to assert and deny one and the same thing, and other similar principles?3And if it is concerned with substance, (3.) is there one science which deals with all substances, or more than one; and if more than one, are they all cognate, or should we call some of them "kinds of Wisdom" and others something different?4This too is a question which demands inquiry: (iv.) should we hold that only sensible substances exist, or that there are other besides? And should we hold that there is only one class of non-sensible substances, or more than one (as do those who posit the Forms and the mathematical objects as intermediate between the Forms and sensible things)?5These questions, then, as I say, must be considered; and also (v.) whether our study is concerned only with substances, [20] or also with the essential attributes of substance;and further, with regard to Same and Other, and Like and Unlike and Contrariety, and Prior and Posterior, and all other such terms which dialecticians try to investigate, basing their inquiry merely upon popular opinions; we must consider whose province it is to study all of these.Further, we must consider all the essential attributes of these same things, and not merely what each one of them is, but also whether each one has one opposite6; and (vi.) whether the first principles and elements of things are the genera under which they fall or the pre-existent parts into which each thing is divided; and if the genera, whether they are those which are predicated ultimately of individuals, or the primary genera—e.g., whether "animal" or "man" is the first principle and the more independent of the individual.7

Above all we must consider and apply ourselves to the question (7.) whether there is any other cause per se besides matter, and if so whether it is dissociable from matter, and whether it is numerically one or several; and whether there is anything apart from the concrete thing (by the concrete thing I mean matter together with whatever is predicated of it) or nothing; or whether there is in some cases but not in others; and what these cases are.8 [996a] [1] Further, (8.) we must ask whether the first principles are limited in number or in kind9—both those in the definitions and those in the substrate—and (ix.) whether the principles of perishable and of imperishable things are the same or different; and whether all are imperishable, or those of perishable things are perishable.10Further, there is the hardest and most perplexing question of all: (x.) whether Unity and Being (as the Pythagoreans and Plato maintained) are not distinct, but are the substance of things; or whether this is not so, and the substrate is something distinct11(as Empedocles holds of Love,12 another thinker13 of fire, and another14 of water or air15);and (xi.) whether the first principles are universal or like individual things16; and (12.) whether they exist potentially or actually; and further whether their potentiality or actuality depends upon anything other than motion17; for these questions may involve considerable difficulty.Moreover we must ask (13.) whether numbers and lines and figures and points are substances in any sense, or not; and if they are, whether they are separate from sensible things or inherent in them.18 With regard to these problems not only is it difficult to attain to the truth, but it is not even easy to state all the difficulties adequately.19

(1.) Firstly, then, with respect to the first point raised: whether it is the province of one science or of more than one to study all the kinds of causes. [20] How can one science comprehend the first principles unless they are contraries? Again, in many things they are not all present.How can a principle of motion be in immovable things? or the "nature of the Good"? for everything which is good in itself and of its own nature is an end and thus a cause, because for its sake other things come to be and exist; and the end and purpose is the end of some action, and all actions involve motion; thus it would be impossible either for this principle to exist in motionless things or for there to be any absolute Good.Hence in mathematics too nothing is proved by means of this cause, nor is there any demonstration of the kind "because it is better or worse"; indeed no one takes any such consideration into account.And so for this reason some of the sophists, e.g. Aristippus,20 spurned mathematics, on the ground that in the other arts, even the mechanical ones such as carpentry and cobbling, all explanation is of the kind "because it is better or worse," while mathematics takes no account of good and bad.21 [996b] [1]

On the other hand if there are several sciences of the causes, and a different one for each different principle, which of them shall we consider to be the one which we are seeking, or whom of the masters of these sciences shall we consider to be most learned in the subject which we are investigating?For it is possible for all the kinds of cause to apply to the same object; e.g. in the case of a house the source of motion is the art and the architect; the final cause is the function; the matter is earth and stones, and the form is the definition. Now to judge from our discussion some time ago22 as to which of the sciences should be called Wisdom, there is some case for applying the name to each of them.Inasmuch as Wisdom is the most sovereign and authoritative kind of knowledge, which the other sciences, like slaves, may not contradict, the knowledge of the end and of the Good resembles Wisdom (since everything else is for the sake of the end ); but inasmuch as it has been defined as knowledge of the first principles and of the most knowable, the knowledge of the essence will resemble Wisdom.For while there are many ways of understanding the same thing, we say that the man who recognizes a thing by its being something knows more than he who recognizes it by its not being something; and even in the former case one knows more than another, and most of all he who knows what it is, and not he who knows its size or quality or natural capacity for acting or being acted upon.Further, in all other cases too, even in such as admit of demonstration, [20] we consider that we know a particular thing when we know what it is (e.g. what is the squaring of a rectangle? answer, the finding of a mean proportional to its sides; and similarly in other instances); but in the case of generations and actions and all kinds of change, when we know the source of motion.This is distinct from and opposite to the end . Hence it might be supposed that the study of each of these causes pertained to a different science.23

