LECTURE 10 (206 a 8-b 32)

THE INFINITE EXISTS, NOT AS BEING IN ACT, BUT AS BEING IN POTENCY. DIFFERENT INFINITES ARE COMPARED WITH EACH OTHER

             370. After the Philosopher has treated the infinite by means of disputation, he begins here to determine the truth.

             First he shows whether there is an infinite, and secondly, what it is, where he says, 'The infinite turns out to be . . .' (206 b 33).

             The first part is divided into two parts. First he shows how the infinite exists. Secondly, where he says, 'The infinite exhibits itself . . .' (206 a 25), he compares different infinites with each other.

             Concerning the first part he makes three points. First he shows that in one way the infinite exists and in another way it does not. Secondly, where he says, 'We must keep in mind . . .' (206 a 14), he shows that the infinite exists in potency but does not exist as a being in act. Thirdly, where he says, 'But the phrase . . .' (206 a 18), he shows how the infinite exists in potency.

             371. He says, therefore, first that it is clear from what has been said above that there is no infinite body in act. It is also clear from what was said above that, if the infinite simply is not, many impossible consequences follow. One of these is that time will have a beginning and an end, which is thought to be absurd by those who hold that the world is eternal.

             Furthermore, it would follow that magnitude is not always divisible into magnitudes, but sometimes we would, by the division of magnitudes, arrive at certain things which are not magnitudes. But every magnitude is divisible.

             Moreover, it would follow that number may not be increased to infinity.

             Therefore, since according to what has already been established neither of these seems to be acceptable, i.e., that there is an infinite in act or that the infinite simply does not exist, it is necessary to say that the infinite exists in one way and in another way does not.

             372. Next where he says, 'We must keep in mind . . .' (206 a 14), he shows that the infinite exists as being in potency.

             He says that to exist in act is one thing and to exist in potency is another. Now the infinite is said to exist either by addition, as in numbers, or by subtraction, as in magnitudes. For it was shown that a magnitude is not infinite in act. And thus in magnitudes an infinite by addition is not found. However, the infinite is found in magnitudes by division. For it is not difficult to refute the opinion of those who hold that lines are indivisible. Or according to another text, it is not difficult 'to divide indivisible lines', i.e., to show that the lines which some say are indivisible are divisible. Hence a thing is said to be infinite by addition or division insofar as it can be added to or divided. It follows, therefore, that the infinite exists as a being in potency.

             373. Next where he says, 'But the phrase . . .' (206 a 18), he shows how the infinite exists in potency.

             A thing is found to be in potency in two ways. First a thing is in potency in the sense that the whole can be reduced to act. Thus it is possible for this bronze to be a statue because it will at sometime be a statue. However, the infinite is not said to be in potency in the sense that at some later time the whole is in act. Secondly a thing is said to be in potency in the sense that at some later time it comes to be in act, not at once as a whole, but successively.

             For a thing is said 'to be' in many ways, either because it is a whole all at once, as a man and a house, or because one part of it always comes to be after another. In this latter way a day and the Agonalian games are said to be.

             And in this latter way the infinite is said to be both in potency and in act at the same time. For all things of this sort are simultaneously in potency with respect to one part and in act with respect to another. For the Olympic games, i.e., the Agonalian feasts which are celebrated on Mount Olympus, are said to be and to endure insofar as the public games can come to be and do come to be in act. For as long as these feasts continued, part of these games were occurring and another part was, as it were, about to occur.

             374. Next where he says, 'The infinite exhibits itself . . .' (206 a 25), he compares different infinites with each other.

             First he compares the infinite of time and generation with the infinite which is in magnitudes. Secondly, where he says, 'In a way the infinite . . .' (206 b 3), he compares the infinite in respect to addition with the infinite in respect to division in magnitudes.

             Concerning the first part he makes three points. First he states his intention. He says that he will show that the infinite in time and in the generation of men is different than the infinite in the division of magnitudes.

             375. Secondly, where he says, 'For generally the infinite . . .' (206 a 26), he explains what is common to all infinites.

             He says that in all infinites it is always and universally found that the infinite is always different in respect to some succession, so that whatever is designated in act in the infinite is a finite whole.

             Hence, it is not necessary to say that the infinite is a whole existing all at once as a designated 'this', like a man or a house. Rather the infinite exists successively, as does a day and the Agonalian games, which do not exist in the way in which a whole, complete substance exists in act.

             Moreover, even if generation and corruption go on to infinity, that which is found to be in act is always finite. For in the whole course of generation, even if it goes on to infinity, all the men who are found to be simultaneously in act are finite in number. And this finite group must always be different insofar as certain men succeed others.

             376. Thirdly, where he says, 'But when this takes place . . .' (206 b 1), he explains the difference. He says that the finite thing which we take in magnitudes, either by adding or dividing, remains and is not corrupted. But the finite things which are found in the infinite course of time and of human generation are corrupted, so that in this way time and generation do not cease.

