SIXTH TRACTATE.

ON NUMBERS.

1. It is suggested that multiplicity is a falling away from The Unity, infinity being the complete departure, an innumerable multiplicity, and that this is why unlimit is an evil and we evil at the stage of multiplicity.

A thing, in fact, becomes a manifold when, unable to remain self-centred, it flows outward and by that dissipation takes extension: utterly losing unity it becomes a manifold since there is nothing to bind part to part; when, with all this outflowing, it becomes something definite, there is a magnitude.

But what is there so grievous in magnitude?

Given consciousness, there will be, since the thing must feel its exile, its sundrance from its essence. Everything seeks not the alien but itself; in that outward moving there is frustration or compulsion; a thing most exists not when it takes multiplicity or extension but when it holds to its own being, that is when its movement is inward. Desire towards extension is ignorance of the authentically great, a movement not on the appropriate path but towards the strange; to the possession of the self the way is inward.

Consider the thing that has taken extension; broken into so many independent items, it is now those several parts and not the thing it was; if that original is to persist, the members must stand collected to their total; in other words, a thing is itself not by being extended but by remaining, in its degree, a unity: through expansion and in the measure of the expansion, it is less itself; retaining unity, it retains its essential being.

Yet the universe has at once extension and beauty?

Yes; because it has not been allowed to slip away into the limitless but is held fast by unity; and it has beauty in virtue of Beauty not of Magnitude; it needed Beauty to parry that magnitude; in the degree of its extension it was void of beauty and to that degree ugly. Thus extension serves as Matter to Beauty since what calls for its ordering is a multiplicity. The greater the expansion, the greater the disorder and ugliness.

2. What, then, of the “Number of the Infinite”?

To begin with, how is Number consistent with infinity?

Objects of sense are not unlimited and therefore the Number applying to them cannot be so. Nor is an enumerator able to number to infinity; though we double, multiply over and over again, we still end with a finite number; though we range over past and future, and consider them, even, as a totality, we still end with the finite.

Are we then to dismiss absolute limitlessness and think merely that there is always something beyond?

No; that more is not in the reckoner’s power to produce; the total stands already defined.

In the Intellectual the Beings are determined and with them Number, the number corresponding to their total; in this sphere of our own — as we make a man a multiple by counting up his various characteristics, his beauty and the rest — we take each image of Being and form a corresponding image of number; we multiply a non-existent in and so produce multiple numbers; if we number years we draw on the numbers in our own minds and apply them to the years; these numbers are still our possession.

3. And there is the question How can the infinite have existence and remain unlimited: whatever is in actual existence is by that very fact determined numerically.

But, first, if multiplicity holds a true place among Beings, how can it be an evil?

As existent it possesses unity; it is a unit-multiple, saved from stark multiplicity; but it is of a lessened unity and, by that inwoven multiplicity, it is evil in comparison with unity pure. No longer steadfast in that nature, but fallen, it is the less, while in virtue of the unity thence retained it keeps some value; multiplicity has value in so far as it tends to return to, unity.

But how explain the unlimited? It would seem that either it is among beings and so is limited or, if unlimited, is not among beings but, at best, among things of process such as Time. To be brought to limit it must be unlimited; not the limited but the unlimited is the subject of limitation, since between the limited and the unlimited there is no intermediate to accept the principle of limitation. The unlimited recoils by very nature from the Idea of limit, though it may be caught and held by it from without: — the recoil, of course, is not from one place to another; the limitless can have nothing to do with place which arises only with the limiting of the unlimited. Hence what is known as the flux of the unlimited is not to be understood as local change; nor does any other sort of recognisable motion belong to it in itself; therefore the limitless cannot move: neither can it be at rest: in what, since all place is later? Its movement means little more than that it is not fixed in rest.

Is it, then, suspended at some one point, or rocking to and fro?

No; any such poising, with or without side motion, could be known only by place [which Matter precedes].

How, then, are we to form any conception of its being?

We must fasten on the bare notion and take what that gives us — opposites that still are not opposed: we think of large and small and the unlimited becomes either, of stationary and moving, and it will be either of these. But primarily it can be neither in any defined degree, or at once it is under limit. Limitless in this unlimited and undefined way, it is able to appear as either of a pair of opposites: draw near, taking care to throw no net of limit over it, and you have something that slips away; you come upon no unity for so it would be defined; approach the thing as a unit, and you find it manifold; call it a manifold, and again you falsify, for when the single thing is not a unity neither is the total a manifold. In one manifestation it takes the appearance of movement, in another of rest, as the mind envisages it.

And there is movement in its lack of consciousness; it has passed out of Intellectual-Principle, slid away. That it cannot break free but is under compulsion from without to keep to its circling with no possibility of advance, in this would be its rest. Thus it is not true to speak of Matter as being solely in flux.

