1 The Dichotomy The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics, 239b11) This paradox is known as the ‘dichotomy because it involves repeated division into two (like the second paradox of plurality). Like the other paradoxes of motion we have it from Aristotle, who sought to refute it.Suppose a very fast runner??”such as mythical Atalanta??”needs to run for the bus. Clearly before she eaches the bus stop she must run half-way, as Aristotle says. There’s no problem there; supposing a constant motion it will take her 1/2 the time to run half-way there and 1/2 the time to run the rest of the way. Now she must also run half-way to the half-way point??”i. e.

, a 1/4 of the total distance??”before she reaches the half-way point, but again she is left with a finite number of finite lengths to run, and plenty of time to do it.And before she reaches 1/4 of the way she must reach 1/2 of 1/4 = 1/8 of the way; and before that a 1/16; and so on. There is no problem at any finite point in his series, but what if the halving is carried out infinitely many times? The resulting series contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. However it does contain a final distance, namely 1/2 of the way; and a penultimate distance, 1/4 of the way; and a third to last distance, 1/8 of the way; and so on.Thus the series of distances that Atalanta is required to run is: ..

. , then 1/16 of the way, then 1/8 of the way, then 1/4 of the way, and finally 1/2 of the way (of course we are not suggesting that she stops at he end of each segment and then starts running at the beginning of the next??”we are thinking of her continuous run being composed of such parts). And now there is a problem, for this description of her run has her travelling an infinite number of finite distances, which, Zeno would have us conclude, must take an infinite time, which is to say it is never completed.And since the argument does not depend on the distance or who or what the mover is, it follows that no finite distance can ever be traveled, which is to say that all motion is impossible. (Note that the paradox could asily be generated in the other direction so that Atalanta must first run half way, then half the remaining way, then half of that and so on, so that she must run the following endless sequence of fractions of the total distance: 1/2, then 1/4, then 1/8, then .

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… ) A couple of common responses are not adequate. One might??”as Simplicius ((a) On Aristotle’s Physics, 1012. 2) tells us Diogenes the Cynic did by silently standing and walking??”point out that it is a matter of the most common experience that things in fact do move, and that we know very well that Atalanta would have no trouble reaching her bus stop. But this would not impress Zeno, who, as a paid up Parmenidean, held that many things are not as they appear: it may appear that Diogenes is walking or that Atalanta is running, but appearances can be deceptive and surely we have a logical proof that they are in fact not moving at all.Alternatively if one doesn’t accept that Zeno has given a proof that motion is illusory??”as we hopefully do not??”one then owes an account of what is wrong with his argument: he has given reasons why motion is impossible, and so an adequate response must show why those reasons are not sufficient.

And it wont do simply to point out that here are some ways of cutting up Atalanta’s run??”into Just two halves, say??”in which there is no problem.For if you accept all of the steps in Zeno’s argument then you must accept his conclusion (assuming that he has reasoned in a logically deductive way): it’s not enough to show an unproblematic division, you must also show why the given division is unproblematic. Another response??”given by Aristotle himself??”is to point out that as we divide the distances run, we should also divide the total time taken: there is 1/2 the time for the final 1/2, a 1/4 of the time for the previous 1/4, an /8 of the time for the 1/8 of the run and so on.Thus each fractional distance has Just the right fraction of the finite total time for Atalanta to complete it, and thus the distance can be completed in a finite time. Aristotle felt that this reply should satisfy Zeno, however he also realized (Physics, 263a1 5) that it could not be the end of the matter. For now we are saying that the time Atalanta takes to reach the bus stop is composed of an infinite number of finite pieces??”…

1/8, 1/4, and 1/2 of the total time ??”and isn’t that an infinite time? Of course, one could again claim that some infinite ums have finite totals, and in particular that the sum of these pieces is 1 x the total time, which is of course finite (and again a complete solution would demand a rigorous account of infinite summation, like Cauchy’s). However, Aristotle did not make such a move. Instead he drew a sharp distinction between what he termed a ‘continuous’ line and a line divided into parts.

Consider a simple division of a line into two: on the one hand there is the undivided line, and on the other the line with a mid-point selected as the boundary of the two halves. Aristotle claims that these are wo distinct things: and that the latter is only ‘potentially derivable from the former. Next, Aristotle takes the common-sense view that time is like a geometric line, and considers the time it takes to complete the run. We can again distinguish the two cases: there is the continuous interval from start to finish, and there is the interval divided into Zeno’s infinity of half-runs.The former is ‘potentially infinite’ in the sense that it could be divided into the latter ‘actual infinity. Here’s the crucial step: Aristotle thinks that since these intervals are geometrically distinct they must be physically istinct.

