Aristotle · a new plain-English translation from the original language
1| We have said what the substance of sensible things is — in the treatise on natural science, concerning matter, and later concerning substance in accordance with actuality. But since the inquiry is whether there is, besides the sensible substances, some substance that is unmoved and eternal, or whether there is not, and if there is, what it is, we must first examine what others say, so that, if they say anything not well, we may not be liable to the same faults, and if there is some doctrine common to us and to them, we should not for that reason be privately annoyed at ourselves; for it is something to be content with, if one states some things better and others no worse. There are two opinions about these matters: some say that the mathematical objects — numbers and lines and things akin to these — are substances, and again there are the forms. Since some make these two classes, the forms and the mathematical numbers, while others make a single nature of both, and certain others say that only the mathematical objects are substances, we must examine first the mathematical objects, adding to them no other nature — for instance, not asking whether they happen to be forms or not, and whether they are principles and substances of the things that are or not, but treating them only as mathematical objects, asking whether they exist or not, and if they exist, how they exist.
2| Then after this, separately, we must examine the forms themselves, simply and only so far as customary usage requires; for much has already been said repeatedly even in the popular discussions, and besides, the greater part of the discussion must bear on that other inquiry, when we examine whether the substances and principles of the things that are, are numbers and forms; for after the forms this is left as a third inquiry. Now it is necessary, if the mathematical objects exist, that they exist either in sensible things, as some say, or separated from sensible things (some say this too); or if neither, then either they do not exist, or they exist in some other way. So our dispute will not be about whether they exist, but about the manner of their existence. That it is impossible for them to exist within sensible things, and that the account given for this view is at the same time a fabrication, has been said in the difficulties, namely that it is impossible for two solids to be in the same place at once,
3| [1076b] and further that by the same argument the other powers and natures too would have to be in sensible things, and none separated. This has indeed been said before; but besides these points it is clear that it is impossible for any body whatever to be divided in this way: for it will be divided along a plane, and this along a line, and this along a point; so that if it is impossible to divide the point, then it is impossible to divide the line too, and if this, the other things too. What difference, then, does it make whether these are natures of this kind themselves, or whether they themselves are not such, but natures of this kind exist within them? The same result will follow: for when the sensible things are divided, these will be divided too, or else not even the sensible things will be divisible. But surely it is not possible either for natures of this kind to exist as separated. For if there are to be solids apart from the sensible solids, separated from these and prior to the sensible ones, clearly there must also be, apart from the planes of these, other planes, separated, and points and lines (for the argument is the same); and if these, then again, apart from the planes and lines and points of the mathematical solid, there must be other, separated ones (for the incomposite is prior to the composite: and if bodies not sensible are prior to sensible ones, by the same argument the planes that belong to the unmoved solids are, in their own right, prior to those that belong to the moved ones), so that these would be different planes and lines from those that exist together with the separated solids —
4| for some exist together with the mathematical solids, and others are prior to the mathematical solids. Again, then, there will be lines belonging to these planes, and prior to these there will have to be, by the same argument, other lines and points; and of these — the ones belonging to the prior lines — there will have to be other, prior points, than which there are no further prior ones. Now the accumulation becomes absurd: for it turns out that there is only a single class of solids apart from the sensible ones, but three classes of planes apart from the sensible ones — those apart from the sensible ones, and those in the mathematical solids, and those apart from the ones in these — and four classes of lines, and five of points; so which of these will the mathematical sciences be about? For surely not about the planes and lines and points in the unmoved solid — for a science is always about the prior things. And the same argument applies to numbers too: for apart from each set of points there will be different units, and apart from each class of the things that are — the sensible things, then the intelligible ones — so that there will be classes of mathematical numbers. Further, how is it possible to resolve the very difficulties we went over among the puzzles?
5| [1077a] For the objects astronomy studies will likewise exist apart from the sensible things, and so will the objects geometry studies; but how is it possible for there to be a heaven and its parts, or anything else that has motion? And likewise too the objects of optics and of harmonics: for there will be voice and sight apart from the sensible things and the particulars, so that it is clear that the other senses too, and the other objects of sense, will exist apart as well — for why these rather than those? And if this, there will be animals too, if indeed there are perceptions. Further, some universal propositions are written down by the mathematicians applying beyond these substances. There will, then, be some other substance too, an intermediate one, separated both from the forms and from the intermediates, which is neither number nor points nor magnitude nor time. But if this is impossible, clearly it is also impossible for those to be separated from sensible things. And in general the opposite of both the truth and the customary supposition results, if one posits the mathematical objects to exist in this way, as certain separated natures. For it is necessary, because they exist in this way, that they be prior to sensible magnitudes, but according to the truth, posterior: for the incomplete magnitude is prior in generation, but posterior in substance — as the soulless is prior to the ensouled.
