Lecture 13 — A natural and demonstrative argument showing no natural body can be infinite

79 Infinite weight is therefore impossible, and the same reasoning applies also to infinite lightness. Bodies then of infinite weight and of infinite lightness are equally impossible. That there is no infinite body may be shown, as we have shown it, by a detailed consideration of the various cases.

80 But it may also be shown universally, not only by such reasoning as we advanced in our discussion of principles (though in that passage we have already determined universally the sense in which the existence of an infinite is to be asserted or denied), but also suitably to our present purpose in the following way.

81 That will lead us to a further question. Even if the total mass is not infinite, it may yet be great enough to admit a plurality of universes. The question might possibly be raised whether there is any obstacle to our believing that there are other universes composed on the pattern of our own, more than one, though stopping short of infinity. First, however, let us treat of the infinite universally.

82 Every body must necessarily be either finite or infinite, and if infinite, either of similar or of dissimilar parts. If its parts are dissimilar, they must represent either a finite or an infinite number of kinds.

83 That the kinds cannot be infinite is evident, if our original presuppositions remain unchallenged. For the primary movements being finite in number, the kinds of simple body are necessarily also finite, since the movement of a simple body is simple, and the simple movements are finite, and every natural body must always have its proper motion.

84 Now if the infinite body is to be composed of a finite number of kinds, then each of its parts must necessarily be infinite in quantity, that is to say, the water, fire, etc., which compose it. But this is impossible, because, as we have already shown, infinite weight and lightness do not exist.

85 Moreover it would be necessary also that their places should be infinite in extent, so that the movements too of all these bodies would be infinite. But this is not possible, if we are to hold to the truth of our original presuppositions and to the view that neither that which moves downward, nor, by the same reasoning, that which moves upward, can prolong its movement to infinity. For it is true in regard to quality, quantity, and place alike that any process of change is impossible which can have no end. I mean that if it is impossible for a thing to have come to be white, or a cubit long, or in Egypt, it is also impossible for it to be in process of coming to be any of these. It is thus impossible for a thing to be moving to a place at which in its motion it can never by any possibility arrive.

86 Again, suppose the body to exist in dispersion, it may be maintained none the less that the total of all these scattered particles, say, of fire, is infinite.

87 But a body we saw to be that which has extension every way. How can there be several dissimilar elements, each infinite? Each would have to be infinitely extended every way.

88 It is no more conceivable, again, that the infinite should exist as a whole of similar parts. For, in the first place, there is no other (straight) movement beyond those mentioned: we must therefore give it one of them. And if so, we shall have to admit either infinite weight or infinite lightness. Nor, secondly, could the body whose movement is circular be infinite, since it is impossible for the infinite to move in a circle. This, indeed, would be as good as saying that the heavens are infinite, which we have shown to be impossible.

89 Moreover, in general, it is impossible that the infinite should move at all. If it did, it would move either naturally or by constraint: and if by constraint, it possesses also a natural motion, that is to say, there is another place, infinite like itself, to which it will move. But that is impossible.

124. After showing that no single natural body is infinite, the Philosopher here shows by a general argument that no natural body is infinite—for a proof through a common medium causes more perfect science.

About this, therefore, he does two things:

first he mentions his intention;

secondly, he proves his proposition, at every body must necessarily (274a30; [128]).

125. As to the first he does three things.

First he shows (as if summarizing) what has been previously said. And he states that for those who think according to the lines already laid down, it is clear that there is no infinite body, by a detailed consideration of the various cases, i.e., on account of the reasons applied to the individual parts of the universe, namely, to the body that is moved circularly and to the bodies that are move upward or downward.

126. Secondly, at but it may also be shown universally (274a19), he shows what immediately remains to be said. And he says that the same thing can be clear if someone looks at it universally, i.e., by a common medium. And this is in addition to those general arguments given in the book of the Physics where the common principles of all natural bodies were discussed—for in Physics III is a universal treatment of the infinite, as to how it exists and how not, it being shown there that the infinite exists in potency but not in act. Now, however, the infinite has to be treated in another way, by showing universally that no sensible body can be infinite in act.

