VI.6 On Numbers

The aspiration to be great in this way belongs to someone who does not know what is really great and who hurries not where he should, but toward the outside; whereas what is turned toward itself was the inside. The evidence is what has come to be in magnitude. If it is broken up so that each of its parts exists on its own, then those parts are each things, but not the original thing itself; if it is to be itself, all its parts must be directed toward one. So it is itself when it is in some way one, not great. Through magnitude, then, and to the extent of its magnitude, it comes to be lost to itself; but insofar as it has one, it has itself. And yet the All is surely great and beautiful. Or rather: it is because it was not let go to flee into unlimitedness, but was encompassed by one; and it is beautiful not by being great but by the beautiful; and it needed the beautiful because it had become great.

For bereft of this, the greater it was, the uglier it would have appeared; and so magnitude is matter for the beautiful, because what needs ordering is much. So the great is more disordered and more ugly.

Or perhaps: in the intelligible world, number, like the beings, is limited, as many as the beings are. But just as we make man many by applying him many times—and beauty and the rest likewise—so along with each image we also generate an image of number; and just as we multiply a city that does not exist, in the same way we make the numbers many times greater. And if we count times, we bring to bear on the times the numbers we have, while those numbers remain in us.

Or rather, even if it is limited, it is for that very reason unlimited: for it is not the limit that is limited, but the unlimited, since there is nothing else between limit and unlimited to receive the nature of a boundary. This unlimited, for its part, flees the Form of limit, but it is caught when encompassed from outside. It does not flee from one place into another—it does not even have a place—but when it is caught, place came into existence. That is why we must not take its so-called motion to be motion in place, nor take any of the other motions attributed to it to belong to it of itself; so it would not move. But neither does it stand still: where would it stand, when the “where” comes later?

And if you come close without throwing some limit over it like a net, you will have it slipping out from under you, and you will find no single one; for then you [would] already have limited it. If you approach something in it as one, it will appear many; and if you say “many,” you will be wrong again, for since each is not one, all of them together are not many either. And this nature of it is motion in respect of one of its appearances, and rest in respect of the way imagination approached it. And its inability to see itself through itself is a motion away from intellect, a slipping away; while its having no way to escape, being fenced in from outside and all around and not allowed to advance, would be rest. So one cannot say that it only moves.

Plato, when he says that human beings came to the notion of number through the alternation of days and nights, attributes the thinking of it to the otherness of things. He might, then, be saying that countable things first produce number through their otherness, and that number is constituted in the passage of a soul going on from one thing to another, and comes to be at the moment the soul counts—that is, when it goes through them and says to itself “this one,” and then “that one,” whereas so long as it thinks something the same, with nothing other after it, it says “one.” But when he speaks of [being] in the true number, and of number in being, he would again be speaking of a certain existence of number by itself, one that does not come into existence in the counting soul; rather, the notion of number is stirred up in the soul by the alternation in perceptible things.note

For in this way there will be a multiplicity of unities, but no number that goes into one, apart from the simple one—unless someone were to say that a dyad is that thing, or rather what is observed in the thing, which has two powers comprehended together, like something composite made into one; or numbers of the kind the Pythagoreans spoke of, who seemed to speak of numbers by analogy, calling justice a tetrad, for instance, and so on in other cases. But rather in this way: to the multiplicity of the thing, though the thing is one, they paired a number too, a one of that size—a decad, for instance. And yet that is not how we speak of ten things; we bring together even things that are separate and call them ten. Or perhaps we call them “ten” in this way, but when one comes to be from many, we call it “a decad,” as is the case There too. But if so, will number still have existence in things, if it is merely observed in them?

“But what prevents,” someone might say, “white too, observed in things, from having existence in the things? Motion too, observed in being, had existence as a motion that is in being.” But number is not like motion. Rather, because motion is something, it is in this way that “one” was observed in it; so it is said. Then again, existence of this kind removes number from being a substance and makes it rather an accident.

