Commentary Aristotle Physics
Lecture 7
The meanings of “infinite”
Belief in the existence of the infinite comes mainly from five considerations:
from the nature of time, which is infinite;
from the division of magnitudes—for the mathematicians also use the notion of the infinite;
from coming to be and passing away—for if they do not give out, it is only because that from which things come to be is infinite;
from the limited always finding its limit in something, such that there must be no limit, if everything is always limited by something different from itself;
most of all, from a reason that is peculiarly appropriate and presents the difficulty that is felt by everybody—not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite, because they never give out in our thought.
The last fact (that what is outside is infinite) leads people to suppose that body also is infinite, and that there is an infinite number of worlds. Why should there be body in one part of the void rather than in another? Grant only that mass is anywhere and it follows that it must be everywhere. Also, if void and place are infinite, there must be infinite body too, for in the case of eternal things, what may be must be.
But the problem of the infinite is difficult: many contradictions result, whether we suppose it to exist or not to exist. If it exists, we have still to ask how it exists—as a substance or as the essential attribute of some entity. Or in neither way, yet none the less is there something that is infinite or some things that are infinitely many?
The problem that specially belongs to the physicist, however, is to investigate whether there is a sensible magnitude that is infinite.
We must begin by distinguishing the various senses in which the term “infinite” is used.
In one way, what is “infinite” is incapable of being gone through, because it is not in its nature to be gone through (the sense in which the voice is “invisible”).
In another way, it admits of being gone through but the process has no termination, or it scarcely admits of being gone through.
In another way, the “infinite” naturally admits of being gone through but is not actually gone through or does not actually reach an end.
Further, everything that is infinite may be so in respect of addition or division or both.
Now, it is impossible that the infinite should be a thing that is itself infinite, separable from sensible objects. If the infinite is neither a magnitude nor an aggregate, but is itself a substance and not an accident, it will be indivisible, for the divisible must be either a magnitude or an aggregate.
But, if it is indivisible, then it is not infinite, except in the sense in which the voice is “invisible.”
But this is not the sense in which it is used by those who say that the infinite exists, nor that in which we are investigating it, namely as “that which could not be passed through.”
But, if the infinite exists as an accident, it would not be, qua infinite, an element in substances any more than the invisible would be an element of speech, though the voice is invisible.
Further, how can the infinite be itself any thing unless both number and magnitude, of which it is an essential attribute, exist in that way? If they are not substances, then, all the more, the infinite is not.
It is plain, too, that the infinite cannot be an actual thing and a substance and principle. For any part of it that is taken will be infinite, if it has parts, for that which is infinite and the nature of the infinite are the same if it is a substance and not predicated of a subject. Hence, it will be either indivisible or divisible into infinites.
But the same thing cannot be many infinites. (Yet, just as part of air is air, so a part of the infinite would be infinite, if it is supposed to be a substance and principle.) Therefore, the infinite must be without parts and indivisible.
But this cannot be true of what is infinite in full completion, for it must be a definite quantity. Suppose, then, that infinity belongs to substance as an accident. But, if so, it cannot, as we have said, be described as a principle, but rather that of which it is an accident—the air or the even number.
Thus, the view of those who speak after the manner of the Pythagoreans is absurd. With the same breath they treat the infinite as substance and divide it into parts.
This discussion, however, involves the more general question whether the infinite can be present in mathematical objects and things that are intelligible and do not have extension as well as among sensible objects.
Our inquiry is limited to physics’ special subject matter, the objects of sense, and we have to ask whether there is or is not among them a body that is infinite in the direction of increase.
336. Having listed the opinions of the earlier philosophers on the infinite, Aristotle now begins to inquire into the truth of the matter.
First, he objects to both sides of the question;
second, he solves the objections, at “it is plain from these arguments” (206a6; [370]).
About the first, he does two things:
first, he gives reasons to show that the infinite exists;
second, to show that it does not exist, at “but the problem” (203b30; [342]).
337. In regard to the first, he gives five reasons to show that the infinite exists.
The first of these is based on time, which, according to the common opinion of the earlier philosophers, was infinite. For Plato alone supposed that time was generated, as will be shown in book 8.
He says first that there are five arguments for the infinite’s existence.
And the first of these is taken from time, which is infinite according to those who held that time always was and always will be.