(2.) Again, with respect to the demonstrative principles as well, it may be disputed whether they too are the objects of one science24 or of several.25By demonstrative I mean the axioms from which all demonstration proceeds, e.g. "everything must be either affirmed or denied," and "it is impossible at once to be and not to be," and all other such premisses. Is there one science both of these principles and of substance, or two distinct sciences? and if there is not one, which of the two should we consider to be the one which we are now seeking?

It is not probable that both subjects belong to one science; for why should the claim to understand these principles be peculiar to geometry rather than to any other science? Then if it pertains equally to any science, and yet cannot pertain to all, [997a] [1] comprehension of these principles is no more peculiar to the science which investigates substances than to any other science.Besides, in what sense can there in be a science of these principles? We know already just what each of them is; at any rate other sciences employ them as being known to us.26 If, however there is a demonstrative science of them, there will have to be some underlying genus, and some of the principles will be derived from axioms, and others will be unproved(for there cannot be demonstration of everything), since demonstration must proceed from something, and have some subject matter, and prove something. Thus it follows that there is some one genus of demonstrable things; for all the demonstrative sciences employ axioms.

On the other hand, if the science of substance is distinct from the science of these principles, which is of its own nature the more authoritative and ultimate?The axioms are most universal, and are the first principles of everything. And whose province will it be, if not the philosopher's, to study truth and error with respect to them?27

(3.) And in general, is there one science of all substances, or more than one?28 if there is not one, with what sort of substance must we assume that this science is concerned?On the other hand, it is not probable that there is one science of all substances; for then there would be one demonstrative of all attributes—assuming that every demonstrative science [20] proceeds from accepted beliefs and studies the essential attributes concerned with some definite subject matter.Thus to study the essential attributes connected with the same genus is the province of the same science proceeding from the same beliefs. For the subject matter belongs to one science, and so do the axioms, whether to the same science or to a different one; hence so do the attributes, whether they are studied by these sciences themselves or by one derived from them.29

(v.) Further, is this study concerned only with substances, or with their attributes as well?30 I mean, e.g., if the solid is a kind of substance, and so too lines and planes, is it the province of the same science to investigate both these and their attributes, in every class of objects about which mathematics demonstrates anything, or of a different science?If of the same, then the science of substance too would be in some sense demonstrative; but it does not seem that there is any demonstration of the "what is it?" And if of a different science, what will be the science which studies the attributes of substance? This is a very difficult question to answer.31

(iv.) Further, are we to say that only sensible substances exist, or that others do as well? and is there really only one kind of substance, or more than one [997b] [1] (as they hold who speak of the Forms and the Intermediates, which they maintain to be the objects of the mathematical sciences)?In what sense we Platonists hold the Forms to be both causes and independent substances has been stated32 in our original discussion on this subject. But while they involve difficulty in many respects, not the least absurdity is the doctrine that there are certain entities apart from those in the sensible universe, and that these are the same as sensible things except in that the former are eternal and the latter perishable.33For Platonists say nothing more or less than that there is an absolute Man, and Horse, and Health; in which they closely resemble those who state that there are Gods, but of human form; for as the latter invented nothing more or less than eternal men, so the former simply make the Forms eternal sensibles.