And so it is prior to motion too, if motion too is one, and prior to being, so that being too may obtain being one. I don’t mean that One which we say is beyond being, but this one which is predicated of each of the Forms. And so the decad, too, is prior to what “decad” is predicated of; and this will be the Decad itself, since the thing in which a decad is observed will not be the Decad itself. But was it born together with the beings, and did it come into existence together with them? If it was born together with them as an accident, like health in a man, it must also exist by itself; and if the one is like an element of a compound, there must first be one, the one itself, so that it can be with something else. Then, mixed with something else, it will make the product that came to be through it one falsely, making it two. And how is it with the decad?

For what need of the decad will that thing have which is to be a decad by virtue of so great a power? But if the decad will give it form, as if it were matter, and the thing will be ten and a decad by the presence of a decad, then the decad must first exist by itself, being nothing other than a decad alone.

Justice did not come to be because he thought what justice is, nor did motion come into existence because he thought what motion is. For in that case this thought would have to be both later than the very thing thought—the thinking of justice later than justice itself—and, again, the thinking would be prior to what came into existence from the thinking, if it came into existence by being thought. And if justice is the same as a thinking of this kind, then, first, it is absurd that justice should be nothing but, as it were, its definition. For what is it to have thought justice or motion, except to grasp what they are? And this is the same as grasping the account of a thing that does not exist—which is impossible.

So the thinking of motion did not make Motion Itself; Motion Itself made the thinking, so that it made itself both motion and thinking. For the motion There is also the thinking of that motion; and it is itself motion because it is first—there is no other before it—and real motion, because it is not an accident of something else, but the activity of what moves, which is in activity. So it is also substance, though the concept of being is different. And justice is not the thinking of justice, but, as it were, a disposition of Intellect—or rather an activity of a certain kind—whose face is truly beautiful, and “neither the evening star [nor the morning star is so beautiful],” nor anything at all among perceptible things. It is like an intellectual statue, standing, as it were, out of itself and appearing in itself—or rather, being in itself.note

We convince those who are amazed at this by appeal to the things that participate in it. Its greatness and beauty are reflected in soul’s love for it, in other things’ love for soul because of a nature like this, and in soul’s having been made like it in some respect by possessing it. And indeed it would be absurd for there to be a beautiful living being, while the Living Being Itself was not of wondrous and inexpressible beauty. The complete Living Being, then, is made up of all living beings—or rather contains all living beings in itself—and is one, as great as all of them together, just as this All too is one and, containing all the visible, contains all that is in the visible.note

First, then, we must let go of all perception, and contemplate Intellect by intellect, and bear in mind that in us too life and intellect are not in bulk but in a power without bulk; and that true being has stripped these off and is a power standing on its own—not some feeble, shadowy thing, but the most living and most intellectual of all. Nothing is more living than it, nothing more intellectual, nothing more fully being; and whatever touches it has these qualities in proportion to the contact, what is near more nearly, what is far more distantly. If, then, being is desirable, what is most being is more so, and so is what is most intellect, if thinking is desirable at all; and likewise with life.

If, then, we must take being first, as first, then Intellect, then the Living Being—for this already seems to contain all things—and Intellect is second, since it is the activity of being, then number would not correspond to the Living Being—for one and two existed already before it—nor to Intellect; for before Intellect there was being, which was one and many.

But if the one is one being and two are two beings, then the one will precede being, and number will precede the beings. In conception and in attention, then? Or in existence as well? Let us consider it this way. When someone thinks of a man who exists and is beautiful as one, he surely thinks the one afterwards, in each case; and likewise when he thinks of a horse and a dog—and clearly the two comes later here. But if he were to generate a man, and generate a horse and a dog, or bring them forth when they are in him, and neither generate nor bring them forth at random, will he not say: “I must go to one, then pass on to another one and make two, and make another along with myself”?

For since being came from the One, as that was one, being must in this way be number. That is why they called the Forms both unities and numbers. And this is substantial number; the other, called monadic number, is an image of it. Substantial number is, on the one hand, observed in the Forms and helps to generate them; but primarily it is the number in being, with being, and before the beings. The beings have in it their base, source, root and principle. For the one is principle for being, and being rests on it—otherwise it would be scattered; but the one does not rest on being. For then it would already be one before obtaining the one, and what obtains the decad would already be a decad before obtaining the decad.