127. Thirdly, at that will lead us to a further question (274a24), he shows what must be determined immediately after these questions. And he says that after proving what has been proposed, our aim will be to inquire (on the supposition that the whole body of the universe is not infinite) whether the whole body is of such size that there can be made from it several heavens, i.e., many worlds. For perhaps someone could wonder whether it is possible that, just as our world is established about us, there might be other worlds, i.e., more than one though not an infinitude. But before dealing with that question, we shall speak universally of the infinite and show that from common reasons no body is infinite.

128. Then at every body must necessarily (274a30) he proves the proposition:

first, by natural demonstrative arguments;

secondly, by logical arguments the question may also be examined (L. 15; 275b12).

Now I call "demonstrative" and "natural" those arguments that are taken from the proper principles of natural science, whose consideration concerns motion, and action and passion which reside in motion, as is said in Physics III.

First, therefore, at (274a30) he shows that no body is infinite from the side of local motion, which is the first and most common of motions;

secondly, universally on the part of action and passion, at that in general it is impossible (L. 14; 274b33).

As to the first he does two things:

first he presents certain divisions (274a30; [129]);

secondly, he examines the members individually, at that the kinds (274a33; [130]).

129. Therefore he first (274a30) presents three divisions.

The first of these is that every body must be either finite or infinite. If it is finite, we have our proposition; but if it is infinite, a second division remains, namely, that it be a heterogeneous whole, i.e., having dissimilar parts, as an animal body which is composed of flesh, bones and sinews; or it is a homogeneous whole, i.e., having like parts, such as water, each part of which is water. But if it is a whole of dissimilar parts, a third division remains, namely, whether the species of the parts of such a body are finite or infinite in number. If it is proved that they are not infinite, nor again finite, and further that no body of parts that are alike is infinite, it will have been proved universally that no body is infinite.

130. Then, at that the kinds (274a33) he pursues each member. He does three things about this:

first he shows that it is not possible in a body of unlike parts for the species of its parts to be infinite;

secondly, that it is not possible for an infinite body of unlike parts to be such that the species of its parts be finite, at now if the infinite body (274b5; [131]);

thirdly, he shows that there can be no infinite body having parts that are alike, at it is no more conceivable (274b22; [135]).

He says therefore first (274a33) that it is plainly not possible for an infinite body to be constituted from an infinite species of parts, so long as one is loyal to the first hypotheses, i.e., the previously made suppositions that there are only three species of simple motion. For if the first motions, i.e. the simple motions, are finite, then the species of simple bodies must be finite, for the motion of a simple body is itself simple, as was had above. But it was also held above that simple motions are finite: for there are three, namely, motion to the middle, motion from the middle, and motion around the middle. Now the reason why simple bodies are finite, if simple motions are finite, is that every natural body must have its own proper motion—but if there were an infinite species of bodies, while the number of motions was finite, there would have to be some species of bodies without motions, which is impossible.

Consequently, from the fact that simple motions are finite, it is sufficiently proved that the species of simple bodies are finite. Now it is from simple bodies that mixed bodies are composed. Hence, if there were a whole having unlike parts and composed of an infinite species of mixed bodies, the species of the first components would still have to be finite—though it does not even seem possible that mixtures from finite elements should be infinitely diversified. Neither can any compound body be called a mixture of all like parts, because, even if its quantitative parts be specifically alike, as each part of a stone is stone, yet its essential parts are specifically diverse, for the substance of a mixed body is composed of simple bodies.

131. Then, at now if the infinite body (274b5), he shows that it is impossible to have an infinite body of unlike parts, the species of which parts are finite. And he arrives at this with four arguments.

The first is that, if a body of unlike parts is infinite and composed of parts that are finite with respect to species, each of the parts would have to be infinite in magnitude. For example, if a mixed body were infinite and composed of elements that were finite, air would have to be infinite, and so would the water and the fire. But this is impossible, because, since each of these is either heavy or light, it would follow according to what was previously said that its heaviness or lightness would be infinite. But it has been proved that no heaviness or lightness can be infinite. Therefore, it is not possible for an infinite body of unlike parts to be composed of a finite species of parts.

However, someone could object that it does not follow from this argument that each of the parts is infinite: for it could be possible for the whole to be infinite if one part were infinite in magnitude and the others finite. But this was rejected in Physics III—for if one part were infinite it would consume the other finite parts on account of its excessive power. Likewise it can be said that even in that case the same impossibility will follow, namely, that there would be an infinite heaviness or lightness. And therefore Aristotle was not concerned with it.

