Guthrie-Plotinus: numbers

Thus a worthy man, perceiving in a youth the character of virtue, is agreeably impressed, because he observes that the youth harmonizes with the true type of virtue which he bears within himself. Thus also the beauty of color, though simple in form, reduces under its sway that obscurity of matter, by the presence of the light, which is something incorporeal, a reason, and a form. Likewise, fire surpasses all other bodies in beauty, because it stands to all other elements in the relation of a form; it occupies the highest regions; it is the subtlest of bodies because it most approaches the incorporeal beings; without permitting itself to be penetrated by other bodies, it penetrates them all; without itself cooling, it communicates to them its heat; by its own essence it possesses color, and communicates it to others; it shines and coruscates, because it is a form. The body in which it does not dominate, shows but a discolored hue, and ceases being beautiful, merely because it does not participate in the whole form of color. Once more, thus do the hidden harmonies of sound produce audible harmonies, and also yield to the soul the idea of beauty, though showing it in another order of things. Audible harmonies can be expressed in numbers; not indeed in any kind of numbers, but only in such as can serve to produce form, and to make it dominate. [Ennead I,6 (1) 3]

The qualities that are natural, quantities, numbers, magnitudes, states, actions and natural experiences, movements and recuperations, either general or particular, are among the contents of the intelligible world, where time is replaced by eternity, and space is replaced by the “telescoping” of intelligible entities (that are within each other). As all entities are together in the intelligible world, whatever entity you select (by itself) is intellectual and living “being,” identity and difference, movement and rest; it is what moves, and what is at rest; it is “being,” and quality; that is, it is all. There every essence is in actualization, instead of merely being in potentiality; consequently it is not separated from quality. [Ennead V,9 (5) 10]

Last, we have the divine Plato, who has said so many beautiful things about the soul. In his dialogues he often spoke of the descent of the soul into the body, so that we have the right to expect from him something clearer. Unfortunately, he is not always sufficiently in agreement with himself to enable one to. follow his thought. In general, he depreciates corporeal things; he deplores the dealings between the soul and the body; insists that the soul is chained down to it, and that she is buried in it as in a tomb. He attaches much importance to the maxim taught in the mysteries that the soul here below is as in a prison. What Plato calls the “cavern” and Empedocles   calls the “grotto,” means no doubt the sense-world. To break her chains, and to issue from the cavern, means the soul’s rising to the intelligible world. In the Phaedrus  , Plato asserts that the cause of the fall of the soul is the loss of her wings; that after having once more ascended on high, she is brought back here below by the periods; that there are souls sent down into this world by judgments, fates, conditions, and necessity; still, at the same time, he finds fault with the “descent” of the soul into the body. But, speaking of the universe in the Timaeus  , he praises the world, and calls it a blissful divinity. He states that the demiurgic creator, being good, gave it a soul to make it intelligent, because without the soul, the universe could not have been as intelligent as it ought to have been. Consequently, the purpose of the introduction of the universal Soul into the world, and similarly of each of our souls was only to achieve the perfection of the world; for it was necessary for the sense-world to contain animals equal in kind and numbers to those contained in the intelligible world. [Ennead IV,8 (6) 1]

If the generating principle were intelligence, what it begot would have to be inferior to intelligence, and nevertheless approximate it, and resemble it more than anything else. Now as the generating principle is superior to intelligence, the first begotten thing is necessarily intelligence. Why, however, is the generating principle not intelligence? Because the act of intelligence is thought, and thought consists in seeing the intelligible; for it is only by its conversion towards it that intelligence achieves a complete and perfect existence. In itself, intelligence is only an indeterminate power to see; only by contemplation of the intelligible does it achieve the state of being determined. This is the reason of the saying, “The ideas and numbers, that is, intelligence, are born from the indefinite doubleness, and the One.” Consequently, instead of being simple, intelligence is multiple. It is composed of several elements; these are doubtless intelligible, but what intelligence sees is none the less multiple. In any case, intelligence is simultaneously the object thought, and the thinking subject; it is therefore already double. [Ennead V,4 (7) 2]

