Platonismo

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====== Tratado 34 (VI, 6, 4-16) - Of Numbers. (Guthrie) ======




**HOW OTHER NUMBERS FORM PART OF THE INTELLIGIBLE WORLD.**

4. Let us now examine how the numbers form part of the intelligible world. Are they inherent in the other forms? Or are they, since all eternity, the consequences of the existence of these forms? In the latter case, as the very essence possessed primary existence, we would first conceive the monad; then, as movement and stability emanated from it, we would have the triad; and each one of the remaining intelligible entities would lead to the conception of some of the other numbers. If it were not so, if a unity were inherent in each intelligible entity, the unity inherent in the first Essence would be the monad; the unity inherent in what followed it, if there be an order in the intelligible entities, would be the "pair"; last, the unity inhering in some other intelligible entity, such as, for instance, in ten, would be the decad. Nevertheless this could not yet be so, each number being conceived as existing in itself. In this case, will we be compelled to admit that number is anterior to the other intelligible entities, or posterior thereto? On this subject Plato says that men have arrived to the notion of number by the succession of days and nights, and he thus refers the conception of number to the diversity of (objective) things. He therefore seems to teach that it is first the numbered objects that by their diversity produce numbers, that number results from movement of the soul, which passes from one object to another, and that it is thus begotten when the soul enumerates; that is, when she says to herself, Here is one object, and there is another; while, so long as she thinks of one and the same object, she affirms nothing but unity. But when Plato says that being is in the veritable number, and that the number is in the being, he intends to teach that by itself number possesses a hypostatic substantial existence, that it is not begotten in the soul which enumerates, but that the variety of sense-objects merely recalls to the soul the notion of number.

**PYTHAGOREAN INTELLIGIBLE NUMBERS DISCUSSED.**

5. What then is the nature of number? Is it a consequence, and partially an aspect of each being, like man and one-man, essence and one-essence? Can the same be said for all the intelligibles, and is that the origin of all numbers? If so, how is it that on high (in the intelligible world) the pair and triad exist? How are all things considered within unity, and how will it be possible to reduce number to unity, since it has a similar nature ? There would thus be a multitude of unities, but no other number would be reduced to unity, except the absolute One. It might be objected that a pair is the thing, or rather the aspect of the thing which possesses two powers joined together, such as is a composite reduced to unity, or such as the Pythagoreans conceived the numbers, which they seem to have predicated of other objects, by analogy. For instance, they referred to justice as the (Tetrad, or) group-of-four, and likewise for everything else. Thus a number, as for instance a group-of-ten, would be considered as a single (group of) unity, and would be connected with the manifold contained in the single object. This, however, is an inadequate account of our conception of "ten"; we speak of the objects after gathering (ten) separate objects. Later, indeed, if these ten objects constitute a new unity, we call the group a "decad." The same state of affairs must obtain with intelligible Numbers. If such were the state of affairs (answers Plotinos), if number were considered only within objects, would it possess hypostatic existence? It might be objected, What then would hinder that, though we consider white within things, that nevertheless the White should (besides) have a hypostatic substantial existence? For movement is indeed considered within essence, and yet (it is agreed that) movement possesses a "hypostatic" substantial existence within essence. The case of number, however, is not similar to that of movement; for we have demonstrated that movement thus considered in itself is something unitary. Moreover, if no more than such a hypostatic substantial existence be predicated of number, it ceases to be a being, and becomes an accident, though it would not even then be a pure accident; for what is an accident must be something before becoming the accident (of some substance). Though being inseparable therefrom, it must possess its own individual nature in itself, like whiteness; and before being predicated of something else, it already is what it is posited. Consequently, if one be in every (being), one man is not identical with man; if "one" be something different from "man" and from every other (being), if it be something common to all (beings), one must be anterior to all men and to all other (beings), so that man and all other beings may be one. The one is therefore anterior to movement, since movement is one, and likewise anterior to essence, to allow for essence also being one. This of course does not refer to the absolute Unity that is recognized as superior to essence, but of the unity which is predicated of every intelligible form. Likewise, above that of which the decad is predicated subsists the "Decad in itself," for that in which the decad is recognized could not be the Decad in itself.

**THE INTELLIGIBLE UNITY AND DECAD EXIST BEFORE ALL NUMBERS ONE OR TEN.**

Does unity therefore inhere in essences, and does it subsist with them ? If it inhere in essences, or if it be an accident, as health is an accident of man, it must be something individual (like health). If unity be an element of the composite, it will first have to exist (individually), and be an unity in itself, so as to be able to unify itself to something else; then, being blended with this other thing that it has unified, it will not longer remain really one, and will thereby even become double. Besides, how would that apply to the decad? What need of the (intelligible) Decad has that which is already a decad, by virtue of the power it possesses? Will it receive its form from that Decad? If it be its matter, if it be ten and decad only because of the presence of the Decad, the Decad will have first to exist in itself, in the pure and simple state of (being a) Decad.

