Commentary Aristotle Physics
Lecture 6. The opinions of the ancients concerning the infinite
The science of nature is concerned with spatial magnitudes and motion and time, and each of these at least is necessarily infinite or finite, even if some things dealt with by the science are not, such as a point and a passion.
It is not necessary, perhaps, that such things should be put under either head. Hence, it is incumbent on the person who specializes in physics to discuss the infinite and to inquire whether there is such a thing or not, and if there is, what it is.
The appropriateness of this problem to the science is clearly indicated. All who have touched on this kind of science in a way worth considering have formulated views about the infinite, and indeed, to a man, make it a principle of things. Some, as the Pythagoreans and Plato, make the infinite a principle in the sense of a self-subsistent substance, and not as a mere accident of some other thing.
Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these) and assert that what is outside the heaven is infinite.
Plato, on the other hand, holds that there is no body outside—the Forms are not outside because they are nowhere—yet that the infinite is present not only in the objects of sense but in these also.
Further, the Pythagoreans identify the infinite with the even. For this, they say, when it is cut off and shut in by the odd, provides things with the element of infinity. An indication of this is what happens with numbers.
By the addition of the gnomons to one and outside it, sometimes the figure that results is different, sometimes it is the same. But Plato has two infinites, the great and the small.
The physicists, on the other hand, all of them, always regard the infinite as an attribute of a substance that is different from it and belongs to the class of the so-called elements—water or air or what is intermediate between them. Those who make them limited in number never make them infinite in amount.
But those who make the elements infinite in number, as Anaxagoras and Democritus do, say that the infinite is continuous by contact—compounded of the homogeneous parts according to the one, and of the seed-mass of the atomic shapes according to the other.
Further, Anaxagoras held that any part is a mixture in the same way as the all, on the ground of the observed fact that anything comes out of anything.
For it is probably for this reason that he maintains that, once upon a time, all things were together.
(This flesh and this bone were together, and so of any thing; therefore, all things were, and at the same time, too.)
For there is a beginning of separation not only for each thing, but for all. Each thing that comes to be comes from a similar body, and there is a generation of all things, though not, it is true, at the same time.
Hence, there must also be an origin of generation. One such source there is that he calls mind, and mind begins its work of thinking from some starting point. So necessarily all things must have been together at a certain time, and must have begun to be moved at a certain time.
Democritus, for his part, asserts the contrary, namely, that no element arises from another element. Nevertheless, for him, the common body is a source of all things, differing from part to part in size and in shape. It is then clear from these considerations that the inquiry concerns the physicist.
Nor is it without reason that they all make it a principle. We cannot say that the infinite is in vain, and the only effectiveness that we can ascribe to it is that of a principle. Everything is either a principle or derived from a principle. But there cannot be a principle of the infinite or limitless, for that would be an end of it.
Further, as it is a beginning, it is both uncreatable and indestructible. For there must be a point at which what has come to be reaches its end, and also an end of all passing away.
That is why, as we say, there is no principle of this, but it is this that is held to be the principle of other things, and to encompass all and to steer all, as those assert who do not recognize, alongside the infinite, other causes, such as mind or friendship.
Further, they identify it with the divine, for it is “deathless and imperishable,” as Anaximander says, with the majority of the physicists.
326. After settling motion, the Philosopher now begins to treat of the infinite.
First, he shows that natural science should treat of the infinite;
second, he begins to treat of the infinite, at “belief in the existence” (203b15; [336]).
As to the first, he does two things:
first, he shows that it pertains to natural science to treat of the infinite;
second, he gives the opinions of the earlier philosophers concerning the infinite, at “and indeed” (203a3; [329]).
327. He proves the first point with an argument and a sign. The argument is as follows. Natural science studies magnitudes and time and motion. But, in such things, the finite and infinite are necessarily found, for every magnitude or motion or time is contained under one or the other, that is, either the finite or the infinite. Therefore, it pertains to natural science to consider the infinite as to whether it exists and as to what it is.
But, because it could be objected that consideration of the infinite pertains to first philosophy on account of its general character, he counters this by saying that not every being has to be either finite or infinite. For a point and a passion—that is, passible quality—are not contained under either, but the objects of consideration in first philosophy are things that follow upon being inasmuch as it is being, and not upon some definite genus of being.