6| Further, by what, and when, will mathematical magnitudes be one? For things here are reasonably one by soul, or a part of soul, or something else (otherwise they are many, and dissolve); but for those, being divisible and quantities, what is the cause of their being one and holding together? Further, the processes of generation make this clear. For first a thing comes to be in length, then in breadth, and finally into depth, and then it has reached its end. If, then, what is posterior in generation is prior in substance, body would be prior to plane and to length; and in this respect it is also more complete and more a whole, because it becomes ensouled: but how could a line or a plane be ensouled? The claim would be beyond our powers of perception. Further, body is a kind of substance (for it already has, in some way, completeness), but how are lines substances? Neither as a form, some shape — as, say, the soul might be something of this kind — nor as matter, like body: for nothing appears able to be composed out of lines, or planes, or points, whereas if it were a kind of material substance, this would appear to be able to happen. In definition, then, let them be prior,
7| [1077b] But not everything that is prior in account is also prior in substance. For prior in substance are those things which, when separated, exceed others in being; prior in account are those whose accounts are composed out of other accounts — and these two do not hold together. For if the attributes do not exist apart from the substances (for example something moving, or something white), then the white is prior in account to the white man, but not prior in substance; for it cannot exist separated, but always exists together with the concrete whole (and by concrete whole I mean the white man), so that it is clear that neither what comes from abstraction is prior, nor what comes from addition is posterior; for the white man is spoken of by addition to the white. That, then, mathematical objects are neither substances more than bodies are, nor prior in being to perceptible things but prior only in account, and that it is not possible for them to exist separated anywhere, has been said sufficiently. But since it was not possible for them to exist within perceptible things either, it is clear that either they do not exist at all, or they exist in a certain way and for this reason do not exist without qualification; for we speak of being in many ways. For just as the universals in mathematics too are not about things separated apart from magnitudes and numbers, but are about these — not, however, insofar as they are of such a kind as to have magnitude or be divisible — clearly it is also possible for there to be accounts and demonstrations about perceptible magnitudes, not insofar as they are perceptible but insofar as they are of a certain kind.
8| For just as, insofar as things are only moving, there are many accounts of them apart from what each of them is and apart from the accidents belonging to them, and it is not necessary on this account either that something moving exist separated from perceptible things, or that some determinate nature exist within these — so too in the case of moving things there will be accounts and sciences of them, not insofar as they are moving but insofar as they are only bodies, and again insofar as they are only surfaces and insofar as they are only lengths, and insofar as they are divisible and insofar as they are indivisible but possess position, and insofar as they are only indivisible. So that since it is true to say without qualification not only that separable things exist but also that things which are not separable exist (for example, that moving things exist), it is also true without qualification to say that mathematical objects exist, and that they are of just the sort mathematicians say. And just as it is true to say of the other sciences too, without qualification, that each is a science of this — not of the accidental (for example, of the white, if the healthy happens to be white, and the science is of the healthy)
9| [1078a] but of that of which the science strictly is — if of the healthy insofar as healthy, if of man insofar as man — so too with geometry. If it happens that the things it is a science of are perceptible, though it is not a science of them insofar as they are perceptible, its mathematical sciences will not on that account be sciences of perceptible things — nor, however, will they be sciences of other things separated apart from these. Many attributes belong in their own right to things insofar as each is a certain kind of thing: since even insofar as an animal is female and insofar as it is male there are attributes peculiar to it (and yet there is no female or male existing separated among animals), so too there are attributes belonging to things insofar as they are only lengths and insofar as they are only surfaces. And the more a subject is prior in account and simpler, the more exactness a science dealing with it has (for this simplicity just is exactness), so that a science is more exact without magnitude than with magnitude, and most exact of all without motion; but if it does involve motion, most of all with the primary motion, for this is the simplest, and of this motion the uniform is simplest again. The same account holds also for harmonics and optics; for neither of these studies its subject insofar as it is sight or insofar as it is sound, but insofar as sight and sound involve lines and numbers (these being, however, attributes proper to sight and sound), and mechanics likewise. So that if someone, positing things as separated from their accidents, examines something about them insofar as they are of that kind, he will not for this reason be asserting anything false — just as when someone draws a line on the ground and calls a line a foot long when it is not a foot long,
10| for the falsehood is not in the premises. Each thing would be best studied in this way: if one posits as separated what is not separated, which is exactly what the arithmetician and the geometer do. For man, insofar as he is man, is one and indivisible; and the arithmetician posited him as one indivisible thing, and then examined whether anything belongs to man insofar as he is indivisible. But the geometer considers him neither insofar as he is man nor insofar as he is indivisible, but insofar as he is a solid. For whatever would have belonged to him even if he had not, somehow, been indivisible, clearly can belong to him also apart from these — so that for this reason geometers speak correctly, and discuss things that are, and what they discuss are beings; for being is spoken of in two ways, the one in actuality, the other materially. Since the good and the fine are different (for the good is always found in action, while the fine is found also in unmoved things), those who assert that the mathematical sciences say nothing about the fine or the good are mistaken. For they do speak of these, and demonstrate them most of all: it is not the case that, because they do not name them, but exhibit their works and their ratios, they are not speaking of them. And the greatest forms of the fine are order and symmetry and the determinate,