4. We have to enquire into the existence of the Numbers in the Intellectual. Are they Ideas added to the other Ideas? Or are they no more than necessary concomitants to the Ideas?

In the latter case, Being, as the first [in the Intellectual] would give us the conception of the Monad; then since Being produces motion and rest, Three exists; and so on for all the other members of the realm of Being. Or perhaps there is one monad for each member, or a monad for the first, with a dyad for its next, since there exists a series, and a corresponding number for every successive total, decad for ten, and so on.

If, on the contrary, Number is a direct production of the Intellectual-Principle [an Idea in itself], there is the question whether it preceded or followed the other Ideas.

Plato, where he says that men arrived at the conception of Number by way of the changes of day and night — thus making the concept depend upon variation among things — seems to hold that the things numerable precede and by their differences produce number: Number then would consist in a process within the human mind passing onwards from thing to thing; it results by the fact that the mind takes count, that is when the mind traverses things and reports their differences; observing pure identity unbroken by difference, it says One. But there is the passage where he tells us that the veritable Number has Being, is a Being; this is the opposed view that Number is no product of the reckoning mind but a reality in itself, the concept of which is reawakened in the mind by changes in things of sense.

5. What then is the veritable nature of Number?

Is it an accompaniment upon each substance, something seen in the things as in a man we see one man, in a being one being and in the total of presentations the total of number?

But how explain the dyad and triad? How comes the total to be unitary and any particular number to be brought under unity? The theory offers a multiplicity of units, and no number is reducible to unity but the simple “one.” It might be suggested that a dyad is that thing — or rather what is observed upon that thing — which has two powers combined, a compound thing related to a unity: or numbers might be what the Pythagoreans seem to hold them in their symbolic system in which Justice, for example, is a Tetrad: but this is rather to add the number, a number of manifold unity like the decad, to the multiplicity of the thing which yet is one thing. Now it is not so that we treat the ten things; we bring them together and apply the figure ten to the several items. Or rather in that case we say ten, but when the several items form a unity we say decad. This would apply in the Intellectual as in the sensible.

But how then can number, observed upon things, rank among Real Beings?

One answer might be that whiteness is similarly observed upon things and yet is real, just as movement is observed upon things and there is still a real existence of movement. But movement is not on a par with number: it is because movement is an entity that unity can be observed upon it. Besides, the kind of real existence thus implied annuls the reality of number, making it no more than an attribute; but that cannot be since an attribute must exist before it can be attributed; it may be inseparable from the subject but still must in itself be something, some entity as whiteness is; to be a predicate it must be that which is to be predicated. Thus if unity is observed in every subject, and “one man” says more than “man’s oneness being different from the manness and common to all things — then this oneness must be something prior to man and to all the rest: only so can the unity come to apply to each and to all: it must therefore be prior also to even movement, prior to Being, since without unity these could not be each one thing: of course what is here meant is not the unity postulated as transcending Being but the unity predicable of the Ideas which constitute each several thing. So too there is a decad prior to the subject in which we affirm it; this prior would be the decad absolute, for certainly the thing in which the decad is observed is not that absolute.

Is this unity, then, connate and coexistent to the Beings? Suppose it coexistent merely as an accidental, like health in man, it still must exist of itself; suppose it present as an element in a compound, there must first exist unity and the unity absolute that can thus enter into composition; moreover if it were compounded with an object brought into being by its agency it would make that object only spuriously a unity; its entry would produce a duality.

But what of the decad? Where lies the need of decad to a thing which, by totalling to that power, is decad already?

The need may be like that of Form to Matter; ten and decad may exist by its virtue; and, once more, the decad must previously exist of its own existence, decad unattached.

6. Granted, then, that there exist, apart from things, a unity absolute and a decad absolute in other words, that the Intellectual beings, together with their characteristic essence have also their order, Henads, Dyads, Triads, what is the nature of these numerical entities and how does it come into being? We cannot but think that some reason accounts for their origin.

As a beginning, what is the origin of the Ideas in general? It is not that the thinking principle thought of each Idea and by that act of thought procured their several existences; not because Justice and Movement were thus thought did they come to be; that would imply that while the thought is later than the thing — the concept of Justice must be later than Justice itself — yet the thought precedes what, as founded on the thinking, owes its existence to it. Besides, if justice is only a certain definite thought we have the absurdity that Justice is nothing more than a definition of Justice. Thinking of Justice or Movement is but grasping their nature; this would mean grasping the non-existent, an impossibility.

We may be reminded that in immaterial objects the knowledge is identical with the thing; but we must not misapply that statement; it does not say that the knowledge is the thing known, or that the reason surveying the thing is the thing, but that the immaterial thing, being an Intellectual object is also a thought; this does not imply a definition or conception of the object; the thing itself, as belonging to the Intellectual, can be nothing else than Intellect or knowledge. This is not a case of knowledge self-directed; it is that the thing in the Intellectual transmutes the knowledge, which is not fixed like the knowledge of material things; in other words it makes it true knowledge, that is to say no image of the thing but the thing directly.

