Lecture 11 — The arguments of Zeno, who tried to deny all motion, are answered

Zeno’s reasoning, however, is fallacious when he says that, if everything is at rest when it occupies an equal space and if what is in locomotion is always occupying such a space at any moment, then the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.

Zeno’s arguments about motion, which cause so much disquietude to those who try to solve the problems that they present, are four in number.

The first asserts the non-existence of motion on the ground that what is in locomotion must arrive at the half-way stage before it arrives at the goal. This we have discussed above.

The second is the so-called “Achilles,” and it amounts to this: in a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, such that the slower must always hold a lead.

This argument is the same in principle as that which depends on bisection, though it differs from it in that the spaces with which we successively have to deal are not divided into halves.

The result of the argument is that the slower is not overtaken, but it proceeds along the same lines as the bisection-argument (for in both, a division of the space in a certain way leads to the result that the goal is not reached, though the “Achilles” goes further, in that it affirms that even the quickest runner in tragic tradition must fail in his pursuit of the slowest), such that the solution must be the same.

And the axiom that that which holds a lead is never overtaken is false: it is not overtaken, it is true, while it holds a lead, but it is overtaken nevertheless if it is granted that it traverses the finite distance prescribed. These, then, are two of his arguments.

The third is that already given above, to the effect that the flying arrow is at rest, which result follows from the assumption that time is composed of moments; but if this assumption is not granted, the conclusion will not follow.

The fourth argument is that concerning bodies moved in an equal manner on a race course with equal velocity in opposite directions—the one originally between the goal and the midpoint, and the other between the midpoint and the start.

This, he thinks, involves the conclusion that half a given time is equal to double that time.

The fallacy of the reasoning lies in the assumption that a body occupies an equal time in passing with equal velocity a body that is in motion and a body of equal size that is at rest, which is false.

For instance (so runs the argument), let AAA be the bodies of equal size which are standing still, and let BBB the bodies equal in number and in size to AAA and originally occupying the half of the course from the starting post to the middle of AAA, and let CCC be those originally occupying the other half from the goal to the middle of AAA, equal in number, size, and velocity to BBB.

Then three consequences follow: first, as BBB and CCC pass one another, the first B reaches the last C at the same moment as the first C reaches the last B.

Second, at this moment, the first C has passed all of AAA, whereas the first B has passed only half of AAA, and has consequently occupied only half the time occupied by the first C, since each of the two occupies an equal time in passing each A.

Third, at the same moment, all of BBB has passed all of CCC, for the first C and the first B will simultaneously reach the opposite ends of the course, since (so says Zeno) the time occupied by the first C in passing each B is equal to that occupied by it in passing each A, because an equal time is occupied by both the first B and the first C in passing all of AAA.

This is the argument, but it presupposed the previously stated fallacious assumption.

Nor, in reference to contradictory change, shall we find anything unanswerable in the argument that, if a thing is changing, say, from not-white to white and is in neither condition, then it will be neither white nor not-white. For the fact that it is not wholly in either condition will not preclude us from calling it white or not-white.

We call a thing white or not-white not necessarily because it is wholly either in one or the other, but because most of its parts or the most essential parts of it are so: not being in a certain condition is different from not being wholly in that condition.

It is the same in the case of being and non-being and all other conditions that stand in a contradictory relation: while the changing thing must of necessity be in one of the two opposites, it is never wholly in either.

Again, in the case of circles and spheres and everything whose motion is confined within the space that it occupies, it is not true to say the motion can be nothing but rest on the ground that such things in motion—themselves and their parts—will occupy the same position for a period of time, and that therefore they will be at once at rest and in motion.

For in the first place, the parts do not occupy the same position for any period of time; in the second place, the whole also is always changing to a different position;

for if we take the orbit as described from a point A on a circumference, it will not be the same as the orbit as described from B or C or any other point on the same circumference, except in an accidental sense—the sense, that is to say, in which a musical man is the same as a man.

Thus, one orbit is always changing into another, and the thing will never be at rest. And it is the same with the sphere and all other things that are moved within themselves.

860. After finishing with the division of motion and of rest, the Philosopher now refutes certain opinions that have been the source of error in regard to motion.

About this, he does three things:

first, he answers the arguments of Zeno, who denies that motion exists at all;

second, he shows that an indivisible is not moved, against Democritus, who said that they are always in motion, at our next point is that what is (240b8; [872]).

third, he shows that all change is finite, against Heraclitus, who said that all things are eternally moved, at our next point is that no process (241a26; [879]).

About the first, he does two things:

first, he gives and rejects one of Zeno’s arguments, which pertains to what Zeno had accepted about motion;

second, he explains all his arguments in order, at Zeno’s arguments (239b9; [862]).