11| [1078b] which the mathematical sciences exhibit most of all. And since these appear to be causes of many things — I mean, for example, order and the determinate — it is clear that they would also acknowledge this sort of cause, the one whereby the fine is in some way a cause. But we shall speak more intelligibly about these matters elsewhere. Concerning mathematical objects, then, let this much be said: that they are beings, and how they are beings, and how they are prior and how they are not prior. Concerning the forms, we must first examine the doctrine of the form itself, in its own right, connecting it in no way with the nature of numbers, but taking it as those who first said that forms exist originally supposed it. Now the doctrine about the forms occurred to those who stated it because they had been persuaded of the truth of the Heraclitean arguments, that all perceptible things are always flowing, so that if there is to be any knowledge or practical wisdom of anything, there must be some other, abiding natures besides the perceptible ones; for there is no knowledge of things that are flowing. But Socrates was occupied with the ethical virtues, and was the first to seek to define these universally — for among the natural philosophers only Democritus touched on this to a small degree, and in some way defined the hot and the cold,
12| while the Pythagoreans before him had dealt with a few things only, whose accounts they attached to numbers — for example what opportunity is, or the just, or marriage. Socrates, reasonably enough, was seeking the what-it-is; for he was seeking to reason by syllogism, and the starting point of syllogisms is the what-it-is. For the strength of dialectic did not yet exist at that time, such that one could, even apart from the what-it-is, investigate contraries, and ask whether the same science applies to contraries. For there are two things one might justly credit to Socrates: inductive arguments and universal definition; for both of these concern the starting point of scientific knowledge. But Socrates did not make the universals separate, nor the definitions; his successors, however, did separate them, and gave this sort of being the name forms, so that it followed for them, by pretty much the same argument, that there are forms of everything that is spoken of universally — much as if someone wanting to count things thought he could not count them while they were fewer, but could count them once he had made them more; for the forms are, one might say, more numerous than the particular perceptible things,
13| [1079a] in seeking the causes of which they proceeded from these to those. For in the case of each particular thing there is something homonymous with it existing apart from the substances, and likewise for the other things too, there is one thing over many, both among things here and among eternal things. Further, by none of the ways by which it is shown that the forms exist does this appear to follow: for from some of them no syllogism need result, and from others forms result even of things of which they do not think there are forms. For according to the arguments from the sciences, there will be forms of everything of which there are sciences; and according to the one-over-many argument, there will be forms even of negations; and according to the argument from thinking of something once it has perished, there will be forms of perishable things, for there is some image of these. Further, the most exact of the arguments either produce forms of relatives, of which they say there is no genus in its own right, or they state the third-man argument. And in general the arguments about the forms do away with things which those who speak of forms want to exist even more than they want the forms themselves to exist; for it follows that the dyad is not first, but number is, and that the relative is prior to number, and that which is in its own right is prior to the relative — and so too in all the other respects in which certain people, by following the opinions about the forms, ran counter to the first principles.
14| Further, on the assumption by which they say the Ideas exist, there will be forms not only of substances but of many other things as well (for the concept is one not only in the case of substances but also in the case of non-substances, and there are sciences not only of substance; and countless other such consequences follow). But by what is necessary, and by the opinions held about them, if the Forms are participated in, there must be Ideas of substances only; for they are not participated in accidentally, but each thing must participate in a Form in that respect in which the Form is not said of a subject (I mean, for example, if something participates in the double itself, this also participates in the eternal, but accidentally; for it is accidental to the double to be eternal), so that the Forms will be substance. And the same things signify substance here as there; or else what would it mean to say that there is something besides these particulars, the one over the many? And if the form of the Ideas and of the things that participate in them is the same, there will be something common to both (for why should the dyad be one and the same in the case of perishable dyads and of the many but eternal dyads, any more than in the case of the dyad itself and a particular dyad?); but if the form is not the same,
15| [1079b] they would be homonymous, and it would be like calling both Callias and a piece of wood "man," without noting any community between them at all. But if we posit that the common definitions apply to the Forms in other respects — for example, that "plane figure" and the rest of the parts of the definition apply to the circle itself — while "what it is" will be added on besides, we must consider whether this addition is not entirely empty. For to what will it be added? To the middle term, or to "plane," or to the whole of it? For all the elements in the substance are themselves Ideas — for example, animal and two-footed. Further, it is clear that this added element must itself be something, like "plane," some nature that will be present in all the Forms as a genus. But above all one might raise the difficulty: what do the Forms ever contribute, either to the eternal perceptible things or to the things that come to be and perish? For they are the cause of no motion and of no change at all in these things. Moreover they are of no help toward the knowledge of the other things either (for they are not even the substance of those things — otherwise they would be present in them), nor toward their being, at least not if they are not present in the things that participate in them. For if they were present, they might perhaps seem to be causes in the way that white mixed into a white thing is a cause; but this argument is very easily overturned — the one that Anaxagoras used first and Eudoxus later, in raising the difficulty, and certain others too (for it is easy to muster up many impossibilities against a view of that kind).