338. The second reason is taken from the infinite divisibility of magnitude. For even mathematicians use the infinite in their demonstrations. This, however, would not happen if there were no infinite at all. Hence, the infinite exists.
339. The third reason is based on the perpetual processes of generation and corruption according to the opinion of many. For if the infinite were denied, generation and corruption could not endure indefinitely. Hence, it would have to be admitted that generation would sometime cease, which is against the opinion of many. Therefore, it is necessary to posit the infinite.
340. The fourth reason is based on the apparent account of the finite, which seems to many to consist in this, that it is something always included by something else, because we observe that every finite reaches into something else. Let a body be pointed out; if it be infinite, then the infinite exists; if it be finite, it must be terminated at something else, and this latter, if it in turn be finite, at something else. We must either proceed thus to infinity or come to a body that is infinite. In either case, the infinite exists. Hence, there can be no end to bodies, if every finite body is always included by some other.
341. The fifth reason is taken from the apprehension of the intellect or of the imagination. Hence, he says that what chiefly constitutes the common difficulty that induces men to posit the infinite is that the intellect never is exhausted, but can always add something to any given finite amount. Now, the earlier philosophers supposed that things corresponded to the intellect’s or senses’ apprehension of them. Hence, because they said that whatever appeared to be is true, as stated in Metaphysics 4, they believed that, even in reality, there exists an infinite. Hence, number seems to be infinite, because the intellect can always create a new number simply by adding unity to a given number. For the same reason, mathematical magnitudes, which exist in the imagination, seem to be infinite, because, given any definite magnitude, we can imagine a greater. And, for the same reason, there seems to be an infinite space beyond the heavens, because we can imagine certain dimensions existing beyond the heavens to infinity.
Now, if there be infinite space beyond the heavens, it seems that there is an infinite body and even infinite worlds.
This is so for two reasons.
The first is that, if the totality of space be considered infinite, that totality will be uniform; hence, there is no reason why that space should be devoid of body in one part rather than in another. Therefore, if there is found in one part of that space the bodily magnitude of this world, then there should be found in each part of that space some bodily magnitude comparable to that of this world. Thus, body must be infinite in the same way as space or there must even exist infinite worlds, as Democritus supposed.
Another reason proving the same point is that, if there be infinite space, it is either empty or full. If it is full, we have our point that there is infinite body; but, if it is empty, then, since the empty is a place not filled with a body but capable of being so filled, it follows that, if space is infinite, there is infinite place capable of being filled with body. Thus, there must be infinite body, for in eternal things, there is no difference between what can be and what is. Hence, if infinite place can be filled with body, it must be admitted that it is filled with infinite body. Therefore, it seems necessary to say that there is infinite body.
342. Then, at “but the problem” (203b30), he takes the opposite position.
And, in regard to this, he does three things:
first, he shows that the matter is debatable, lest anyone suppose that the aforementioned reasons are unassailable;
second, he gives the various meanings of the word “infinite,” at “we must begin” (204a2; [344]);
third, he gives reasons showing that the infinite does not exist, at “now, it is impossible” (204a8; [345]).
343. He first says, therefore (203b30), that there is a question about whether the infinite exists or not. For on the one hand, many impossibilities follow upon holding that it does not, such as those listed above. On the other hand, there are also difficulties attendant upon holding that the infinite does exist, as will be clear subsequently. There is doubt also as to its manner of existence. Does it exist as a substance does, or as an accident belonging essentially to some nature? Or, if neither as a substance nor as an essential accident, but as an accident nevertheless, is there some infinite continuum, and are there things infinite in number? Now, it very much pertains to the philosopher of nature to discuss whether there exists such a thing as an infinite sensible magnitude, for a sensible magnitude is a natural magnitude.
344. Then, at we must begin (204a2), he shows in how many ways “infinite” is said and lists two divisions of the infinite. The first division is common to the infinite and to all things said privatively.
For invisible is said in three ways:
either as denoting what of its very nature is not apt to be seen, such as a sound, which is not in the genus of visible things;
or what is difficult to see, as what is seen in the dark or from a distance;
or what is apt to be seen but is not, as something in total darkness.
Thus in one way “infinite” is said of what of its very nature is not apt to be passed over (for the infinite is the same as that which cannot be gone through). This is because it belongs to the genus of intransible things, as are indivisibles, such as a point and a form; in this way, a sound was called invisible.