             377. Next where he says, 'In a way the infinite . . .' (206 b 3), he compares the two infinites which are present in magnitudes, i.e., according to addition and according to division.

             Concerning this he makes three points. First, he explains the agreement between these two infinites. Secondly, where he says, 'Yet the sum of the parts . . .' (206 b 18), he explains the difference. Thirdly, where he says, 'But in respect of addition . . .' (206 b 20), he draws a certain conclusion from what he has said.

             He says, therefore, first that in a certain way the infinite by addition is the same as the infinite by division. For the infinite by addition comes to be in a manner converse to that of the infinite by division. For insofar as something is divided to infinity, it also seems that a determinate quantity can be added to ad infinitum.

             378. He then explains how the infinite by division exists in magnitude. He says that if in a finite magnitude, when a determinate part has been taken by division, one always takes, when dividing, other parts according to the same ratio [ratio], i.e., proportion, but not according to the same quantity in the same proportion, one will not by dividing pass through that finite thing. For example, if from a line which is a cubit long one takes a half, and from the remainder half again, it is thus possible to proceed to infinity. For the same proportion will be preserved in the subtraction, but not the same quantity of what is subtracted. For in respect to quantity half of a half is less than half of the whole.

             But if one would always take the same quantity, the proportion must always grow larger and larger. For example, if from a quantity of ten cubits, one cubit is subtracted, then that which is subtracted is related to the whole in a proportion of one to ten. If, however, one cubit is again subtracted from the remainder, what was subtracted will have a greater proportion, for one cubit is exceeded less by nine than by ten. Therefore, just as by preserving the same proportion the quantity is diminished, so also by taking the same quantity the proportion is increased. If, therefore, in subtracting from a finite magnitude one would always increase the proportion by taking the same quantity, then he will pass through the finite magnitude by dividing. For example, from a line one hundred cubits long one always subtracts one cubit. This is so because any finite thing is consumed if any finite amount is always taken away.

             Therefore, the infinite by division exists only in potency. And it is simultaneously in act and in potency, as was said of the day and the public games. And since the infinite is always in potency, it is like matter, which is always in potency. It does not exist in itself as a whole in act, as the finite exists in act. And as the infinite by division is simultaneously in potency and in act, the same must be said of the infinite by addition, which in a way is the same as the infinite by division, as was said. Hence it is clear that the infinite by addition is in potency, because it is always possible to take another by addition.

             379. Next where he says, 'Yet the sum of the parts . . .' (206 b 18), he explains the difference between the infinite by addition and the infinite by division.

             He says that the infinite by addition is not greater than every given finite magnitude. But the infinite by division is smaller than every determinate smallness.

             For let us take some determinate smallness, for example, one inch. If a line one hundred cubits long is divided to infinity by always taking away a half, one will come to something less than one inch.

             But in adding to infinity, contrary to what takes place in division, it will be possible to give some finite quantity which will never be surpassed. For let there be given two magnitudes each of which is ten cubits long and a third which is twenty. If, then, that which I subtract to infinity, by always taking one half from one magnitude of ten cubits length, is added to the other which is also ten cubits long, I will never arrive by adding to infinity at that measure of quantity which is twenty cubits long. For to the extent that something will remain in the magnitude from which the subtraction was made, to that extent there will be a deficit of the given measure in the quantity to which the addition is made.

             380. Next where he says, 'But in respect of addition . . .' (206 b 20), he draws his conclusion from what has been said.

             First he draws the conclusion. Secondly, where he says, 'It is for this reason . . .' (206 b 27), he clarifies it with a remark from Plato.

             He says, therefore, first that since addition to infinity does not transcend every determinate quantity, it is not possible, not even in potency, for every determinate quantity to be exceeded by addition. For if there were in nature a potency for an addition transcending every quantity, it would follow that there would be an infinite in act, and this infinite would be an accident of some nature. Thus the natural philosophers said that outside the body of this world which we see there is a certain infinite whose substance is air or some other such thing. If, therefore, it is not possible for there to be an infinite sensible body in act, as has been shown, it follows that there is not in nature a potency for an addition transcending every magnitude, but only a potency for infinite addition as the contrary of division, as was said. Therefore, if there were a potency for an infinite addition transcending every magnitude, it would follow that there would be an infinite body in act. But from an infinite addition in numbers which transcends every number it does not follow that there is an infinite number in act, as will be shown below.

             381. Next where he says, 'It is for this reason that Plato . . .' (206 b 27), he clarifies what he has said with a remark from Plato.

             He says that since the infinite in the addition of magnitudes is opposed to the infinite by division, Plato posited two infinites, namely, the great, which pertains to addition, and the small, which pertains to division. For the infinite seems to excel, both by addition toward increase, and by division toward decrease, or by tending toward nothing.

             But although Plato posited two infinites, he does not use them. For he said that number is the substance of all things. But the infinite by division is not found in numbers, since the smallest number is unity. Nor according to him is there an infinite by addition. For he said that the species of numbers vary only up to ten, and thereafter number is reduced to unity by counting eleven and twelve, etc.