Thus it is not the conception of movement that brings movement to be; movement absolute produces that conception; it produces itself as at once movement and the concept of movement, for movement as it exists There, bound up with Being, is a concept. It is movement absolute because it is the first movement — there can be none till this exist — and it is the authentic Movement since it is not accidental to something else but is the activity of actual Being in motion. Thus it is a real existent, though the notion of Being is different.

Justice therefore is not the thought of Justice but, as we may put it, a state of the Intellectual-Principle, or rather an activity of it — an appearance so lovely that neither evening nor dawn is so fair, nor anything else in all the realm of sense, an Intellectual manifestation self-rising, self-seen, or, rather, self-being.

7. It is inevitably necessary to think of all as contained within one nature; one nature must hold and encompass all; there cannot be as in the realm of sense thing apart from thing, here a sun and elsewhere something else; all must be mutually present within a unity. This is the very nature of the Intellectual-Principle as we may know from soul which reproduces it and from what we call Nature under which and by which the things of process are brought into their disjointed being while that Nature itself remains indissolubly one.

But within the unity There, the several entities have each its own distinct existence; the all-embracing Intellect sees what is in it, what is within Being; it need not look out upon them since it contains them, need not separate them since they stand for ever distinct within it.

Against doubters we cite the fact of participation; the greatness and beauty of the Intellectual-Principle we know by the soul’s longing towards it; the longing of the rest towards soul is set up by its likeness to its higher and to the possibility open to them of attaining resemblance through it.

It is surely inconceivable that any living thing be beautiful failing a Life-Absolute of a wonderful, an ineffable, beauty: this must be the Collective Life, made up of all living things, or embracing all, forming a unity coextensive with all, as our universe is a unity embracing all the visible.

8. As then there is a Life-Form primal — which therefore is the Life-Form Absolute — and there is Intellectual-Principle or Being, Authentic Being, these, we affirm, contain all living things and all Number, and Absolute Justice and Beauty and all of that order; for we ascribe an existence of their own to Absolute Man, Absolute Number, Absolute Justice. It remains to discover, in so far as such knowledge is possible, how these distinct entities come to be and what is the manner of their being.

At the outset we must lay aside all sense-perception; by Intellectual-Principle we know Intellectual-Principle. We reflect within ourselves there is life, there is intellect, not in extension but as power without magnitude, issue of Authentic Being which is power self-existing, no vacuity but a thing most living and intellective — nothing more living, more intelligent, more real — and producing its effect by contact and in the ratio of the contact, closely to the close, more remotely to the remote. If Being is to be sought, then most be sought is Being at its intensest; so too the intensest of Intellect if the Intellectual act has worth; and so, too, of Life.

First, then, we take Being as first in order; then Intellectual-Principle; then the Living-Form considered as containing all things: Intellectual-Principle, as the Act of Real Being, is a second.

Thus it is clear that Number cannot be dependent upon the Living-Form since unity and duality existed before that; nor does it rise in the Intellectual-Principle since before that there existed Real Being which is both one and numerous.

9. It remains then to consider whether Being by its distinction produced Number or Number produced that distinction. It is certain that either Number was the cause of Being, movement, rest, identity and difference, or these the cause of Number.

The first question is whether Number can exist in and of itself or is dependent upon things — Two being something observed in two things, Three in three; and so of the arithmetical One, for if this could exist apart from numbered objects it could exist also before the divisions of Being.

But could it precede Being itself?

For the present we must take it that Being precedes Number, is its source. But if One means one being and the duality two beings, then unity precedes Being, and Number precedes the Beings.

Mentally, to our approach? Yes: and in reality of existence as well.

Let us consider: When we think of the existence and the fine appearance of a man as forming one thing, that unity is certainly thought of as subsequent to a precedent duality; when we group a horse with a dog, the duality is obviously the subsequent. But think of that which brings man or horse or dog into being or produces them, with full intention, from where they lie latent within itself: the producer must say “I begin with a first, I pass on to a second; that makes two; counting myself there are three.” Of course there was no such numbering even of Beings for their production, since the due number was known from the very beginning; but this consideration serves to show that all Number precedes the very Beings themselves.

But if Number thus preceded the Beings, then it is not included among them?

The truth is that it existed within the Authentic Being but not as applying to it, for Being was still unparted; the potentiality of Number existed and so produced the division within Being, put in travail with multiplicity; Number must be either the substance of Being or its Activity; the Life-Form as such and the Intellectual-Principle must be Number. Clearly Being is to be, thought of as Number Collective, while the Beings are Number unfolded: the Intellectual-Principle is Number moving within itself, while the Living-Form is Number container of the universe. Even Being is the outcome of the Unity, and, since the prior is unity, the secondary must be Number.