In a second way, infinity is ascribed to what could be passed over but its passage is impossible for us—thus, we say that the depth of the sea is infinite. Or, if it could be passed through, it would be with difficulty, as if we should say that a trip to India is infinite. Both of these belong to that which is difficultly traversable.
In a third way, infinity is ascribed to what is passable but there is no passage to its terminus, such as a line without an end or any other such quantity without limits. This is the proper sense of the word “infinite.”
He then gives the other division of infinite, at “further, everything that is infinite” (204a6), saying that infinity is spoken either by addition, as in numbers, or according to division, as in magnitudes, or both ways, as in time.
345. Then, at “now, it is impossible” (204a8), he lays down the arguments leading to an exclusion of the infinite:
first, those excluding a separated infinite, such as laid down by the Platonists;
second, those excluding the infinite from sensible things, at we may begin (204b4; [349]).
With respect to the first, he lays down three reasons.
As to the first of these, he says that it is impossible for the infinite to be separated from sensible things in such a way that the infinite should be something existing of itself, as the Platonists laid down. For if the infinite is asserted to be something separated, either it has a certain quantity (namely, continuous, which is size, or discrete, which is number), or not. If it is a substance without either the accident of size or that of number, then the infinite must be indivisible, since whatever is divisible is either number or size. But, if something is indivisible, it will not be infinite except in the first way—namely, as something is called “infinite” that is not by nature susceptible to being passed through, in the same way that a sound is said to be “invisible,” but this is not what is intended in the present inquiry concerning the infinite, nor by those who posited the infinite.
For they did not intend to assert that the infinite is something indivisible, but that it is that which could not be passed through, that is, as being susceptible to such but with the passage having no completion.
If, however, the infinite should not only be a substance but also should have an accident that is size or number to which the infinite belongs (in such a way that the infinite would be inherent in the substance in the manner of that accident), then the principle of existing things will not be infinite as such, as the ancients asserted, just as we do not say that the principle of speech is invisible, although invisibility is an accident of the voice, which is the principle of speech.
346. The second reason, given at “further, how can the infinite” (204a17), is as follows. A passion is less separable and able to exist of itself than a subject. But the infinite is a passion of size and number, which cannot be separated and exist of themselves, as is proved in the Metaphysics. Therefore, neither can this be so of the infinite.
347. The third argument, given at “it is plain” (204a20), is as follows. He states that it is clear that the infinite cannot be asserted to be being in act, and a certain substance, and the principle of things. For the infinite is either divisible or indivisible. If it is indeed divisible, every one of its parts will have to be infinite, on the supposition that the infinite is a substance.
For if it is a substance and is not predicated of any subject as an accident, then that which is infinite and the nature of the infinite—that is, the essence and account of the infinite—will have to be the same.
For that which is white and the nature of white are not the same, but man and the nature of man are. Hence, it will be necessary that the infinite, if it be a substance, be either indivisible, or divided into parts that are infinite—which is impossible, since it is impossible to compose some same thing out of many infinities, as this would involve one infinite being terminated by another infinite.
It likewise appears not only from argument but also from likeness that, if the infinite is a substance and is divided, it is necessary that each and every part of it be infinite. For just as every part of air is air, so too every part of the infinite will be infinite if the infinite is a substance and a principle. For if it is a principle, the infinite has to be a simple substance, not composed out of differing parts, as in the case of man whose every part is not man. Since, therefore, it is impossible for every part of some infinite to be infinite, the infinite must then be indivisible and unable to be reduced to parts. But what is indivisible cannot be infinite in act, since whatever is infinite in act is quantified, and everything quantified is divisible. It therefore follows that, if there be any infinite in act, it is not after the manner of substance, but has the account of the accident that is quantity. And, if this be infinite, it will not be a principle, but that to which the infinite occurs, whether it be some sensible substance, such as air, or some intelligible substance, such as the even, as the Pythagoreans asserted. From this, it is plain that the Pythagoreans did not speak sensibly, positing the infinite as a substance and at the same time holding it as divisible, since it follows that every part of it would be infinite—which is impossible, as said above.
348. Finally, he says that this question—whether there be an infinite in mathematical quantities and in intelligible things not having magnitude—is a more general one than the present question. For our question concerns sensible things, which natural science treats: whether among natural things there be a body infinite in size, such as the early philosophers posited.