861. He says, therefore (239b5), that Zeno reasoned badly and used what had only the appearance of a syllogism to show that nothing is being moved—even what seems to be in rapid motion, as an arrow in flight. And this was his argument. Anything that is in a place equal to itself is either being moved or is at rest. But whatever is being moved is at each instant in a place equal to itself. Therefore, even at each instant, it is either in motion or at rest. But it is not in motion; therefore, it is at rest. But if it is not in motion at any instant but at rest, as it seems, then it is at rest throughout the entire time and not in motion.

Now, this argument could be answered by appealing to something already proved: in an instant, there is neither motion nor rest. (Lect. 5, nn. 794–95.) But such a solution would not cripple Zeno’s intention, for he is satisfied to show that, through the entire time, there is no motion. This seems to follow if there is no motion at any instant of the time. Therefore, Aristotle answers in a different manner and says that the conclusion both is false and does not follow from the premises.

For in order that something be moved in a given period of time, it has to be moved in each part of the time. But instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. (Lect. 1.) Hence, it does not follow that a thing is not in motion in a given time just because it is not in motion in any instant of that time.

862. Then, at Zeno’s arguments (239b9), he lists in order all the arguments that Zeno used for destroying motion.

About this, he does three things:

first, he shows how he destroyed local motion with his arguments;

second, he shows how he destroyed the other types of change, at nor, in reference to (240a19; [870]).

third, how in particular he destroyed circular motion, at again, in the case (240a29; [871]).

863. In regard to the first, he lists four reasons. Zeno used four arguments against motion that have caused difficulty for many of those who tried to answer them. The first is this. If anything is being moved through a certain space, it must reach the middle before it reaches the end. But, since the first half is divisible, half of it must be first traversed and so on indefinitely, since a magnitude can be divided to infinity. Infinites, however, cannot be traversed in finite time. Therefore, nothing can be moved.

Therefore, Aristotle says that he has already answered this argument (in the beginning of this book 6 (Lect. 4.)), when he proved that time is divided to infinity in the same way as a magnitude is. This answer is directed less to the immediate question than to one who asks whether infinites can be traversed in finite time, as he will say in book 8, (Lect. 17.) where he answers this argument by showing that a mobile does not use the infinites that exist in a magnitude as though they were actually existing, but only as existing potentially. For a thing in motion uses a point in space as actually existing when it uses it as a beginning and as an end, and it is then that the mobile must be at rest, as will be explained in book 8. But, if it had to traverse infinites that were actually existing, then it would never reach the end.

864. The second argument is given at the second is (239b14), and he says that they called this one the Achilles, as though it were invincible and unanswerable. The argument was this. If anything is being moved, it follows that a slower thing, if it started earlier, will never be caught by anything moving most rapidly. And it was proved in the following way. If a slower object began to be moved for some time before a very swift one, then, in that time, it has traversed some distance. Therefore, before the very swift one in pursuit could reach the slower, which is still running, it must leave the place first left by the pursued and reach the place that the pursued reached during the time the pursuer was not in motion. But the very fast pursuer must traverse this space in some time, during which the slower has meanwhile traversed a certain space, and so on forever. Therefore, the slower must always hold a lead—that is, is always some distance ahead of the most swift pursuer—and so the swifter will never catch the slower. But this is unacceptable. Therefore, it is better to say that nothing is moved.

865. In solving this argument, he says that it is the same as the first, which proceeded by dividing the distance into two halves and then continually halving one part of the remainder.

But the difference between them is that, in the second, the given magnitude of space is not divided into halves, but according to the difference between the swift and the slower motion.

For in the first period of time, in which only the slower was in motion, there is a greater magnitude involved; in the second period (in which the faster traversed the distance covered by the slower between its start and the start of the faster), which is a shorter time period, a smaller magnitude was traversed by the slower, and so on forever.

Hence, the time and the magnitude are always being divided and that seems to be the reason why the slower is never caught by the swifter.

But this tends to the same thing as what was said of the division of the magnitude into halves: in both arguments, it seems that the mobile cannot reach a certain goal on account of the magnitude’s being infinitely divided, no matter how it happens to be divided—that is, whether according to halves, as happens in the first argument, or according to the excess of the faster over the slower, as in the second argument. However, in this second argument, it is further added that the very swift cannot reach the slower that it is pursuing. This tragic phraseology employs inflated language in order to excite wonder, but it does not do anything to the force of the argument.