**WHAT IS THE NATURE OF THESE INTELLIGIBLE NUMBERS?**

6. But if, independently of the things themselves, there be an One in itself, and a Decad in itself; and if the intelligible entities be unities, pairs, or triads, independently of what they are by their being, what then is the nature of these Numbers? What is their constitution? It must be admitted that a certain Reason presides over the generation of these Numbers. It is therefore necessary clearly to understand that in general, if intelligible forms at all exist, it is not because the thinking principle first thought each of them, and thereby gave them hypostatic existence. Justice, for instance, was not born because the thinking principle thought what justice was; nor movement, because it thought what movement was. Thus thought had to be posterior to the thing thought, and the thought of justice to justice itself. On the other hand, thought is anterior to the thing that owes its existence to thought, since this thing exists only because it is thought. If then justice were identical with such a thought, it would be absurd that justice should be nothing else than its definition; for in this case, the thinking of justice or movement, would amount to a conception of these objects (by a definition). Now this would be tantamount to conceiving the definition of a thing that did not exist, which is impossible.

**JUSTICE, LIKE AN INTELLECTUAL STATUE, WAS BORN OF ITSELF.**

The statement that in what is immaterial, knowledge and the known thing coincide, must not be understood to mean that it is the knowledge of the thing which is the thing itself, nor that the reason which contemplates an object is this object itself, but rather, conversely, that it is the thing which, existing without matter, is purely intelligible and intellection. I do not here mean the intellection which is neither a definition nor an intuition of a thing; but I say that the thing itself, such as it exists in the intelligible world, is exclusively intelligence and knowledge. It is not (the kind of) knowledge that applies itself to the intelligible, it is the (actual) thing itself which keeps that knowledge (thereof possessed by reason) from remaining different from it, just as the knowledge of a material object remains different from that object; but it is a veritable (kind of) knowledge, that is, a knowledge which is not merely a simple image of the known thing, but really is the thing itself. It is not therefore the thought of the movement which produced movement in itself, but the movement in itself which produced the thought, so that the thought thinks itself as movement, and as thought. On the one hand, intelligible movement is thought by the intelligible Essence; on the other hand, it is movement in itself because it is first—for there is no movement anterior thereto; it is real movement, because it is not the accident of a subject, but because it is the actualization of the essence which moves, and possesses actualized (existence); it is therefore "being," though it be conceived as different from essence.

Justice, for instance, is not the simple thought of justice; it is a certain disposition of Intelligence, or rather it is an actualization of a determinate nature. The face of Justice is more beautiful than the evening or morning stars, and than all visible beauty. Justice may be imagined as an intellectual statue which has issued from itself and which has manifested itself such as it is in itself; or rather, which subsists essentially in itself.

**INTELLIGENCE THINKS THINGS NOT BECAUSE THEY EXIST, BUT BECAUSE IT POSSESSES THEM.**

7. We must, in fact, conceive intelligible essences as subsisting in one nature, and one single nature as possessing and embracing all (things). There no one thing is separated from the others, as in the sense-world, where the sun, moon, and other objects each occupy a different locality; but all things exist together in one unity; such is the nature of intelligence. The (universal) Soul imitates it, in this respect, as does also the power called Nature, conformably to which, and by virtue of which individuals are begotten each in a different place, while she remains in herself. But, although all things exist together (in the unity of Intelligence), each of them is none the less different from the others. Now, these things which subsist in Intelligence and "being," are seen by the Intelligence that possesses them, not because it observes them, but because it possesses them without feeling the need of distinguishing them from each other; because from all eternity they have dwelt within it distinct from each other. We believe in the existence of these things on the faith of those who admire them, because they have participated therein. As to the magnitude and beauty of the intelligible world, we can judge of it by the love which the Soul feels for it, and if other things feel love for the Soul, it is because she herself possesses an intellectual nature, and that by her the other things can, to some extent, become assimilated to Intelligence. How indeed could we admit that here below was some organism gifted with beauty, without recognizing that the Organism itself (the intelligible world) possesses an admirable and really unspeakable beauty? Further, the perfect Organism is composed of all the organisms; or rather it embraces all the organisms; just as our Universe is one, yet simultaneously is visible, because it contains all the things which are in the visible universe.