Let us further examine if the indeterminate, or infinite, be an accident, or an attribute of some other nature; how it comes to be an accident, and whether privation ever can become an accident. The things that are numbers and reasons are exempt from all indetermination, because they are determinations, orders, and principles of order for the rest. Now these principles do not order objects already ordered, nor do they order orders. The thing that receives an order is different from that which gives an order, and the principles from which the order is derived are determination, limitation and reason. In this case, that which receives the order and the determination must necessarily be the infinite (as thought Plato). Now that which receives the order is matter, with all the things which, without being matter, participate therein, and play the part of matter. Therefore matter is the infinite itself. Not accidentally is it the infinite; for the infinite is no accident. Indeed, every accident must be a reason; now of what being can the infinite be an accident? Of determination, or of that which is determined? Now matter is neither of these two. Further, the infinite could not unite with the determinate without destroying its nature. The infinite, therefore, is no accident of matter (but is its nature, or “being”). Matter is the infinite itself. Even in the intelligible world, matter is the infinite. [Ennead II,4 (12) 15]

Let us first explain how there can be a plurality of intelligences, souls, and essences. If we consider the things that proceed from the first principles, as they are numbers and not magnitudes, we shall also have to ask ourselves how they fill the universe. This plurality which thus arises from the first principles does not in any way help us to solve our question, since we have granted that essence is multiple because of the difference (of the beings that proceed from it), and not by place; for though it be multiple, it is simultaneously entire; “essence everywhere touches essence,” and it is everywhere entirely present. Intelligence likewise is manifold by the difference (of the intelligences that proceed therefrom), and not by space; it is entire everywhere. It is so also with souls; even their part which is divisible in the bodies is indivisible by its nature. But the bodies possess extension because the soul is present with them; or rather, it is because there are bodies in the sense-world; it is because the power of the Soul (that is universal) which is in them manifests itself in all their parts, that the Soul herself seems to have parts. What proves that she is not divided as they are, and with them, that she is entirely present everywhere, is that by nature she is essentially one and indivisible. Thus, the unity of the Soul does not exclude the plurality of souls, any more than the unity of essence excludes the plurality of (beings), or that the plurality of intelligibles does not disagree with the existence of the One. It is not necessary to admit that the Soul imparts life to the bodies by the plurality of souls, nor that that plurality derives from the extension of the body (of the world). Before there ever were any bodies, there was already one (universal) Soul and several (individual) souls. The individual souls existed already in the universal Soul, not potentially, but each in actuality. The unity of the universal Soul does not hinder the multitude of the individual souls contained within her; the multitude of the individual souls does not hinder the unity of the universal Soul. They are distinct without being separated by any interval; they are present to each other instead of being foreign to each other; for they are not separated from each other by any limits, any more than different sciences are within a single soul. The Soul is such that in her unity she contains all the souls. Such a nature is, therefore, infinite. [Ennead VI,4 (22) 4]

Further, treating of incorporeal things, “parts” is taken in several senses. Speaking of numbers, we may say that two is a part of ten (referring exclusively to abstract numbers). We may also say that a certain extension is a part of a circle or line. Further, a notion is said to be a part of science. [Ennead IV,3 (27) 2]

When dealing with numbers and geometrical figures, as well as with bodies, it is evident that the whole is necessarily diminished by its division into parts, and that each part is smaller than the whole. Rightly, these things should be susceptible to increase or diminution, as their nature is that of definite quantities, not quantity in itself. It is surely not in this sense that, when referring to the soul, we speak of quantities. The soul is not a quantity such as a “dozen,” which forms a whole divisible into unities; otherwise, we would end in a host of absurdities, since a group of ten is not a genuine unity. Either each one of the unities would have to be soul, or the Soul herself result from a sum of inanimate unities. [Ennead IV,3 (27) 2]