But how could that be? He claims that the runner must do something at the end of each half-run to make it distinct from the next: she must stop, making the run itself discontinuous. (It’s not clear why some other action wouldn’t suffice to divide the interval. Then Aristotle’s full answer to the paradox is that the question of whether the infinite series of runs is possible or not is ambiguous: the potentially infinite series of halves in a continuous run is possible, while an actual infinity of discontinuous half runs is not??”Zeno does identify an impossibility, but it does not escribe the usual way of running down tracks! It is hard??”from our modern perspective perhaps??”to see how this answer could be completely satisfactory. In the first place it assumes that a clear distinction can be drawn between potential and actual infinities, something that was never fully achieved.

Second, suppose that Zeno’s problem turns on the claim that infinite sums of finite quantities are invariably infinite. Then Aristotle’s distinction will only help if he can explain why potentially infinite sums are in fact finite (and couldn’t I potentially add 1 +1 +1 + which does ot have a finite total); or if he can give a reason why potentially infinite sums Just don’t exist. Or perhaps Aristotle did not see infinite sums as the problem, but rather whether completing an infinity of finite actions is metaphysically and conceptually and physically possible, an idea discussed at length in recent years: see ‘Supertasks’ below.In this case we need an account of actions that makes precise the sense in which the continuous run is indeed a single action (using rest to individuate motions seems problematic, for humans are probably never completely still, and yet we erform distinct motions??”breathing, eating, skipping and so on).

Finally, the distinction between potential and actual infinities has played no role in mathematics since Cantor tamed the transfinite numbers??”certainly the potential infinite has played no role in the modern mathematical solutions discussed here.One last point: Zeno’s argument seeks most obviously to establish the impossibility of motion, but he also intended it (and the following arguments) as further refutations of plurality??” certainly, Plato interprets Zeno’s intentions in this way. How might the argument seek o establish this conclusion? Presumably Zeno has in mind the view that spatial (and perhaps temporal) distances have a plurality of parts; parts which are infinitely divisible into two. Given that assumption, supposedly finite distances (or times) can be decomposed into an infinity of finite parts with no first (or alternatively, last) one.And how can such distances be finite after all? And if the pluralist also believes in motion, how can such a distance be traversed? It seems it cannot be. 3. 2 Achilles and the Tortoise The [second] argument was called “Achilles,” accordingly, from the fact that Achilles as taken [as a character] in it, and the argument says that it is impossible for him to overtake the tortoise when pursuing it.

For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth.In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which what is pursuing advanced . And in the time again in which what is pursuing will traverse this [interval] which what is fleeing advanced, in this time again what is fleeing will traverse some amount . And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount. (Simplicius(b) On Aristotle’s Physics, 1014. 0) This paradox turns on much the same considerations as the last. Imagine Achilles chasing a tortoise, and suppose that Achilles is running at 1 m/s, that the tortoise is crawling at 0.

1 m/s and that the tortoise starts out 0. 9 m ahead of Achilles. On the face of it Achilles should catch the tortoise after Is, at a distance of 1m from where he starts (and so 0. 1m from where the Tortoise starts). We could break Achilles’ motion up as we did Atalanta’s, into halves, or we could do it as follows: before Achilles can catch the tortoise he must reach the point where the tortoise started.But in the time he takes to do this the tortoise crawls a little further forward. So next Achilles must reach this new point.

But in the time it takes Achilles to achieve this the tortoise crawls forward a tiny bit further. And so on to infinity: every time that Achilles reaches the place where the tortoise was, the tortoise has had enough time to get a little bit further, and so Achilles has another run to make, and so Achilles has n infinite number of finite catch-ups to do before he can catch the tortoise, and so, Zeno concludes, he never catches the tortoise.One aspect of the paradox is thus that Achilles must traverse the following infinite series of distances before he catches the tortoise: first 0. 9m, then an additional 0. 09m, then 0.

009m, . These are the series of distances ahead that the tortoise reaches at the start of each of Achilles’ catch-ups. Looked at this way the puzzle is identical to the Dichotomy, for it is Just to say that that which is in locomotion must arrive [nine tenths of the way] before it arrives at the goal’. And so everything we said above applies here too.But what the paradox in this form brings out most vividly is the problem of completing a series of actions that has no final member??”in this case the infinite series of catch-ups before Achilles reaches the tortoise. But Just what is the problem? Perhaps the following: Achilles’ run to the point at which he should reach the tortoise can, it seems, be completely decomposed into the series of catch-ups, none of which take him to the tortoise. Therefore, nowhere in his run does he reach the tortoise after all. But if this is what Zeno had in mind it won’t do.