16| Moreover, the other things do not come from the Forms in any of the senses in which "from" is customarily used. And to say that the Forms are patterns and that the other things participate in them is to speak empty words and to utter poetic metaphors. For what is it that works while looking to the Ideas? For it is possible for anything whatever both to be and to come to be without being modeled on anything, so that whether Socrates exists or not, someone like Socrates could come to be; and it is likewise clear that this would hold even if Socrates were eternal. And there will be several patterns of the same thing, and so several Forms — for example, of man, animal and two-footed, and at the same time man-itself as well. Further, the Forms will be patterns not only of perceptible things but of the Forms themselves too — for example, the genus, being the genus of species that are themselves forms, will be a pattern of those species; so that the same thing will be both pattern and image. Further, it would seem impossible for the substance and that of which it is the substance to exist apart:
17| [1080a] so how could the Ideas, being the substances of things, exist apart from them? Now in the Phaedo it is said in this way, that the Forms are causes both of being and of coming to be; and yet, even though the Forms exist, a thing still does not come to be unless there is something to cause the motion, and many other things come to be, such as a house and a ring, of which they say there are no Forms; so it is clear that it is possible for those very things too, of which they say there are Ideas, both to exist and to come to be through causes of the same kind as those just mentioned, and not through the Forms. But about the Ideas, in this way and also by more logical and more precise arguments, one can gather together many points similar to those already examined. Since these matters have been determined, it is well to examine in turn what happens with regard to numbers for those who say that numbers are separate substances and the primary causes of the things that are. Now if number is a certain nature, and its substance is nothing else but this very thing, as some say, then of necessity either one part of it is first and another next to it, each part differing in kind, and this holds directly of the units themselves — so that any unit whatever is incomparable with any other unit — or else all the units are directly in succession and comparable with one another, as they say mathematical number is (for in the mathematical case no unit differs at all from another):
18| or else some units are comparable and others are not (for example, if after the one the dyad comes first, then the triad, and so on for the rest of number, and the units within each number are comparable with one another — those in the first dyad with one another, and those in the first triad with one another, and so on for the other numbers — but the units in the dyad itself are incomparable with those in the triad itself, and likewise for the other numbers in succession; hence mathematical number is counted, after the one, as two, another one added to the previous one, and three as another one added to these two, and so on for the rest; but this other number, after the one, is two further ones without the first one, and the triad is without the dyad, and likewise for the rest of number); or else one kind of number is such as was first described, another such as the mathematicians speak of, and a third is the kind last mentioned. Further, these numbers must either be separate from things,
19| [1080b] or not separate but present in perceptible things (not in the way we first examined, but as though perceptible things were composed out of numbers present within them), or else one kind of them exists and the other does not, or all kinds exist. These, then, are of necessity the only ways in which it is possible for them to exist, and pretty much all who say that the one is a principle and substance and element of all things, and that number arises from this together with something else, have each stated one or another of these ways, except for the view that all units are incomparable. And this has happened reasonably; for it is not possible for there to be still another way besides those stated. Some, then, say that both kinds of number exist, the one having priority and posteriority being the Ideas, and the mathematical being distinct from both the Ideas and perceptible things, both kinds separate from perceptible things; others say that mathematical number alone exists, as the first of the things that are, separate from perceptible things. And the Pythagoreans too posit one kind, the mathematical, except that they do not hold it separate but say that perceptible substances are composed out of it; for they construct the whole heaven out of numbers — not, however, numbers composed of units, but they suppose the units to have magnitude:
20| but as to how the first one, having magnitude, was composed, they appear to be at a loss. Someone else says that the first number, that of the Forms, is one and the same, while some say that mathematical number is this very same thing too. And likewise concerning lengths, planes, and solids. For some make the mathematical objects different from those that come after the Ideas; and of those who speak otherwise, some speak of the mathematical objects and speak of them in a properly mathematical way — namely those who do not make the Ideas numbers, nor say that Ideas exist at all — while others speak of the mathematical objects, but not in a properly mathematical way; for they say that not every magnitude is divisible into magnitudes, nor do just any two units make a dyad. But all posit that numbers are composed of units, except the Pythagoreans, who say that the one is an element and principle of the things that are; but they hold the units to have magnitude, as has been said before. In how many ways, then, it is possible to speak about these matters, and that all the ways stated have now been enumerated, is clear from this; and all of them are impossible, though perhaps some more so than others. First, then, we must examine whether the units are comparable or incomparable, and if incomparable, in which of the ways we distinguished.
21| [1081a] For it is possible for any unit whatever to be incomparable with any other unit whatever, or for the units in the dyad itself to be incomparable with those in the triad itself, and thus for the units in each of the primary numbers to be incomparable with one another. Now if all the units are comparable and undifferentiated, the number that results is the mathematical number, and only one such number, and the forms cannot be numbers (for what number will man-itself be, or animal-itself, or any other of the forms? For there is one form of each thing—one form of man-itself, and another single form of animal-itself; but numbers that are similar and undifferentiated are unlimited in number, so that this triad is no more man-itself than any other triad whatever), but if the forms are not numbers, then it is not possible for them to exist at all (for out of what principles will the forms come? Number is out of the one and the indefinite dyad, and the principles and elements are said to be principles and elements of number, and it is not possible to rank the forms either as prior to numbers or as posterior to them). But if the units are incomparable, and incomparable in such a way that any one is incomparable with any other, then this cannot be the mathematical number (for the mathematical number is composed of undifferentiated units, and what is demonstrated of it fits it as being of that kind), nor can it be the number of the forms.
22| For the first dyad will not be from the one and the indefinite dyad, with the successive numbers following after it as they are called—dyad, triad, tetrad—since the units in the first dyad come into being together, whether, as the first person to say so held, they come from unequals (for they came to be when the unequals were equalized), or in some other way—since, if one unit is to be prior to the other, it will also be prior to the dyad composed of them: for whenever one thing is prior and another posterior, then what is composed of them will also be prior in respect of the one and posterior in respect of the other. Further, since there is first the one itself, and then among the others there is some one that is first, and a second after that, and again a third that is second after the second one but third after the first one, it follows that the units would be prior to the numbers they are said to belong to: for example, in the dyad there will be a third unit before three exists, and in the triad a fourth and a fifth unit before those numbers exist. Now no one has said in so many words that their units are incomparable, but on their principles it is reasonable that it should turn out so, though in point of truth it is impossible.
