Ben Brinckerhoff

Problems in Philosophy

12-14-00

Denying the Obvious: Zeno=s Racetrack Paradox

All movement is impossible. Logically, you should not be able to move from the living room to the kitchen, wave your arm, or even blink your eyes. This is the startling conclusion proposed by Zeno in his Racetrack paradox. Zeno points out that, to reach the finish line, a runner must first run half the track, then half the remaining distance, then half that distance, and so on and so forth. Since every conceivable distance can be cut in half, it is clear that the runner must run an infinite number of (admittedly very small) distances. Zeno concludes that since no one can do an infinite amount of things in finite time, the runner will never reach the finish line anything less than infinite time . Although this paradox seems perplexing at first, it can be partially resolved by attacking Zeno=s reasoning concerning infinity.

Before attempting to solve the paradox, it is important to formalize its premises and conclusion, as can be seen below.

Premise One - Traveling from any point P (the starting point) to any other point P* (the endpoint) requires completing an infinite number of journeys. Specifically, one must travel from P to P₁, the point halfway between P and P*, then to P₂, the point halfway between P₁ and P*, then P₃, P₄, P₅, etc., up to P_n, where n gets infinitely large.

One can see the required journeys above. The

first journey is from the starting point, P to P₁.

Premise Two - It is logically impossible for anyone to complete an infinite amount of journeys in finite time.

Conclusion - Traveling from P to P* is therefore impossible. Since P and P* can be any two points in space, all motion is impossible.

All paradoxes, by definition, reach unacceptable conclusions from apparently acceptable premises and reasoning. Some paradoxes deal with ideas that are abstract or philosophical in nature. For instance, the Liar paradox shows that there is a hole in our concepts about the truth value of sentences. While this is interesting, it is not an issue most people deal with directly. The Racetrack paradox, on the other hand, is possibly one of the most frustrating paradoxes because it forces us to justify our spatial concepts, ideas that we have taken for granted as true.

What makes this paradox so frustrating is that its conclusion is so obviously wrong. Everyone knows that motion is possible. Consequently, a natural response may be to try and refute the conclusion by looking at the problem another way. For instance, one could simply move from one spot to another and claim that, by moving, s/he has proved that motion is possible. Unfortunately, all this does is reinforce the earlier intuition that the paradox=s conclusion is clearly unacceptable. It does not however, defeat the paradox. To solve the Racetrack paradox, one must show why Zeno=s way of thinking about motion is wrong. As with all paradoxes, one has three choices: attack the premises, attack the reasoning, or accept the conclusion.

Accepting the conclusion is out of the question, since this would mean one would have to admit that motion is impossible, a belief which is very difficult (and arguably impossible) to hold while still living a life in which one interacts with the world.

Premise One seems acceptable initially. It=s true Zeno=s conception of motion is unorthodox - people do not, in fact usually think of movement as a infinite series of increasingly smaller journeys. Despite this, there is no apparent problem with conceptualizing movement in this way.

The problem lies in Premise Two. At first, it appears that it is impossible to complete infinite journeys in finite time. For instance, if you had a journey that took 1 second to complete, it is clear that completing this journey infinitely many times would take infinite time. We can illustrate this below.

1 second + 1 second + 1 second + 1 second +1 second +1 second + 1 second . . .

As we add more journeys, the time to complete these journeys increases without limit, and therefore one would need an infinite amount of time to complete the entire distance.

To find a solution to the paradox, one must realize that the above computation does not describe what the runner is doing, because each journey the runner travels does not take the same amount of time. Let=s assume for the moment that the runner is moving at a constant, or very near constant, speed. Since speed is simply distance/time, it follows that as each successive journey is one half as long as the last journey (as described in Premise One), the time needed to complete that journey is also cut in half. So, as n gets large (in fact, infinitely large), and the distance between P_n and P_n+1 gets infinitely small, the time to travel between point P_n and P_n+1 gets infinitely small as well.

To illustrate this point , let=s assume the first journey (from P to P₁) takes 1 second (a short racetrack and a fast runner), with each successive journey taking half as long to complete as the previous journey. Then the total time would start to add up as follows.

1 + .5 + .25 +.125 + .0625 + .03125 + .015625 = 1.984375

No matter how many more times I add to the total time, it will never reach, say, 5 seconds (in fact, it will never even reach 2 seconds), since each time must necessarily be less than .015625/2, which would be the time to complete the next journey. One could add up the times to complete infinitely many journeys and would still never reach a total time even close to 5 seconds. So, as long as the distances traveled (and therefore the times required to travel those distances) are getting smaller and smaller, one can travel an infinite number of distances in a finite amount of time. In this case, 5 seconds would be more than enough to complete an infinite series of journeys.

The time to complete each of the runner's journeys is shown above.

One can see that the total time will come close to 2 seconds, but will

never reach anything more than 2 seconds.

While this argument shows that Premise Two unacceptable, there are weaknesses in it. One could pose the following question: Even if the runner can travel an infinite number of ever-decreasing journey=s in finite time, would s/he ever reach P*? Premise One states that the runner moves from P₁ to P₂, and from P₂ to P₃, etc., but it never says anything about moving from any point to P*. This is because each P point is defined as the point halfway between the last P point and P*, and hence P* is not included in the points through which the runner travels. It seems that even after traveling infinite journeys, the runner still cannot be at P*.

