Lecture 10 — The infinite exists not as being in act, but as being in potency

It is plain from these arguments that there is no body that is actually infinite. But, on the other hand, to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and an end of time, a magnitude will not be divisible into magnitudes, number will not be infinite.

If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in, and clearly there is a sense in which the infinite exists and another in which it does not.

We must keep in mind that the word “is” means either what potentially is or what fully is. Further, a thing is infinite either by addition or by division.

Now, as we have seen, magnitude is not actually infinite. But, by division, it is infinite. (There is no difficulty in refuting the theory of indivisible lines.) The alternative, then, remains that the infinite has a potential existence.

But the phrase “potential existence” is ambiguous. When we speak of the potential existence of a statue, we mean that there will be an actual statue. It is not so with the infinite. There will not be an actual infinite.

The word “is” has many senses, and we say that the infinite “is” in the sense in which we say “it is day” or “it is the games,” because one thing after another is always coming into existence.

For of these things too, the distinction between potential and actual existence holds. We say that there are Olympic games both in the sense that they may occur and that they are actually occurring.

The infinite exhibits itself in different ways—in time, in the generations of man, and in the division of magnitudes.

For generally, the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different.

Again, “being” has more than one sense, such that we must not regard the infinite as this substance, such as a man or a horse, but must suppose it to exist in the sense in which we speak of the day or the games as existing things whose being has not come to them like that of a substance, but consists in a process of generation or passing away—definite, if you like, at each stage, yet always different.

But, when this takes place in spatial magnitudes, what is taken persists, while in the succession of time and of men, it takes place by the passing away of these in such a way that the source of supply never gives out.

In a way, the infinite by addition is the same thing as the infinite by division. In a finite magnitude, the infinite by addition comes about in a way inverse to that of the other. For as we see division going on, in the same proportion we see addition being made to what is already marked off.

For if we take a determinate part of a finite magnitude and add another part determined by the same ratio (not taking in the same amount of the original whole), and so on, we shall not traverse the given magnitude.

But, if we increase the ratio of the part, so as always to take in the same amount, we shall traverse the magnitude, for every finite magnitude is exhausted by means of any determinate quantity however small.

The infinite, then, exists in no other way, but in this way it does exist, potentially and by reduction. It exists fully in the sense in which we say “it is day” or “it is the games,” and potentially, as matter exists, not independently, as what is finite does.

By addition, then, there is also potentially an infinite, namely, what we have described as being in a sense the same as the infinite in respect of division. For it will always be possible to take something more.

Yet the sum of the parts taken will not exceed every determinate magnitude, just as, in the direction of division, every determinate magnitude is surpassed in smallness and there will be a smaller part.

But, in respect of addition, there cannot be an infinite that even potentially exceeds every assignable magnitude, unless it has the accident of being actually infinite, as the physicists hold to be true of the body that is outside the world, whose essential nature is air or something of the kind.

But, if there cannot be in this way a sensible body that is infinite in the full sense, evidently there can no more be a body that is potentially infinite in respect of addition, except as the inverse of the infinite by division, as we have said.

It is for this reason that Plato also made the infinites two in number, because it is supposed to be possible to exceed all limits and to proceed infinitely in the direction both of increase and of reduction.

Yet, though he makes the infinites two, he does not use them. For in the numbers, the infinite in the direction of reduction is not present, as the monad is the smallest; nor is the infinite in the direction of increase, for the parts number only up to ten.

370. After discussing the infinite dialectically, the Philosopher now begins to determine the truth.

First, he determines whether there is an infinite;

second, what it is, at the infinite turns out to be (206b33; [382]).

The first is divided into two parts:

in the first, he shows how the infinite exists;

in the second, he compares various infinites to one another, at the infinite exhibits itself (206a25; [374]).

About the first he does three things:

first, he shows that the infinite in a way exists and in a way it does not;

second, he shows that it is in potency and is not as a being in act, at we must keep in mind (206a14; [372]);

third, he shows how it is in potency, at but the phrase (206a18; [373]).