Hence it is that the Forms have been described as Henads and Numbers. This is the authentic Number; the other, the “monadic” is its image. The Authentic is that made manifest in the Forms and helping to bring them to be; primally it is the Number in the Authentic Being, inherent to it and preceding the Beings, serving to them as root, fount, first principle.

For the Unity is source to Being; Being’s Being is stayed upon the Unity as its safeguard from dissolution; the Unity cannot rest upon Being which at that would be a unity before possessing unity; and so with the decad before possessing decadhood.

10. When it takes lot with multiplicity, Being becomes Number by the fact of awakening to manifoldness; — before, it was a preparation, so to speak, of the Beings, their fore-promise, a total of henads offering a stay for what was to be based upon them.

Here with us a man will say “I wish I had such and such a quantity of gold” — or “such and such a number of houses.” Gold is one thing: the wish is not to bring the numerical quantity into gold but to bring the gold to quantity; the quantity, already present in the mind, is to be passed on to the gold so that it acquire that numerical value.

If the Beings preceded the number and this were discerned upon them at the stirring, to such and such a total, of the numbering principle, then the actual number of the Beings would be a chance not a choice; since that total is not a matter of chance, Number is a causing principle preceding that determined total.

Number then pre-exists and is the cause by which produced things participate in quantity.

The single thing derives its unity by participation in Unity-Absolute; its being it derives from Being-Absolute, which holds its Being from itself alone; a unity is a unity in virtue of Being; the particular unity — where the unity is a multiple unity — is one thing only as the Triad is; the collective Being is a unity of this kind, the unity not of the monad but of the myriad or any such collective number.

Take a man affirming the presence of ten thousand things; it is he that produces the number; he does not tell us that the ten thousand have uttered it; they merely exhibit their several forms; the enumerator’s mind supplies the total which would never be known if the mind kept still.

How does the mind pronounce?

By being able to enumerate; that is by knowing Number: but in order to this, Number must be in existence, and that that Principle should not know its own total content is absurd, impossible.

It is with Number as with Good. When we pronounce things to be good either we mean that they are in their own nature so or we affirm goodness as an accidental in them. Dealing with the primals, the goodness we have in mind is that First Hypostasis; where the goodness is an accidental we imply the existence of a Principle of Good as a necessary condition of the accidental presence; there must be some source of that good which is observed elsewhere, whether this source be an Absolute Good or something that of its own nature produces the good. Similarly with number; in attributing the decad to things we affirm either the truly existent decad or, where the decadhood is accidental, we necessarily posit the self-subsistent decad, decad not associated; if things are to be described as forming a decad, then either they must be of themselves the decad or be preceded by that which has no other being than that of decadhood.

It must be urged as a general truth that anything affirmed of a subject not itself either found its way in from outside or is the characteristic Act of that subject; and supposing the predicated attribute to show no variation of presence and absence but to be always present, then, if the subject is a Real Being so also is the accidental in an equal degree; or, failing Real Being, it at least belongs to the existents, it exists. In the case when the subject can be thought of as remaining without its Act, yet that Act is inbound with it even though to our minds it appears as a later; when on the contrary the subject cannot be conceived without the attribute-man, for example, without unity — then the attribute is either not later but concomitant or, being essential to the existence, is precedent. In our view, Unity and Number are precedent.

11. It may be suggested that the decad is nothing more than so many henads; admitting the one henad why should we reject the ten? As the one is a real existence why not the rest? We are certainly not compelled to attach that one henad to some one thing and so deprive all the rest of the means to unity: since every existent must be one thing, the unity is obviously common to all. This means one principle applying to many, the principle whose existence within itself we affirmed to be presupposed by its manifestation outside.

But if a henad exists in some given object and further is observed in something else, then that first henad being real, there cannot be only one henad in existence; there must be a multiplicity of henads.

Supposing that first henad alone to exist, it must obviously be lodged either in the thing of completest Being or at all events in the thing most completely a unity. If in the thing of completest Being, then the other henads are but nominal and cannot be ranked with the first henad, or else Number becomes a collection of unlike monads and there are differences among monads [an impossibility]. If that first henad is to be taken as lodged in the thing of completest unity, there is the question why that most perfect unity should require the first henad to give it unity.

Since all this is impossible, then, before any particular can be thought of as a unit, there must exist a unity bare, unrelated by very essence. If in that realm also there must be a unity apart from anything that can be called one thing, why should there not exist another unity as well?