Rising therefore to the One, we must add nothing to Him; we must rest in Him, and take care not to withdraw from Him, and fall into the manifold. Without this precaution there will be an occurrence of duality, which cannot offer us unity, because duality is posterior to Unity. The One cannot be enumerated along with anything, not even with uniqueness (the monad), nor with anything else. He cannot be enumerated in any way; for He is measure, without Himself being measured; He is not in the same rank with other things, and cannot be added to other things (being incommensurable). Otherwise, He would have something in common with the beings along with which He would be enumerated; consequently, He would be inferior to this common element, while on the contrary He must have nothing above Him (if He is to be the one first Being). Neither essential (that is, intelligible) Number, nor the lower number which refers to quantity, can be predicated of the unique; I repeat, neither the essential intelligible Number, whose essence is identical with thought, nor the quantative number, which, because all number is quantity, constitutes quantity concurrently with, or independently of other genera. Besides, quantative number, by imitating the former (essential intelligible) Numbers in their relation to the Unique, which is their principle, finds its existence in its relation to real Unity, which it neither shares nor divides. Even when the dyad (or “pair”) is born, (it does not alter) the priority of the Monad (or Uniqueness). Nor is this Uniqueness either of the unities that constitute the pair, nor either of them alone; for why should it be one of them rather than the other? If then the Monad or Uniqueness be neither of the two unities which constitute the pair, it must be superior to them, and though abiding within itself, does not do so. In what then do these unities differ from the Uniqueness (or Monad)? What is the unity of the “pair”? Is the unity formed by the “pair” the same as that which is contained in each of the two unities constituting the “pair”? The unities (which constitute the “pair”) participate in the primary Unity, but differ from it. So far as it is one, the “pair” also participates in unity, but in different ways; for there is no similarity between the unity of a house and the unity of an army. In its relation to continuity, therefore, the “pair” is not the same so far as it is one, and so far as it is a single quantity. Are the unities contained in a group of five in a relation to unity different from that of the unities contained in a group of ten? (To answer this we must distinguish two kinds of unity.) The unity which obtains between a small and a great ship, and between one town and another, and between one army and another, obtains also between these two groups of five and of ten. A unity which would be denied as between these various objects would also have to be denied as obtaining between these two groups. (Enough of this here); further considerations will be studied later. [Ennead V,5 (32) 4]

Returning to our former assertion that the First ever remains identical, even though giving birth to other beings, the generation of numbers may be explained by the immanence of Unity, and by the action of another principle which forms them, as images of unity. So much the more must the Principle superior to beings be immanent Unity; but here it is the First himself who begets the beings, and not another principle who produces beings in the image of the First while this First would abide within Himself. Likewise the form of unity, which is the principle of numbers, exists within all in different degrees, because the numbers posterior to unity participate therein unequally. Likewise, the beings inferior to the First contain something of His nature, which something constitutes their form. Numbers derive their quantity from their participation in unity. Likewise here beings owe their being to their containing the trace of the One, so that their being is the trace of the One. Not far from the truth would we be in holding that essence, which is the (more common or) plainer nomenclature of being, is derived from the word “hen,” which means one. Indeed essence proceeded immediately from the One, and has differentiated from Him but very little. Turning towards its own basis, it has settled, and both became and is the “being” of all. When a man pronounces essence (“on”), and emphasizes it, he unconsciously approximates the sound meaning one (“hen”), demonstrating that essence proceeds from unity, as indeed is indicated, so far as possible, by the word “on,” which means essence. That is why “being” (“ousia”) and essence (“einai”) imitate so far as they can the principle of the Power from which they have emanated. The human mind, observing these similarities, and guided by their contemplation, imitated what it grasped by uttering the words “on,” “einai,” “ousia,” and “hestia.” Indeed, these sounds try to express the nature of what has been begotten by unity, by means of the very effort made by the speaker so as to imitate as well as possible the generation of being. [Ennead V,5 (32) 5]