If, then, they are not so many at random, number, existing beforehand, is the cause of their being so many. That is, number already existing, the things that came to be participated in “so many,” and each participated in “one” in order to be one. Each is a being from being, since being too is being from itself; and one from the one. And each is one—if the one applied to them was many together—as a triad is one, and all the beings are one in this way: not as the one of the unit is one, but as ten thousand, or some other number, is one.

For someone who speaks of ten thousand things that have already come to be—if the one counting said “ten thousand”—does not claim that the “ten thousand” is announced by the things themselves, displaying it as they display their colors, but that discursive thought says “so many.” For if it did not say so, it would not know how great the multitude is. How then will it say so?

Necessarily, then, if he calls the beings a decad, either they themselves are a decad, or before them there is another decad which is nothing other than just this, a decad. In general, then, it must be shown that whatever is predicated of something else either has come into it from something else or is its activity. And if it is such as not to be present at one time and absent at another, but is always with it, then if that thing is substance, it too is substance, and that thing is no more substance than it is; and if one will not grant that it is substance, at any rate it is among the beings and is a being.

And if that thing can be thought without its activity, the activity is nonetheless simultaneous with it, but placed later in conception by us. But if it is not possible to think the thing without it—man without the one, for instance—then either it is not later than the thing but coexists with it, or it is prior to it, so that the thing exists through it. And we say that the one and number are prior.

But if there is a unity in this thing and again one observed in another, then if that one too exists, not just one unity will have existence, and so there will be a multiplicity of unities. But if only that first one exists, it is together either with what is most being or with what is in every way most one. If it is with what is most being, the other unities would be unities in name only and would not be ranked with the first; or else number will be made of unlike units, and there will be differences among units insofar as they are units. And if it is with what is most one, why would what is most one need this unit in order to be one? If these things are impossible, there must be a one that is nothing other than bare one, isolated in its own being, before each thing is called and thought one.

If, then, the one exists There too without the thing that is called one, why will another one not come into existence as well? Taken separately, each of them gives many units, which are also many ones. But if nature were to generate in sequence, as it were—or rather, were to have generated, not stopping at each one of the things it generated, as though making a continuous one—then by marking off a boundary and stopping sooner in its procession it would generate the smaller numbers, and by moving further—not upon other things, but in the motions themselves—it would bring the larger numbers into existence. And so it would fit each multiplicity and each of the beings to each number, knowing that if each were not fitted to each number, it would either not exist at all, or would have stepped aside into something else, become numberless and irrational.

And there are many other such things, they say, like “right” and “above” and their opposites. For what is there in “right” that amounts to existence, except that one person stands or sits here and another there? And likewise with “above”: one thing has a certain position and is more in that part of the All which we call “above,” another in what is called “below.” To arguments like these, the first thing to say is that there is some existence of what has been mentioned in each of these cases—though not the same in all of them, neither for these compared with one another nor for all of them compared with the one. But we must attend to each of the points made separately.

For unless discursive thought had first settled on one, it would say neither “other” nor “different.” And when it says “alone,” it says “one alone,” so it says “one” before “alone.” Then too, what speaks is one before it says “one” of anything else, and what it speaks about is one before anyone says or thinks anything about it. For it is either one, or more than one and many; and if many, one must exist before.

For there cannot be a “more” among things that are not. We predicate substance of each perceptible thing, and predicate it of the intelligibles as well, but we make the predication more properly of the intelligibles, placing the “more” and the “more properly” among the things that are; and being belongs more to substance, even perceptible substance, than to the other kinds. In the same way, when we see the one present more and more properly, differing by degree both among the perceptibles themselves and among the intelligibles, we must say that it is there in all these ways, but always with reference to one.

Just as substance and being are intelligible and not perceptible, even if the perceptible participates in them, so too the one might be contemplated in the perceptible by participation. But discursive thought grasps it as intelligible, and intelligibly. So from one thing it thinks another, which it does not see. Then it knew it beforehand.