Again, if anyone posits Intermediates distinct from Forms and sensible things, he will have many difficulties;because obviously not only will there be lines apart from both Ideal and sensible lines, but it will be the same with each of the other classes.34 Thus since astronomy is one of the mathematical sciences, there will have to be a heaven besides the sensible heaven, and a sun and moon, and all the other heavenly bodies.But how are we to believe this? Nor is it reasonable that the heaven should be immovable; but that it should move [20] is utterly impossible.35 It is the same with the objects of optics and the mathematical theory of harmony; these too, for the same reasons, cannot exist apart from sensible objects. Because if there are intermediate objects of sense and sensations, clearly there will also be animals intermediate between the Ideal animals and the perishable animals.36

One might also raise the question with respect to what kind of objects we are to look for these sciences. For if we are to take it that the only difference between mensuration and geometry is that the one is concerned with things which we can perceive and the other with things which we cannot, clearly there will be a science parallel to medicine (and to each of the other sciences), intermediate between Ideal medicine and the medicine which we know.Yet how is this possible? for then there would be a class of healthy things apart from those which are sensible and from the Ideally healthy. Nor, at the same time, is it true that mensuration is concerned with sensible and perishable magnitudes; for then it would perish as they do. Nor, again, can astronomy be concerned with sensible magnitudes or with this heaven of ours; [998a] [1] for as sensible lines are not like those of which the geometrician speaks (since there is nothing sensible which is straight or curved in that sense; the circle37 touches the ruler not at a point, but <along a line> as Protagoras used to say in refuting the geometricians), so the paths and orbits of our heaven are not like those which astronomy discusses, nor have the symbols of the astronomer the same nature as the stars.

Some, however, say that these so-called Intermediates between Forms and sensibles do exist: not indeed separately from the sensibles, but in them. It would take too long to consider in detail all the impossible consequences of this theory, but it will be sufficient to observe the following.On this view it is not logical that only this should be so; in clearly it would be possible for the Forms also to be in sensible things; for the same argument applies to both. Further, it follows necessarily that two solids must occupy the same space; and that the Forms cannot be immovable, being present in sensible things, which move.And in general, what is the object of assuming that Intermediates exist, but only in sensible things? The same absurdities as before will result: there will be a heaven besides the sensible one, only not apart from it, but in the same place; which is still more impossible.38

[20] Thus it is very difficult to say, not only what view we should adopt in the foregoing questions in order to arrive at the truth, but also in the case of the first principles (vi.) whether we should assume that the genera, or the simplest constituents of each particular thing, are more truly the elements and first principles of existing things. E.g., it is generally agreed that the elements and first principles of speech are those things of which, in their simplest form, all speech is composed; and not the common term "speech"; and in the case of geometrical propositions we call those the "elements"39 whose proofs are embodied in the proofs of all or most of the rest.Again, in the case of bodies, both those who hold that there are several elements and those who hold that there is one call the things of which bodies are composed and constituted first principles. E.g., Empedocles states that fire and water and the other things associated with them are the elements which are present in things and of which things are composed; he does not speak of them as genera of things.Moreover in the case of other things too, if a man wishes to examine their nature [998b] [1] he observes, e.g., of what parts a bed consists and how they are put together; and then he comprehends its nature. Thus to judge from these arguments the first principles will not be the genera of things.

But from the point of view that it is through definitions that we get to know each particular thing, and that the genera are the first principles of definitions, the genera must also be the first principles of the things defined.And if to gain scientific knowledge of things is to gain it of the species after which things are named, the genera are first principles of the species. And apparently some even of those40 who call Unity or Being or the Great and Small elements of things treat them as genera.

Nor again is it possible to speak of the first principles in both senses.The formula of substance is one; but the definition by genera will be different from that which tells us of what parts a thing is composed.

Moreover, assuming that the genera are first principles in the truest sense, are we to consider the primary genera to be first principles, or the final terms predicated of individuals? This question too involves some dispute.For if universals are always more truly first principles, clearly the answer will be "the highest genera," since these are predicated of everything. Then there will be as many first principles of things [20] as there are primary genera, and so both Unity and Being will be first principles and substances, since they are in the highest degree predicated of all things.But it is impossible for either Unity or Being to be one genus of existing things. For there must be differentiae of each genus, and each differentia must be one41; but it is impossible either for the species of the genus to be predicated of the specific differentiae, or for the genus to be predicated without its species.42 Hence if Unity or Being is a genus, there will be no differentia Being or Unity.But if they are not genera, neither will they be first principles, assuming that it is the genera that are first principles. And further, the intermediate terms, taken together with the differentiae, will be genera, down to the individuals; but in point of fact, although some are thought to be such, others are not. Moreover the differentiae are more truly principles than are the genera; and if they also are principles, we get an almost infinite number of principles, especially if one makes the ultimate genus a principle. [999a] [1]