Each particular, considered in itself, would be a manifold of monads, totalling to a collective unity. If however Nature produces continuously — or rather has produced once for all — not halting at the first production but bringing a sort of continuous unity into being, then it produces the minor numbers by the sheer fact of setting an early limit to its advance: outgoing to a greater extent — not in the sense of moving from point to point but in its inner changes — it would produce the larger numbers; to each number so emerging it would attach the due quantities and the appropriate thing, knowing that without this adaptation to Number the thing could not exist or would be a stray, something outside, at once, of both Number and Reason.

12. We may be told that unity and monad have no real existence, that the only unity is some definite object that is one thing, so that all comes to an attitude of the mind towards things considered singly.

But, to begin with, why at this should not the affirmation of Being pass equally as an attitude of mind so that Being too must disappear? No doubt Being strikes and stings and gives the impression of reality; but we find ourselves just as vividly struck and impressed in the presence of unity. Besides, is this attitude, this concept itself, a unity or a manifold? When we deny the unity of an object, clearly the unity mentioned is not supplied by the object, since we are saying it has none; the unity therefore is within ourselves, something latent in our minds independently of any concrete one thing.

[An objector speaks-] “But the unity we thus possess comes by our acceptance of a certain idea or impression from things external; it is a notion derived from an object. Those that take the notion of numbers and of unity to be but one species of the notions held to be inherent in the mind must allow to numbers and to unity the reality they ascribe to any of the others, and upon occasion they must be met; but no such real existence can be posited when the concept is taken to be an attitude or notion rising in us as a by-product of the objects; this happens when we say “This,” “What,” and still more obviously in the affirmations “Crowd,” “Festival,” “Army,” “Multiplicity.” As multiplicity is nothing apart from certain constituent items and the festival nothing apart from the people gathered happily at the rites, so when we affirm unity we are not thinking of some Oneness self-standing, unrelated. And there are many other such cases; for instance “on the right,” “Above” and their opposites; what is there of reality about this “On-the-right-ness” but the fact that two different positions are occupied? So with “Above”: “Above” and “Below” are a mere matter of position and have no significance outside of this sphere.

Now in answer to this series of objections our first remark is that there does exist an actuality implicit in each one of the relations cited; though this is not the same for all or the same for correlatives or the same for every reference to unity.

But these objections must be taken singly.

13. It cannot reasonably be thought that the notion of unity is derived from the object since this is physical — man, animal, even stone, a presentation of that order is something very different from unity [which must be a thing of the Intellectual]; if that presentation were unity, the mind could never affirm unity unless of that given thing, man, for example.

Then again, just as in the case of “On the right” or other such affirmation of relation, the mind does not affirm in some caprice but from observation of contrasted position, so here it affirms unity in virtue of perceiving something real; assuredly the assertion of unity is not a bare attitude towards something non-existent. It is not enough that a thing be alone and be itself and not something else: and that very “something else” tells of another unity. Besides Otherness and Difference are later; unless the mind has first rested upon unity it cannot affirm Otherness or Difference; when it affirms Aloneness it affirms unity-with-aloneness; thus unity is presupposed in Aloneness.

Besides, that in us which asserts unity of some object is first a unity, itself; and the object is a unity before any outside affirmation or conception.

A thing must be either one thing or more than one, manifold: and if there is to be a manifold there must be a precedent unity. To talk of a manifold is to talk of what has something added to unity; to think of an army is to think of a multitude under arms and brought to unity. In refusing to allow the manifold to remain manifold, the mind makes the truth clear; it draws a separate many into one, either supplying a unity not present or keen to perceive the unity brought about by the ordering of the parts; in an army, even, the unity is not a fiction but as real as that of a building erected from many stones, though of course the unity of the house is more compact.

If, then, unity is more pronounced in the continuous, and more again where there is no separation by part, this is clearly because there exists, in real existence, something which is a Nature or Principle of Unity. There cannot be a greater and less in the non-existent: as we predicate Substance of everything in sense, but predicate it also of the Intellectual order and more strictly there — since we hold that the greater and more sovereign substantiality belongs to the Real Beings and that Being is more marked in Substance, even sensible Substance, than in the other Kinds — so, finding unity to exhibit degree of more and less, differing in sense-things as well as in the Intellectual, we must similarly admit that Unity exists under all forms though still by reference, only, to that primal Unity.

As Substance and Real Being, despite the participation of the sensible, are still of the Intellectual and not the sensible order, so too the unity observed present in things of sense by participation remains still an Intellectual and to be grasped by an Intellectual Act. The mind, from a thing present to it, comes to knowledge of something else, a thing not presented; that is, it has a prior knowledge. By this prior knowledge it recognises Being in a particular being; similarly when a thing is one it can affirm unity as it can affirm also duality and multiplicity.