What opinion should we hold of that which is called the number of infinity? We must begin by examining how it can be a number, if it be infinite. Indeed, sense-objects are not infinite; consequently, the number which inheres in them could not be infinite, and he who numbers them, does not number infinity. Even if they were multiplied by two, or by more, they still could always be determined; if they were multiplied in respect of the past or the future, they would still be determined. It might be objected that number is not infinite in an absolute manner, but only (in a relative manner) in this sense, that it is always possible to add thereto. But he who numbers does not create numbers; they were already determined, and they existed (before being conceived by him who was numbering them). As beings in the intelligible world are determined, their number is also determined by the quantity of beings. Just as we make man manifold by adding to him the beautiful, and other things of the kind, we can make an image of number correspond to the image of every intelligible being. Just as, in thought, we can multiply a town that does not exist, so can we multiply numbers. When we number the parts of time, we limit ourselves to applying to them the numbers that we have in ourselves, and which, merely on that account, do not cease remaining in us. [Ennead VI,6 (34) 2]

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. [Ennead VI,6 (34) 4]

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. [Ennead VI,6 (34) 5]

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. [Ennead VI,6 (34) 6]

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. [Ennead VI,6 (34) 8]

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. [Ennead VI,6 (34) 9]

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, hypostatically 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. [Ennead VI,6 (34) 9]

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. [Ennead VI,6 (34) 11]

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 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. [Ennead VI,6 (34) 14]

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. [Ennead VI,6 (34) 14]

(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 organisms are also in the living Organism, so far as each of them is a living organism. [Ennead VI,6 (34) 15]

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. [Ennead VI,6 (34) 15]

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? [Ennead VI,6 (34) 15]

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. [Ennead VI,6 (34) 16]

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. [Ennead VI,6 (34) 16]

Thus, in the intelligible world, every number is finite. But we can conceive of a number greater than any assigned number, and thus it is that our mind, while considering the numbers, produces the (notion of the) infinite. On the contrary, in the intelligible world, it is impossible to conceive a number greater than the Number conceived (by divine Intelligence); for on high Number exists eternally; no Number is lacking, or could ever lack, so that one could never add anything thereto. [Ennead VI,6 (34) 18]

Very ancient philosophers have investigated the number and kinds of essences. Some said there was but one; others, that there was a limited number of them; others still, an infinite number. Besides, those who recognized but a single (essence) have advanced opinions very different, as is also the case with those who recognized a limited or unlimited number of essences. As the opinions of these philosophers have been sufficiently examined by their successors, we shall not busy ourselves therewith. We shall study the doctrine of those who, after having examined the opinions of their predecessors, decided on determinate numbers (of essences); admitting neither a single essence, because they recognized that there was a multiplicity even in the intelligibles; nor an infinite number of essences, because such an infinity could not exist, and would render all science impossible; but who, classifying the essences whose number is limited, and seeing that these classifications could not be considered elements, looked on them as “kinds.” Of these, some (the Peripatetic Aristotelians) proposed ten, while others proposed a lesser number (the Stoics taught four), or a greater number (the Pythagorean “oppositions,” for instance). As to the kinds, there is also difference of opinions: some looked upon the kinds as principle (Plotinos himself); while others (Aristotle) held that they formed classes. [Ennead VI,1 (42) 1]

If continuous quantity be quantity as far as it is continuous, then definite quantity will no longer be quantity. If, on the contrary, continuous quantity be quantity only accidentally, then there is nothing in common between continuous and definite quantity. We will grant that numbers are quantities, although if their nature of being quantities were plain, one would not see why they should be given that name. As to the line, the surface, and the body, they are called sizes and not quantities; and the latter name is given them only when they are estimated numerically; as when, for instance, they are measured by two or three feet. A body is a quantity only in so far as it is measured, just as space is a quantity only by accident, and not by its spatiality. We must here not consider what is quantity by accident, but by its quantitativeness, quantity itself. Three oxen are not a quantity; in this case, the quantity is the number found in them. Indeed, three oxen belong already to two categories. The case is similar with the line, and the surface, both of which possess such quantity. But if the quantity of surface be quantity itself, why would surface itself be a quantity? It is no doubt only when determined by three or four lines that the surface is called a quantity. [Ennead VI,1 (42) 4]

