by A. A. Long
With these objections out of the way, let us examine Zeno’s reasoning. As given by Aristotle, the argument has three premises.
1. It is always necessary to cross half the distance.
2. These [namely, the half-distances] are infinite.
3. It is impossible to get through things infinite in number. Therefore,
4. It is impossible to cross the whole distance.
The expanded reading at the beginning of this section restates the premises so that the conclusion (4) follows validly. As earlier, I take it that Zeno’s opponents will agree to premises (1) and (2). What of premise (3)?
At this point it will be useful to bring in the notion of the mathematical infinite. For Zeno’s description of the situation, so far from involving any logical impossibility, reveals some features of the infinite – features that may strike us as odd and counter-intuitive, and that are false for finite collections but that are inevitable consequences of describing the finite interval AB as being composed of an infinite number of subintervals.
There is an infinite number of natural numbers, 1, 2, 3,…. In the Dichotomy the sequence of intervals needing to be crossed can be put into one-to-one correspondence with the sequence of natural numbers. The first interval, AA1 (the halfway distance from A to B), corresponds to the number 1, the second interval, A1 A2 (the halfway distance from A1 to B), to 2, and so on. There is one natural number for each interval and one interval for each natural number. Now, however far we count the natural numbers, there are still more, and likewise, however many intervals we cross, there are still more. There is no highest number and no last interval. If saying “one” or “two” and so on is an act of counting, then there is no last act of counting that exhausts the natural numbers. Likewise, there is no act of crossing an interval that is the last such act in crossing AB. We cannot get through either of these sequences by going through its members one by one.
Further, in certain cases the sum of an infinite sequence of numbers is finite. In particular, consider the sequence mirrored in the Dichotomy, 1/2, 1/4, 1/8 …, and call this sequence T. It is an infinite sequence, in that it has an infinite number of members. These members correspond to the lengths of the intervals that need to be crossed in crossing the stadium. Call the sum of the first n members of T the nth partial sum of T and designate the nth partial sum of T as S.n. Then Sn. Then S1 = 1/2, S2 = 3/4, and so on. Let S stand for the sequence S1, S2, S3,…. The members of S correspond to the total distance travelled after each move: 1/2 the stadium after the first move, 3/4 after the second, and so on. There is no last member of T or (therefore) of S. Since the members of T are all greater than zero, as n increases Sn keeps increasing. But since each member of T is only half the size of the previous one, the amount by which Sn increases each time is only half the size of the amount it increased the previous time. In fact, all the partial sums are less then 1. This is precisely like the situation Zeno describes. However many intervals we may cross, we have not yet reached the finish line (i.e., for all n, Sn < 1). Also, as n increases, Sn gets as close to 1 as we like (in the precise sense that for any given x no matter how small, there is a y such that 1 – x < Sy). In these circumstances, mathematicians define the limit of Sn as n approaches infinity as 1. This means precisely that as n gets larger and larger (or as n approaches infinity) Sn gets as close as we like to 1. It does not mean that n ever reaches infinity or that Sn ever reaches 1, and so, it does not require us to speak of completing an infinite number of tasks.
Now this description applies straightforwardly to the motion across the stadium: no matter how many intervals we have crossed, we have not reached the finish line. But since the partial sums correspond to the total distance covered after crossing each successive interval, the limit of the partial sums corresponds to the total distance to be covered, the entire length of the stadium. The more intervals we cross, the closer we are to the finish line. We can get as close to the finish line as we like in the sense that for any given distance from the finish line, no matter how small, there is a definite number of intervals such that once we have crossed them, we are less than the given distance from the finish line, even though there is no interval such that when we cross that interval we reach the finish line.
We can now return to premise (3). Zeno’s claim that it is impossible to get through things infinite in number is correct, in that we cannot get through them if we take them one by one. There is no last interval in the infinite sequence of intervals, so there is no last one to take. In other words, there is no interval in the infinite sequence such that by crossing it we finish crossing the stadium. But this does not entail that we cannot cross the stadium at all. The illusion that it does, comes from our tendency to think in finite terms. If it takes 100 steps to cross the stadium, then crossing the stadium requires us to complete all 100 steps; we finish crossing the stadium by taking the last step. So we expect that if crossing the stadium involves crossing an infinite sequence of intervals, we finish crossing the stadium by crossing the last interval. Since there is no last interval, it seems to follow that we cannot manage to cross the stadium. Likewise, we complete a journey of 100 steps by taking the hundredth step, so we expect that we must complete a journey across an infinite sequence of intervals by crossing the infinitieth interval. But since we cannot do an infinite number of tasks one after another, it seems to follow that we cannot complete the journey.
But these results do not follow. In the case at hand we cross the stadium by taking 100 steps. Since the stadium can be described as an infinite sequence of intervals, crossing the stadium involves crossing all of them. Consequently, when we have crossed the stadium by taking 100 steps, we have crossed the infinite sequence of intervals – all of them. This is simply a consequence of Zeno’s description of the motion. But it does not imply that we have crossed the (nonexistent) last interval.