It is impossible to name or conceive anything not making one or two or some number; equally impossible that the thing should not exist without which nothing can possibly be named or conceived; impossible to deny the reality of that whose existence is a necessary condition of naming or affirming anything; what is a first need, universally, to the formation of every concept and every proposition must exist before reasoning and thinking; only as an existent can it be cited to account for the stirring of thought. If Unity is necessary to the substantial existence of all that really is — and nothing exists which is not one — Unity must precede Reality and be its author. It is therefore, an existent Unity, not an existent that develops Unity; considered as Being-with-Unity it would be a manifold, whereas in the pure Unity there is no Being save in so far as Unity attends to producing it. As regards the word “This,” it is nat a bare word; it affirms an indicated existence without using the name, it tells of a certain presence, whether a substance or some other existent; any This must be significant; it is no attitude of the mind applying itself to a non-existent; the This shows a thing present, as much as if we used the strict name of the object.

14. To the argument touching relation we have an answer surely legitimate:

The Unity is not of a nature to lose its own manner of being only because something else stands in a state which it does not itself share; to stray from its unity it must itself suffer division into duality or the still wider plurality.

If by division the one identical mass can become a duality without loss of quantity, clearly the unity it possessed and by this destructive division lost was something distinct. What may be alternatively present and absent to the same subject must be classed among Real-Beings, regardless of position; an accidental elsewhere, it must have reality in itself whether it be manifested in things of sense or in the Intellectual — an accidental in the Laters but self-existent in the higher, especially in the First in its aspect of Unity developing into Being. We may be told that Unity may lose that character without change in itself, becoming duality by association with something else; but this is not true; unity does not become two things; neither the added nor what takes the addition becomes two; each remains the one thing it was; the duality is predicable of the group only, the unity remaining unchanged in each of those unchanged constituents.

Two and the Dyad are not essentially relative: if the only condition to the construction of duality were meeting and association such a relation might perhaps constitute Twoness and Duality; but in fact we see Duality produced by the very opposite process, by the splitting apart of a unity. This shows that duality — or any other such numerical form — is no relation produced either by scission or association. If one configuration produces a certain thing it is impossible that the opposite should produce the same so that the thing may be identified with the relation.

What then is the actual cause?

Unity is due to the presence of Unity; duality to that of Duality; it is precisely as things are white by Whiteness, just by Justice, beautiful by Beauty. Otherwise we must reject these universals and call in relation here also: justice would arise from a certain attitude in a given situation, Beauty from a certain pattern of the person with nothing present able to produce the beauty, nothing coming from without to effect that agreeable appearance.

You see something which you pronounce to be a unity; that thing possesses also size, form, and a host of other characteristics you might name; size, bulk, sweetness, bitterness and other Ideas are actually present in the thing; it surely cannot be thought that, while every conceivable quality has Real-Being, quantity [Number] has not and that while continuous quantity exists, discrete quantity does not and this though continuous quantity is measured by the discrete. No: as size by the presence of Magnitude, and Oneness by the presence of Unity, so with Duality and all the other numerical modes.

As to the How of participation, the enquiry is that of all participation in Ideal Forms; we must note, however, that the presence of the Decad in the looser totals is different from its presence in the continuous; there is difference again in its presence within many powers where multiplicity is concentred in unity; arrived at the Intellectuals, there too we discover Number, the Authentic Number, no longer entering the alien, Decad-Absolute not Decad of some particular Intellectual group.

15. We must repeat: The Collective Being, the Authentic, There, is at once Being and Intellectual-Principle and the Complete Living Form; thus it includes the total of living things; the Unity There is reproduced by the unity of this living universe in the degree possible to it — for the sense-nature as such cannot compass that transcendental unity — thus that Living-All is inevitably Number-Entire: if the Number were not complete, the All would be deficient to the extent of some number, and if every number applicable to living things were not contained in it, it would not be the all-comprehending Life-Form. Therefore, Number exists before every living thing, before the collective Life-Form.

Again: Man exists in the Intellectual and with him all other living things, both by possession of Real-Being and because that is the Life-Form Complete. Even the man of this sphere is a member of the Intellectual since that is the Life-Form Complete; every living thing by virtue of having life, is There, There in the Life-form, and man is There also, in the Intellectual, in so far as he is intellect, for all intelligences are severally members of That. Now all this means Number There. Yet even in Intellect Number is not present primally; its presence There is the reckoning of the Acts of Intellectual-Principle; it tallies with the justice in Intellectual-Principle, its moral wisdom, its virtues, its knowledge, all whose possession makes That Principle what it is.

But knowledge — must not this imply presence to the alien? No; knowledge, known and knower are an identity; so with all the rest; every member of Intellectual-Principle is therefore present to it primally; justice, for example, is not accidental to it as to soul in its character as soul, where these virtues are mainly potential becoming actual by the intention towards Intellectual-Principle and association with it.

