## zeno's paradoxes

Zeno of Elea was an ancient Greek (born around 490 B.C.) who lived in what is now southern Italy. He became a disciple of Parmenides, a philosopher who maintained that reality was an absolute, unchanging whole, and that therefore many things we take for granted, such as motion and plurality, were simply illusions. This kind of thing must no doubt have brought on ridicule from more common-sensical thinkers, and so Zeno set out to defend his master’s position by inventing ingenious problems for the common-sense view. Ever since then, Zeno’s paradoxes have been debated by philosophers and mathematicians.

Zeno's writings have not survived, so his paradoxes are known to us chiefly through Aristotle's criticisms of them. Aristotle analyzed four paradoxes of motion: the Racetrack (or Dichotomy), Achilles and the Tortoise, the Arrow, and the Stadium (or Moving Rows). However, based on Aristotle's description of it, it is much less clear what Zeno intended by the Stadium paradox than by the other three. I have therefore left out this fourth paradox.

The Racetrack (or Dichotomy)

One can never reach the end of a racecourse, for in order to do so one would first have to reach the halfway mark, then the halfway mark of the remaining half, then the halfway mark of the final fourth, then of the final eighth, and so on

*ad infinitum*. Since this series of fractions is infinite, one can never hope to get through the entire length of the track (at least not in a finite time).

Start ____________________1/2__________3/4_____7/8__15/16... Finnish

But things get even worse than this. Just as one cannot reach the end of the racecourse, one cannot even

*begin*to run. For before one could reach the halfway point, one would have to reach the 1/4 mark, and before that the 1/8 mark, etc., etc. As there is no first point in this series, one can never really get started (this modern twist on Zeno's paradox is known as the Reverse Dichotomy).

Achilles and the Tortoise

Achilles and the Tortoise is similar. Suppose that the swift Achilles is having a race with a tortoise. Since the tortoise is much slower, she gets a head start. When the tortoise has reached a given point

*a*, Achilles starts. But by the time Achilles reaches

*a*, the tortoise has already moved beyond point

*a*, to point

*b*. And by the time Achilles reaches

*b*the tortoise has already moved a little bit farther along, to point

*c*. Since this process goes on indefinitely, Achilles can never catch up to the tortoise.

The Arrow

An arrow in flight is really at rest. For at every point in its flight, the arrow must occupy a length of space exactly equal to its own length. After all, it cannot occupy a greater length, nor a lesser one. But the arrow cannot move within this length it occupies. It would need extra space in which to move, and it of course has none. So at every point in its flight, the arrow is at rest. And if it is at rest at every moment in its flight, then it follows that it is at rest during the entire flight.

A brief analysis of the motion paradoxes

The Arrow does not present as serious a problem as the others. One can concede to Zeno that at each instant the arrow does not move. But it doesn't follow that the arrow does not move at all. The concept of motion can simply be understood as "occupying different points of space at different points in time." That the arrow is, in a sense, motionless

*at each instant*is beside the point.

The Racetrack and the Achilles are more difficult. (These are discussed together, for they are essentially the same paradox — that is, they generate the same basic difficulty.)

Nowadays, the standard solution to these paradoxes relies on the claim that (contrary to Zeno's assumption) an infinite series can in fact be completed. Thanks to advances in mathematics, we now know that the infinite series of fractions involved in e.g., the Racetrack, has a finite sum: (1/2 + 1/4 + 1/8 + ...) = 1. Hence one will of course reach the end of the track.

While I agree that the solution must depend in some way on this fact, I'm not so sure that no problems remain. One can imagine Zeno replying to the proposed solution as follows:

"Of course half the length, plus one fourth, plus one eighth, and so on, add up to the whole length. And that's just the point. The whole length contains an infinite number of finite parts. In order to traverse it, therefore, a runner would have to complete an infinite number of tasks. But how can such a thing to be possible?"

Some modern philosophers have argued that there are indeed serious problems with the notion of completing an infinite number of tasks. The best-known example of a current-day Zeno-type paradox is the Thomson Lamp, named after James F. Thomson.

The Thomson Lamp

Suppose you have a lamp with a simple on/off switch. Press the switch when it is off and the lamp will be turned on, press it again and it will be turned off. Now suppose you run the following experiment. You turn the lamp on at the start of a minute. Thirty seconds later, you turn it off. In another fifteen seconds, you turn it back on, then 7 1/2 seconds later back off again, and so on throughout the midpoints of whatever time remains. Now the question is this. At the end of the minute, will the lamp be on or off?

Since the lamp has been turned on and off an infinite number of times, for every time it has been turned on, it has been turned off, and vice versa. At the end of the minute, therefore, it can be neither on nor off. But it must be one or the other.

