Lecture 12 — The meanings of what is said about the infinite

It is reasonable that there should not be held to be an infinite in respect of addition such as to surpass every magnitude, but that there should be thought to be such an infinite in the direction of division. For the matter and the infinite are contained inside what contains them, while it is the form that contains.

It is natural to suppose that, in number, there is a limit in the direction of the minimum and that, in the other direction, every assigned number is surpassed. In magnitude, on the contrary, every assigned magnitude is surpassed in the direction of smallness, but in the other direction, there is no infinite magnitude. The reason is that what is one is indivisible, whatever it may be—for instance, man is one man, not many. Number, on the other hand, is a plurality of “ones” and a certain quantity of them. Hence, number must stop at the indivisible, for “two” and “three” are merely derivative terms, and so with each of the other numbers.

But, in the direction of largeness, it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite.

Hence, this infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned multitude. But this number is not a number independent of division, and its infinity is not a permanent act but consists in a process of coming to be, like time and the number of time.

With magnitudes, the contrary holds. What is continuous is divided to infinity, but there is no infinite in the direction of increase. For the size that it can potentially be, it can also actually be. Hence, since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible, there would be something bigger than the heavens.

The infinite is not the same in magnitude and motion and time, in the sense of a single nature, but its secondary sense depends on its primary sense; that is, motion is called infinite in virtue of the magnitude covered by the motion (or alteration or growth), and time because of the motion. (I use these terms for the moment. Later I shall explain what each of them means, and also why every magnitude is divisible into magnitudes.)

Our account does not rob the mathematicians of their science by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable.

In point of fact, they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish.

It is possible to have divided in the same ratio as the largest quantity another magnitude of any size you like.

Hence, for the purposes of proof, it will make no difference to them to have such an infinite instead, while its existence will be in the sphere of real magnitudes.

In the fourfold scheme of causes, it is plain that the infinite is a cause in the sense of matter, and that the being of the infinite is privation, the subject as such being the sensible continuum. All the other thinkers, too, evidently treat the infinite as matter—that is why it is inconsistent in them to make it what contains, and not what is contained.

390. After giving a definition of the infinite, the Philosopher now assigns reasons for the things that are said about the infinite:

first, the reason for what is said about addition and division of the infinite;

second, the reason for saying that the infinite is found in different things according to a certain order, at the infinite is not the same (207b21; [397]);

third, the reason for saying that mathematicians use the infinite, at our account does not (207b27; [398]);

fourth, the reason that the infinite is called a principle, at in the fourfold scheme (207b34; [399]).

About the first, he does two things:

first, he presents the reason for what is said about the infinite in relation to division and addition in magnitudes;

second, the reason for what is said of it in numbers by comparison to magnitudes, at it is natural to suppose (207b1; [392]).

391. It was said above that addition to infinity in magnitudes takes place in such a way that the resulting magnitude does not become greater than any given magnitude. (Lect. 10, n. 379.) But division to infinity in magnitudes results in reaching a quantity that is smaller than any preassigned quantity, as was expounded above. (Lect. 10, n. 379.) However, he states (207a32) that this occurs reasonably. For since the infinite is like matter, it is contained inside something, just as matter is, while that which contains is the species and form. Now, it is clear from what was said in book 2 that the whole is like form and the parts are like matter. (Lect. 5, nn. 183–84.) Since, therefore, the division of a magnitude proceeds from the whole to the parts, it is reasonable that no limit be found there that is not transcended through infinite division. But the process of addition goes from the parts to the whole, which is like a form that contains and terminates; hence, it is reasonable that there be some definite quantity that infinite addition does not exceed.

392. Then, at it is natural (207b1), he explains infinity in numbers by comparison to magnitudes. For it was said that, in number, there is a smallest terminus below which division does not go, but there is no maximum limit that it cannot exceed, because it is possible through addition to exceed any given number. The opposite, however, takes place in magnitudes, as was said.

The reason is that every unity, inasmuch as it is a unity, is indivisible, as indivisible man is one man and not many men. Now, every number can be resolved into unity, as is evident from the account of number. For number signifies that there are more things than one, and any plurality exceeding one to a greater or lesser degree constitutes a definite species of number. Hence, since unity pertains to the account of number and indivisibility pertains to the account of unity, it follows that the division of number should halt at an indivisible terminus. This statement that it is in the account of number to be more than unity he explains by appealing to the species of number, for two and three and every other number are denominated by unity. Therefore, it is said in Metaphysics 5 that the substance of six consists in its being six times one and not two times three, or three times two. (Aristotle, Metaphysics Δ.14; cf. St. Thomas, Commentary on Metaphysics, bk. 5, lect. 16.) Otherwise, it would follow that, of the same thing, there would be more than one definition and more than one nature, since, starting from different parts, the same number would come about in different ways.

393. Then, at but, in the direction (207b10), he gives the reason that, in numbers, addition exceeds any predetermined multitude. And he says that we can always think of a number greater than any given number, for the reason that magnitude is divided to infinity. For it is plain that division causes multitude; hence, the more magnitude is divided, the greater is the multitude that results, and upon the infinite division of magnitudes there follows the infinite addition of numbers. Therefore, just as infinite division of magnitude is not in act but in potency and exceeds every determinate quantity in smallness, as was said, (Cf. n. 391; lect. 10, n. 372.) so the infinite addition of numbers is not in act but in potency and exceeds every determinate multitude. But this number that is thus multiplied to infinity is not a number independent of division of magnitudes.