That is why it is one being, and not being first and then one. In “being and one” there would be many, but in “one” being is not contained, unless the one should make it by inclining toward its generation. Nor is “this” empty. In place of the thing’s name it speaks of a real existence being pointed out, a kind of presence, a substance or something else among the things that are. So “this” would signify nothing empty, nor is it an affection of discursive thought referring to nothing that is; there is a real underlying thing, just as if one were to say its proper name.

We must also say that it is accidental to these things but exists by itself, both in the perceptibles, when it appears there, and in the intelligibles: an accident in the things that come later, but on its own in the intelligibles, in the first, when it is one and then being. If someone says that the one, too, will no longer be one but two when something else is added to it, though it has not been affected, he will be wrong. For the one did not become two, neither the thing to which something was added nor what was added. Each remains one, as it was. “Two” is predicated of both together, while “one” is predicated separately of each, which remains as it is.

So two and the dyad are not by nature in relation. If they depended on coming together, and being in a coming-together were the same as making two, then perhaps such a relation would be two and the dyad. But as it is, the dyad is seen again in the opposite affection as well: when some one thing is split, it becomes two. So two is neither coming together nor splitting, as it would have to be if it were a relation. The same argument holds for every number. When it is a relation that generates something, the opposite relation cannot generate the same thing; only then could the thing be the relation.

Surely it will not be that every quality whatever exists while quantity has no place among the things that are; nor that continuous quantity exists while discrete quantity does not, when the continuous uses the discrete as its measure. So just as a thing is large by the presence of largeness, it is one by the presence of the one and two by that of the dyad, and so on with the rest. To ask how things participate is to ask the question of participation common to all the Forms. But we must say that the decad is seen present in one way in discrete things, in another in continuous things, and in yet another in the powers that are so many and gathered into one. And we must say that we have now ascended to the intelligibles; and further, that There the numbers are no longer seen in other things but exist themselves by themselves, and these are the truest numbers: the decad itself, not a decad of some intelligibles.

Man, then, is in the intelligible, and so are the other living beings, insofar as each is what it is, and insofar as that is an all-complete living being. For man here too, insofar as the All is a living being, is a part of it; and each, insofar as it is a living being, is There in the Living Being. But in Intellect, insofar as it is Intellect, all the individual intellects are like parts, and of these too there is a number. So number is not primarily in Intellect either. It is in Intellect in the way all the activities of Intellect are, and as Intellect’s justice and moderation and the other virtues are, and knowledge, and all those things by having which Intellect really is Intellect. How, then, is knowledge not in something else?

After this, then, comes being, and in it number, with which being generates the beings, moving according to number and setting the numbers before itself as the ground of their real existence. In the same way its own one links being itself to the First, while the numbers do not go on to link the other things to the First: it is enough that being is linked. Being, once it has become number, links the beings to itself. For it is split not insofar as it is one; its one remains. But splitting according to its own nature into as many as it wished, it saw into how many, and so generated the number, which was, then, in it. For it was split by the powers of number, and generated as many beings as the number was. So the first and true number is the principle and source of real existence for the beings.

That is why here too the coming-to-be of each thing is with numbers, and if something takes on a different number, it either generates something else or comes to nothing. These are the first numbers, as numbered. The numbers in other things have both characters: insofar as they come from these, they are numbered; insofar as they are according to these, they measure other things by counting, both numbers and numbered things. For by what would they say “ten” except by the numbers in themselves?

We must begin from here, then, and say about these things that when—but first we must make the argument about perceptibles. When you take one thing with another and say “two,” a dog and a man, say, or two men or more, and say “ten” and “a decad of men,” this number is not a substance, not even in the way substance is among perceptibles, but purely a quantity. And when you divide it by ones and make the ones parts of this decad, you are making and positing the one as a principle of quantity; for one of the ten is not one by itself.