23| [1081b] For it is reasonable that the units should be prior and posterior, given that there is a certain first unit and a first one, and likewise for dyads, given that there is a first dyad: for after the first it is reasonable and necessary that there be a second, and if a second, a third, and so on with the rest in sequence (but to assert both at once—that the unit after the one is first and also second, and that there is a first dyad—is impossible). But they make a first unit, a first one, yet no second or third beyond it, and a first dyad, yet no second or third beyond it. It is also clear that, if all units are incomparable, it is not possible for there to be a dyad-itself and a triad-itself, and likewise the other numbers. For whether the units are undifferentiated or each differs from each, the number must be counted by addition—the dyad by adding one further unit to the one, the triad by adding another one to the two, and the tetrad in the same way: but if this is so, it is impossible for the generation of the numbers to be as they generate them, out of the dyad and the one. For on this view the dyad becomes a part of the triad, and the triad a part of the tetrad, and the same thing happens with the numbers that follow.
24| But they say the tetrad came to be out of the first dyad and the indefinite dyad, as two dyads distinct from the dyad itself: otherwise, the dyad itself would be a part of the tetrad, and a second, different dyad would be added to it. And the dyad in turn will be composed of the one itself and another one; but if so, the other element cannot be the indefinite dyad, since it generates a single unit, not a determinate dyad. Further, besides the triad-itself and the dyad-itself, how will there be other triads and dyads? And in what way are they composed of prior and posterior units? All this is absurd and fabricated, and it is impossible for there to be a first dyad, and then a triad-itself. Yet it is necessary, given that the one and the indefinite dyad are to be the elements. But if the consequences are impossible, then it is also impossible for these to be the principles. If, then, the units differ, any one from any other, these and other such consequences necessarily follow: but if the units in one number differ from those in another, while the units within the same number alone are undifferentiated from one another, even so the difficulties that arise are no fewer.
25| [1082a] For example, in the decad-itself there are ten units present, and the decad is composed both of these units and of two pentads. But since the decad-itself is not just any random number, nor is it composed of random pentads, any more than of random units, the units in this decad must differ from those in other decads. For if they did not differ, neither would the pentads of which the decad consists differ; but since they do differ, the units will differ too. But if they differ, will there be no other pentads present in it besides these two, or will there be? If none will be present, that is absurd; but if some will be present, what sort of decad will result from them? For there is no other decad within the decad besides itself. But it is also necessary that the tetrad not be composed of random dyads: for the indefinite dyad, as they say, took the determinate dyad and made two dyads, since it was capable of doubling whatever it took. Further, how is it possible for the dyad to be some nature over and above the two units, and the triad over and above the three units? Either one will partake of the other, as a white man is something over and above white and man (for he partakes of these), or it will be as when one thing is a differentia of another, as man is something over and above animal and biped.
26| Further, some things are one by contact, others by mixture, others by position: none of these can belong to the units of which the dyad and the triad are composed: but just as two men are not some one thing over and above both of them, so it must be with the units too. And it is not because they are indivisible that a further unity will arise from them on that account: points too are indivisible, but all the same the dyad formed of two points is nothing other than the two of them. But this too must not escape notice: that it follows that there are prior and posterior dyads, and likewise the other numbers. For let the dyads in the tetrad be simultaneous with one another; but these are prior to those in the octad, and just as the first dyad generated these, so these generated the tetrads in the octad itself, so that if the first dyad is a form, these too will be forms of a sort. The same account applies to the units as well: for the units in the first dyad generate the four units in the tetrad, so that all the units become forms, and a form will be composed of forms: so that it is clear that the very things of which these happen to be the forms will also be composite—as if one were to say that animals are composed of animals, if there are forms of these.
27| [1082b] —And in general, to make the units differ in any way whatever is absurd and fictitious (I call fictitious whatever is forced to fit a hypothesis): for we observe no unit differing from another either in quantity or in quality, and a number must be either equal or unequal — every number, but above all a number made of units — so that if it is neither more nor less, it is equal; and things that are equal and wholly undifferentiated we take to be the same, in the case of numbers. But if this is not so, then not even the twos within the decad itself will be undifferentiated, though they are equal: for what cause will the person who claims they are undifferentiated be able to give? Further, if every unit together with another unit makes two, then the unit from the dyad-itself and the dyad from the triad-itself will be made of differing units — and is the latter prior to the triad or posterior to it? For it seems more necessary that it be prior: since one of its units coexists with the triad, the other with the dyad. And we ourselves suppose, quite generally, that one and one, whether they are equal or unequal, make two — for example, the good and the bad, or a man and a horse; but those who speak this way do not allow it even for units. And if the number belonging to the triad-itself turns out not to be greater than that belonging to the dyad-itself, that is astonishing —
28| but if it is greater, clearly there is also present in it a number equal to the dyad, so that this equal part is undifferentiated from the dyad itself. But that is not possible, if there is a first number and a second. Nor, again, will the Forms be numbers. For this very point is made correctly by those who insist that the units differ, if indeed there are to be Forms, as has been said before: for the form is one, but if the units are undifferentiated, then the twos and the threes will also be undifferentiated. That is why they are forced to say that counting proceeds like this — one, two — without anything being added to what already exists (for on their view the generation of number cannot come from the indefinite dyad, nor can there be a Form of it: for one Form would then exist within another, and all the forms would be parts of one). So with respect to their own hypothesis they speak correctly, but not correctly without qualification: for they destroy many things, since they themselves will admit that this very point raises a difficulty — whether, when we count and say one, two, three, we count by adding, or by taking portions. We do both; that is why it is absurd to trace this difference back to so great a difference in substance.