394. On this point, it must be remembered that division, as was stated, causes multitude. But division is of two kinds:

one is formal, which is through opposites;

the other is according to quantity.

Now, the first division causes that multitude that is a transcendental, accordingly as being is divided into “one” and “many.”

But the division of continuous quantity causes number, which is a species of quantity, insofar as it has the account of measure. And this number can grow to infinity, just as magnitude is divisible to infinity.

But the multitude that arises from formal division cannot grow to infinity. For the species of things are determined, just as there is a determined quantity of the universe. That is why he says that the number that grows to infinity is not separated from the division of the continuum. Nor is this number infinite in the sense of something permanent. Rather, it is as something always in a state of becoming, inasmuch as additions may be successively made to any given number, as is evident in the case of time and the number of time. For the number of time increases successively by the addition of day to day, but not all days existed at once.

395. Then, at with magnitudes (207b15), he shows that the opposite occurs in magnitudes. For although a continuum be divided to infinity, as was said, the size cannot grow indefinitely even potentially. For as great as a thing is in potency, so great can it be in act. If, therefore, it were in the potency of nature that a magnitude grow to infinity, it would follow that there would actually be some infinite sensible magnitude—which is false, as stated. (Lect. 8–9.) The consequence is, therefore, that addition of magnitudes cannot go on to infinity so as to exceed every predetermined quantity; for otherwise there would be something greater than the heavens.

396. From the foregoing, it is plain that the claim of some that, in prime matter, there is a potency to every quantity is false; for in prime matter, there is a potency only to determined quantity.

It is plain also from the foregoing why number does not have to be as great in act as it is potentially, as is said here of magnitude, for addition occurs in number as a consequence of the division of the continuum, by which one passes from a whole to what is in potency to number. Hence, one need not arrive at some act terminating the potency. But the addition of magnitudes arrives at act, as was said.

The Commentator, however, assigns another reason: potency to addition in magnitude is in one and the same magnitude, but the potency to addition in numbers is in various numbers, inasmuch as something can be added to any number. But this reason has little value, for just as addition produces varying species of number, so also varying species of measure are called species of quantity, such as “two cubits long” and “three cubits long.” Moreover, whatever is added to a higher number is added to the lower. Accordingly, there is a potency to infinite addition in one and the same number, such as two or three.

397. Then, at the infinite is not the same (207b21), he shows how the infinite is found in diverse ways in diverse things. And he says that the idea of the infinite is not the same in motion and magnitude and time, as if it were one nature being predicated univocally in all three cases.

Rather, it is said of the subsequent member in terms of its antecedent: of motion by reference to the magnitude in which motion takes place (whether it be local motion, alteration, or augmentation), and of time by reference to motion.

This happens because the infinite pertains to quantity, and motion is quantified by reference to magnitude, while time is quantified by reference to motion, as will be evident below. (Bk. 4, lect. 17.) And therefore he says that we are now mentioning these, but later what each of them is will be explained, as well as that every magnitude is divisible into magnitudes. (Bk. 6, lect. 1.)

398. Then, at our account does not (207b27), he explains how mathematicians make use of the infinite and says that the argument that there is no actually infinite magnitude (Cf. lect. 8–9.) does not destroy the consideration of the mathematicians, who use the infinite—for example, as when the geometer says, “let this line be infinite.” For they do not need the infinite in act for their demonstrations, nor do they use it, but they need only some finite line of sufficient quantity for their needs, so as to be able to subtract from it so much as they wish. For their purpose, it is enough that there exist some maximum magnitude that can be divided according to any proportion in respect to another given magnitude. Hence, for purposes of demonstration, it makes no difference whether this maximum magnitude be one way or the other, namely, finite or infinite. But, as to the being of things, it makes a great difference whether it is one or the other.

399. Then, at in the fourfold scheme (207b34), he shows how the infinite is a principle. And he says that, since there are four genera of causes, as was said above, (Bk. 2, lect. 5.) it is clear from the foregoing (Cf. n. 391; lect. 10, n. 372.) that the infinite is a cause in the manner of matter. For the infinite has being in potency, which is proper to matter. Now, matter is sometimes under a form and sometimes under privation. However, the infinite has the account of matter, not insofar as matter lies under a form, but inasmuch as matter has privation—for the infinite implies the lack of perfection and term.

That is why the Philosopher adds that the being of the infinite is privation; that is, the account of the infinite consists in privation. And, lest anyone suppose that the infinite is matter like prime matter, he adds that the per se subject of the privation that constitutes the account of the infinite is the sensible continuum. That this is so is clear from the fact that the infinite found in numbers is caused from the infinite division of magnitude; similarly, the infinite in time and motion are caused by magnitude. Hence, the first subject of the infinite is a continuum. And since really existing magnitude is not separated from sensible things, it follows that the subject of the infinite is sensible. And, on this point, all the earlier philosophers agree who use the infinite as a material principle. Therefore, they improperly attributed to the infinite the capacity to contain, for matter does not contain, but rather is contained.