**WHAT AND HOW IS EVERY INTELLIGIBLE ENTITY.**

8. Since then the (universal) Organism possesses primary existence, since it is simultaneously organism, intelligence, and veritable "Being"; and as we state that it contains all organisms, numbers, justice, beauty, and the other similar beings—for we mean something different by the Man himself, and Number itself, and Justice itself—we have to determine, so far as it is possible in such things, what is the condition and nature of each intelligible entity.

**NUMBER MUST EXIST IN THE PRIMARY ESSENCE.**

(To solve this problem) let us begin by setting aside sensation, and let us contemplate Intelligence by our intelligence exclusively. Above all, let us clearly understand that, as in us life and intelligence do not consist of a corporeal mass, but in a power without mass, likewise veritable "Being" is deprived of all corporeal extension, and constitutes a power founded on itself. It does not indeed consist in something without force, but in a power sovereignly vital and intellectual, which possesses life in the highest degree, intelligence, and being. Consequently, whatever touches this power participates in the same characteristics according to the manner of its touch; in a higher degree, if the touch be close; in a lower degree, if the touch be distant. If existence be desirable, the com-pletest existence (or, essence) is more desirable still. Likewise, if intelligence deserve to be desired, perfect Intelligence deserves to be desired above everything; and the same state of affairs prevails in respect to life. If then we must grant that the Essence is the first, and if we must assign the first rank to Essence, the second to Intelligence, and the third to the Organism, as the latter seems already to contain all things, and Intelligence justly occupies the second rank, because it is the actualization of "Being"—then number could not enter into the Organism, for before the organism already existed one and two ("Being" and Intelligence). Nor could number exist in Intelligence, for before Intelligence was "Being," which is both one and manifold. (Number therefore must exist, or originate, in the primary Being.)

**NUMBER FOLLOWS AND PROCEEDS FROM ESSENCE.**

9. It remains for us to discover whether it were "Being," in the process of division, that begat number, or whether it be the number that divided "Being." (This is the alternative:) either "being," movement, stability, difference and identity produced number, or it is number that produced all these (categories, or) genera. Our discussion must start thus. Is it possible that number should exist in itself, or must we contemplate two in two objects, three in three objects, and so forth? The same question arises about unity as considered within numbers; for if number can exist in itself independently of numbered things, it can also exist previously to the essences. Can number therefore exist before the essences? It might be well preliminarily to assert that number is posterior to the Essence, and proceeds therefrom. But then if essence be one essence, and if two essences be two essences, one will precede essence, and the other numbers will precede the essences. (Would number then precede the essences) only in thought and conception, or also in the hypostatic existence? We should think as follows. When you think of a man as being one, or the beautiful as being one, the one that is thus conceived in both (beings) is something that is thought only afterward. Likewise, when you simultaneously consider a dog and a horse, here also two is evidently something posterior. But if you beget the man, if you beget the horse or the dog, or if you produce them outside when they already exist in you, without begetting them, nor producing them by mere chance (of seeing them), you will say, "We should go towards one (being), then pass to another, and thus get two; then make one more being, by adding my person." Likewise, (beings) were not numbered after they were created, but before they were created, when (the creator) decided how many should be created.

**NUMBER SPLIT THE UNITY INTO PLURALITY; PYTHAGOREAN IDENTIFICATION OF IDEAS AND NUMBERS.**

The universal Number therefore existed before the essences (were created); consequently, Number was not the essences. Doubtless, Number was in Essence; but it was not yet the number of Essence; for Essence still was one. But the power of Number, hypostati-cally existing within it, divided it, and made it beget the manifold. Number is either the being or actualization (of Essence); the very Organism and Intelligence are number. Essence is therefore the unified number, while the essences are developed number; Intelligence is the number which moves itself, and the Organism is the number that contains. Since therefore Essence was born from Unity, Essence, as it existed within Unity, must be Number. That is why (the Pythagoreans) called the ideas unities and numbers.

**TWO KINDS OF NUMBER: ESSENTIAL AND UNITARY.**

Such then is "essential" Number (number that is "Being"). The other kind of number, which is called a number composed of digits, or "unities," is only an image of the former. The essential Number is contemplated in the intelligible forms, and assists in producing them; on the other hand, it exists primitively in essence, with essence, and before the essences. The latter find therein their foundation, source, root and principle. Indeed, Number is the principle of Essence, and rests in it, otherwise it would split up. On the contrary, the One does not rest upon essence; otherwise essence would be one before participating in the One; likewise, what participates in the decad would be the decad already before participating in the decad.