One way of stating this point is that to get through either a finite or an infinite sequence of moves, we must get through them all. (When we reach the finish line, we have taken all 100 steps and have crossed the entire infinite sequence of intervals.) But whereas, getting through a finite sequence involves making a last move (the hundredth step), getting through an infinite sequence does not. This means that there is no way to get through an infinite sequence of moves taking them one by one. However, if there is another way to take them, it may be possible to get through them all. This is to say that premise (3) holds for cases where we take the “infinite things” one by one but not necessarily for other ways of taking them. In the present case, we get through the infinite sequence of intervals as the result of taking 100 ordinary steps, so premise (3) does not apply.
The Dichotomy fails. It attempts to show that our ordinary beliefs about motion lead to a contradiction: we believe we can cross the stadium, but premises (1), (2), and (3) entail that we cannot. Our ordinary beliefs commit us to accepting (1) and (2). But the plausibility of (3) depends on a particular way of tackling infinite sequences of tasks. If there are other ways, and in particular if there is another way entailed by crossing the stadium in a finite number of (finite-sized) steps, we do not need to concede Zeno’s point. In fact, there is such a way. We can accept his redescription of the motion (premises (1) and (2)) and show that so far from disproving the existence of motion, it is entirely compatible with it. (And this welcome result obtains.) In crossing the stadium in 100 steps, we finish crossing Zeno’s first interval after 50 steps. After 25 more we finish crossing Zeno’s second interval. By the time we have taken 13 more, we have finished crossing the third interval. (We have gone 88 m. and the third interval ends at 87.5 m.) Likewise, we have finished crossing the fourth, fifth, and sixth intervals by the time we have taken a total of 94, 97, and 99 steps, respectively. By the time we have taken 100 steps, we have finished crossing all the remaining intervals – an infinite number. It is possible to get through things that are infinite in this way, which is precisely what we need to refute the Dichotomy. And unlike the earli
er attempt at a solution, which stacked the deck unfairly against Zeno, the present solution does not simply put up an alternative description of motion that does not involve impossibility. Instead, it shows that Zeno’s redescription of motion not only does not entail any impossibility, but actually yields consequences consistent with the existence of motion – consequences that would cause serious difficulties if they did not follow from Zeno’s first two premises.
THE ACHILLES
An expanded reading of the Achilles follows.
Achilles will never catch the tortoise even though he runs faster than the tortoise. By the time he reaches the tortoise’s starting point (A) the tortoise will have moved some distance, however small, to a new point (A1). By the time Achilles reaches A1 the tortoise will have moved on to a further point (A2), and so on. Each time Achilles reaches a point where the tortoise has been, the tortoise is no longer there; the tortoise is always ahead, so Achilles never catches up.
As I remarked earlier, this paradox turns on “never” and “always,” not on properties of infinite sequences, even though in Zeno’s description the race consists of an infinite sequence of stages or subtasks. The paradox is stated from Achilles’ point of view as he runs the race. Achilles will never get through all the subtasks needed to complete the original task in the sense that however many subtasks he may have completed at any moment, there are always still more left to do. In these circumstances, it does no good to point out that he is getting as close as he likes (in the sense defined on p. 147) to the tortoise, or that he will catch the tortoise after running a distance equal to XY/(Y – Z) and after running for a time equal to X/(Y – Z), where X is the original head start, Y is Achilles’ speed, and Z is the tortoise’s speed. The problem is not that there is no time at which he will catch the tortoise, or that there is no point at which he catches it, but that reaching that point (and reaching that time, for that matter) requires doing something impossible.
On a natural way of construing always and never, “the tortoise is always ahead” means “at every time the tortoise is ahead,” and “Achilles never catches the tortoise” means “there is no time at which Achilles catches the tortoise.” However, the paradox does not establish these claims, but requires us to take the two sentences differently, “the tortoise is always ahead” as claiming “at every time during the race (i.e., while Achilles is catching up) the tortoise is ahead,” and “Achilles never catches the tortoise” as claiming “there is no time during the race when Achilles catches the tortoise.” Clearly, “the tortoise is always ahead while Achilles is catching up” does not entail that the tortoise is always ahead, and yet the harmless former claim is all the paradox proves, whereas it purports to prove the latter, and it is the latter, not the former, that contradicts our ordinary views about motion. It would be dismaying to be told that Achilles will never catch up with the tortoise at all, but quite welcome, indeed unsurprising and virtually tautologous to be told that he does not catch up at any time before the race is over, that is, before he catches up.
The Achilles fails because it trades on ambiguity. Moreover, it sets up an infinite sequence of tasks that is subject to the same analysis as the sequence of intervals in the Dichotomy. The sequence has no final element and cannot be completed by taking the tasks one by one (as the paradox depicts Achilles trying to do). But just as in the previous discussion, the hundredth (constant-length) step we took in crossing the stadium put us in the position of having crossed the entire infinite sequence of intervals that the Dichotomy shows we must cross, so, when Achilles has taken his final (constant-length) pace, he will have completed the entire infinite sequence of tasks that the paradox shows he must complete. Again, a welcome situation that vindicates rather than undermines our ordinary beliefs about motion.
