Aristotle · a new plain-English translation from the original language
1| So much, then, let it be said about this kind of substance. Now everyone makes the principles contraries, as they do in the works on nature, and similarly with regard to the unmoved substances. But if nothing can be prior to the principle of all things, it would be impossible for the principle, being something else as well, to be a principle — for example, if someone were to say that the white is a principle not insofar as it is something else but insofar as it is white, yet that it holds of some substrate and, being something else as well, is white: for that substrate will be prior. But indeed everything comes to be out of contraries as out of some substrate; therefore this — having a substrate — must belong above all to the contraries.
2| [1087b] Therefore all contraries always hold of a substrate, and none is separate; but, just as it appears that nothing is contrary to substance, so reason bears this out too. None of the contraries, then, is in the strict sense the principle of all things, but something else is. But they make one of the two contraries matter: some pair the unequal with the one, i.e. with the equal, taking the unequal to be the nature of plurality; others pair plurality with the one (for on the one view numbers are generated from the dyad of the unequal, the great and the small, while on the other they are generated from plurality, but in both cases through the substance of the one): for even the one who calls the unequal and the one the elements, and the unequal a dyad of the great and small, speaks of the unequal and the great and small as being one, and does not distinguish that this is so in account but not in number. Moreover, they do not correctly render the principles they call elements: those who speak of the great and the small along with the one make these three the elements of numbers, the two serving as matter and the one as form, while others speak instead of the many and the few, on the grounds that the great and the small are by nature more proper to magnitude, and others again use a more general term for these, the exceeding and the exceeded.
3| None of these differences make, so to speak, any difference with regard to some of the consequences, but only with regard to the logical difficulties, which they are guarding against because they themselves offer logical demonstrations. Except that it is by the same reasoning that the exceeding and the exceeded, rather than the great and the small, should be the principles, and that number should be prior to the dyad composed of the elements: for both are more universal. But as it is, they assert the one and do not assert the other. Others set the different and the other over against the one, others set plurality and the one over against each other. But if, as they wish, the things that are come from contraries, and to the one either nothing is contrary, or if indeed anything is to be, it is plurality — while the unequal is contrary to the equal, and the different to the same, and the other to itself — then it is above all those who oppose the one to plurality who hold to some plausible view, yet even they do not hold it adequately: for on this view the one will be few: for plurality is opposed to fewness, and the many to the few. But that the one signifies a measure is evident. And in every case there is some other underlying thing — for example, in harmony, the quarter-tone, in magnitude, the finger or the foot or some such unit, and in rhythms, the beat or the syllable;
4| and likewise in weight there is some definite standard weight; and in the same way in every case, [1088a] in qualities some particular quality, in quantities some particular quantity, and the measure is indivisible, in one case in kind, in another to perception — since the one is not, in its own right, some substance. And this is reasonable: for the one signifies that it is the measure of some plurality, and number signifies a measured plurality and a plurality of measures (and so it is also reasonable that the one is not a number: for a measure is not measures, but a measure, like the one, is a starting point). The measure must always be something the same belonging to all the things measured — for example, if they are horses, the measure is a horse, and if men, a man. But if it is a man and a horse and a god, the measure will perhaps be an animal, and their number will be a number of animals. But if it is a man and a white thing and a walking thing, there will least of all be a number of these, because they all belong to the same thing, one in number; yet there will still be a number of these as genera, or of some other such designation. But those who treat the unequal as some single thing, and make the dyad of the great and small indeterminate, say things very far removed from what is plausible and possible: for these — the many and the few of number, and the great and the small of magnitude — are affections and accidents of numbers and magnitudes rather than substrates for them, just as even and odd, smooth and rough, straight and curved are;
5| Further, in addition to this error, the great and the small and all such things must also be relative; and the relative is, of all the categories, least of all some nature or substance, and is posterior to quality and quantity: and the relative is an affection of quantity, as was said, but not matter — if indeed it is at all distinct — belonging to what is relative in general, and to its parts and species alike. For nothing is great or small, many or few, or relative at all, which is not, in being something else, many or few or great or small or relative. A sign that the relative is least of all some substance and some being is that it alone has no coming-to-be, nor passing-away, nor motion belonging to it, as there is, corresponding to quantity, growth and diminution, corresponding to quality, alteration, corresponding to place, locomotion, and corresponding to substance, unqualified coming-to-be and passing-away — but there is none of these corresponding to the relative: for without itself being moved, a thing will at one time be greater, at another smaller or equal, simply because the other thing has changed in respect of quantity.