29| [1083a] First of all, it is well to determine what the differentia of number is, and of the unit, if it has one. Now it must differ either in quantity or in quality; but neither of these appears possible for it to have. But insofar as it is a number, it differs in quantity. If, then, the units also differed in quantity, one number would differ from another number equal to it in the plurality of its units. Further, are the first units greater or smaller, and do the later ones increase, or the opposite? All this is irrational. But nor can they differ in quality either. For no affection can belong to them: indeed they themselves say that quality belongs to numbers only later than quantity. Further, this could not come to the units either from the one or from the dyad: for the one is not qualitative, while the dyad is quantity-making, since this nature is the cause of the things that are being many. If, then, it is somehow otherwise, this above all needs to be stated at the outset, and the differentia of the unit needs to be determined — above all, why it must exist; and if it does not, what differentia do they mean? It is clear, then, that if the Forms are numbers, it is not possible for all the units to be comparable with one another, nor is it possible for them to be incomparable with one another — neither of the two ways will do:
30| but nor, again, is what certain others say about numbers said correctly. These are those who think there are no Forms, either without qualification or as being certain numbers, but hold that the objects of mathematics exist, and that numbers are the first of the things that are, and that the one itself is their principle. For it is absurd that the one should be something first among ones, as they say, while a dyad is not first among dyads, nor a triad among triads; for all these belong to the same account. If, then, this is how matters stand with number, and one posits that only the mathematical number exists, the one is not a principle (for this particular one must differ from the other units; and if so, there must also be some dyad that is first among dyads, and likewise the other numbers in succession). But if the one is a principle, then matters must rather stand, regarding numbers, as Plato used to say — there must be a first dyad and a first triad, and the numbers must not be comparable with one another. But if, on the other hand, one posits this, it has been said that many impossibilities follow. Yet it is necessary that things stand either the one way or the other, so that if neither, it would not be possible for number to be separate.
31| [1083b] —It is clear from this, too, that the third way of speaking is the worst: that the number of the Forms is the same as the mathematical number. For on this single opinion two errors are bound to fall together: this cannot be the mathematical number in that way, but one who posits it must, by laying down hypotheses of his own, draw things out at length; and he must also state everything that follows for those who speak of number as the Forms. The Pythagorean way has fewer difficulties than those already mentioned, but has peculiar difficulties of its own. For not making number separate removes many of the impossibilities; but that bodies should be composed of numbers, and that this number should be the mathematical one, is impossible. For it is not true to speak of indivisible magnitudes; and even granting that this holds in the fullest sense, the units at least do not have magnitude — and how can magnitude possibly be composed of indivisibles? But the number of arithmetic, at any rate, is made of units. Yet the Pythagoreans say that number is the things that are: at any rate they attach their theorems to bodies as though the numbers were what those bodies are made of. If, then, it is necessary — supposing that number is something that exists in its own right among the things that are — that it be one of the ways stated, and none of these is possible, it is clear that number has no such nature as those construct who make it separate.
32| Further, is each unit composed of the great and the small, equalized, or is one unit from the small and another from the great? If the latter, then each unit is neither composed of all the elements, nor are the units undifferentiated (for the great is present in the one, the small in the other, which is contrary in nature); further, how do matters stand for the units within the triad itself? For one of them is odd — but perhaps it is for this reason that they make the one itself the middle term within the odd. But if each of the units is from both, equalized, how will the dyad, being some single nature, be composed of the great and the small? Or how will it differ from the unit? Further, the unit is prior to the dyad (for when the unit is destroyed, the dyad is destroyed): so the unit must be a Form of a Form, since it is prior to a Form, and must have come to be prior. From what, then, did it come to be? For the indefinite dyad was what made things two. Further, number must necessarily be either infinite or finite: for they make number separate, so that it is not possible for neither of these to hold.
33| [1084a] That number cannot be infinite, then, is clear (for the infinite is neither odd nor even, but the generation of number is always the generation either of an odd or of an even number: in one way, when the one falls upon what is even, an odd number results; in another way, when the dyad is imposed, the number doubled from the one results; and in another way, from odd numbers a different even number results. Further, if every form is the form of something, and numbers are forms, then the infinite too will be the form of something, either of perceptible things or of something else; yet this is possible neither on their own positing nor by their own reasoning, given the way they arrange the forms) — but if number is finite, up to what point? For this must be stated not only as a fact but also with the reason why. But if number goes only up to the decad, as some say, then in the first place the forms will quickly give out — for example, if the triad is man-itself, what number will be horse-itself? For each number up to the decad is itself the form of something; so necessarily some among these numbers must be horse-itself (for these numbers are substances and forms); but still the forms will give out, for the species of animal will exceed them. At the same time it is clear that if the triad is man-itself in this way, so are the other triads, since triads within the same numbers are alike; so there will be infinitely many men — each man himself, if each triad is a form, or at least all men, even if not.
34| And if the smaller number is a part of the greater when it is composed of units commensurable with those in the same number, then, if the tetrad itself is the form of something, say of horse or of white, man will be a part of horse, since man is the dyad. It is also absurd that there should be a form of the decad but none of the hendecad, nor of the numbers that follow it. Further, some things both exist and come to be of which there are no forms; so why are there no forms of these as well? The forms, then, are not the causes. Further, it is absurd if the number up to the decad has more being and is more truly a form than the decad itself, and yet there is no generation of the series as a single thing, while there is of the decad. But they proceed as though the number up to the decad were the complete number. At any rate they generate the derivative items — the void, proportion, the odd, and other things of this kind — within the decad; for some things they assign to the first principles, such as motion and rest, good and bad, and the rest they assign to the numbers. That is why the one is the odd; but if oddness lies in the triad, how is the pentad odd? Further, magnitudes and things of that sort go up to a certain amount —
35| [1084b] for example the first line, the indivisible one, then the dyad, and these too go up to the decad. Further, if number is separate, one might raise the puzzle whether the one is prior, or the triad and the dyad are. Insofar as number is composite, the one is prior; but insofar as the universal and the form are prior, number is prior; for each of the units is a part of number as its matter, while number stands to them as form. And in one sense the right angle is prior to the acute, because it is determinate and definable by an account; but in another sense the acute is prior, because it is a part, and the right angle is divided into it. As matter, then, the acute angle, the element, and the unit are prior; but as form, and as the substance defined by the account, the right angle is prior, and so is the whole composed of matter and form, for what is composed of both is nearer to the form and to that of which there is an account, though it is posterior in generation. How, then, is the one a principle? Because it is not divisible, they say; but the universal too is indivisible, and so is the particular thing, and so is the element — only in a different way, the one indivisible in account, the other in time. In which of these ways, then, is the one a principle?