Next we come to Being, fully realized, and this is the seat of Number; by Number, Being brings forth the Beings; its movement is planned to Number; it establishes the numbers of its offspring before bringing them to be, in the same way as it establishes its own unity by linking pure Being to the First: the numbers do not link the lower to the First; it suffices that Being is so linked; for Being, in taking form as Number, binds its members to itself. As a unity, it suffers no division, remaining self-constant; as a thing of division, containing its chosen total of members, it knows that total and so brings forth Number, a phase therefore of its content: its development of part is ruled by the powers of Number, and the Beings it produces sum to that Number. Thus Number, the primal and true, is Principle and source of actuality to the Beings.

Hence it is that in our sphere, also, Number accompanies the coming to be of particular things and to suppose another number than the actual is to suppose the production of something else or of nothing.

These then are the primal numbers; they are numerable; the numbers of the other order are of a double character; as derived from the first numbers they are themselves numerable but as acting for those first they are measures of the rest of things, numbering numbers and numerables. For how could they declare a Decad save in the light of numbers within themselves?

16. But here we may be questioned about these numbers which we describe as the primal and authentic:

“Where do you place these numbers, in what genus among Beings? To everyone they seem to come under Quantity and you have certainly brought Quantity in, where you say that discrete Quantity equally with the continuous holds place among Beings; but you go on to say that there are the numbers belonging to the Firsts and then talk of other numbers quite distinct, those of reckoning; tell us how you arrange all this, for there is difficulty here. And then, the unity in sense-things — is that a quantity or is quantity here just so many units brought together, the unity being the starting-point of quantity but not quantity itself? And, if the starting-point, is it a kindred thing or of another genus? All this you owe it to us to make clear.”

Be it so; we begin by pointing out a distinction:

You take one thing with another — for we must first deal with objects of sense — a dog and a man, or two men; or you take a group and affirm ten, a decad of men: in this case the number affirmed is not a Reality, even as Reality goes in the sphere of sense, but is purely Quantity: similarly when you resolve into units, breaking up the decad, those units are your principle of Quantity since the single individual is not a unity absolute.

But the case is different when you consider one man in himself and affirm a certain number, duality, for example, in that he is at once living and reasoning.

By this analysis and totalling, you get quantity; but there are two objects under consideration and each of these is one; each of the unities contributes to the complete being and the oneness is inherent in each; this is another kind of number; number essential; even the duality so formed is no posterior; it does not signify a quantity apart from the thing but the quantity in the essence which holds the thing together. The number here is no mere result of your detailing; the things exist of themselves and are not brought together by your reckoning, but what has it to do with essential reality that you count one man in with another? There is here no resultant unity such as that of a choir — the decad is real only to you who count the ten; in the ten of your reckoning there cannot be a decad without a unitary basis; it is you that make the ten by your counting, by fixing that tenness down to quantity; in choir and army there is something more than that, something not of your placing.

But how do you come to have a number to place?

The Number inherent apart from any enumeration has its own manner of being, but the other, that resulting upon the appearance of an external to be appraised by the Number within yourself, is either an Act of these inherent numbers or an Act in accordance with them; in counting we produce number and so bring quantity into being just as in walking we bring a certain movement into being.

But what of that “Number within us having its own manner of being”?

It is the Number of our essence. “Our essence” we read “partakes of Number and harmony and, also, is Number and harmony.” “Neither body nor magnitude,” someone says: soul, then, is Number since it is essence. The number belonging to body is an essence of the order of body; the number belonging to soul constitutes the essences of souls.

In the Intellectuals, all, if the Absolute Living-Form, there is a multiple — a triad, let us say — that Triad of the Living-Form is of the nature of essence: and the Triad prior to any living thing, Triad in the realm of Being, is a principle of essence.

When you enumerate two things — say, animal and beauty — each of these remains one thing; the number is your production; it lay within yourself; it is you that elaborate quantity, here the dyad. But when you declare virtue to be a Tetrad, you are affirming a Tetrad which does actually exist; the parts, so to speak, make one thing; you are taking as the object of your act a Unity — Tetrad to which you accommodate the Tetrad within yourself.

17. But what of the Infinite Number we hear of; does not all this reasoning set it under limit?

And rightly so if the thing is to be a number; limitlessness and number are in contradiction.

How, then, do we come to use the term? Is it that we think of Number as we think of an infinite line, not with the idea that any such lire exists but that even the very greatest — that of the [path of the] universe, for example — may be thought of as still greater? So it might be with number; let it be fixed, yet we still are free to think of its double, though not of course to produce the doubled quantity since it is impossible to join to the actual what is no more than a conception, a phantasm, private to ourselves.

It is our view that there does exist an infinite line, among the Intellectual Beings: for There a line would not be quantitative and being without quantity could be numerically infinite. This however would be in another mode than that of limitless extension. In what mode then? In that the conception of the Absolute Line does not include the conception of limit.