Moreover, if Unity is really more of the nature of a principle, and the indivisible is a unity, and every thing indivisible is such either in quantity or in kind, and the indivisible in kind is prior to the divisible, and the genera are divisible into species, then it is rather the lowest predicate that will be a unity (for "man" is not the genus43 of individual men).Further, in the case of things which admit of priority and posteriority, that which is predicated of the things cannot exist apart from them. E.g., if 2 is the first number, there will be no Number apart from the species of number; and similarly there will be no Figure apart from the species of figures. But if the genera do not exist apart from the species in these cases, they will scarcely do so in others; because it is assumed that genera are most likely to exist in these cases.In individuals, however, there is no priority and posteriority. Further, where there is a question of better or worse, the better is always prior; so there will be no genus in these cases either.

From these considerations it seems that it is the terms predicated of individuals, rather than the genera, that are the first principles. But again on the other hand it is not easy to say in what sense we are to understand these to be principles;for the first principle and cause must be apart from the things of which it is a principle, and must be able to exist when separated from them. But why should we assume that such a thing exists [20] alongside of the individual, except in that it is predicated universally and of all the terms? And indeed if this is a sufficient reason, it is the more universal concepts that should rather be considered to be principles; and so the primary genera will be the principles.44

In this connection there is a difficulty which is the hardest and yet the most necessary of all to investigate, and with which our inquiry is now concerned. (7.) If nothing exists apart from individual things, and these are infinite in number, how is it possible to obtain knowledge of the numerically infinite? For we acquire our knowledge of all things only in so far as they contain something universal, some one and identical characteristic.But if this is essential, and there must be something apart from individual things, it must be the genera; either the lowest or the highest; but we have just concluded that this is impossible.45

Further, assuming that when something is predicated of matter there is in the fullest sense something apart from the concrete whole, if there is something, must it exist apart from all concrete wholes, or apart from some but not others, or apart from none? [999b] [1] If nothing exists apart from individual things, nothing will be intelligible; everything will be sensible, and there will be no knowledge of anything—unless it be maintained that sense-perception is knowledge. Nor again will anything be eternal or immovable, since sensible things are all perishable and in motion.Again, if nothing is eternal, even generation is impossible; for there must be something which becomes something, i.e. out of which something is generated, and of this series the ultimate term must be ungenerated; that is if there is any end to the series and generation cannot take place out of nothing.Further, if there is generation and motion, there must be limit too. For (a) no motion is infinite, but every one has an end; (b) that which cannot be completely generated cannot begin to be generated, and that which has been generated must be as soon as it has been generated.Further, if matter exists apart in virtue of being ungenerated, it is still more probable that the substance, i.e. that which the matter is at any given time becoming, should exist. And if neither one nor the other exists, nothing will exist at all. But if this is impossible, there must be something, the shape or form, apart from the concrete whole.

But again, if we assume this, there is a difficulty: in what cases shall we, and in what shall we not, assume it? Clearly it cannot be done in all cases; for we should not assume that a particular house exists apart from particular houses. [20] Moreover, are we to regard the essence of all things, e.g. of men, as one? This is absurd; for all things whose essence is one are one.Then is it many and diverse? This too is illogical. And besides, how does the matter become each individual one of these things, and how is the concrete whole both matter and form?46

(8.) Further, the following difficulty might be raised about the first principles. If they are one in kind, none of them will be one in number, not even the Idea of Unity or of Being. And how can there be knowledge unless there is some universal term?47On the other hand if they are numerically one, and each of the principles is one, and not, as in the case of sensible things, different in different instances (e.g. since a given syllable is always the same in kind, its first principles are always the same in kind, but only in kind, since they are essentially different in number)—if the first principles are one, not in this sense, but numerically, there will be nothing else apart from the elements; for "numerically one" and "individual" are identical in meaning. This is what we mean by "individual": the numerically one; but by "universal" we mean what is predicable of individuals. [1000a] [1] Hence just as, if the elements of language48 were limited in number, the whole of literature would be no more than those elements—that is, if there were not two nor more than two of the same <so it would be in the case of existing things and their principles>.49

(ix.) There is a difficulty, as serious as any, which has been left out of account both by present thinkers and by their predecessors: whether the first principles of perishable and imperishable things are the same or different. For if they are the same, how is it that some things are perishable and others imperishable, and for what cause?The school of Hesiod, and all the cosmologists, considered only what was convincing to themselves, and gave no consideration to us. For they make the first principles Gods or generated from Gods, and say that whatever did not taste of the nectar and ambrosia became mortal—clearly using these terms in a sense significant to themselves;but as regards the actual applications of these causes their statements are beyond our comprehension. For if it is for pleasure that the Gods partake of them, the nectar and ambrosia are in no sense causes of their existence; but if it is to support life, how can Gods who require nourishment be eternal?

