LECTURE 12 (207 a 32-208 a 4)

THE MEANINGS OF THINGS WHICH ARE SAID ABOUT THE INFINITE

             390. Having set forth the definition of the infinite, he here explains from this definition the meaning [ratio] of the things which are said about the infinite.

             First he explains the meaning of what is said about the addition and division of the infinite.

             Secondly, where he says, 'The infinite is not the same . . .' (207 b 21), he explains why the infinite is found in different things according to an order.

             Thirdly, where he says, 'Our account does not . . .' (207 b 28), he explains why mathematicians use the infinite.

             Fourthly, where he says, 'In the four-fold scheme . . .' (207 b 35), he explains why the infinite is posited as a principle.

             Concerning the first part he makes two points. First he explains the meaning [ratio] of what is said about the infinite with reference to division or addition in magnitudes. Secondly, where he says, 'It is natural too . . .' (207 b 1), he explains the meaning of what is said about numbers by comparing them with magnitudes.

             391. Now it was said above that addition to infinity is found in magnitudes in such a way that a determinate magnitude is not exceeded by this addition. But division to infinity is found in magnitude in such a way that any quantity is surpassed in smallness by dividing, as was explained above.

             He says, moreover, that it occurs according to reason [ratio]. For since the infinite has the nature [ratio] of matter, it is contained within as matter, whereas that which contains is the species and the form. Furthermore, it is clear from what was said in Book II that the whole has the nature [ratio] of form whereas parts have the nature [ratio] of matter. Since, therefore, in magnitudes we go from the whole to the parts by division, it is reasonable that no terminus is found there which is not surpassed by infinite division. But in addition we go from the parts to the whole, which has the nature [ratio] of form which contains and terminates. Hence it is reasonable for there to be some determinate quantity which infinite addition does not surpass.

             392. Next where he says, 'It is natural too . . .' (207 b 1), he explains the meaning [ratio] of the infinite [as found] in numbers by comparing them with magnitudes.

             It is said that in number a limit in smallness is found which is not surpassed by division. But there is not found any limit in greatness. For by addition we find another number which is greater than any given number. Yet the converse is true in magnitudes, as was said.

             He explains the reason [ratio] for this. First he explains why in number some limit is found which is not surpassed in smallness by division. The reason [ratio] for this is as follows. Every one thing, insofar as it is one, is indivisible, e.g., an indivisible man is one man, not many. Moreover, it is necessary to reduce every number to one, as is clear from the very nature [ratio] of number. For 'number' signifies that there is a plurality of ones. Now every plurality, which surpasses one more or less, is a determinate species of number. Hence, since unity pertains to the nature [ratio] of number, and since indivisibility pertains to the nature [ratio] of unity, it follows that the division of number stops at the indivisible limit.

             What he has said about the nature [ratio] of number, namely that numbers are pluralities of ones, he clarifies by means of the species [of number]. For two or three or any other number is denominated by one. Hence it is said in Metaphysics, V, /CS> that the substance of six lies in this, that it is six times one, and not in this, that it is twice three or three times two. For otherwise it would follow that there would be many definitions and many substances of one thing, since one number arises in different ways from different parts.

             393. Next where he says, 'But in the direction . . .' (207 b 10), he explains the reason why addition in numbers surpasses every determinate multitude.

             He says that we can always think of a number which is greater than any given number, because of the fact that magnitude is divided to infinity. For it is clear that division causes multitude. Hence the more a magnitude is divided, the greater is the multitude which arises. And thus the infinite addition of numbers follows upon the infinite division of magnitudes. And so, just as the infinite division of magnitude does not exist in act, but in potency, and exceeds every determinate quantity in smallness, as was said, so also the infinite addition of numbers does not exist in act, but in potency, and exceeds every determinate multitude. But this number which is thus multiplied to infinity is not a number separated from the division of magnitudes.

             394. And with reference to this it must be noted that division, as was said, causes multitude. However, division is twofold; one is formal, which is by opposites, and the other is according to quantity.

             Now the first division causes multitude, which pertains to the transcendentals insofar as being is divided into the one and the many. But the division of continuous quantity causes number, which is a species of quantity insofar as it has the nature [ratio] of a measure. And this number can be multiplied to infinity just as magnitude is divisible to infinity. But the multitude which follows from the formal division of things is not multiplied to infinity. For the species of things are determinate, just as there is also a determinate quantity in the universe. And thus he says that this number which is multiplied to infinity is not separated from the division of the continuous.