Attempts to find fault in this paradox often attack some irrelevant aspect of the argument. Thus one sometimes hears the criticism that this situation is physically impossible, since no mechanism could operate indefinitely fast. The on/off switch would not be able to keep up. As a counter argument to this type of criticism, I offer the following simplified version of the Thomson Lamp:

The Odd/Even Paradox

Suppose a point P is moving between points A and B (just like in the original Racetrack). And suppose also that we stipulate that P is in the state "even" for the first half of the journey, "odd" for the next 1/4, "even" for the next 1/8, and so on. That is, we simply decide to classify P based on where along the journey it is, such that it alternates between what we call an "even" and an "odd" state. It seems that we can in addition stipulate that once it is in one state it remains in that state unless it gets switched according to the above rule.

What state will P be in at B? Just as with Thomson's lamp, it cannot be in either, yet it must be in one or the other. The only solution to this paradox, it seems, is to claim that there is something wrong with the way it is set up. The stipulated conditions simply cannot form a consistent set. But why not?

A Solution?

One thing that Aristotle pointed out is that the amount of time to reach the end of the racecourse is not infinite. Just as the racecourse is the finite sum of the infinite series (1/2 + 1/4 + ...), the time to reach the end of the racecourse is the finite sum of a similar infinite series. The runner might take, say, 1/2 a minute to run the first half, 1/4 of a minute to run the next 1/4 of the track, and so on, thus reaching the end in one minute. Thus an infinite number of tasks can be accomplished in a finite time.

Now if we ask where the runner is at the end of one minute, it seems obvious that he can only be at the finish line. For he cannot be somewhere short of the line, since every point prior to it was reached before one minute. Nor can he be anywhere beyond the finish line, since he hasn't had enough time for that yet. Therefore, contrary to what Zeno said, he has to reach the finish line. (And essentially the same thing can be said regarding Achilles and the Tortoise.)

This is perhaps at least a partial solution, since it seems to show that the runner has to reach the finish line. At least it shows this provided one assumes that he can reach the

*time*one minute after the start of the race. But Zeno might in turn ask how

*that*is possible. Therefore the basic problem remains. It still has not been explained how an infinite number of tasks can be performed.

[Note: I no longer agree with the above. See The Paradoxes of Denying Infinity]

Benardete's Paradox of the Gods

Another modern version of the Racetrack that brings out the conceptual difficulties inherent in infinite tasks is the paradox of the gods. Suppose Achilles wants to run the length of a racetrack but there are an infinite number of gods who have the following intentions: the first god intends to paralyze Achilles if he reaches the halfway mark; the second intends to paralyze Achilles if he reaches the quarter mark; the third, if he reaches the one-eighth mark; and so on. As in the Reverse Dichotomy, Achilles cannot even start running: to do so would violate the intentions of an infinite number of gods. However, it is not clear why he cannot start running, for until he does, no god has actually paralyzed him.

A couple of additional Zeno-style paradoxes:

The Paradox of the Spaceship

Suppose that a spaceship travels in a straight line for half a minute and then doubles its speed, then a quarter minute later doubles its speed again, and so on

*ad infinitum*. Where will it be at the end of the minute? It must be infinitely far away. But does that make sense?

The Flying Bird Paradox

Suppose two people, Alice and Bob, are walking towards one another and a bird is flying back and forth between them. When it reaches Alice, it turns around and flies until it reaches Bob, then turns around again until it reaches Alice, and so forth. (Assume the bird takes no time to turn around.) Which direction is the bird facing when Alice and Bob meet? This is just like Thomson's lamp again: it seems it can't be facing either way.

Suppose that Alice and Bob begin walking apart again and the bird begins flying between them in the same way as before. If the two of them travel at 1 Km/h and the bird at 10 Km/h, where will the bird be when they are 1 Km apart?

Answer: the bird can be anywhere along the path! Proof: place the bird anywhere along the path and reverse the process. No matter where the bird started from, it will end up at the same place, in the same amount of time.

Zeno's Plurality Paradoxes

Zeno also argued against the notion that there is a plurality of objects, for the common sense world of spatially extended objects is supposedly an illusion. Two of the better-known plurality paradoxes are:

(1) If something is divisible, then it is infinitely divisible. Now if each part has zero size, then the total has zero size, for an infinite number of zero lenghts add up to zero. If on the other hand each part has some finite size, then the total is infinite, for an infinite number of finite lenghts, however minuscule, must add up to an infinite total. So something divisible is either infinite or else has no size at all. Thus something finite is not divisible.

(2) The total number of things is both finite and infinite. It is finite because, if there are many things, then there must be as many as there are "neither more nor less". And in that case their number is limited, hence finite. But on the other hand if there are many things, they must be infinite in number, for between any two there must always be others, and between those others still, and so on. (This paradox apparently is meant to apply to spatial points, rather than to physical objects.)

The Paradox of the Divided Stick

A modern version of a plurality paradox asks what would happen if an infinitely divisible stick were cut in two, then half a minute later each half were again cut in two, then a quarter of a minute later each fourth cut in two, and so on

*ad infinitum*. At the end of one minute what would be left? An infinite number of pieces? Would each piece have any length?

©2000 Franz Kiekeben