However, it is not worth while to consider seriously the subtleties of mythologists; we must ascertain [20] by cross-examining those who offer demonstration of their statements why exactly things which are derived from the same principles are some of an eternal nature and some perishable. And since these thinkers state no reason for this view, and it is unreasonable that things should be so, obviously the causes and principles of things cannot be the same.Even the thinker who might be supposed to speak most consistently, Empedocles, is in the same case; for he posits Strife as a kind of principle which is the cause of destruction, but none the less Strife would seem to produce everything except the One; for everything except God50 proceeds from it.At any rate he says

From which grew all that was and is and shall be

In time to come: the trees, and men and women,

The beasts and birds and water-nurtured fish,

And the long-living Gods.51

And it is obvious even apart from this; [1000b] [1] for if there had not been Strife in things, all things would have been one, he says; for when they came together "then Strife came to stand outermost."52 Hence it follows on his theory that God, the most blessed being, is less wise than the others, since He does not know all the elements; for He has no Strife in Him, and knowledge is of like by like:

By earth (he says) we earth perceive, by water water,

By air bright air, by fire consuming fire,

Love too by love, and strife by grievous strife.53

But—and this is the point from which we started—thus much is clear: that it follows on his theory that Strife is no more the cause of destruction than it is of Being. Nor, similarly, is Love the cause of Being; for in combining things into one it destroys everything else.54Moreover, of the actual process of change he gives no explanation, except that it is so by nature:

But when Strife waxing great among the members55

Sprang up to honor as the time came round

Appointed them in turn by a mighty oath,56

as though change were a necessity; but he exhibits no cause for the necessity.However, thus much of his theory is consistent: he does not represent some things to be perishable and others imperishable, but makes everything [20] perishable except the elements. But the difficulty now being stated is why some things are perishable and others not, assuming that they are derived from the same principles.

The foregoing remarks may suffice to show that the principles cannot be the same.If however they are different, one difficulty is whether they too are to be regarded as imperishable or as perishable. For if they are perishable, it is clearly necessary that they too must be derived from something else, since everything passes upon dissolution into that from which it is derived. Hence it follows that there are other principles prior to the first principles;but this is impossible, whether the series stops or proceeds to infinity. And further, how can perishable things exist if their principles are abolished? On the other hand if the principles are imperishable, why should some imperishable principles produce perishable things, and others imperishable things? This is not reasonable; either it is impossible or it requires much explanation.Further, no one has so much as attempted to maintain different principles; they maintain the same principles for everything. [1001a] [1] But they swallow down the difficulty which we raised first57 as though they took it to be trifling.58

But the hardest question of all to investigate and also the most important with a view to the discovery of the truth, is whether after all Being and Unity are substances of existing things, and each of them is nothing else than Being and Unity respectively, or whether we should inquire what exactly Being and Unity are, there being some other nature underlying them.Some take the former, others the latter view of the nature of Being and Unity. Plato and the Pythagoreans hold that neither Being nor Unity is anything else than itself, and that this is their nature, their essence being simply Being and Unity.But the physicists, e.g. Empedocles, explain what Unity is by reducing it to something, as it were, more intelligible—or it would seem that by Love Empedocles means Unity; at any rate Love is the cause of Unity in all things. Others identify fire and others air with this Unity and Being of which things consist and from which they have been generated.Those who posit more numerous elements also hold the same view; for they too must identify Unity and Being with all the principles which they recognize. [20] And it follows that unless one assumes Unity and Being to be substance in some sense, no other universal term can be substance; for Unity and Being are the most universal of all terms,and if there is no absolute Unity or absolute Being, no other concept can well exist apart from the so-called particulars. Further, if Unity is not substance, clearly number cannot be a separate characteristic of things; for number is units, and the unit is simply a particular kind of one.