6| [1088b] Every matter must be that which is potentially the thing in question, and this holds of substances too; but a relative is neither potentially nor actually a substance. It is odd, then—or rather impossible—to make an element and a prior thing out of what is not a substance rather than a substance; for all the categories are posterior to substance. Further, the elements are not predicated of the things of which they are elements, but the many and the few, and apart and together, are predicated of number, and the long and the short of a line, and the plane is both broad and narrow. Now if indeed there is some plurality of which one part is always the few—as with the dyad (for if it were many, the one would be few)—while another is many without qualification, as the decad is many, or anything beyond it, or the myriad, then how will number be composed in this way out of the few and the many? For either both ought to be predicated of it, or neither; but as it is, only one of the two is predicated. But we must consider the matter simply: is it possible for the eternal things to be composed of elements? For then they will have matter, since everything composed of elements is composite. If, then, it is necessary that what a thing is made of—if it is eternal, and even if it came to be, it would come to be out of this same thing—and everything that comes to be comes to be out of what is potentially the very thing that comes to be (for it could not have come to be, nor have existed, out of what is incapable of it), and what is potential admits of both being actual and not being actual, then even if number, or anything else that has matter, is as eternal as can be, it would still admit of not being, just as a thing that lasts one day, or any number of years, admits of not being:
7| And if this is so, so too is the whole length of time which has no limit. They would not, then, be eternal, if indeed what admits of not being is not eternal—as it happened to be argued elsewhere. And if what is now being said is universally true, that no substance is eternal unless it is actuality, while the elements are the matter of substance, then no eternal substance would have elements present in it out of which it is composed. Now there are some who make the indefinite dyad, along with the one, the element, and who reasonably balk at the unequal because of the impossibilities that follow; but they have only rid themselves of as many difficulties as necessarily follow, for those who say this, from making the unequal and the relative an element; those difficulties that are independent of this doctrine must belong to them as well, whether they construct the ideal number or the mathematical number out of these. Now there are many causes of this deviation toward these causes,
8| [1089a] but most of all the raising of the difficulty in an old-fashioned way. For it seemed to them that all things that are would be one—being itself—unless someone should resolve and come to grips with Parmenides' argument: "For this shall never be proved, that things that are not, are"; rather one must show that what is not, is, in some sense—for only in this way, out of being and something else, could the things that are come to be, if indeed they are many. And yet, in the first place, if being is spoken of in many ways (for one sense signifies substance, another quality, another quantity, and so with the other categories), then in what sense will all the things that are be one, if not-being is not to be? Will it be the substances, or the affections and the rest likewise, or all of them—so that the this and the such and the so-much and everything else that signifies some one thing will be one? But it is odd—or rather impossible—that a single nature, once it has come to be, should be the cause of being's being, in one case, a this, in another a such, in another a so-much, in another a somewhere. Next, out of what sort of not-being and being are the things that are? For not-being too is spoken of in many ways, since being also is:
9| Not being a man signifies not being this; not being straight signifies not being such; not being three cubits long signifies not being so much. Out of what sort of being and not-being, then, are the many things that are? Now what they mean is that falsity, and this nature, is the not-being out of which, together with being, the many things that are arise; that is why it was also said that one must posit a falsehood, just as geometers posit that the line which is not a foot long is a foot long. But it is impossible for things to stand this way: geometers posit no falsehood (for the premise is not part of the syllogism), nor do the things that are come to be, or perish, out of not-being in that sense. But since not-being, taken according to its cases, is spoken of in as many ways as there are categories, and beside this, not-being is spoken of as falsity, and also as what exists potentially, it is out of this latter that generation takes place—man out of what is not a man but is potentially a man, white out of what is not white but is potentially white—alike whether one thing comes to be or many. The inquiry evidently concerns how being, spoken of in respect of substances, is many:
10| For what is generated is numbers, lengths, and bodies. It is odd, then, to inquire how being in the sense of the what-it-is is many, but not to inquire how it is many in quality or quantity. For surely the indefinite dyad, or the great and the small, is not the cause of there being two white things, or many colors, flavors, or shapes:
11| [1089b] for then these too would be numbers and units. But if in fact they had gone into these cases, they would have seen the cause at work there as well; for the same cause, or its analogue, is the cause. For this same digression is also the cause of those who, in seeking the opposite of being and of the one, out of which together with these the things that are arise, posit the relative and the unequal—which is neither the contrary nor the negation of these, but a single nature among the things that are, just as the what and the of-what-kind are. And they ought also to have inquired into this: how the relatives are many and not one; but as it is, how there are many units besides the primary one is inquired into, while how there are many unequal things besides the unequal is no longer inquired into. And yet they use and speak of great and small, many and few, out of which come numbers; long and short, out of which comes length; broad and narrow, out of which comes the plane; deep and shallow, out of which come solids; and they speak further of still more kinds of the relative. What, then, is the cause of these being many? It is necessary, then, as we say, to posit for each thing what is potentially it (and the man who says these things went on to declare in addition what the potential this and substance is—that it is not a being in its own right but is the relative, as if he had said the of-what-kind—which is neither potentially the one or being, nor a negation of the one or of being, but one among the things that are); and it would have been far more to the purpose, as was said, if, when inquiring how the things that are are many, he had not inquired into things within the same category—how there are many substances or many qualities—but how the things that are, taken as a whole, are many:
12| For some things are substances, some affections, some relatives. Now in the case of the other categories there is another difficulty as well as to how they are many (for because they are not separable, it is through the subject's becoming many, and existing, that there come to be many qualities and many quantities as well; and yet there must be some matter for each genus, only it is impossible for it to be separable from substances). But in the case of thises, there is some point in asking how the this is many, unless something is going to be both a this and such a nature at once—and this difficulty is really rather that of how there can be many substances in actuality and not one. But further, even if the this and the quantity are not the same, one is not told how or why the things that are are many, but only how quantities are many; for every number signifies a quantity, and so does the unit, if it is not a measure and something indivisible in respect of quantity. If, then, the quantity and the what-it-is are different, one is not told out of what the what-it-is is, nor how it is many:
13| [1090a] But if it is the same, the one who says so faces many contrary difficulties. One might also fix one's inquiry on the question of numbers: from what source we ought to get conviction that they exist. For the one who posits Forms provides a certain cause for their existing, if indeed each of the numbers is a Form, and the Form is in some way or other a cause of being for the other things (let this be assumed as underlying for them). But for the one who does not think this way, because he sees the difficulties inherent in the Forms, so that on account of these he does not posit numbers as Forms, but instead posits mathematical number — from what source must one get conviction that such a number exists, and what use is it to other things? For the one who says it exists does not say it is useful for anything, but speaks of it as being a certain nature in its own right, and it does not appear to be a cause of anything: for all the theorems of arithmetic will hold also of perceptible things, just as was said. Now those who posit the Forms as existing, and hold that they themselves are numbers, do try, by taking the one thing over and above the many in accordance with the abstraction made in each case, in some way to say why number exists; but nonetheless, since these claims are neither necessary nor possible, one must not on that basis say that number exists on their account.
14| The Pythagoreans, because they saw many affections of numbers belonging to perceptible bodies, made the things that are to be numbers — not separate numbers, but the things that are out of numbers. But why? Because the affections of numbers hold in harmony, and in the heaven, and in many other things. But for those who say that number is only mathematical, nothing of this sort can be said, according to their hypotheses, but it was said that the sciences would not belong to them. We, however, say that number exists, just as we said before. And it is clear that mathematical objects are not separate: for if they were separate, their affections would not belong to bodies. Now the Pythagoreans are open to no objection on this score; but insofar as they make natural bodies out of numbers — out of things having no weight or lightness, making things that have lightness and weight — they seem to be talking about some other heaven and about bodies other than the perceptible ones. But those who make number separate, because the axioms will not hold of perceptible things, while the things said about number are true and delight the soul, suppose that numbers exist and are separate; and likewise also the mathematical magnitudes.