On the other hand, after a finite time, the runner cannot possibly be at any P point, because, by definition, there is always another P point to the right of any P point. Stated another way, for every P_n,, there is always a P_n+1. Hence, after traveling infinite journeys in finite time, it seems we are left the contradictory statement that the runner cannot be at P*, yet s/he cannot be at any other place along the racetrack either. By solving the Racetrack paradox, we only discover another, different hole in our spatial concepts.

A partial response to this problem is to attack Premise One, which appeared to be acceptable at first. Zeno has asked us to use mathematics to try to describe and justify our spatial concepts, specifically, to think of a distance as a series of points, and of motion as simply traveling through those points. But as the above problem shows, there is not always a perfect match between the mathematical idea and our spatial concept. Our spatial concept of the racetrack is a distance with two ends, yet our series of P points has no last member, that is, for any point, there is yet another point to its right that is halfway between it and P*. This difference shows a mathematical series of points can not perfectly describe a racetrack, and therefore may confuse our thinking about spatial concepts instead of clarifying it.

Along the same lines, when we say that traveling all the P points would not let the runner reach P*, what do we really mean? It is really meant that a mathematical point traveling all the P points would not be at the same place as the P*, because he have a mathematical structure, a series of points, to describe motion. This description is imperfect, however, because the runner is not a mathematical point. Instead, s/he has dimensions and occupies space.

At P₁, both the mathematical point and the runner are clearly not at P*. At

P₅, however, the mathematical point is still clearly not at P*, yet the runner is.

One can see in the diagram above that after traveling through a relatively few number of P points, the runner is close enough to P*, because some part of her/him will occupy P* even before s/he has traveled through all the P points. It is therefore clear that the runner can occupy P* for all intents and purposes even if s/he does not logically arrive there in the strict mathematical sense.

The fact that the runner takes up space also gives support to the previous, rather non-intuitive conclusion that it is possible to complete infinite journeys in finite time. Earlier, it was established that the time to complete the infinite journeys was just under 2 seconds. Since we used a mathematical structure to describe the runner=s total journey (from P to P*), the calculation gives us the total time of a mathematical point traveling through all the P points, not a runner with volume. The runner could actually travel these distances in less time than a point, since s/he completes different journeys simultaneously. This is true because the distance between the P points get smaller, and therefore the runner, who has a constant length, can cover more and more journeys with each step. For instance, in the diagram above, one can see that at P₁, the runner is finishing the first journey (the one from P to P₁) and starting the second journey (from P₁ to P₂) at the same time. Similarly, at P₅, the runner is simultaneously finishing the journey from P₃ to P₄ and traveling all the journeys after P₄. Thus, one can see that the total journey can be broken down conceptually into a series of smaller and smaller journeys, but, since the runner has volume, s/he can travel most of the final distances almost instantly. This shows that the idea of traveling infinite journeys in finite time is not abstract and extraordinary, but rather very conceivable indeed.

Zeno could, however, modify his paradox to defeat the above argument about the volume of the runner. The revised Racetrack paradox could state that the frontmost particle (atom, electron, quark, whatever) of the runner could never move from point P to P*, for the same reasons that the stopped the runner. Zeno could claim that now this Atip of the runner@ would never be close enough to P* to count as being at P*, and likewise could not travel multiple distances simultaneously. Essentially, he would just be trying to get a piece of the runner to behave more like a mathematical point, since his paradox confounds us the most if it is a point that travels all the journeys.

A partial response to this revision would be to point out that the distances are getting infinitely smaller, and hence eventually will be smaller than even the smallest particle of matter. As soon as the distance between two points is smaller than the particle, we could restate that the particle would be close enough to P* and could travel multiple distance at once, just like the runner could. As long as Athe runner@ (that is, whatever it is that goes through the points, whether that be a person, a shoe, or an atom) has volume, the above arguments are valid. The reason this is only a partial response is because it depends upon the existence such a particle (one that is indivisible, and therefore is the smallest possible). If, however, no such particle exits (i.e. matter is infinitely divisible), Zeno can rightly claim matter can be treated like a mathematical point, and hence his paradox will stand up to our objections.

In fact, all of the attacks on the premises of the Racetrack paradox are somewhat vulnerable. We claimed above that a series of points does not correctly model the racetrack because a racetrack has two distinct points and that the journeys of a mathematical point does not correctly model the journeys of a runner. Both of these arguments are weak because they favor our intuitive spatial concepts over mathematical ones when the two disagree. They are thus open to the following objection: the only way we can make sense of our spatial concepts is through mathematics, and therefore if we cannot find a good match between spatial concept, such as distance, and a mathematical structure, then our spatial concept itself is not justified. We can therefore only strengthen the arguments by justifying of our preference of intuitive spatial concepts over mathematical ones, which is a challenging task within itself.

Zeno=s paradox is an extremely interesting yet frustrating paradox. One can partially solve the paradox by understanding that Zeno=s mathematical concepts concerning infinity are flawed. Although the proposed solution can show that motion is possible, the weaknesses in the solution show there are other holes in our spatial concepts. Since this is precisely what the Racetrack paradox exposed, it seems Zeno continues to challenge us to think more clearly and justify the spatial concepts we have come to rely on, even after we have defeated his paradox.