On the other hand, if there is absolute Unity and Being, their substance must be Unity and Being; for no other term is predicated universally of Unity and Being, but only these terms themselves. Again, if there is to be absolute Being and absolute Unity, it is very hard to see how there can be anything else besides these; I mean, how things can be more than one.For that which is other than what is, is not; and so by Parmenides' argument59 it must follow that all things are one, i.e. Being. [1001b] [1] In either case there is a difficulty; for whether Unity is not a substance or whether there is absolute Unity, number cannot be a substance.It has already been stated why this is so if Unity is not a substance; and if it is, there is the same difficulty as about Being. For whence, if not from the absolute One or Unity, can there be another one? It must be not-one; but all things are either one, or many of which each is one. Further, if absolute Unity is indivisible, by Zeno's axiom it will be nothing.For that which neither when added makes a thing greater nor when subtracted makes it smaller is not an existent thing, he says60; clearly assuming that what exists is spatial magnitude. And if it is a spatial magnitude it is corporeal, since the corporeal exists in all dimensions, whereas the other magnitudes, the plane or line, when added to a thing in one way will increase it, but when added in another will not; and the point or unit will not increase a thing in any way whatever.But since Zeno's view is unsound, and it is possible for a thing to be indivisible in such a way that it can be defended even against his argument (for such a thing61 when added will increase a thing in number though not in size)—still how can a magnitude be composed of one or more such indivisible things? It is like saying that the line is composed of points.Moreover, even if one supposes the case to be [20] such that number is generated, as some say, from the One itself and from something else which is not one, we must none the less inquire why and how it is that the thing generated will be at one time number and at another magnitude, if the not-one was inequality and the same principle in both cases.62 For it is not clear how magnitude can be generated either from One and this principle, or from a number and this principle.63

(13.) Out of this arises the question whether numbers, bodies, planes and points are substances or not. If not, the question of what Being is, what the substances of things are, baffles us; for modifications and motions and relations and dispositions and ratios do not seem to indicate the substance of anything; they are all predicated of a substrate, and none of them is a definite thing.As for those things which might be especially supposed to indicate substance—water, earth, fire and air, of which composite bodies are composed— [1002a] [1] their heat and cold and the like are modifications, not substances; and it is only the body which undergoes these modifications that persists as something real and a kind of substance.Again, the body is less truly substance than the plane, and the plane than the line, and the line than the unit or point; for it is by these that the body is defined, and it seems that they are possible without the body, but that the body cannot exist without them.This is why the vulgar and the earlier thinkers supposed that substance and Being are Body, and everything else the modifications of Body; and hence also that the first principles of bodies are the first principles of existing things; whereas later thinkers with a greater reputation for wisdom supposed that substance and Being are numbers.

As we have said, then, if these things are not substance, there is no substance or Being at all; for the attributes of these things surely have no right to be called existent things. On the other hand, if it be agreed that lines and points are more truly substance than bodies are, yet unless we can see to what kind of bodies they belong (for they cannot be in sensible bodies) there will still be no substance.Further, it is apparent that all these lines are divisions of Body, either in breadth [20] or in depth or in length. Moreover every kind of shape is equally present in a solid, so that if "Hermes is not in the stone,"64 neither is the half-cube in the cube as a determinate shape.Hence neither is the plane; for if any kind of plane were in it, so would that plane be which defines the half-cube. The same argument applies to the line and to the point or unit. Hence however true it may be that body is substance, if planes, lines and points are more truly substance than Body is, and these are not substance in any sense, the question of what Being is and what is the substance of things baffles us.Because, in addition to the above arguments, absurd results follow from a consideration of generation and destruction; for it seems that if substance, not having existed before, now exists, or having existed before, subsequently does not exist it suffers these changes in the process of generation and destruction. But points, lines and planes, although they exist at one time and at another do not, cannot be in process of being either generated or destroyed;for whenever bodies are joined or divided, [1002b] [1] at one time, when they are joined one surface is instantaneously produced, and at another, when they are divided, two. Thus when the bodies are combined the surface does not exist but has perished; and when they are divided, surfaces exist which did not exist before. (The indivisible point is of course never divided into two.) And if they are generated and destroyed, from what are they generated?It is very much the same with "the present moment" in time. This too cannot be generated and destroyed; but nevertheless it seems always to be different, not being a substance. And obviously it is the same with points, lines and planes, for the argument is the same; they are all similarly either limits or divisions.65