132. The second argument is presented at moreover it would be necessary (274b8). For if the parts of an infinite whole were infinite in magnitude, their places would have to be infinite in magnitude, because places are necessarily equal to the things in them. But motion is measured according to the magnitude of the place into which it passes, as is proved in Physics VI. Therefore it follows that the motions of all these parts would be infinite. But this is impossible, if what we supposed above is true, namely, that nothing can be moved downward infinitely, nor upward either—because "down" is determinate, since it is the middle, and for the same reason "up" is determinate (for if one contrary is determinate, so is the other).

And he also proves this by what is common to all motions. For in the transmutation according to substance, we see that it is impossible for a thing to become what it cannot be, as, for example, there cannot be made a rational ass, since it is impossible for an ass to be such. And the same goes for a motion in such, i.e., with respect to quality and for a motion in so much, i.e., with respect to quantity, and for a motion in where, i.e., with respect to place. For if it is impossible for something black ever to have been made white, as a raven, it is impossible for it ever to become white. And if it is impossible for anything to be a foot long, as an ant, it is impossible for it to be moving toward that; and if it is impossible for something to be in Egypt, as the Danube, it is impossible for it to be moving thither. The reason for this is that nature does nothing in vain. But it would be in vain for a thing to be tending to what is impossible for it to reach. Consequently, it is impossible for a thing to be locally moved to a place where it cannot arrive. But it is impossible to traverse an infinite place. If, therefore, places were infinite, there would be no motion. But since that is impossible, it cannot be that the parts of an infinite body of unlike parts be infinite in magnitude.

133. He presents the third argument at again, suppose the body to exist in dispersion (274b18). For someone could say that there is no infinite continuous unit, but that there are yet certain parts, disconnected and not continued, which are infinite, as Democritus posited infinite indivisible bodies, and as Anaxagoras posited infinite parts all similar to each other. But Aristotle says that this position leads no less to an impossibility: for if infinite parts of fire are not joined, there is nothing to prevent all of them from joining and thus making one infinite fire from all of them.

134. The fourth argument he presents at but a body (274b19). For when something is said to be infinite, the term should be taken according to its proper meaning. For example, if we say that a line is infinite, we understand it to be infinite in length; while, if we say that a surface is infinite, we understand that it is infinite in length and width. But a body stretches in every direction, because it has three dimensions, as was said above. Consequently, if a body is said to be infinite, it will have to be infinite in every direction, and so in no direction will there be anything outside it. It is therefore not possible that there be in an infinite body many things that are unlike, each of which is infinite, for according to the foregoing it is not possible for there to be a number of infinites.

135. Then, at it is no more conceivable (274b22) he shows that there cannot be an infinite body having like parts — and this with two arguments.

The first of these is that every natural body must have some local motion; but there is no other except those mentioned above, one of which is around the middle, another from the middle, and a third to the middle. It follows, therefore, that it has one of these. But this is impossible—for if it moves upward or downward, it will be heavy or light, and, consequently, its heaviness or lightness will be infinite, which is impossible according to what has gone before. Likewise it cannot be moved circularly, because it is impossible for the infinite to turn in a circle. For there is no difference between saying this and saying that the heaven is infinite—which is impossible, as was proved above. Therefore a whole infinite body cannot be homogeneous.

136. The second argument is set down at moreover, in general (274b29), and it follows from the common notion of local motion. For if there should be an infinite body of parts that are alike, it follows that it cannot be moved at all. If it is moved, it will be moved either according to nature, or by compulsion. But if it has a compulsory motion, then must be a motion natural to it, because a compulsory motion is contrary to a natural motion, as was had above. But if there is a motion natural to it, it follows that there is a place equal to it, into which it is naturally moved, for natural motion belongs to what is moved to its own place. This, however, is impossible, because it would follow that there would be two infinite corporeal places, which is as impossible as that there should be two infinite bodies, for, just as an infinite body is infinite in every direction, so too is an infinite place. Therefore it is not possible for an infinite body to be moved. But if every natural body is moved, it therefore follows that no natural body is infinite. It should be noted that this argument applies only to straight motion, for what is moved circularly does not change its place as to subject, but only in conception, as is proved in Physics VI. But that an infinite body cannot be moved circularly has already been proved above in many ways.