15| [1090b] It is clear, then, that the opposing argument will also say the opposite things, and that what was just raised as a difficulty must be resolved by those who speak this way: why, when the affections in no way belong to perceptible things, do their affections nevertheless belong in perceptible things? There are some who, from the fact that points, planes, and solids are limits and extremities — the point being the limit of a line, this of a plane, this of a solid — think it necessary that such natures exist. One must examine this argument too, lest it be too weak. For extremities are not substances but rather all these are limits (since there is also a limit of walking and of motion in general: this, then, will be a this-something and a certain substance — but that is absurd); and moreover, even if they do exist, they will all belong to these perceptible things (for the argument has spoken of these): why then will they be separate? Further, one who is not too easygoing might raise a difficulty about number as a whole and about mathematical objects: that the earlier terms contribute nothing to the later ones (for if number did not exist, magnitudes would exist no less for those who say only mathematical objects exist, and if these did not exist, soul and the perceptible bodies would exist no less);
16| but nature does not seem to be episodic, put together out of appearances, like a bad tragedy. Those who posit the Forms escape this: for they make magnitudes out of matter and number — lengths out of the dyad, planes perhaps out of the triad, solids out of the tetrad, or else out of other numbers (for it makes no difference) — but will these, then, be Forms, or what is their manner of being, and what do they contribute to the things that are? They contribute nothing, just as the mathematical objects do not either; nor indeed does any theorem hold of them, unless someone wants to disturb mathematics and construct opinions of his own. But it is not difficult, taking any hypotheses whatever, to spin things out at length and string them together. These thinkers, then, err in this way by dragging mathematical objects into connection with the Forms; but the first to make two kinds of number, the number of the Forms and the mathematical number, have neither said nor could they say how and from what the mathematical number will exist. For they make it intermediate between the number of the Forms and the perceptible. For if it is from the great and the small, it will be the same as that number of the Forms (but from some other great and small, since it makes magnitudes);
17| [1091a] but if he will say it is something different, he will be speaking of more elements; and if the principle of each is some one thing, then the one will be something common to these, and one must inquire how the one can also be these many things at once, and how, according to him, number can come to be otherwise than from the one and an indefinite dyad — which is impossible. All these claims, then, are irrational, and they conflict both with each other and with reasonable claims, and there seems to be in them the long speech of Simonides: for the long speech arises just as that of slaves does, when they say nothing sound. And the very elements, the great and the small, seem to cry out, as if being dragged along by force: for they are in no way able to generate number except the number produced by continually doubling from one. And it is also absurd to posit generation for things that are eternal — or rather, this is one of the impossibilities. As for the Pythagoreans, there is no need to be in doubt whether they posit generation or not: for they plainly say that when the one was constituted — whether out of planes, or out of color, or out of seed, or out of things they are at a loss to state — immediately the part nearest the unlimited began to be drawn in and limited by the limit. But since they construct a cosmology and want to speak in the manner of natural science, it is fair to examine them somewhat with regard to nature, but to let them off from the present inquiry;
18| for we are seeking the principles that hold among unchanging things, so that we must also examine the generation of numbers of this sort. Now they say there is no generation of the odd, which implies clearly that there is generation of the even; and some construct the even, the first even, out of unequals, the great and the small being equalized. So it is necessary that inequality belong to them prior to their being equalized; but if they were always equalized, they would not have been unequal before (for there is nothing prior to the always), so that it is clear they do not construct the generation of numbers for the sake of theoretical understanding alone. And there is a further difficulty, and one that raises a reproach even against someone who resolves it well: how the elements and the principles stand in relation to the good and the fine. The difficulty is this: whether any of those principles is what we mean by the good itself and the best, or not, but rather is generated later. Among the theologians it seems to be agreed by some of those now living, who say it is not so, but that only as the nature of beings advances do the good and the fine come to appear (and they do this out of caution about a genuine difficulty which befalls those who say, as some do, that the one is a principle:
19| [1091b] But the difficulty is not because they attribute the good to the principle as belonging to it, but because they make the one a principle -- and a principle as an element -- and number as coming from the one. The ancient poets agree with this in a similar way, in that they say it is not the first beings who reign and rule -- such as Night and Heaven or Chaos or Ocean -- but Zeus. Yet it is because the rulers among beings change that these thinkers come to say such things; since those among them who are mixed in kind, and who do not say everything mythically -- such as Pherecydes and certain others -- posit that what generates first is best, and so do the Magi, and among the later wise men, such as Empedocles and Anaxagoras, of whom the one makes love an element and the other makes intellect a principle. But of those who say that there are unmoved substances, some say that the one itself is the good itself; they thought, however, that above all it was the one that was the substance of it. This, then, is the difficulty -- which way one ought to speak: it would be strange if to the first, eternal, and most self-sufficient thing this very thing -- self-sufficiency and preservation -- did not belong first, as a good. But surely it is imperishable for no other reason than that it is in a good state, nor self-sufficient for any other reason; so that while it is reasonable that it is true to say that the principle is of this sort, to say that this is the one -- or, if not this, then at least an element, and an element of numbers -- is impossible.