328. Then, at “the appropriateness” (202b36), he establishes the same point through a sign taken from the practice of the natural philosophers. For all who have reasonably discussed this—natural philosophy—have mentioned the infinite. This fact is a probable argument, based on the authority of wise men, that it belongs to natural philosophy to treat of the infinite.
329. Then, at “and indeed” (203a3), he gives the opinion of the earlier philosophers about the infinite.
First, he shows in what they differ;
second, he shows in what they all agreed, at “nor is it without reason” (203b4; [335]).
As to the first, he does two things:
first, he gives the opinions on the infinite of those philosophers who were not natural, namely, the Pythagoreans and the Platonists;
second, he gives the opinions of the natural philosophers, at “the physicists” (203a16; [333]).
As to the first, he does two things:
first, he shows the points of agreement between the Pythagoreans and the Platonists;
second, their points of disagreement, at “only the Pythagoreans” (203a6; [331]).
330. He therefore says first (203a3) that, while all the philosophers posited the infinite as a certain principle of things, only the Pythagoreans and Platonists asserted that the infinite is not something accidental to some nature, but something existing of itself. This is not surprising, because it is in keeping with their claim that numbers and quantities are the substances of things. Now, the infinite is found in quantity; hence they posited that the infinite exists of itself.
331. Then, at “only the Pythagoreans” (203a6), he shows the difference between Plato and the Pythagoreans:
first, as to the laying down of the infinite;
second, as to the basis thereof, at “further, the Pythagoreans” (203a10; [332]).
Plato differed in two respects from the Pythagoreans with regard to his position on the infinite.
For the Pythagoreans did not posit an infinite except in sensible things. Since the infinite belongs to quantity and the first quantity is number, the Pythagoreans, not asserting number to be separated from sensible things, but rather stating number to be the substance of sensible things, consequently did not posit any infinite except in sensible things.
Likewise, Pythagoras considered that the sensible beings that are within the confines of the heavens are circumscribed by the heavens, and thus the infinite cannot be in them. Hence, he asserted that the infinite was in the sensible things outside the heavens.
But Plato, by contrast, asserted that nothing is outside the heavens. For neither did he say that there was any sensible body outside the heavens, since he maintained that the heavens contained all sensible things; nor did he say that the ideas and species of things, which he laid down as being separated, were outside the heavens, since “inside of” and “outside of” signify place, while the ideas, according to him, are not in any place, since place is of corporeal things.
Plato likewise said that the infinite is not only in sensible things, but also in the separated ideas, there being even in these, the separated numbers, something formal, such as unity, and something material, such as duality, out of which all numbers are composed.
332. Then, at “further, the Pythagoreans” (203a10), he shows the difference between them as to the basis of the infinite. And he says that the Pythagoreans attributed the infinite to a basis that was “even number.”
And they demonstrated this in two ways.
The first was an argument. That which is enclosed by another and is terminated by another has the account of the infinite; but that which encloses and terminates has the account of a term. Now, even number is comprehended and included under odd number. For if some even number is proposed, it is seen as in every way divisible. But, when it is reduced to an odd number by the addition of unity, it now takes on a certain indivisibility, as though even was compressed under odd. Hence, it seems as though “even” is infinite in itself, and causes infinity in others.
The same is shown by an example. To follow it, one must know that, in geometry, “gnomon” is the name for a square on the diameter with two supplements. If a square is added to this gnomon, a square is constituted. From this likeness, those numbers may be called “gnomons” that are added to certain numbers.
Here one should notice that, if one takes the odd numbers according to the order of natural progression and one adds the first odd number—namely, three—to unity, which is a square as to power (since one times one is one), there will be constituted four, which is a squared number, since twice two is four. If now to this second square there is added the second odd number—namely, five—one obtains nine, which is the square of three, since three times three is nine. Then, if to this third square there is added the third odd number—namely, seven—one obtains sixteen, which is the square of four. And thus, following the ordered addition of odd numbers, there always arises the same form in those numbers: a square.
By the addition of even numbers, however, there is always produced a different shape. For if the first even number—namely, two—be added to unity, there arises three, which has a triangular figure; if to this there next be added the second even number—namely, four—one has seven, which is in the shape of a heptagon. And thus, in this way the figure of the resulting numbers constantly varies with the addition of even numbers.