But what sort of thing is the Line in the Intellectual and what place does it hold?

It is later than Number since unity is observed in it; it rises at one point and traverses one course and simply lacks the quantity that would be the measure of the distance.

But where does this thing lie? Is it existent only in the defining thought, so to speak?

No; it is also a thing, though a thing of the Intellectual. All that belongs to that order is at once an Intellectual and in some degree the concrete thing. There is a position, as well as a manner of being, for all configurations, for surface, for solid. And certainly the configurations are not of our devising; for example, the configurations of the universe are obviously antecedent to ourselves; so it must be with all the configurations of the things of nature; before the bodily reproductions all must exist There, without configuration, primal configurations. For these primals are not shapes in something; self-belonging, they are perfect without extension; only the extended needs the external. In the sphere of Real-Being the configuration is always a unity; it becomes discrete either in the Living-Form or immediately before: I say “becomes discrete” not in the sense that it takes magnitude There but that it is broken apart for the purpose of the Living-Form and is allotted to the bodies within that Form — for instance, to Fire There, the Intellectual Pyramid. And because the Ideal-Form is There, the fire of this sphere seeks to produce that configuration against the check of Matter: and so of all the rest as we read in the account of the realm of sense.

But does the Life-Form contain the configurations by the mere fact of its life?

They are in the Intellectual-Principle previously but they also exist in the Living-Form; if this be considered as including the Intellectual-Principle, then they are primally in the Life-Form, but if that Principle comes first then they are previously in that. And if the Life-Form entire contains also souls, it must certainly be subsequent to the Intellectual-Principle.

No doubt there is the passage “Whatever Intellect sees in the entire Life-Form”; thus seeing, must not the Intellectual-Principle be the later?

No; the seeing may imply merely that the reality comes into being by the fact of that seeing; the Intellectual-Principle is not external to the Life-Form; all is one; the Act of the Intellectual-Principle possesses itself of bare sphere, while the Life-Form holds the sphere as sphere of a living total.

18. It appears then that Number in that realm is definite; it is we that can conceive the “More than is present”; the infinity lies in our counting: in the Real is no conceiving more than has been conceived; all stands entire; no number has been or could be omitted to make addition possible. It might be described as infinite in the sense that it has not been measured — who is there to measure it? — but it is solely its own, a concentrated unit, entire, not ringed round by any boundary; its manner of being is settled for it by itself alone. None of the Real-Beings is under limit; what is limited, measured, is what needs measure to prevent it running away into the unbounded. There every being is Measure; and therefore it is that all is beautiful. Because that is a living thing it is beautiful, holding the highest life, the complete, a life not tainted towards death, nothing mortal there, nothing dying. Nor is the life of that Absolute Living-Form some feeble flickering; it is primal, the brightest, holding all that life has of radiance; it is that first light which the souls There draw upon for their life and bring with them when they come here. It knows for what purpose it lives, towards What it lives, from Whence it lives; for the Whence of its life is the Whither... and close above it stands the wisdom of all, the collective Intellectual-Principle, knit into it, one with it, colouring it to a higher goodness, by kneading wisdom into it, making its beauty still more august. Even here the august and veritably beautiful life is the life in wisdom, here dimly seen, There purely. For There wisdom gives sight to the seer and power for the fuller living and in that tenser life both to see and to become what is seen.

Here attention is set for the most part upon the unliving and, in the living, upon what is lifeless in them; the inner life is taken only with alloy: There, all are Living Beings, living wholly, unalloyed; however you may choose to study one of them apart from its life, in a moment that life is flashed out upon you: once you have known the Essence that pervades them, conferring that unchangeable life upon them, once you perceive the judgement and wisdom and knowledge that are theirs, you can but smile at all the lower nature with its pretention to Reality.

In virtue of this Essence it is that life endures, that the Intellectual-Principle endures, that the Beings stand in their eternity; nothing alters it, turns it, moves it; nothing, indeed, is in being besides it to touch it; anything that is must be its product; anything opposed to it could not affect it. Being itself could not make such an opposite into Being; that would require a prior to both and that prior would then be Being; so that Parmenides was right when he taught the identity of Being and Unity. Being is thus beyond contact not because it stands alone but because it is Being. For Being alone has Being in its own right.

How then can we deny to it either Being or anything at all that may exist effectively, anything that may derive from it?

As long as it exists it produces: but it exists for ever; so, therefore, do its products. And so great is it in power and beauty that it remains the allurer, all things of the universe depending from it and rejoicing to hold their trace of it and through that to seek their good. To us, existence is before the good; all this world desires life and wisdom in order to Being; every soul and every intellect seeks to be its Being, but Being is sufficient to itself.