Commentary Aristotle Physics
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, 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.
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 to 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.