**ESSENCE IS A LOCATION FOR THE THINGS YET TO BE PRODUCED.**

10. Subsisting therefore in the manifold, Essence therefore became Number when it was aroused to multiplicity, because it already contained within itself a sort of preformation or representation of the essences which it was ready to produce, offering the essences, as it were, a locality for the things whose foundation they were to be. When we say, "so much gold," or, "so many other objects," gold is one, and one does not thereby intend to make gold out of the number, but to make a number out of the gold; it is because one already possesses the number that one seeks to apply it to gold, so as to determine its quality. If essences were anterior to Number, and if Number were contemplated in them when the enumerating power enumerates the objects, the number of the (beings), whatever it is, would be accidental, instead of being determined in advance. If this be not the case, then must number, preceding (the beings) determine how many of them must exist; which means that, by the mere fact of the primitive existence of the Number, the (beings) which are produced undergo the condition of being so many, and each of them participates in unity whenever they are one. Now every essence comes from Essence because essence, by itself, is fcssence; likewise, the One is one by itself. If every (being) be one, and if the multitude of (beings) taken together form the unity that is in them, they are one as the triad is one, and all beings also are one; not as is the Monad (or Unity), but as is a thousand, or any other number. He who, while enumerating, produced things, proclaims that there are a thousand of them, claims to do no more than to tell out what he learns from the things, as if he was indicating their colors, while really he is only expressing a condition of his reason; without which, he would not know how much of a multitude was present there. Why then does he speak so? Because he knows how to enumerate; which indeed he knows if he know the number, and this he can know only if the number exist. But not to know what is the number, at least under the respect of quantity, would be ridiculous, and even impossible.

**AN OBJECT'S EXISTENCE IMPLIES A PREVIOUS MODEL IN ITSELF.**

When one speaks of good things, one either designates objects whach are such by themselves, or asserts that the good is their attribute. If one designate the goods of the first order, one is speaking of the first Hypostasis, or rank of existence; if one designate the things of which the good is the attribute, this implies the existence of a nature of the good which has been attributed to them, or which produces this characteristic within them, or which is the Good in itself, or which, producing the good, nevertheless dwells in its own nature. Likewise, when, in connection with (beings), we speak of a decad, (or, group of ten), one is either referring to the Decad in itself, or, referring to the things of which the decad is an attribute, one is forced to recognize the existence of a Decad in itself, whose being is that of a decad. Consequently, the conferring of the name "decad" implies either that these (beings) are the Decad in itself, or above them in another Decad whose being is that of being a Decad in itself.

**UNITY AND NUMBER PRECEDE THE ONE AND THE MANY BEINGS.**

In general, everything which is predicated of an object either comes to it from without, or is its actualization. Unless by nature it be inconstant, being present now, and absent then, if it be always present, it is a being when the object is a being. If it be denied that its nature were that of a being, it will surely be granted that it is a part cf the essences, and that it is an essence. Now, if the object can be conceived without the thing which is its actualization, this thing nevertheless exists contemporaneously with it, even though in thought it be conceived posteriorily. If the object cannot be conceived without this thing, as man cannot be conceived of without one, in this case one is not posterior to man, but is simultaneous, or even anterior, since the man's subsistence is entirely dependent thereon. As to us, we recognize that Unity and Number precede (Essence and the essences).

**UNITY MUST EXIST IN THE INTELLIGIBLE BEFORE BEING APPLIED TO MULTIPLE BEINGS.**

11. It may be objected that the decad is nothing else than ten unities. If the existence of the One be granted, why should we not also grant the existence of ten unities? Since the supreme Unity (the unity of the first Essence), possesses hypostatic existence, why should the case not be the same with the other unities (the complex unities contained within each of the essences) ? It must not be supposed that the supreme Unity is bound up with a single essence; for in this case each of the other (beings) would no longer be one. If each of the other (beings) must be one, then unity is common to all the (beings); that is that single nature which may be predicated of the multiple (beings), and which must, as we have explained it, subsist in itself (in the primary essence) before the unity which resides in the multiple (beings).