371. Accordingly, he says first (206a6) that, from the foregoing, (Lect. 8–9.) it is manifest that there is no infinite body in act. It is also clear from what has been said, (Lect. 7, n. 337ff.) that, if the infinite absolutely does not exist, many impossibilities arise. One is that time will have a beginning and an end, which is considered impossible by those holding for the eternity of the world. Another is that it would follow that a magnitude would not be always divisible into further magnitudes, but eventually one would arrive, through division of magnitudes, at certain things that are not magnitudes. But every magnitude is divisible. Likewise, it would follow that number could not increase to infinity. Since, therefore, according to what has been said, (Lect. 7–9.) neither seems to occur—namely, either an infinite in act or simply no infinite—it must be said that the infinite somehow is and somehow is not.

372. Then, at we must keep in mind (206a14), he shows that the infinite is as a being in potency. And he says that something is said to be in act and something is said to be in potency. Now, the infinite is said to come about either by addition, as in numbers, or by subtraction, as in magnitudes. Now, it has been shown that magnitude is not infinite in act; (Lect. 8–9.) hence, in magnitudes, an infinite through addition is not found, but there is found in them an infinite through division. For it is easy to destroy the opinion that posits lines as indivisibles; or, according to another reading, it is easy to divide indivisible lines, that is, to show that lines held indivisible by some are divisible. Now the infinite in addition or division is spoken of insofar as something can be added or divided. It therefore follows that the infinite is as a being in potency.

373. Then, at but the phrase (206a18), he shows how the infinite exists in potency.

For something is found to be in potency in two ways.

One way is in the sense that the whole can be reduced to act, as it is possible for this bronze to be a statue because, at some time, it will be a statue. But the infinite in potency is not so meant as that which later will be entirely in act.

In another way, something is said to be in potency in such a way that later it will be in act, not, indeed, all at once, but part after part.

For there are many ways in which a thing is said to be, either because the whole exists at the same time, as in the case of a man or a house, or because one part of it always comes to be after another part, in the way that a day is said to exist and a competition exists.

It is in this latter way that the infinite is said to be at once in potency and in act. For all successive things are at once in potency as to one part and in act as to another part. For the Olympic games (the festive contests held on Mt. Olympus) are said to be and to continue as long as the contests are scheduled and as long as the schedule is being carried out. For as long as those games lasted, one part of the schedule was taking place at the time and another was to take place later.

374. Then, at the infinite exhibits itself (206a25), he compares various infinites one to another.

First, he compares the infinite of time and of generation to the infinite that is in magnitudes;

second, he compares the infinite according to addition to the infinite according to division in the case of magnitudes, at in a way, the infinite (206b3; [377]).

In regard to the first, he does three things:

first, he proposes his intention and says that the infinite in the generation of man and in time must be explained in a manner different from that of the infinite in the division of magnitudes.

375. Second, at for generally, the infinite (206a26), he shows what is common to all infinites, saying that, in all of them, it is universally found that the infinite consists in always taking one thing followed by another according to some certain succession, in such a way that the whole of whatever of the infinite is taken to be in act is finite. Hence, one must not suppose the infinite to be some whole existing all at once, as this substance that can be pointed out, such as a man or a house. Rather, the infinite must be taken as in the case of successive things, such as a day or a tournament, whose existence is not that of a perfect substance actually existing as a complete whole all at once. Now, in generation and corruption, even if the process continues to infinity, whatever is taken in act is finite. For in the whole course of generation, even should it proceed to infinity, both all the men existing at a given time are finite in number and this finite amount must be taken as different, accordingly as men succeed one another in time.

376. Third, at but, when this takes place (206a33), he shows how they differ, saying that the finite actually present in magnitudes as a result of adding or of dividing is permanent and is not corrupted, but the finites considered in the infinite course of time and of human generation are corrupted, although in such a way that time and generation themselves do not fail.

377. Then, at in a way, the infinite (206b3), he compares the two types of infinite that are found in magnitudes: the infinite according to addition and the infinite according to division.

About this, he does three things:

first, he shows their points of agreement;

second, he shows wherein they differ, at yet the sum of the parts (206b17; [379]);

third, he draws a conclusion from what has been said, at but, in respect of addition (206b20; [380]).

In regard to the first (206b3), he says that, in some sense, the infinite resulting from addition is the same as the one resulting from division, because the former comes to be as a converse of the latter. For it is insofar as something is divided to infinity that additions to infinity seem to be able to be made to some determinate quantity.