In general one might wonder why we should seek for other entities apart from sensible things and the Intermediates:66 e.g., for the Forms which we Platonists assume.If it is for the reason that the objects of mathematics, while differing from the things in our world in another respect, resemble them in being a plurality of objects similar in form, so that their principles cannot be numerically determined (just as the principles of all language in this world of ours are determinate not in number but in kind—unless one takes such and such a particular syllable [20] or sound, for the principles of these are determinate in number too—and similarly with the Intermediates, for in their case too there is an infinity of objects similar in form), then if there is not another set of objects apart from sensible and mathematical objects, such as the Forms are said to be, there will be no substance which is one both in kind and in number, nor will the principles of things be determinate in number, but in kind only.Thus if this is necessarily so, it is necessary for this reason to posit the Forms also. For even if their exponents do not articulate their theory properly, still this is what they are trying to express, and it must be that they maintain the Forms on the ground that each of them is a substance, and none of them exists by accident.On the other hand, if we are to assume that the Forms exist, and that the first principles are one in number but not in kind, we have already stated67 the impossible consequences which must follow.68

(12.) Closely connected with these questions is the problem whether the elements exist potentially or in some other sense.If in some other sense, there will be something else prior to the first principles. [1003a] [1] For the potentiality is prior to the actual cause, and the potential need not necessarily always become actual. On the other hand, if the elements exist potentially, it is possible for nothing to exist; for even that which does not yet exist is capable of existing. That which does not exist may come to be, but nothing which cannot exist comes to be.69

(xi.) Besides the foregoing problems about the first principles we must also raise the question whether they are universal or such as we describe the particulars to be. For if they are universal, there will be no substances; for no common term denotes an individual thing, but a type; and substance is an individual thing.But if the common predicate be hypostatized as an individual thing, Socrates will be several beings: himself, and Man, and Animal—that is, if each predicate denotes one particular thing.These then are the consequences if the principles are universal. If on the other hand they are not universal but like particulars, they will not be knowable; for the knowledge of everything is universal. Hence there will have to be other universally predicated principles prior to the first principles, if there is to be any knowledge of them.70

1 The principles and causes referred to in Book I.

2 The problem is discussed Aristot. Met. 3.2.1-10, and answered Aristot. Met. 4.1.

3 Discussed Aristot. Met. 3.2.10-15; answered Aristot. Met. 4.2.

4 Discussed Aristot. Met. 3.2.15-17; answered Aristot. Met. 4.2.9-10, Aristot. Met. 6.1.

5 Discussed Aristot. Met. 3.2.20-30 answered Aristot. Met. 12.6-10, and also by the refutation of the Platonic Ideas and Intermediates in Books 13 and 14.

6 Discussed Aristot. Met. 3.2.18-19; answered Aristot. Met. 4.2.8-25.

7 DiscussedAristot. Met. 3.3; answered Aristot. Met. 7.10, 12-13

8 Discussed iv. 1-8. For answers to these questions see Aristot. Met. 7.8, 13-14; Aristot. Met. 12.6-10; Aristot. Met. 13.10.

9 Discussed Aristot. Met. 3.4.8-10; answered Aristot. Met. 12.4-5, Aristot. Met. 13.10.

10 Discussed Aristot. Met. 3.4.11-23; for Aristotle's general views on the subject see Aristot. Met. 7.7-10, Aristot. Met. 12.1-7.

11 Discussed Aristot. Met. 3.4.24-34; answered Aristot. Met. 7.16.3-4, Aristot. Met. 10.2.

12 Actually Love was no more the universal substrate than was any other of Empedocles' elements; Aristotle appears to select it on account of its unifying function.

13 Heraclitus.

14 Thales.

15 Anaximenes.

16 Discussed Aristot. Met. 3.6.7-9; for the answer see Aristot. Met. 7.13-15, Aristot. Met. 13.10.

17 Discussed Aristot. Met. 3.6.5-6; for the relation of potentiality to actuality see Aristot. Met. 9.1-9; for actuality and motion see Aristot. Met. 12.6-7.

18 Discussed Aristot. Met. 3.5; answered Aristot. Met. 13.1-3, 6-9; Aristot. Met. 14.1-3, 5, 6.

19 For another statement of the problems sketched in this chapter see Aristot. Met. 9.1, 2.

20 Founder of the Cyrenaic school in the early fourth century.

21 For a defense of mathematics see Aristot. Met. 13.3.10-12.

22 Cf. Aristot. Met. 1.2.5-6.

23 See Aristot. Met. 4.1

24 sc. the science which studies the four causes.