20| For a great difficulty results -- one which some, fleeing from it, have given up on: those who agree that the one is the first principle and element, but only of mathematical number -- for then every unit turns out to be some good, and there comes to be a great abundance of goods. Further, if the forms are numbers, all the forms are some good; but then let one posit Ideas of whatever one wishes: for if of goods only, the Ideas will not be substances; but if also of substances, then all animals and plants, and the things that participate in them, are good. These consequences, then, are absurd; and so is the opposite element, whether it is plurality, or the unequal and the great-and-small, being the bad itself (which is why one thinker avoided attaching the good to the one, on the ground that this was necessary, since generation is from contraries, so that the bad would be the nature of plurality; while others say that the unequal is the nature of the bad): it follows, then, that all things that exist partake of the bad except the one itself, and that numbers partake of it more unmixed than magnitudes do,
21| [1092a] and that the bad is the region of the good, and that it partakes of and desires what is destructive of it -- for the contrary is destructive of the contrary. And if, as we said, matter is what each thing is potentially -- as, in relation to fire in actuality, potential fire is what corresponds to it -- then the bad will be the potential good itself. All these consequences, then, follow: partly because they make every principle an element, partly because they make the contraries principles, partly because they make the one a principle, and partly because they make numbers the first substances, separate, and forms. If, then, both not positing the good among the principles, and positing it in this way, are equally impossible, it is clear that the principles are not being correctly stated, nor are the first substances. Nor is one's supposition correct if one likens the principles of the whole to those of animals and plants, on the ground that the more complete things always come from the indefinite and the incomplete -- which is why he says the same holds of the first things too, so that the one itself is not even something that exists. For even in that case the principles from which these come are themselves complete: for a man begets a man, and the seed is not first. It is absurd, too, to bring place into being together with the mathematical solids (for place is peculiar to particular things, which is why they are separable in respect of place, whereas mathematical objects are not anywhere), and to say that a thing will be somewhere without saying what place is.
22| Now those who say that beings are made of elements, and that numbers are the first of beings, ought -- having distinguished the ways in which one thing comes from another -- to explain in this way in what manner number comes from the principles. Is it by mixture? But not everything admits of mixture, and what results from mixture is something different, and the one will not be separable, nor will it be a distinct nature -- yet they want it to be. Is it, then, by composition, as with a syllable? But then position must belong to it, and one who thinks of the one and of plurality will think of them separately. This, then, will be what number is: unit and plurality, or the one and the unequal. And since being made out of certain things is in one sense being made out of things that persist in it and in another sense not, in which of these senses is number so made? For it cannot be made in the sense of things persisting in it, except in the case of things of which there is generation. Is it, then, as from seed? But it is not possible for anything to come away from what is indivisible. Is it, then, as from a contrary that does not persist? But whatever comes to be in this way also comes from some other thing that does persist. Since, then, one thinker posits the one as contrary to plurality,
23| [1092b] while another posits it as contrary to the unequal, treating the one as equal -- in either case number would be from contraries: there must therefore be some other thing, over and above these, from which, as persisting, and from the other as well, number consists or has come to be. Further, why is it that all the other things that come from contraries, or that have contraries, perish -- even if made of them entirely -- while number does not? For nothing is said about this. And yet the contrary destroys, whether it is present in the thing or not present in it -- as strife destroys the mixture (though it need not, for strife is not the contrary of that). Nothing has been determined, either, as to in which of two ways numbers are the causes of substances and of being -- whether as boundaries (as points are of magnitudes, and as Eurytus assigned which number belonged to which thing, this one to man, this one to horse, in the way that those who bring numbers into shapes make a triangle and a square, thus likening with pebbles the forms of plants), or because a ratio is the concordance of numbers, and likewise man too, and each of the other things? But how are affections -- the white, the sweet, the hot -- numbers? That numbers are not the substance, nor the causes of form, is clear:
24| for it is the ratio that is the substance, while number is matter. For example, the substance of flesh or of bone is a number in this sense -- three parts of fire and two of earth: and the number, whatever it is, is always a number of certain things -- either of fire, or of earth, or of units -- but the substance is the being of so much in relation to so much according to the mixture: and this is no longer a number but a ratio of the mixture of bodily numbers, or of whatever kind. Number, then, is not a cause either by producing, or as number generally, or as a number of units, nor as matter, nor as the ratio and form of things. Nor, again, is it a cause as the end for the sake of which. One might also raise the question what the good is that comes from numbers by virtue of the mixture being in a number -- whether in one that is easily reckoned or in an odd one. For as it stands, honey-water is no more healthful if it is mixed three times three, but it would do more good if it were watery and in no fixed ratio than if it were unmixed and in a fixed number. Further, the ratios of mixtures consist in an addition of numbers, not in numbers as such -- for example, three to two, not three times two. For the same kind must hold throughout the multiplications, so that the series on which A, B, C stand must be measured by A, and the series D, E, F by D:
25| So all things will belong to the same number. Then the number of fire will not be BEGZ, nor that of water twice three. [1093a] But if all things must share in number, then many things must turn out to be the same, and the same number must belong to this thing and to another. Is this, then, the cause, and is the thing what it is because of this, or is this unclear? For instance, there is a certain number belonging to the sun's revolutions, and again one belonging to the moon's, and one belonging to the life and age of each kind of animal: what, then, prevents some of these numbers from being squares, others cubes, some equal, others double? Nothing prevents it — but it is necessary that things move within these limits, if all things share in number. And it was possible for different things to fall under the same number; so that if the same number happened to belong to several things, those things, having the same form of number, would be the same as one another — for instance, the sun and the moon would be the same. But why are these numbers causes? There are seven vowels, seven strings to the musical scale, the Pleiades are seven, animals shed their teeth at seven (some do, some do not), and there were seven who marched against Thebes. Is it, then, because the number is naturally of this sort, that those men came to be seven, or that the Pleiades consist of seven stars? Or rather, is it that the former are seven because of the gates, or some other cause, while the latter we simply count that way — as we count the Bear as twelve stars, though others count more? Since indeed people say that Ξ, Ψ, Ζ correspond to the musical concords, and that, because those concords are three, these letters too are three.
26| But that there could be countless other correspondences of this kind makes no difference (for Γ and Ρ could equally well be counted as one sign); and if it is said that each of these double letters is double the others while no other letter is, and that the cause is that, there being three places of articulation, one element is added to the sigma at each of the three, and that this is why there are only three — that is not because the concords are three, since the concords are actually more than three, whereas here, with the letters, there cannot be more. These men are like the ancient Homeric scholars, who see small resemblances and overlook large ones. Some say there are many such correspondences — for instance, that the two middle strings are one at nine and the other at eight, and that the epic verse has seventeen syllables, equal in number to these, and that it is scanned with nine syllables in the right half-line and eight in the left.
27| [1093b] And they say that the interval is equal both among the letters, from Α to Ω, and, among flutes, from the lowest note to the highest, whose number is equal to the whole embodied system of the heavens. But one must see that no one would have any trouble either stating or discovering such correspondences among things eternal, since they can be found among perishable things too. But the properties praised in numbers, and their opposites, and in general the things found in mathematics — as some people speak of them and make them causes of nature — seem, when looked at in this way, to slip away as causes (for on none of the ways distinguished concerning the principles is any of them a cause). Still, in a sense they do make it clear that goodness is present, and that the odd, the straight, the equal-times-equal, and the powers of certain numbers belong to the same series as the fine; for at the same time as the seasons come round, a number of a certain kind comes round too. And all the other correspondences they gather from mathematical theorems all have this same character. That is why they look like coincidences: for they are accidental, yet all akin to one another, and one by analogy — for in each category of being there is the analogous term: as the straight is in length, so the level is in breadth, the odd, perhaps, is in number, and the white is in color. Again, the numbers found in the forms are not the causes of harmonic relations and the like (for those numbers which are equal in form differ from one another — for even the units do); so that for these reasons at least, forms should not be posited. These, then, are the things that follow from the theory, and still more could be gathered; but it seems to be evidence that these thinkers suffer much difficulty over the generation of these numbers, and are in no way able to weave together an account, that mathematical objects are not separate from perceptible things, as some say, nor are these the principles.