Shall we then say that numbers alone are quantity? Shall we attribute this privilege to Numbers in themselves, which are beings, because they exist in themselves? Shall we grant the same privilege to numbers existing in things which participate in them, and which serve to number, not unities, but ten oxen, for example, or ten horses? First, it would seem absurd that these numbers should not be beings, if the former ones be such. Then, it will seem equally absurd that they should exist within the things they measure, without existing outside them, as the rules and instruments which serve to measure exist outside of the objects they measure. On the other hand, if these numbers that exist in themselves serve to measure, and nevertheless do not exist within the objects that they measure, the result will be that these objects will not be quantities since they will not participate in quantity itself. [Ennead VI,1 (42) 4]

Why should these numbers be considered quantities? Doubtless because they are measures. But are these measures quantities, or quantity itself? As they are in the order of beings, even if they should not apply to any of the other things, the numbers will nevertheless remain what they are, and they will be found in quantity. Indeed, their unity designates an object, since it applies to another; then the number expresses how many objects there are, and the soul makes use of number to measure plurality. Now, when measuring thus, the soul does not measure the “whatness” (or, quality) of the object, since she says “one,” “two,” whatever be their objects, even if of opposite nature; she does not determine the character of each thing, for instance, if it be warm or beautiful; she limits herself to estimating its quantity. Consequently, whether we take Number in itself, or in the objects which participate therein, quantity exists not in these objects, but in the number; quantity finds itself not in the object three feet long, but in the number three. [Ennead VI,1 (42) 4]

Why then should sizes also be quantities? Probably because they approximate quantities, and because we call quantities all objects that contain quantities, even though we do not measure them with quantity in itself. We call large what numerically participates in much; and small what participates in little. Greatness and smallness are quantities, not absolute, but relative; nevertheless the Aristotelians say that they are relative quantities so far as they seem to be quantities. That is a question to be studied; for, in this doctrine, number is a kind apart, while sizes would hold second rank; it is not exactly a kind, but a category which gathers things which are near each other, and which may hold first or second rank. As to us, we shall have to examine if the Numbers which exist in themselves be only substances, or if they be also quantities. In either case, there is nothing in common between the Numbers of which we speak, and those which exist in things which participate therein. [Ennead VI,1 (42) 4]

In what sense, therefore, could each of the elements of essence be called “one”? In that it is something unitary, without being unity itself; for what is a “certain one” is already manifold. No species is “one” except figuratively; for in itself it is manifold. It is in the same sense that, in this sense-world, we say that an army, or a choric ballet, constitute a unity. Not in such things is absolute unity; and therefore it may not be said that unity is something common. Neither does unity reside in essence itself, nor in the individual essences; therefore, it is not a genus. When a genus is predicated of something, it is impossible to predicate of the same thing contrary properties; but of each of the elements of universal essence it is possible to assert both unity and its opposite. Consequently (if we have called unity a genus), after having predicated of some essence unity as a genus, we would have affirmed, of the same essence, that unity was not a genus. Unity, therefore, could not be considered one of the primary genera; for essence is no more one than it is manifold. As to the other genera, none of them is one without being manifold; much less could unity be predicated of the secondary genera of which each is quite manifold. Besides, no genus, considered in its totality, is unitary; so that if unity were a genus, it would merely thereby cease being unity; for unity is not a number, and nevertheless it would become a number in becoming a genus. Of course, numbers include an alleged unity, as soon as we try to erect it into a genus, it is no longer a unity, in a strict sense. Among numbers unity is not applied to them as would have been a genus; of such unity it is merely said that it is among numbers, not that it is a genus; likewise, if unity were among the essences, it would not be there as genus of essence, nor of anything else, nor of all things. Again, just as the simple is the principle of the composite without being considered a genus in respect to it — then it would be simultaneously simple and composite — so, if one were considered to be a principle, it could not be a genus in respect to things subsumed under it; and therefore will be a genus neither for essence, nor for other (categories or things). [Ennead VI,2 (43) 10]