**THE SUPREME UNITY ADJUSTS ALL LOWER GROUP UNITIES.**

As unity is seen in some one (being), and then in some other, if the second unity possess hypostatic existence also, then the supreme Unity (of the first Essence) will not alone possess hypostatic existence, and there will be thus a multitude of unities (as there is a multitude of beings). If the hypostatic existence of the first Unity be alone acknowledged, this will exist either in the Essence in itself, or in the One in itself. If it exist in the Essence in itself, the other unities (which exist in the other beings) will then be such merely by figure of speech, and will no longer be subordinated to the primary unity; or number will be composed of dissimilar unities, and the unities will differ from each other in so far as they are unities. If the primary unity exist already in the Unity in itself, what need would that Unity in itself have of that unity to be one? If all that be impossible, we shall have to recognize the existence of the One which is purely and simply one, which, by its "being" is entirely independent of all the other beings, which is named the chief Unity, and is conceived of as such. If unity exist on high (in the intelligible world) without any object that may be called one, why might not another One (the one of the first Being) subsist on high also? Why would not all the (beings), each being a separate unity, not constitute a multitude of unities, which might be the "multiple unity"? As the nature (of the first Being) begets, or rather, as it has begotten (from all eternity); or at least, as it has not limited itself to one of the things it has begotten, thus rendering the unity (of the first Being) somewhat continuous; if it circumscribe (what it produces) and promptly ceases in its procession, it begets small numbers; if it advance further, moving alone not in foreign matters, but in itself, it begets large numbers. It thus harmonizes every plurality and every being with every number, knowing well that, if each of the (beings) were not in harmony with some number, either they would not exist, or they would bear neither proportion, measure, nor reason.

**ONE AND UNITY ARE WITHIN US; INDEPENDENTLY OF THE ONE OUTSIDE.**

12. (Aristotle) objects that "One" and "Unity" have no hypostatic (or, genuine) existence. Everywhere the One is something that is one. That is nothing but a simple modification experienced in our soul in presence of each essence. We might as easily affirm that when we assert "essence," this is but a simple modification of our soul, Essence (in itself) being absolutely nothing. If it be insisted that Essence exists because it excites and strikes our soul, which then represents it to herself, we see that the soul is equally impressed by the One, and represents Him to herself. Besides, we should ask (Aristotle) if this modification or conception of our soul do not bear to us the aspect of unity or the manifold? So much the more, we often say that an object is not one; evidently we then are not deriving the notion of unity from the object, because we are affirming that there is no unity in it. Unity therefore dwells within us, and it is in us without the object of which we predicate that it is some one thing."

**THERE IS INDEED A UNITARY MODE OF EXISTENCE IN OUTSIDE OBJECTS.**

It may be objected that having this unity in our soul depends on receiving from the exterior object a notion and an image, which is a conception furnished by this object. As the philosophers who profess this opinion do not differentiate the species of one and of number, and as they allow them no other hypostatic existence (than to be conceived by our soul), if they (practically do) allow them any sort of hypostatic existence, it will be very interesting to scrutinize the opinions of these. They then say that the notion or conception that we have of the one or of the number derives from the objects themselves, is a notion as much "a posteriori" as those of "that," "something," "crowd," "festival," "army," or of "multitude"; for, just as the manifold is nothing without the multiple objects, nor a festival without the men gathered to celebrate the religious ceremony, thus "the One" is nothing without the one object, when we posit the one, conceiving it alone, having made an abstraction of everything else. The partisans of this opinion will cite many examples of the same kind, as the "right hand side," "the upper part," and their contraries. What reality indeed (to speak as they do), can the "right hand side" possess outside of a person who stands or sits here or there? The case is similar with "the upper side," which refers to a certain part of the universe, and the "lower side" to another. Our first answer to this argument is that we will allow that there is a certain kind of existence in the things themselves of which we have just spoken; but that this mode of existence is not identical in all things, considered either in respect to each other, or each in respect to the One which is in all. Further, we intend to refute one by one these arguments that have been opposed to us.

**THE NOTION OF THE SUBJECT ONE DOES NOT COME FROM THE SUBJECT ITSELF.**

13. To begin with, it is unreasonable to insist that ithe notion of the subject one comes to us from the subject itself (which is one), from the visible man, for instance, or from some other animal, or even some stone. Evidently the visible man and the One are things entirely different, which could not be identified; otherwise, our judgment would not be able '(as it is) to predicate unity of the non-man. Besides, as the judgment does not operate on emptiness for the right side, and other such things, seeing a difference of position when it tells us that an object is here, or there; likewise, it also sees something when it says that an object is one; for it does not experience there an affection that is vain, and it does not affirm unity without some foundation. It cannot be believed that the judgment says that an object is one because it sees that it is alone, and that there is no other; for, while saying that there is no other, the judgment implicitly asserts that the other is one. Further, the notions of "other" .and "different" are notions posterior to that of unity; if the judgment did not rise to unity, it would not assert either the "other" nor the "different"; when it .affirms that an object is alone, it says, "there is one only object"; and therefore predicates unity before "only." Besides, the judgment which affirms is itself a substantial (being) before affirming unity of some other (being); and the (being) of which it speaks is one likewise before the judgment either asserts or conceives anything about it. Thus (being) must be one or many; if it be many, the one is necessarily anterior, since, when the judgment asserts that plurality is present, it evidently asserts that there is more than one; likewise, when it says that an army is a multitude, it conceives of the soldiers as arranged in one single corps. By this last example, it is plain that the judgment (in saying one body), does not let the multitude remain multitude, and that it thus reveals the existence of unity; for, whether by giving to the multitude a unity which it does not possess, or by rapidly revealing unity in the arrangement (which makes the body of the multitude), the judgment reduces multitude to unity. It does not err here about unity, any more than when it says of a building formed by a multitude of stones that it is a unity; for, besides, a building is more unified than an army. If, further, unity inhere in a still higher degree in that which is continuous, and in a degree still higher in what is not divisible, evidently that occurs only because the unity has a real nature, and possesses existence; for there is no greater or less in that which does not exist.

