Zeno's Paradox of the Race Course

1. Zeno argues that it is impossible for a runner to traverse a race course. His reason is that
   
   "motion is impossible, because an object in motion must reach the half-way point before it gets to the end" (Aristotle, Physics 239b11-13).
   
2. Why is this a problem? Because the same argument can be made about the first half of the race course: the moving object must first reach the point half way to the half way point. And so on, ad infinitum.
   
3. So a crucial assumption that Zeno makes is that of infinite divisibility: the distance from the starting point (S) to the goal (G) can be divided into an infinite number of parts.
   
4. Terminology:
   

   R	the runner
   S	the starting point (= Z0)
   G	the end point
   Z1	the point halfway between S and G
   Z2	the point halfway between Z1 and G
   Zn	the point halfway between Zn-1 and G
   Z-run	a run that takes the runner from one Z-point to the next Z-point

   
   
   

5. Zeno's argument can now be formulated:
   
   1. In order to get from S to G, R must make infinitely many Z-runs.
   2. It is impossible for R to make infinitely many Z-runs.
   3. Therefore, it is impossible for R to reach G.
      
6. Evaluation of the argument:
   
   1. Is it valid?  Yes: the conclusion follows from the premises.
   2. Is it sound?  I.e., are the premises true?  This is what is at issue.
   3. One might try to object to the first premise, (1), on the grounds that one can get from S to G by making one run, or two (from S to Z₁ and from Z₁ to G). But this is not an adequate response. For according to the definitions above, the runner, if he passes from S to G, will have passed through all the Z-points. But to do that is to make all the Z-runs.
      
      Alternatively, one might object to (1) on the grounds that passing through all the Z-points (even though there are infinitely many of them) does not constitute making an infinite number of Z-runs. The reason might be that after you keep halving and halving the distance, you eventually get to distances that are so small that they are no larger than points. But points have no dimension, so no "run" is needed to "cross" one. But this is a mistake. For every Z-run, no matter how tiny, covers a finite distance (>0). No Z-run is as small as a point.
      
      So we have established that the first premise is true. (Note: this does not establish that R can actually get from S to G. It only establishes that if he does, he will make all the Z-runs.)
      
   4. The crucial premise is (2). Why can't R make infinitely many Z-runs?
      
7. Three possible reasons that can be given in support of (2):
   
   1. To make all the Z-runs R would have to run infinitely far.
   2. To make all the Z-runs R would have to run forever (i.e., for an infinite length of time).
   3. To make all the Z-runs R would have to do something it is logically impossible to do. (I.e., the claim that R makes all the Z-runs leads to a logical contradiction.)
      
8. Which of these reasons did Zeno have in mind? As we will see, there is some reason to think that Zeno believed (a). Aristotle assumed that (b) was what Zeno intended (and he based his refutation on that assumption). As we will see, both (a) and (b) are false.
   
   More recent critics have suggested that Zeno's argument can be made much more interesting if we use (c) to support his second premise.
   
9. Our difficulty here is that Zeno gives no explicit argument here in support of (2). To see why one might think that Zeno had (a) in mind, we will examine a related argument that he actually gave: his argument against plurality. We will then return to the race course.
   

• This page was last updated on 11/12/00.