36| For as has been said, the right angle seems prior to the acute, and the acute seems prior to the right, and each is one. In both ways, then, they make the one a principle. But that is impossible; for in one sense the one is a principle as form and as substance, and in the other sense as part and as matter. Each is in some way one — potentially, in truth, if indeed number is some single thing and not like a heap, but rather distinct in kind from other units, as they say — but not in actuality; each of the two is a unit. The cause of the error that results is that they were hunting at once from mathematics and from arguments about universals, so that from the former they laid down the one and the principle as a point (for the unit is a positionless point; just as certain others too composed the things that are out of the smallest thing, so these men did as well, so that the unit becomes the matter of numbers, and at once prior to the dyad, yet again posterior, inasmuch as the dyad is a kind of whole, a unity, and a form); while because they were seeking the universal, they called what is predicated 'one,' and spoke of it in this way as a part. But it is impossible for both of these to belong to the same thing at once. But if the one itself must be merely positionless (for it differs in nothing except that it is a principle), and the dyad is divisible while the unit is not, the unit would be more like the one itself.
37| But if the unit is more like the one, then the one itself is more like the unit than like the dyad; so that in either case the unit would be prior to the dyad. But they do not say this; at any rate they generate the dyad first. [1085a] Further, if the dyad is itself some one thing, and the triad is likewise itself one thing, then both together are a dyad. From what, then, is this dyad composed? One might also raise the puzzle, since there is no contact among numbers, but there is succession — belonging to those between which no units lie, as with the numbers within the dyad or the triad — whether the dyad is in succession to the one itself or not, and whether the dyad is prior to the numbers in succession, or to whichever of its own units. The same difficulties arise, in like manner, concerning the kinds that come after number — line, plane, and body. For some construct these out of the species of the great and the small: lengths out of the long and the short, planes out of the broad and the narrow, solids out of the deep and the shallow; and these are species of the great and the small. As for the principle corresponding to the one, different thinkers posit it differently for these kinds. And in these matters too countless impossibilities appear, and fictions, and things contrary to everything reasonable. For it turns out that these attributes are disconnected from one another, unless the principles also follow along together, so that the broad-and-narrow is also the long-and-short (and if this is so, the plane will be a line, and the solid a plane —
38| further, how will angles and figures and things of that kind be accounted for?) — and the same thing happens with the numbers as well; for these are attributes of magnitude, but magnitude is not composed out of them, just as length is not composed out of the straight and the curved, nor solids out of the smooth and the rough. Common to all these cases is the same difficulty that arises for forms taken as genera, whenever one posits the universals: whether animal-itself is present within the particular animal, or is something other than the particular animal. If it is not separate, this raises no difficulty; but if it is separate, as those who make these claims about the one and about numbers say, it is not easy to resolve — unless one ought to call the impossible 'not easy.' For whenever one thinks of the one within the dyad, or in general within a number, does one think of that very one, or of a different one? Some, then, generate magnitudes out of matter of this kind, others out of the point (the point seeming to them to be not a one, but something like the one) and out of another kind of matter, one like plurality, but not plurality itself; and concerning these the same difficulties arise no less. For if the matter is one, line and plane and solid will be the same thing, since from the same things the same and a single thing will result.