But when you say that man himself, by himself, is a number, a dyad, say—living being and rational—then there is no longer a single mode here. Insofar as you go through them and count, you produce a quantity. But insofar as the underlying things are two and each is one, if the one in each fills out the substance and the unity is in each, you are speaking of another number, a substantial one. This dyad is not posterior, and it does not merely state a quantity external to the thing; it is what is in the substance and holds the thing’s nature together. For here it is not you who make the number, going through things that exist by themselves and do not come together by being counted.

Or rather, it is the number of our substance. “Participating in number and harmony,” he says; and soul in turn is number and harmony. For it is neither body, someone says, nor magnitude; so soul is a number, if it is a substance. The number of body is substance in the way body is; the number of soul is substances in the way souls are. And in general, in the intelligibles, if the Living Being There is itself many—a triad, say—this triad, the one in the Living Being, is substantial. But the triad that does not yet belong to a living being, simply triad in being, is a principle of substance. If you count “living being” and “beautiful,” each is one, but you generate number in yourself and produce by your activity a quantity, a dyad.note

If, however, you were to say that virtue is four, it is also a kind of tetrad, its parts gathered, as it were, into one; and, as substrate, it is a tetrad that is a unity; and you fit to it the tetrad in you.

Or is it unlimited in another way, not as what cannot be traversed? But how is it unlimited? Perhaps in that, in the rational principle of the line itself, no limit is thought along with it. What, then, is a line There, and where is it? It is posterior to number, for the one is seen in it: it starts from one and runs in one dimension. But it does not have a quantity as the measure of its extension. Where, then, is this? Only in thought, as a kind of defining thought?

When I say “distinguished,” I do not mean that it acquired magnitude, but that each was apportioned to each, as to the living beings, and given to the bodies There: to the fire There, for instance, if you like, the pyramid There. That is why this fire here wants to imitate it, though it cannot because of matter; and the rest analogously, as is said about the things here. But is it in the Living Being insofar as it is a living being? Or rather, before that, in Intellect. It is indeed in the Living Being; so if the Living Being contained Intellect, it would be in the Living Being first, but if Intellect comes first in order, it is first in Intellect. But if there are souls too in the all-complete Living Being, Intellect is prior. Yet he says “Intellect, as many as it sees in the all-complete Living Being.” If, then, it sees, it is posterior.note

Or perhaps “sees” can be said in the sense that the real existence comes about in the seeing. For it is not something other; all is one. Thinking has the bare sphere, and the Living Being has the sphere of a living being.

For insofar as it is a living being, it is beautiful: it has the best life, lacks no life, and its life is not mixed with death; for nothing There is mortal or dying. Nor is the life of the Living Being itself feeble. It is the first life, the most vivid, with all the clarity of living, like the first light, from which the souls There live and which the souls that come here bring with them. It also knows for whose sake it lives and toward what it lives, that from which it lives; for what it lives from it also lives toward.

And the understanding of all things, the whole Intellect, which is upon it and with it and together with it, tinges it with greater goodness and mixes understanding into it, and so makes its beauty more majestic. For even here a life with understanding is what is truly majestic and beautiful, though it is seen dimly; There it is seen purely. It gives the one who sees sight, and the power to live more fully, and, living more intensely, to see more and to become what it sees.

For it is from this being that life remains, and Intellect remains, and the beings stand in eternity. Nothing displaces it, nothing alters it, nothing disturbs it; for there is no being after it that could touch it, and if there were, it would depend on it. And if there were something opposed to it, this would remain unaffected by the opposite itself. And if the opposite were itself being, it would not have made this being; there would be something else prior to it, common to both, and that would be being. So in this respect Parmenides was right to say that being is one. And it is unaffected not because nothing else is there, but because it is being. For to it alone belongs being from itself. How, then, could anyone take away from it being, or anything else of all that really is in activity and all that comes from it?

For as long as it is, it supplies; and it always is; so those things always are too. It is so great in power and beauty that it enchants, and all things depend on it, and having a trace of it from it they cherish it, and after that seek the Good. For being comes before that, as far as we are concerned. This whole cosmos wants to live and to understand in order to be; and every soul and every intellect wants to be what it is. And being is sufficient to itself.