Consequently, in speaking of (beings) other than (essence itself), as, for instance, of man, we say simply “man” (without adding to it the idea of unity); if however we say “a man,” it is to distinguish him from two; if however we use the word one in still another sense, it is by adding to it “some” (as, “someone”). Not so is it with essence; we say, “being one,” conceiving of “being” (“essence”) and one, as if forming a single whole, and in positing essence as one, we emphasize its narrow affinity with the Good. Thus conceived, essence becomes one; and in the one finds its origin and goal. Nevertheless it is not one as unity itself, but rather in a different manner, in this sense that the (unity of essence) admits priority and posteriority. What then is (the unity of essence)? Must it not then be considered similar in all the parts (of essence), as something common to all (and consequently, as forming a genus)? But in the first place, the point is also something common to all the lines, and nevertheless it is not a genus; in the numbers, unity is something common to all, and is not any more of a genus. Indeed, the unity which is found in the monad, in the dyad (or pair), and in other numbers, cannot be confused with unity in itself. Then, nothing hinders there being in essence some anterior, and other posterior parts, both simple and compound ones (which would be impossible for the One in itself). Even if the unity found everywhere in all the parts of essence were everywhere identical, by the mere fact that it would offer no difference, it could not give rise to species, and consequently, it could not be a genus. [Ennead VI,2 (43) 11]

Now why should we not posit quantity among the primary genera? And why not also quality? Quantity is not one of the primary genera like those we have posited, because the primary genera coexist with essence (which is not the case with quantity). Indeed, movement is inseparable from essence; being its actualization and life. Stability is implied in being; while identity and difference are still more inseparable from essence; so that all these (categories) appear to us simultaneously. As to number (which is discrete quantity), it is something posterior. As to (mathematical) numbers, far more are they posterior both to these genera, and themselves; for the numbers follow each other; the second depends on the first, and so forth; the last are contained within the first. Number, therefore, cannot be posited among the primary genera. Indeed, it is permissible to doubt whether quantity may be posited as any kind of a genus. More even than number, extension (which is continuous quantity), shows the characteristics of compositeness, and of posteriority. Along with number, the line enters into the idea of extension. This would make two elements. Then comes surface, which makes three. If then it be from number that continuous dimension derives its quantitativeness, how could this dimension be a genus, when number is not? On the other hand, anteriority and posteriority exist in dimension as well as in numbers. But if both kinds of quantities have in common this, that they are quantities, it will be necessary to discover the nature of quantity. When this will have been found, we shall be able to make of it a secondary genus; but it could not rank with the primary genera. If, then, quantity be a genus without being a primary one, it will still remain for us to discover to which higher genus, whether primary or secondary, it should be subsumed. [Ennead VI,2 (43) 13]

Let us now pass on to quality. Why does quality also fail to appear among the primary genera? Because quality also is posterior to them; it does indeed follow after being. The first Being must have these (quantity and quality) as consequences, though being is neither constituted nor completed thereby; otherwise, being would be posterior to them. Of course, as to the composite beings, formed of several elements, in which are both numbers and qualities, they indeed are differentiated by those different elements which then constitute qualities, though they simultaneously contain common (elements). As to the primary genera, however, the distinction to be established does not proceed from simpleness or compositeness, but of simpleness and what completes being. Notice, I am not saying, “of what completes ‘some one’ being”; for if we were dealing with some one being, there would be nothing unreasonable in asserting that such a being was completed by a quality, since this being would have been in existence already before having the quality, and would receive from the exterior only the property of being such or such. On the contrary, absolute Being must essentially possess all that constitutes it. [Ennead VI,2 (43) 14]