39| [1085b] But if there are several matters — one for the line, another for the plane, and yet another for the solid — either these follow one another or they do not, so that the same results follow either way: for either the plane will not contain a line, or there will be a line. Again, no one has attempted to explain how it is possible for number to come from the one and plurality: but however they put it, the same difficulties follow as follow for those who derive number from the one and the indefinite dyad. For the one party generates number from what is universally predicated, and not from a particular plurality, while the other generates it from a particular plurality, though the first one (for they say the dyad is a first plurality), so that there is virtually no difference; the same puzzles will attend it — mixture, or juxtaposition, or blending, or generation, and all the other such notions. But the question one would raise most of all is this: if each unit is one, out of what is it? For surely each unit is not the One itself. It must, then, come from the One itself and from plurality, or from a part of plurality. But to say that the unit is a sort of plurality is impossible, since it is indivisible; while deriving it from a part involves many other difficulties:
40| for each of the parts must be indivisible (or else it will be a plurality, and the unit will be divisible), and the one and the plurality will not be elements (for then each unit would not be made of plurality and the one); further, one who says this is doing nothing but making another number, since a plurality of indivisibles is a number. Further, one must ask, in the case of those who speak this way, whether the number is infinite or finite. For there was, it appears, also a finite plurality, from which come the finite units and the One; and plurality itself is one thing, and infinite plurality another: which kind of plurality, then, together with the One, is the element? A similar question would be raised about the point, and about the element out of which they construct magnitudes. For surely this is not the only point there is; then out of what does each of the other points come? Certainly not out of some interval and a point itself. Nor again can there be indivisible parts of the interval, as there are units of which the plurality is composed: for number is composed of indivisibles, but magnitudes are not. All this and other such considerations make it clear that it is impossible for number and magnitudes to be separate,
41| [1086a] and further, that the disagreement among the ways of treating numbers is a sign that it is the facts themselves, not being true, that create the confusion for these thinkers. For those who posit only the objects of mathematics, alongside the objects of perception, seeing the difficulty and the artificiality connected with the forms, abandoned the number of forms and made number mathematical instead; while those who wanted to make the forms and numbers the same thing, without seeing how, if one lays down these principles, mathematical number could exist alongside the number of forms, made the number of forms and mathematical number the same in name — since in fact the mathematical number has been done away with (for the hypotheses they state are peculiar to themselves and not the mathematical ones); while the one who first posited that the forms exist, and that the forms are numbers, reasonably kept the forms and the objects of mathematics apart. So it turns out that everyone is right about something, but not right as a whole. And they themselves confirm this, by not saying the same things but contrary ones. The cause is that their hypotheses and principles are false. And it is hard, as Epicharmus says, to speak well from premises that are not well put: for as soon as a thing is said, it is immediately apparent that it is not well put. But on the subject of numbers, enough has been raised in the way of difficulties and enough has been determined (for one who is already persuaded might still be further persuaded by more arguments, but for persuading one who is not yet persuaded nothing more would be gained):
42| as for the first principles and the first causes and elements, whatever those who define only about perceptible substance say — some of this has been stated in the works on nature, and some of it does not belong to the present inquiry; but whatever those say who assert that there are other substances besides the perceptible ones, this is next to be examined, following on what has been said. Since, then, some say that the Forms and the numbers are such substances, and that the elements of these are the elements and principles of the things that are, we must examine what they say about these and how they say it. Those who make only numbers, and these mathematical, must be examined later; but as for those who speak of the Forms, one can at the same time observe both their manner of proceeding and the difficulty involved in it. For they make the Forms universal, and at the same time treat them as substances, and again as separate and as belonging to particulars. That this is not possible has been argued before. The reason those who speak of universal substances joined these two together is that they did not make the same substances as the perceptible ones: they supposed that the particulars among perceptible things flow, and that none of them remains,
43| [1086b] while the universal exists apart from these and is something different. This, as we said before, Socrates set in motion because of his definitions, though he did not separate it from the particulars — and in this he thought rightly, in not separating it. This is clear from the facts: without the universal it is not possible to acquire scientific knowledge, but separating it is the cause of the difficulties that arise concerning the Forms. Others, thinking it necessary, if indeed there are to be any substances besides the perceptible and flowing ones, that they be separate, had no others to point to, and so set forth these, the ones called universal, so that it follows that the universal natures and the particular natures are practically the same. This, then, would be a difficulty belonging to what has been said, in its own right. But there is a point which raises a puzzle both for those who speak of the Forms and for those who do not, and which was stated earlier at the beginning among the puzzles raised — let us now state it. If one does not posit substances as separate, in the way in which particular existing things are said to be, one will do away with substance as we wish to speak of it; but if one posits substances as separate, how will one posit their elements and principles?
44| For if they are particular and not universal, then the things that are will be exactly as many as the elements, and the elements will not be objects of knowledge (let the syllables in speech be substances, and their letters the elements of the substances: then BA must be one thing, and each of the syllables one, if indeed they are not universal and the same in form, but each one in number, and a this, and not something said in many senses; further, they posit each thing itself as being one; and if the syllables are so, then so too are the things of which they are composed: there will not, then, be more than one alpha, nor any other of the elements, on the same reasoning by which no other syllable is the same as another; but if this is so, there will be nothing besides the elements, but only the elements; and further, the elements will not even be objects of knowledge, for they are not universal, and knowledge is of universals: this is clear from demonstrations and definitions, for there is no syllogism to the conclusion that this triangle has its angles equal to two right angles, unless every triangle has its angles equal to two right angles, nor that this man is an animal, unless every man is an animal):
45| [1087a] But then, if the principles are universal, either the substances derived from them are also universal, or there will be a non-substance prior to substance: for the universal is not a substance, but the element and the principle are universal, and the element and the principle are prior to the things of which they are the principle and element. All this follows reasonably enough, when they make the Forms out of elements and, besides the substances that have the same form, claim that there is some single separate Idea. But if nothing prevents there being many alphas and betas, as with the elements of spoken sound, with no alpha-itself or beta-itself existing besides the many, then for this reason there will be an unlimited number of similar syllables. The claim that all knowledge is universal, so that the principles of the things that are must also be universal and not separate substances, presents the greatest difficulty of those mentioned; yet what is said is in one way true, and in another way not true. For knowledge, like knowing, is twofold: one kind potential, the other actual. The potentiality, then, being like matter, universal and indeterminate, is of what is universal and indeterminate; but the actuality is determinate and of something determinate, being a this of a this, but it is only accidentally that sight sees color-in-general, because the particular color it sees is a color, and what the grammarian studies, this particular alpha, is an alpha. For if the principles must be universal, then what comes from them must also be universal, as in the case of demonstrations; and if this is so, nothing will be separate, nor will there be any substance. But it is clear that in one way knowledge is universal, and in another way it is not.