THE FLYING ARROW
If everything is always at rest when it occupies a space equal to itself, and what is moving is always “at a now,” the moving arrow is motionless (Aristotle, Phys. VI.9 239b5-7).
This argument can be analyzed as follows:
1. If something occupies a space equal to itself at time t, it is at rest at t.
2. At each instant (“now”) of its flight, an arrow occupies a space equal to itself.
3. At each instant (“now”) of its flight, the arrow is at rest. (from (1) and (2))
4. What is moving is always “at a now,” that is, the entire duration of its motion consists of instants.
5. During the whole of its flight the arrow is at rest. (from (3) and (4))
Some of these statements need elucidation. The purpose of (1) is to provide a sufficient condition for a thing’s being at rest, but the corresponding view of motion is difficult to make out. (1) implies that if something is not at rest, that is, if it is in motion, it does not occupy a space equal to itself; presumably it occupies a space larger than itself.19 If we construe time t as an instant, then Zeno is claiming that things in motion stretch, so that the purpose of (1) will be to rule out the possibility that things move like a rubber band that originally extends from A to B, then stretches to extend from A to D, and then returns to its original size, coming to extend from C to D. (Thus the distance between A and B equals the distance between C and D and is less than the distance between A and D.) If at instant t the rubber band extends from A to D, it is in motion at t in the sense that it is in the process of ceasing to occupy the interval AB and coming to occupy the interval CD. On this interpretation (3) follows validly from (1) and (2), but it is not clear why Zeno should think that motion necessarily involves stretching. Another possibility is that the time involved, t, is an interval, and Zeno holds that if X, which remains the same size, changes over interval t from occupying interval AB to occupying interval CD, then over the interval t taken as a whole, X occupies the whole interval AD, which is greater than a space equal to X (i.e., AB). This is not to say that at any instant in t, X occupies the whole interval AD, or indeed any space that is not equal to itself. On this interpretation (which is admittedly difficult to get out of the text), we have a more plausible account of what happens during motion, and the corresponding claim about rest, that if X occupies AB throughout t, X is at rest during t, is evidently true. However, the inference via (2) to (3) now becomes invalid, since (2) and (3) are concerned with motion at an instant, not over an interval.
I have supplied (2) and (3); they are not in Aristotle’s text, but are the most plausible way to make the argument go through.20 For in order to have a chance of inferring (5) from (4), we need an additional premise framed in terms of instants (not intervals). In (4) the phrase rendered as “at a now” is usually translated as “in the now”; it means “at an instant.”
This paradox raises deep problems about the nature of motion. I shall discuss two. First, a point in connection with premise (1). Zeno assumes that something can be at rest at an instant, a concept which Aristotle showed to be problematic. Aristotle argued that motion cannot take place at an instant; it occurs over an interval of time. Further, since rest is the absence of motion, rest takes place over intervals too; it is no more possible to be at rest at an instant than it is to move at an instant (Phys. VI.3 234a24-b9). But (as was finally established in the nineteenth century when the foundations of the calculus were put on a sound basis) Aristotle is wrong. We do talk of both motion and rest at an instant (“At precisely 3 minutes and 12 seconds after 8 p.m. I was driving at 65 miles per hour, officer. Also, I was caught in a traffic jam from about 8:10 until about 8:20, so at precisely 15 minutes and π seconds after 8 the car was at rest.”), and such talk is not nonsense. Even if the primary sense of motion (and rest) involves an interval of time for the motion to take place, there is a perfectly good secondary or derivative sense in which we may speak of something being in motion or at rest at an instant – not claiming that anything moves any distance in an instant, but that something may have a velocity at an instant, since anything whose velocity is zero is at rest, and anything whose velocity is different from zero is i
n motion. Velocity over an interval of time is defined as the ratio of the distance covered in that time interval to the length of the interval:
Correspondingly, velocity at instant t is equal to the limit (similarly to the sense defined above, p. 147) of the ratio of the distance covered in time intervals containing t to the length of those intervals, as the length of the intervals approaches zero. If t1 is earlier than t2, and the interval t1 t2 contains t, the velocity at t is the limit as t2 – t1 approaches zero of the ratio between the distance covered between t1 and t2 and the length of the interval between t1 and t2: 21
Second, a point in connection with premise (4). Even if Zeno grants the above point, there remains another problem. At different instants the flying arrow is at different points of its trajectory, but how does it move from one point to another? Aristotle says that Zeno’s conclusion “follows from assuming that time is composed of ‘nows.’ If this is not conceded, the deduction will not go through"(Phys. VI.9 239b30-33). The problem as Aristotle sees it is this. If time is atomic, then there are adjacent instants. If something is in motion, it occupies different places at different instants. If t1 and t2 are successive instants, something in motion over the interval t1t2 occupies different places, d(t1) and d(t2), at those instants. But when does it move from d(t1) to d(t2)? There is no answer, since there is no time, no instant, between t1 and t2 for the motion to take place.