25 Cf. Aristot. Met. 3.1.5.

26 sc. and so there can be no science which defines them.

27 For the answer see Aristot. Met. 4.3.

28 Cf. Aristot. Met. 3.1.6.

29 For the answer see Aristot. Met. 4.2.9-10, Aristot. Met. 6.1.

30 Cf. Aristot. Met. 3.1.8-10.

31 This problem, together with the appendix to it stated in Aristot. Met. 3.1.9-10, is answered in Aristot. Met. 4.2.8-25.

32 Aristot. Met. 1.6.

33 As it stands this is a gross misrepresentation; but Aristotle's objection is probably directed against the conception of Ideas existing independently of their particulars. See Introduction.

34 sc. of objects of mathematical sciences.

35 The reference is to the supposed "intermediate" heaven. A "heaven" (including heavenly bodies) without motion is unthinkable; but a non-sensible heaven can have no motion.

36 If there are "intermediate," i.e. non-sensible, sights and sounds, there must be "intermediate" faculties of sight and hearing, and "intermediate" animals to exercise these faculties; which is absurd.

37 i.e., the visible circle which we draw. Like the ruler, it is geometrically imperfect; thus they touch at more than one point.

38 The problem is dealt with partly in Aristot. Met. 12.6-10, where Aristotle describes the eternal moving principles, and partly in Books 13 and 14, where he argues against the Platonic non-sensible substances.

39 Cf. Aristot. Met. 5.3.3.

40 The Pythagoreans and Plato.

41 i.e., each differentia must have Being and Unity predicated of it.

42 The reasons are given in Aristot. Topica, 144a 36-b11.

43 sc. but the species.

44 For partial solutions to the problem see Aristot. Met. 7.10, 12-13.

45 In Aristot. Met. 3.3.

46 For answers to these questions see Aristot. Met. 7.8, 13-14; Aristot. Met. 12.6-10; Aristot. Met. 13.10.

47 If the principles are one in kind only, particular things cannot be referred to the same principle but only to like principles; i.e., there will be no universal terms, without which there can be no knowledge.

48 Or "letters of the alphabet." Cf. Aristot. Met. 1.9.36n.

49 For the answer to the problem see Aristot. Met. 12.4-5, Aristot. Met. 13.10.

50 The expressions "the One" and "God" refer to Empedocles' Sphere: the universe as ordered and united by Love. Cf. Empedocles, Fr. 26-29 (Diels).

51 Empedocles, Fr. 21. 9-12.

52 Empedocles, Fr. 36. 7.

53 Empedocles, Fr. 109.

54 Cf. Aristot. Met. 1.4.6.

55 i.e., of the Sphere.

56 Empedocles, Fr. 30.

57 i.e., whether all things have the same principles.

58 For Aristotle's views about the principles of perishable and imperishable things see Aristot. Met. 7.7-10, Aristot. Met. 12.1-7.

59 By τὸ ὄν Parmenides meant "what is," i.e. the real universe, which he proved to be one thing because anything else must be "what is not," or non-existent. The Platonists meant by it "being" in the abstract. Aristotle ignores this distinction.

60 Cf. Zeno, Fr. 2, and see Burnet, E.G.P. sects. 157 ff.

61 e.g., a point is indivisible and has no magnitude, yet added to other points it increases their number.

62 The reference is to the Platonists. Cf. Aristot. Met. 14.1.5, 6; Aristot. Met. 14.2.13, 14.

63 For the answer to this problem see Aristot. Met. 7.16.3, 4; Aristot. Met. 10.2; and cf. Aristot. Met. 13.8.

64 Apparently a proverbial expression.

65 For arguments against the substantiality of numbers and mathematical objects see Aristot. Met. 13.1-3, 6-9; Aristot. Met. 14.1-3, 5, 6.

66 Cf. Aristot. Met. 3.2.20ff..

67 Aristot. Met. 3.4.9, 10.

68 This problem is not stated in ch. 1., but is akin to problems 5. and 8., which see.

69 For the relation of potentiality to actuality see Aristot. Met. 9.1-9. The second point raised in this connection in ch. 1 is not discussed here; for actuality and motion see Aristot. Met. 12.6, 7.

70 For the answer to this problem see Aristot. Met. 7.13-15, Aristot. Met. 13.10.