**UNITY, THOUGH BY PARTICIPATION EXISTING IN SENSE-OBJECTS, IS INTELLIGIBLE.**

Just as we predicate being of every sense-thing, as well as of every intelligible thing, we predicate it in a higher degree of intelligible things, attributing a higher degree (of substantiality) to the (beings that are veritable than to sense-objects), and to sense-objects than to other genera (of physical objects); likewise, clearly seeing unity in sense-objects in a degree higher than in the intelligible (essences), we recognize the existence of unity in all its modes, and we refer them all to Unity in itself. Besides, just as "being and essence" are nothing sensual, though sense-objects participate therein, so unity, though by participation it inhere in sense-objects, is not any the less an in-> telligible Unity. Judgment grasps it by an intellectual conception; by seeing one thing (which is sensual) it also conceives another which it does not see (because it is intelligible); it therefore knew this thing in advance; and if judgment knew it in advance, judgment was this thing, and was identical with that whose existence it asserted. When it says, "a certain" object, it asserts the unity, as, when it speaks of "certain" objects, it says that they are two or more. If then one cannot conceive of any object whatever without "one," "two," or some other number, it becomes possible to- insist that the thing without which nothing can be asserted or conceived, does not at all exist. We cannot indeed deny existence to the thing without whose existence we could not assert or conceive anything. Now that which is everywhere necessary to speak and to conceive must be anterior to speech and conception, so as to contribute to their production. If, besides, this thing be necessary to the hypostatic existence of every essence—for there is no essence that lacks unity—it must be anterior to being, and being must be begotten by it. That is why we say "an essence" instead of first positing "essence," and "a" only thereafter, for there must be "one" in essence, to make "several" possible; but (the converse is not true; for) unity does not contain essence, unless unity itself produce it by applying itself to the begetting of it. Likewise, the word "that" (when employed to designate an object) is not meaningless; for instead of naming the object, it proclaims its existence, its presence, its "being," or some other of its kinds of "essence." The word "that" does not therefore express something without reality, it does not proclaim an empty conception, but it designates an object as definitely as some proper name.

**UNITY ONLY AN ACCIDENT IN SENSE-THINGS, BUT SOMETHING IN ITSELF IN THE INTELLIGIBLE.**

14. As to those who consider unity as relative, they might be told that unity could not lose its proper nature merely as a result of the affection experienced by some other being without itself being affected. It cannot cease being one without experiencing the privation of unity by division into two or three. If, on being divided, a mass become double without being destroyed in respect to its being a mass, evidently, besides the subject, there existed unity; and the mass lost it because the unity was destroyed by the division. So tthis same thing which now is present, and now disappears, should be classified among essences wherever it be found; and we must recognize that, though it may be an accident of other objects, it nevertheless exists by itself, whether it manifest in sense-objects, or whether it be present in intelligent entities; it is only an accident in posterior (beings, namely, the sense-objects); but it exists in itself in the intelligible entities, especially in the first Essence, which is One primarily, and only secondarily essence.

**TWO IS NOT AN ADDITION TO ONE, BUT A CHANGE (REFUTATION OF ARISTOTLE).**

The objection that unity, without itself experiencing anything, by the mere addition of "something else, is no longer one, but becomes double, is a mistake. The one has not become two, and is not that which has been added to it, nor that to which something has been added. Each of them remains one, such as it was; but two can be asserted of their totality, and one of each of them separately. Two therefore, not any more than "pair," is by nature a relation. If the pair consisted in the union (of two objects), and if "being united" were identical with "to duplicate," in this case the union, as well as the pair, would constitute two. Now a "pair" appears likewise in a state contrary (to that of -the reunion of two objects); for two may be produced by the division of a single object. Two, therefore, is neither reunion nor division, as it would have to be in order to constitute a relation.