378. He demonstrates, therefore, how the infinite in division exists in magnitude. Thus, he states that, if someone, having taken some determinate part by division in some infinite magnitude, should then continue to take other parts by division, always maintaining the same ratio (that is, proportion), he will not go through that finite magnitude by means of division.

For example, from a line of one cubit, we may take one half, and from the remainder, one-half again. We can proceed in this process to infinity. For the same proportion will be maintained in subtracting, but not the same amount of what is subtracted. The half of the half is less, according to quantity, than the half of the whole. But, if we were to take away always the same amount, the proportion taken away would be continually growing.

For example, if from a quantity of ten cubits, we take away one cubit, the ratio of the part removed to the original is one tenth. If we take from the remainder another cubit, the ratio between the part removed and that which remains will be in a greater proportion. For one cubit is less exceeded by nine than by ten.

Just as, by preserving the same proportion throughout, the quantity subtracted is continually smaller, so also, by taking away the same amount each time, the proportion gets continually larger.

If, therefore, by so subtracting from some finite magnitude, we continually increase the proportion by taking away the same amount, the original magnitude will be exhausted—for example, if from a line of ten cubits we always subtract one cubit. This will happen because every finite thing will be exhausted by continually removing the same finite amount.

Therefore, the infinite that depends on division does not exist except in potency, but with this potency, there exists always something in act, as was said of a day or of a tournament. And, since the infinite is always in potency, it is similar to matter, which likewise is always in potency; and it never exists in act in its entirety, the way that the finite is in act. And, just as the infinite according to division is at once in potency and in act, so too is the infinite according to addition, which has been shown to be in some sense the same as the infinite according to division, as was said. And the reason that the infinite according to addition is in potency is that it can always grow through addition.

379. Then, at yet the sum of the parts (206b17), he shows the difference between the infinite according to addition and the infinite according to division. And he says that the former does not exceed every given finite magnitude, but the latter diminishes beyond every predetermined smallness. For if we take any predetermined smallness—for example, the width of a finger—we can, by repeated halving of a line of ten cubits, arrive at a remainder that is less than the width of a finger. But, in adding to infinity, in distinction from division, there will exist some given finite quantity that will never be gone through. Take two magnitudes each of ten cubits and a third one of twenty cubits. If what I subtract to infinity from one magnitude of ten cubits, always taking a half, is added to the other, which is also of ten cubits, I shall never reach the measure of the quantity of twenty cubits by adding to infinity, since as much as remains in the quantity being divided will be lacking from the given measure in the quantity being added to.

380. Then, at but, in respect of addition (206b20), he draws a conclusion from the foregoing.

First, he draws it;

second, he explains it by a saying of Plato, at it is for this reason (206b27; [381]).

Therefore, he says first (206b20) that, since addition to infinity never actually transcends every determined quantity, it is not possible, even in potency, to transcend every determined quantity by addition. For if there were in nature potency for addition transcending every quantity, it would follow that something actually infinite exists; such an infinite would be an accident of some nature, in the same way that the natural philosophers posit some sort of infinite outside the world we see whose substance is air or something similar. But, if, as was shown, (Lect. 8–9.) no infinite sensible body exists in act, it follows that there is in nature no potency to transcend every magnitude by addition, but only a potency to the infinite addition that is in contrast to division, as was said above. Why the existence of a potency to infinite addition transcending every magnitude would imply a body infinite in act, when, in numbers, infinite addition transcending every number does not imply an actually infinite number, will be explained below. (Lect. 12, n. 396.)

381. Then, at it is for this reason (206b27), he confirms what he has said by a dictum of Plato, saying that, because the infinite resulting from the addition of magnitudes is the reverse of division, Plato therefore posited two infinites: “the large,” which pertains to addition, and “the small,” which pertains to division. For the finite seems to excel both by addition unto increase and by division unto decrease or toward nothing. Yet, although Plato makes two infinites, he does not use them. For in number, which he posited to be the substance of all things, there is no infinite by division, since there is among them something smallest, which is unity; nor is there according to him an infinite according by addition, since he said that the species of number vary only up to ten, and then a return is made to unity when we count eleven, twelve, and so on.