Commentary Aristotle Physics
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. 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.
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 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, it is clear from the foregoing 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.
Lecture 13. Refutation of the statement that the infinite exists in both potency and act
It remains to dispose of the arguments that are supposed to support the view that the infinite exists not only potentially but also as a separate thing. Some have no cogency; others can be met by fresh objections that are valid.
In order that generation should not fail, it is not necessary that there should be a sensible body that is actually infinite. The passing away of one thing may be the generation of another, the all being limited.
There is a difference between to be touched and to be limited. The former is relative to something and is of something (for everything that touches touches something), and further is an accident of some one of the things that are limited.
On the other hand, what is limited is not limited in relation to anything. Again, contact is not necessarily possible between any two things taken at random.
To rely on mere thinking is incorrect, for then the excess or defect is not in the thing, but in the thought. One might think that one of us is a multiple of himself and magnify him to infinity. But it does not follow that he is bigger than the size we are, just because some one thinks he is, but only because he is the size he is. The thought is an accident.
Indeed, time and motion are infinite, and also thinking, in the sense that each part that is taken passes in succession out of existence.
Magnitude is not infinite either in the way of reduction or in that of magnification in thought.
This concludes my account of the way in which the infinite exists, and of the way in which it does not exist, and of what it is.
400. After the Philosopher has used the definition of the infinite to explain the things attributed to it, he now solves the argument presented above to show that the infinite exists.
First, he proposes his intention;
second, he follows it out, at “in order that” (208a8; [401]).
He says first (208a5) that, after speaking of the nature of the infinite, it remains to settle the arguments that appeared to show that the infinite is not only something in potency, as we determined above, but also in act, as are finite and determined things. For some of the arguments do not conclude necessarily, but are entirely false, while others are partially true.
401. Then, at “in order that” (208a8), he solves the five reasons cited above as proving that the infinite exists.
And first, he solves the one based on the fact of generation. For it concluded that, if generation does not cease, then the infinite must be. Now, this argument concludes truly insofar as the infinite is in a potency that is successively reduced to act, as was said above. But it is not necessary that there be some sensible body that is infinite in act in order to account for generation not ceasing, as the earlier philosophers supposed when they said that generation continues to infinity, supposing it to take place by extracting from some body—with the consequence that the process could not be infinite unless that body were infinite. But this is not necessary, for even supposing the whole of sensible body as finite, generation can endure to infinity by the fact that the corruption of one thing is the generation of another.
402. Then, at “there is a difference” (208a11), he solves the argument based on the principle of contact, as though it were necessary for every finite body to touch some other body and so on to infinity.
But he solves this by saying that it is one thing to be touched and another to be limited, because to be touched and enclosed are said in respect to something else, for whatever touches touches something else.
To be limited, however, is said absolutely and does not imply a relationship to something else, because a thing is made finite in itself by its own terminations.
For it is accidental to the finite that it be touching something. Nevertheless, neither is it necessary that everything touched by something should touch something else and that this go on to infinity. Hence, it is evident that this argument does not conclude anything of necessity.
403. Then, at “to rely on mere thinking” (208a14), he solves the argument based on the intellect and the imagination, which the ancients did not distinguish from the intellect. This argument above concluded that there was outside the universe an infinite space, and consequently a place and a body.
But he says that to rely on mere thinking is incorrect, that is, to believe that whatever is apprehended by the imagination or intellect is true, as some of the ancients thought, whose opinion is refuted in Metaphysics 4. For if I apprehend a thing as smaller or larger than it is, it does not thereby follow that there is such an abundance or defect in the object itself, but only in the apprehension of the intellect or imagination. For one might understand some man to be a multiple of himself—that is, two or three times larger than he really is (or any other amount to infinity)—yet there will not be, because of this, a corresponding multiplication of him outside the intellect or outside a definite quantity or magnitude. But, while a thing remains what it is, one can conceive of it in such a manner.
404. Then, at “indeed, time” (208a20), he solves the difficulty based on time and motion. And he says that time and motion are not infinite in act, because nothing of time is actual but the “now,” and nothing of motion is actual except a kind of indivisible. But the intellect apprehends a continuity in time and in motion by apprehending an order of “prior” and “posterior,” but in such a way that what was first taken in time or in motion does not remain in the same state. Hence, it is not necessary to say that the whole of motion is infinite, or that the whole of time is infinite.
405. Then, at “magnitude is not infinite” (208a21), he solves the argument based on magnitude, and he says that magnitude is not infinite in act, either as a result of division or as a result of an intelligible increase, as is evident from what was said above.
Finally, he summarizes by saying that we have completed our study of the infinite.