Hence, it is clear that the solution of the two arguments is the same. For just as a false conclusion was reached in the first argument—namely, that the mobile would never reach the end of the magnitude on account of the infinite division of the magnitude—so also what the second argument tries to conclude is false, namely, that the slower will never be caught by the swifter, which is just another way of saying that a mobile will never reach its goal.

Now, it is true that, as long as the slower is ahead, it is not yet reached by the swifter. But yet it will at some time be reached, if you concede that a finite magnitude can be traversed in finite time. For the swifter pursuing mobile will traverse the whole distance by which the slower is ahead and even more, in less time than the slower was meantime moving farther ahead. Proceeding in this way, the swifter will not only catch the slower, but pass it. These, therefore, are the solutions to two of Zeno’s arguments.

866. The third argument is given at the third is (239b30), and he says that the third argument of Zeno was the one cited above before he began to give the arguments—namely, that an arrow in flight is always at rest. And as was said above, this seems to happen, because Zeno supposed that time is made up of instants. For unless that be granted, the syllogism fails.

867. He sets out the fourth argument at the fourth argument (239b33).

Concerning it, he does three things:

first, he sets out the argument;

second, he gives the solution, at the fallacy of (240a1; [868]).

third, he explains it by an example, at for instance (240a4; [869]).

First, therefore, he says that the fourth argument of Zeno proceeded from some bodies that move in a stadium such that there are two equal magnitudes that are moved in an equal manner (that is, through a space in the stadium equal to both in quantity), and this motion is in opposite directions (that is, one of the equal magnitudes is moved through that space of the stadium toward one part, and the other toward the other part, in such a way, however, that one of the mobile magnitudes begins to move from the terminus of the stadium that is equal to it, and the other begins to move from the middle of the stadium or of a space in the given stadium); both move with equal velocity. This being given, Zeno held that it would result in a half time equaling a double time. Since this is impossible, he intended to infer further that it is impossible for anything to be moved.

868. Then, at the fallacy of (240a1), he gives the solution. He says that Zeno was deceived in that, on the one hand, he held that the mobile is moved according to the moved magnitude and, on the other, he held that it was moved according to a resting magnitude equal to the moved magnitude. Because an equal velocity of the moved bodies is supposed, he wanted to conclude that the motion of equally swift bodies in regard to equal magnitudes, one of which is in motion and the other standing still, is done in equal times. This is seen to be false for the following reason.

When something is moved in relation to a resting magnitude, there is only one motion, but when something is moved in relation to a moving magnitude, there are two motions. If they are moving in the same direction, it takes more time, and if they are moving in opposite directions, it takes less, according to the amount of either motion. If the magnitude in relation to which something mobile is moved is moved in the same direction with an equal velocity, or even a greater velocity, the other moving body can never pass it. If the magnitude moves with less speed, it will pass by it at a certain time, but it will take more time than if it were resting. It is quite the contrary if the magnitude is moved opposite the direction of the other body. The more swiftly the magnitude moves, the less time the other body takes to pass it, because the two motions work together to pass each other.

869. Then, at for instance (240a4), he makes clear what he said in the latter part. Suppose that there are three magnitudes equal to each other, each designated as A, and these magnitudes are standing still, that is, not moved; thus, there might be a space of three cubits, each one of which is marked by an A. There are another three magnitudes all equal and designated as B, as there might be one moving unit of three cubits. These magnitudes begin to move from the middle of the space. There are also three other magnitudes, equal in number, size, and velocity to B, and designated as C. These begin to move from the last space, that is, from the last A.

This being given, it occurs that the first B, by its motion, arrives at the last A, and likewise the first C, by its motion, arrives at the first A, opposite the last. Also, at the same time, when this is at the last B, as though crossing at the same moment, that is, next to all the Bs, which are moved by one another. When this has been done, it is evident that this first C has passed all the As, but B has passed by only half. Since, therefore, B and C are equal in velocity, and an equal velocity passes by a smaller magnitude in less time, it follows that the time in which B travels to the last A is half the time in which C arrives at the first A opposite; but each of them (C and B) occupies an equal time in passing each A, that is, in equal times pass each section of A.

This being supposed—that the time in which B arrives at the last A is half the time in which C arrives at the first A opposite—it must be further considered how Zeno wished to conclude that this half time is equal to its double. For from the supposition that the time of the motion of C is double the time of the motion of B, it is supposed that, in the first half of the time, B was still and C moved, and thus C arrived in that half of the time at the middle of the space, where B was; and then B began to move to one part and C to another. When B arrived at the last A, it had to pass all of C, because, at the same time, the first B and the first C are at contrary ultimates, one at the first A and the other at the last, and as he said, C is next to each B in the same amount of time as it takes to pass each one of the As. This is so, because B and C each pass one A in the same interval of time. Thus, it seems that, if B covers a time equal to that in which it passes C, then C passes B and A in an equal interval of time. Therefore, the interval in which C passes all of B is equal to the time in which it passed all of A. The time in which C passed all of B is equal to the time in which C or B passed the middle of A, as was said. But it was proved that the time in which B passed the middle of A is half the time in which C reached all of A. Therefore, it follows that the half is equal to the double, which is impossible.