And this appears to be a sign that uniformity belongs to odd number, while deformity and variation and the infinite belong to even number. Hence he says that a sign of this—namely, that infinity follows even number—is what occurs in numbers. For by the addition of gnomons, or added numbers, to one, or unity, and outside, that is, to other numbers, sometimes the figure that results is different, another natural form, as when one adds an even number, sometimes it is the same, as when one adds an odd number. From this, it is evident why Pythagoras attributed infinity to even number.
But Plato attributed it to two roots, namely, to the large and the small. For these two, according to him, belong to matter, to which in turn the infinite belongs.
333. Then, at “the physicists” (203a16), he gives the opinions of the natural philosophers about the infinite. He says that all the natural philosophers—those who gave natural principles for things—taught that the infinite does not subsist by itself, as said above, but is an accident of some nature. Hence, those who posited just one material principle (some member of the list of things called elements, such as air or water or something intermediate) said it was infinite. But, of those who posited a finite number of principles, none supposed them to be infinite in quantity: for the very distinction of the elements seemed to conflict with the notion that they could be infinite. But those who posited an infinitude of principles said that from all those infinites was formed one infinite through contact.
334. Those who taught this were Anaxagoras and Democritus, who differed in two respects.
They differed first as to the nature of the infinite principles: Anaxagoras taught that the infinite principles were infinite similar parts of flesh and of bone and so on; but Democritus taught that they were indivisible bodies differing in figure. He said these bodies were the seeds of all of nature.
Another difference was as in the relation of these principles one to the other. For Anaxagoras said that each of these parts was a mixture of all the others, such that, in each part of flesh, there was bone and vice versa and the same for the other parts. He came to this opinion because he saw that anything came from anything; and hence, since he believed that whatever comes to be from something is in it, he concluded that everything is in everything. And, from this, he seems to assert that, at some time, all things were commingled and nothing was distinct from anything else. Just as this flesh and this bone are commingled (which is proved by the generation of one from the other), so is everything else commingled. Therefore, at one time, all things were together. For it is necessary to posit a principle of separation not only in one single thing, but in all things simultaneously.
He proved this thus. Whatever comes to be from something other was previously commingled with it and is produced by being separated from it; but all things are produced, though not all at the same time;
therefore, there must be some one principle generating not only each thing but all things. This one principle he called “intellect,” which alone has the capacity to separate and bring together, because it is itself uncommingled.
Now, whatever comes to be through intellect seems to have a principle, because intellect acts by starting from a definite principle. Therefore, if separation is brought about by intellect, separation must have a principle; hence, he concluded, all things were at some time together and the motion by which things were separated one from the other began in time and did not previously exist. Thus, Anaxagoras laid down one principle as producing another.
But Democritus said that one principle is not derived from another, but that the nature of body, which is common to all indivisible bodies, though different in parts and figure, is the principle of all things according to magnitude, for he posited that all divisible magnitudes are composed of indivisibles.
And thus does Aristotle conclude that to consider the infinite pertains to the natural philosopher.
335. Then, at “nor is it without reason” (203b4), he outlines four points of agreement among the early philosophers in regard to the infinite.
The first is that all posited the infinite as a principle, and this with reason, that is, through a probable argument. If the infinite exists, it is impossible for it to be in vain, to lack some definite standing among the beings of reality. But it can have no power other than that of a principle. For all things in the world are either principles or derived from principles. But it is not fitting for the infinite to have a principle, because what has a principle has an end. Hence it follows that the infinite is a principle. Note, however, that, in this reasoning, principle and end are both used equivocally; for that which is derived from a principle has a principle of origin, but it is incompatible with the infinite to have a beginning and end of its quantity or size.
The second point of agreement is that they denied coming into existence and ceasing to exist to the infinite. This follows from the fact that it is a principle. For whatever is produced must have an ending, just as it has a principle; and, likewise, any process of corruption has an end. But “end” and “infinite” are incompatible; hence, the infinite can be neither generated nor corrupted. Hence, it is clear that the infinite has no principle, but that the infinite is the principle of everything else. This argument, too, uses principle and end equivocally, as above.
The third point of agreement is that they attributed to the infinite the prerogative of containing and governing all things, for this seems to belong to a first principle. And this was the opinion of those who did not grant, in addition to matter (which they said was infinite), other causes—namely, agent causes—as Anaxagoras posited an intellect and Empedocles concord. For to contain and to govern pertain more to an active principle than to matter.
The fourth point of agreement was to attribute divinity to the infinite; for whatever is immortal or incorruptible they called divine. This was the doctrine of Anaximander and a number of the ancient natural philosophers.