Commentary Aristotle Physics
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, it is manifest that there is no infinite body in act. It is also clear from what has been said, 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, 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; 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.