Of course Achilles doesnt reach the tortoise at any point of the sequence, for every run in the sequence occurs before we expect Achilles to reach it! Thinking in terms of the points that Achilles must reach in his run, 1m does not occur in the sequence 0. 9m, 0. 99m, 0. 999m, , so of course he never catches the tortoise during that sequence of runs! The series of catch-ups does not after all completely decompose the run: the final point??”at which Achilles does catch the tortoise??”must be added to it. So is there any puzzle? Arguably yes. Achilles run passes through the sequence of points 0. 9m, 0.

99m, 0. 999m, , 1m.But does such a strange sequence??”comprised of an infinity of members followed by one more??” make sense mathematically? If not then our mathematical description of the run cannot be correct, but then what is? Fortunately the theory of transfinites pioneered by Cantor assures us that such a series is perfectly respectable. It was realized that the order properties of infinite series are much more elaborate than those of finite series.

Any way of arranging the numbers 1, 2 and 3 gives a series in the same pattern, for instance, but there are many distinct ways to order the natural numbers: 1, Z 3, for instance. 1. or 1, 3, 5, . or Z , , 1, which is Just the same kind of series as the positions Achilles must run through. Thus the theory of the transfinites treats not Just ‘cardinal’ numbers??”which depend only on how many things there are??”but also ‘ordinal’ numbers which depend further on how the things are arranged. Since the ordinals are standardly taken to be mathematically legitimate numbers, and since the series of points Achilles must pass has an ordinal number, we shall take it that the series is mathematically legitimate. Again, see ‘Supertasks’ below for another kind of problem that might arise for Achilles’. 3.

3 The Arrow The third is that the flying arrow is at rest, which result follows from the assumption that time is composed of moments . he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless. (Aristotle Physics, 239b. 30) Zeno abolishes motion, saying “What is in motion moves neither in the place it is nor in one in which it is not”. Diogenes Laertius Lives of Famous Philosophers, ix. 72) This argument against motion explicitly turns on a particular kind of assumption of lurality: that time is composed of moments (or ‘nows’) and nothing else. Consider an arrow, apparently in motion, at any instant.

First, Zeno assumes that it travels no distance during that moment??”’it occupies an equal space’ for the whole instant. But the entire period of its motion contains only instants, all of which contain an arrow at rest, and so, Zeno concludes, the arrow cannot be moving.An immediate concern is why Zeno is Justified in assuming that the arrow is at rest during any instant. It follows immediately if one assumes that an instant lasts Os: whatever speed the arrow has, it will get nowhere if it has no time at all. But what if one held that the smallest parts of time are finite??”if tiny??”so that a moving arrow might actually move some distance during an instant? One way of supporting the assumption??”which requires reading quite a lot into the text??”starts by assuming that instants are indivisible.Then suppose that an arrow actually moved during an instant. It would be at different locations at the start and end of the instant, which implies that the instant has a ‘start’ and an ‘end’, which in turn implies that it has at least two parts, and so is divisible, contrary to our assumption. (Note that this argument only stablishes that nothing can move during an instant, not that instants cannot be finite.

) So then, nothing moves during any instant, but time is entirely composed of instants, so nothing ever moves.A first response is to point out that determining the velocity of the arrow means dividing the distance traveled in some time by the length of that time. But??”assuming from now on that instants have zero duration??”this formula makes no sense in the case of an instant: the arrow travels 0m in the Os the instant lasts, but 0/0 m/s is not any number at all. Thus it is fallacious to conclude rom the fact that the arrow doesnt travel any distance in an instant that it is at rest; whether it is in motion at an instant or not depends on whether it travels any distance in a finite interval that includes the instant in question.The answer is correct, but it carries the counter-intuitive implication that motion is not something that happens at any instant, but rather only over finite periods of time. Think about it this way: time, as we said, is composed only of instants. No distance is traveled during any instant.

So when does the arrow actually move? How does it get from one place to another at a later moment?There’s only one answer: the arrow gets from point X at time 1 to point Y at time 2 simply in virtue of being at successive intermediate points at successive intermediate times??”the arrow never changes its position during an instant but only over intervals composed of instants, by the occupation of different positions at different times. In Bergson’s memorable words??” which he thought expressed an absurdity??”’movement is composed of immobilities’ (1911, 308): getting from X to Y is a matter of occupying exactly one place in between at each instant (in the right order of course). For a discussion of this issue see Arntzenius (2000).

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