**OBJECTS PARTICIPATE IN NUMBERS TUST AS THEY PARTICIPATE IN ALL INTELLIGIBLE ENTITIES.**

What then is the principal cause (by virtue of which , objects participate in numbers) ? A being is one by the presence of one; double, because of the presence i of the pair; just as it is white because of the presence of whiteness; beautiful, because of the presence of beauty; and just by that of justice. If that be not admitted, we shall be reduced to asserting that whiteness, beauty and justice are nothing real, and that their only causes are simple relations; that justice consists in some particular relation with some particular being; that beauty has no foundation other than the affection that we feel; that the object which seems beautiful possesses nothing capable of exciting this affection either by nature, or by acquirement. When you see an object that is one, and that you call single, it is simultaneously great, beautiful, and susceptible of receiving a number of other qualifications. Now why should unity not inhere in the object as well as greatness and magnitude, sweetness and bitterness, and other qualities ? We have no right to admit that quality, whatever it be, forms part of the number of beings, whilst quantity is excluded; nor to limit quantity to continuous quantity, while discrete quantity is excluded from the conception of quantity; and that so much the less as continuous quantity is measured by discrete quantity. Thus, just as an object is great because of the presence of magnitude, as it is one by the presence of unity; so is it double because of the presence of being a pair, and so forth.

**THE VERITABLE NUMBERS ARE INTELLIGIBLE ENTITIES.**

Should we be asked to describe the operation of the participation of objects in unity and in numbers, we shall answer that this question connects with the more general problem of the participation of objects in intelligible forms. Besides, we shall have to admit that the decad presents itself under different aspects, according as it is considered to exist either in discrete quantities, or in continuous quantities, or in the reduction of many great forces to unity, or, last, into the intelligible entities to which we are later raised. It is among them, indeed, that are found the veritable Numbers (spoken of by Plato,) which, instead of being considered as discovered in other (beings), exist within themselves; such is the Decad-in-itself, which exists by itself, instead of simply being a decad composed of some intelligible entities.

**NUMBER EXISTS BEFORE EVERY ANIMAL, AND THE UNIVERSAL ANIMAL.**

15. (From the above discussion about the intelligibility of numbers) let us now return to what we said in the beginning. The universal (Being) is veritable Essence, Intelligence, and perfect living Organism; and at the same time contains also all the living organisms. Our universe, which also is an organism, by its unity imitates so far as it can the unity of the perfect living Organism. I say, to the extent of its capacity, because, by its nature, the sense-world has departed from the unity of the intelligible world; otherwise, it would not be the sense-world. Moreover, the universal living Organism must be the universal Number; for if it were not a perfect number, it would lack some number; and if it did not contain the total number of living organisms, it would not be the perfect living Organism. Number therefore exists before every living organism, and before the universal living Organism. Man and the other living organisms are in the intelligible world; so far as they are living organisms, and so far as the intelligible world is the universal living Organism; for man, even here below, is a part of the living Organism, so far as itself is a living organism, and as the living Organism is universal; the other living orgamsms are also in the living Organism, so far as each of them is a living organism.

**THE INTELLIGIBLE AS POTENTIAL AND ACTUALIZED IN THE SOUL.**

Likewise, Intelligence, as such, contains all the individual intelligences as its parts. These, however, form a number. Consequently, the number which is in the Intelligence does not occupy the first degree. So far as the number is in Intelligence, it is equal to the quantity of the actualizations of Intelligence. Now, these actualizations are wisdom, justice, and the other virtues, science, and all the (ideas) whose possession characterizes it as veritable Intelligence. (If then science exist in the Intelligence) how does it happen that it is not there in some principle other than itself? In Intelligence the knower, the known, and science are one and the same thing; and with everything else within it. That is why every (entity) exists in the intelligible world in its highest degree. For instance, within it, Justice is no accident, though it be one in the soul, as such; for intelligible entities are in the soul (only in) potential condition (so long as she remains no more than soul); and they are actualized when the soul rises to Intelligence and dwells with it.

**NUMBER AS THE UNIVERSAL BOND OF THE UNIVERSE.**

Besides Intelligence, and anterior thereto, exists Essence. It contains Number, with which it begets (beings); for it begets them by moving according to number, determining upon .the numbers before giving hypostatic existence to the (beings), just as the unity (of essence) precedes its (existence), and interrelates it with the First (or, absolute Unity). Numbers interrelate nothing else to the First; it suffices for Essence to be interrelated with Him, because Essence, on becoming Number, attaches all (beings) to itself: Essence is divided not so far as it is a unity (for its unity is permanent); but having divided itself conformably to its nature in as many things as it decided on, it saw into how many things it had divided itself; and through this (process) it begat the number that exists within itself; for it divided itself by virtue of the potentialities of number, and it begat as many (beings) as number comported.