             Moreover, this number is not infinite as something permanent, but as always existing in 'coming to be', insofar as we successively add to some given number. And the same is true of time and the number of time. For the number of time grows successively by the addition of day to day and not because all days are present at once.

             395. Next where he says, 'With magnitudes . . .' (207 b 15), he shows that the contrary is true in magnitudes.

             A continuum, as was said, is divided to infinity. But it does not proceed to infinity in greatness, not even in potency. For to the extent that each thing is in potency, to that extent it can be in act. If, therefore, it were in the potency of nature that some magnitude could grow to infinity, it would follow that there would be some infinite sensible magnitude. But this is false, as was said above. It follows, therefore, that there is no potency for the addition of magnitudes to infinity such that every determinate quantity is surpassed. For otherwise it would follow that there would be something greater than the heavens.

             396. From this it is clear that what some say is false, namely, that in primary matter there is potency for every quantity. For in primary matter there is potency only for determinate quantity.

             From what has been said it is also clear why it is not necessary for number to be as great in act as it is in potency, as is said here of magnitude. For the addition of number follows from the division of the continuous by which we go from the whole to that which is in potency to number. Hence one cannot arrive at some act which limits the potency. But the addition of magnitude leads to act, as was said.

             The Commentator, however, gives another explanation [ratio]. The potency for the addition of magnitude is in one and the same magnitude. But the potency for the addition of numbers is indifferent numbers insofar as something can be added to any number.

             But this argument has little value. For just as addition produces different species of number, so also there are different species of measure, insofar as the two-cubit and the three-cubit are said to be species of quantity. Moreover, whatever is added to a greater number is also added to a smaller number. And thus in one and the same number, e.g., two or three, there is potency for infinite addition.

             397. Next where he says, 'The infinite is not the same . . .' (207 b 21), he shows how the infinite is found in different ways in different things.

             He says that the infinite is not found according to the same nature [ratio] in motion and in magnitude and in time, as if there were one univocal nature predicated of all of these. Rather the infinite is predicated of the latter of these by a reference to the prior. Thus the infinite is predicated of motion because of magnitude in which either local motion or alteration or increase occurs. And it is predicated of time because of motion. This is so because the infinite pertains to quantity, and motion has quantity because of magnitude and time because of motion, as will be explained below. And so he says that he is using these terms now and will explain later what each of them is and will show that every magnitude is divisible into magnitudes.

             398. Next where he says, 'Our account does not rob . . .' (207 b 28), he shows how the mathematicians use the infinite.

             He says that the argument set forth above, in which it was stated that there is no infinite magnitude in act, does not destroy the knowledge of mathematicians who use the infinite, e.g., when the geometrician says 'let such and such a line be infinite'. For they do not need an infinite in act for their demonstrations, nor do they use it. They merely require that there be some line which is as great as is necessary for them, so that from it they are able to subtract what they wish. And for this some greatest magnitude is sufficient. For the greatest magnitude can be divided according to any proportion with respect to some other given magnitude. Hence for purposes of demonstration it does not matter whether it be this way or that, i.e., either infinite or the greatest finite quantity. But with reference to the existence of the thing, it matters greatly whether it is one or the other.

             399. Next where he says, 'In the four-fold scheme . . .' (207 b 35), he explains how the infinite is a principle.

             He says that since there are four genera of causes, as was said above, it is clear from what has been said that the infinite is a cause as matter. For the infinite has existence in potency, which is proper to matter. But matter is sometimes under form and sometimes under privation. Now the nature [ratio] of matter does not pertain to the infinite insofar as it is under form, but insofar as it is under privation. For the infinite is named in terms of a removal of perfection and limit. And because of this he adds that 'the being of the infinite is a privation', i.e., the nature [ratio] of the infinite consists in privation.

             And lest anyone think that the infinite is matter as primary matter, he adds that the per se subject of the privation which constitutes the nature [ratio] of the infinite is a sensible continuum.

             And this is apparent because the infinite which is in numbers is caused by the infinite division of magnitude. And in like manner the infinite in time and in motion is caused by magnitude. Hence it follows that the first subject of the infinite is a continuum. And since magnitude according to the order of existence is not separated from sensible things, it follows that the subject of the infinite is the sensible.

             All the ancients who used the infinite as a material principle also agreed on this. Hence it was inconsistent when they said that the infinite contains. For matter does not contain, but rather is contained.