(Of the essences it contains) it possesses the number, as it is both one and many. It is many, that is, (it is) many potentialities, which are admirable powers, full of force and greatness, because they are pure; powers that are vigorous and veritable because they have no goal at which they are forced to stop; consequently being infinite, that is, supreme Infinity, and Greatness. If then we were to scrutinize this greatness and beauty of being, if by the splendor and light which surround it, we were to distinguish what Intelligence contains, then would we see the efflorescing of quality. With the continuity of actualization we would behold greatness, in quiescent condition. As we have seen one (number), two (quality), and three (greatness), greatness, as the third thing, presents itself with universal quantity. Now, as soon as quality and quantity show themselves to us, they unite, blend into one and the same figure (outward appearance). Then comes difference, which divides quality and quantity, whence arise different qualities, and differences of figure. The presence of identity produces equality, and that of difference, inequality, both in quantity, number, and dimension; hence the circle, the quadrilateral, and the figures composed of unequal things; hence numbers that are similar, and different, even and uneven. [Ennead VI,2 (43) 21]

We have already explained that discrete quantity is clearly distinguished from continuous quantity, both by its own definition, and the general definition (for quantity). We may add that numbers are distinguished from each other by being even and odd. If besides there be other differences amidst the even and odd numbers, these differences will have to be referred to the objects in which are the numbers, or to the numbers composed of unities, and not any more to those which exist in sense-beings. If reason separate sense-things from the numbers they contain, nothing hinders us then from attributing to these numbers the same differences (as to the numbers composed of unities). [Ennead VI,3 (44) 13]

What distinctions are admitted by continuous quantity? There is the line, the surface, and the solid; for extension may exist in one, two or three dimensions (and thus count the numerical elements of continuous size) instead of establishing species. In numbers thus considered as anterior or posterior to each other, there is nothing in common, which would constitute a genus. Likewise in the first, second and third increases (of a line, surface, and solid) there is nothing in common; but as far as quantity is found, there is also equality (and inequality), although there be no extension which is quantitative more than any other. However, one may have dimensions greater than another. It is therefore only in so far as they are all numbers, that numbers can have anything in common. Perhaps, indeed, it is not the monad that begets the pair, nor the pair that begets the triad, but it may be the same principle which begets all the numbers. If numbers be not derivative, but exist by themselves, we may, at least within our own thought, consider them as begotten (or, derivative). We conceive of the smaller number as the anterior, the greater as posterior. But numbers, as such, may all be reduced to unity. [Ennead VI,3 (44) 13]

The method of classification adopted for numbers may be applied to sizes, and thus distinguish the line, the surface, and the solid or body, because those are sizes which form different species. If besides each of these species were to be divided, lines might be subdivided into straight, curved and spiral; surfaces into straight and curved; solids into round or polyhedral bodies. Further, as geometers do, may come the triangle, the quadrilateral, and others. [Ennead VI,3 (44) 13]

It may be objected, that nothing would hinder the existence of manifoldness in the actualization of the First, so long as the “being,” or nature, remain unitary. That principle would not be rendered composite by any number of actualizations. This is not the case for two reasons. Either these actualizations are distinct from its nature (“being”), and the First would pass from potentiality to actuality; in which case, without doubt, the First is not manifold, but His nature would not become perfect without actualization. Or the nature (“being”) is, within Him identical to His actualization; in which case, as the actualization is manifold, the nature would be such also. Now we do indeed grant that Intelligence is manifold, since it thinks itself; but we could not grant that the Principle of all things should also be manifold. Unity must exist before the manifold, the reason of whose existence is found in unity; for unity precedes all number. It may be objected that this is true enough for numbers which follow unity, because they are composite; but what is the need of a unitary principle from which manifoldness should proceed when referring (not to numerals, but) to beings? This need is that, without the One, all things would be in a dispersed condition, and their combinations would be no more than a chaos. [Ennead V,3 (49) 12]