**THE GENERATION OF EVERYTHING REGULATED BY NUMBER.**

The first and veritable Number is therefore the source and principle of hypostatic existence for beings. That is the reason that even here below, the classified both discrete and continuous quantity and, with a different number, it is some other thing that is begotten, or nothing more can be begotten. Such are the primary Numbers, so far as they can be numbered. The numbers that subsist in other things play two parts. So far as they proceed from the First, they can be numbered; so far as they are below them, they measure other things, they serve to enumerate both numbers and things which can be enumerated. How indeed could you even say "ten" without the aid of numbers within yourself?

**DIFFICULTIES CONNECTED WITH THESE INTELLIGIBLE NUMBERS.**

16. The first objection might be, Where do you locate, or how do you classify these primary and veritable Numbers? All the philosophers (who follow Aristotle) classify numbers in the genus of quantity. It seems that we have above treated of quantity, and classified both discrete and continuous quantity among other "beings." Here however we seem to say that these Numbers form part of the primary Essences, and add that there are, in addition, numbers that serve for enumerations. We are now asked how we make these statements agree, for they seem to give rise to several questions. Is the unity which is found among sense-beings a quantity? Or is unity a quantity when repeated, while, when considered alone and in itself, it is the principle of quantity, but not a quantity itself? Besides, if unity be the principle of quantity, does it share the nature of quantity, or has it a different nature? Here are a number of points we ought to expound. We shall answer these questions, and here is what we consider our starting-point.

**UNITY CONTAINED IN SENSE-OBJECTS IS NOT UNITY IN ITSELF.**

When, considering visible objects, by which we ought to begin, we combine one (being) with another, as for instance, a horse and a dog, or two men, and say that they form two; or, when considering a greater number of men we say they are ten, and form a group of ten, this number does not constitute being, nor an (accident) among sense-objects; it is purely and simply a quantity. Dividing this group of ten by unity, and making unity of its parts, you obtain and constitute the principle of quantity (unity) for a unity thus derived from a group of ten.

**NUMERALS PREDICATED OF THE MAN IN HIMSELF ARE ESSENTIAL.**

But when you say that the Man considered in himself is a number, as, for instance, a pair, because he is both animal and reasonable, we have here no more than a simple modality. For, while reasoning and enumerating we produce a quantity; but so far as there are here two things (animal and reasonable), and as each of them is one, as each completes the being of the man, and possesses unity; we are here using and proclaiming another kind of number, the essential Number. Here the pair is not posterior to things; it does not limit itself to expressing a quantity which is exterior to essence; it expresses what is in the very being of this essence, and contains its nature.

**COLLECTIVE NOUNS USED AS PROOF OF INDEPENDENT EXISTENCE.**

Indeed, it is not you who here below produce number when you by discursive reason range through things that exist by themselves, and which do not depend for their existence on your enumeration; for you add nothing to the being of a man by enumerating him with another. That is no unity, as in a "choric ballet." When you say, ten men, "ten" exists only in you who are enumerating. We could not assert that "ten" exists in the ten men you are enumerating, because these men are not co-ordinated so as to form a unity; it is you yourself who produce ten by enumerating this group of ten, and by making up a quantity. But when you say, a "choric ballet," an "army," there is something which exists outside of these objects, and within yourself. How are we to understand that the number exists in you? The number which existed in you before you made the enumeration has another mode (of existence) (than the number that you produce by enumeration). As to the number which manifests itself in exterior objects and refers to the number within yourself, it constitutes an actualization of the essential Numbers, or, is conformable to the essential Numbers; for, while enumerating you produce a number, and by this actualization you give hypostatic existence to quantity, as in walking you did to movement.

**THE NUMBER WITHIN IS THE NUMBER CONSTITUTIVE OF OUR BEING.**

In what sense does the number which is within us (before we enumerate) have a mode (of existence) other (than the one we produce in enumeration) ? Because it is the number constitutive of our being, which, as Plato says, participates in number and harmony, and is a number and harmony; for the soul is said to be neither a body nor an extension; she therefore is a number, since she is a being. The number of the body is a being of the same nature as the body; the number of the soul consists in the beings which are incorporeal like souls. Then, for the intelligible entities, if the animal itself be plurality, if it be a triad, the triad that exists in the animal is essential. As to the triad which subsists, not in the animal, but in essence, it is the principle of being. If you enumerate the animal and the beautiful, each of these two in itself is a unity; but (in enumerating them), you beget number in yourself, and you conceive a certain quantity, the pair. If (like the Pythagoreans) you say that virtue is a group of four, or tetrad, it is one so far as its parts (justice, prudence, courage, and temperance) contribute to the formation of a unity; you may add that this group of four, or tetrad, is a unity, so far as it is a kind of substrate; as to you, you connect this tetrad with the one that is inside of you.