This is the argument of Zeno. But he falls into the error previously mentioned: he assumes that C, in the same interval of time, crosses B moving in a counter direction and A resting, which is false, as was said above.

870. Then, at nor, in reference to (240a19), he gives the argument by which Zeno rejected change between contradictories. For he said that whatever is being changed is in neither of the extremities while it is being changed, for while it is in the terminus from which, it is not yet being changed, and while it is in the terminus to which, it has already been changed. Therefore, if something is being changed from one contradictory to another, as from non-white to white, it follows that, while it is being changed, it is neither white nor non-white—which is impossible.

Now, although this strange conclusion would follow for those who posit that an indivisible can be moved, yet for us who posit that whatever is being moved is divisible, nothing impossible follows. For even though it is not totally in one or the other of the extremes, it is not for that reason neither white nor non-white: one part could be white and the other non-white. For a thing is called white not only when all of it is white but also when very many or its main parts are white—that is, the parts to which it is more proper to be such—because it is one thing not to be something at all and another not to be entirely something, for example, white or non-white.

And what has been said of white and non-white is to be understood of unqualified being or non-being and of all things that are contradictorily opposed, as hot and non-hot and so on. For what is being changed must always be in one of the opposites, because it is described in terms of whichever opposite predominates in it. But it does not follow that it is always as a whole in neither of the extremities, as Zeno supposed.

Now, it should be known that this answer is sufficient to refute Zeno’s argument, and that is what Aristotle’s main intention is. But the truth of this matter will be more fully given in book 8. (Lect. 17.) For it is not true in all cases that part is altered or generated after part, but sometimes the whole comes all at once, as was said above. (Lect. 8, n. 840.) In that case, it is not this answer that would apply, but the one in book 8.

871. Then, at again, in the case (240a29), he refutes the argument by which Zeno rejected spherical motion. For he said that it is not possible for anything to be moved circularly or spherically or in such a way that certain things are moved within themselves, that is, the motion is confined within the space occupied by the mobile. And he proved this with the following argument. Anything that is, in its entirety and in respect of its parts, in one and the same place for a period of time is not in motion, but at rest. But all the above-mentioned fulfill these conditions, even when they are apparently in motion. Therefore, they are at once in motion and at rest—which is impossible.

The Philosopher attacks this argument on two points.

He first attacks the statement that the parts of the moving sphere are in the same place for some time. For Zeno was speaking of the place of the whole, and it is true that, while the sphere is in motion, no part passes out of the place of the sphere, but Aristotle speaks of the particular place of each part, according as a part has a place. For it was said in book 4 that the parts of a continuum are in place potentially. (Lect. 7, n. 479.) But it is evident in spherical motion that a part does change its particular place, although it does not leave the place of the whole, because where one part was, another part succeeds.

Second, he attacks the statement that the whole remains in the same place for some time. Against this, Aristotle says that even the whole is changing its place. For in order that two places be not the same, it is not required that one of them be entirely outside the other, but sometimes the second place is partly joined to part of the first and partly divided from it, as is clear in things moved in a straight line. For let a body of one cubit be moved from place AB to place BC—both places being one cubit each. While the mobile is being moved from one place to the other, it must partly desert one place and enter the other; for example, it could leave the portion AD of AB and enter the portion BE of BC. Therefore, it is clear that the place DE is distinct from AB, although not entirely, but only partly separated from it.

But, if it were assumed that that part of the mobile that entered the second place reentered part of the place deserted, there would be two places, yet in no way separated. They would differ not really, but only in account, that is, in the sense that the beginning of the place might be successively called by different letters each time the mobile reentered it—namely, where the beginning of the mobile is, that is, some spot in the mobile that is taken as a beginning, Thus, there would be two places conceptually, but one and the same in reality.

This is how we must understand what Aristotle says here: it is not the same revolution when it is taken as beginning at A and as beginning at B and as beginning at C or any other mark, unless you insist that it is the same revolution as to subject, as in the case of “musical man” and “man,” since one happens to the other. Hence it is clear that the mobile is always being moved from one circular place to another and is not at rest, as Zeno tried to prove. And it is the same with the sphere and everything else whose motion is confined within the space it occupies, as in the case of a potter’s wheel and a (rotating) pillar or anything of that sort.