In a recent piece on Starts With a Bang, Ethan Siegel takes up Zeno’s paradox and argues that it takes physics to solve it. Here I will not so much look at Siegel’s piece as take it as an opportunity to look at the absurdity of Zeno’s paradox seriously. Again this will take us to roads metaphilosophical and metalogical. Such is our lot.
Zeno of Elea was Greek philosopher who argued for a Parmenidean view of reality with a series of paradoxes that aimed to show the impossibility of his opponents’ metaphysical views. Here following Siegel I will only take up one of Zeno’s paradoxes concerning motion – the paradox known as ‘dichotomy’. The idea behind ‘dichotomy’ is simple: in order for a runner, say the mythical Atalanta, to reach her goal, she will first have to reach the half-way point. But, to reach the half-way point, she will have to reach a point half-way between the start and the half-way point i.e. run quarter of the way. Yet, again she must reach the half-way spot to the quarter-of-the-way spot, and so on. Such a division will go on infinitely. Now, if each step along the way takes a finite amount of time (how could it not?) it will take poor Atalanta an infinite time to reach her goal. Moreover, we can make the track she is running arbitrarily short, and still this conclusion will follow. From this we can clearly see that movement is impossible. This is Zeno’s paradox.
A lot of ink has been spilled in attempts to solve this paradox. I think this is a mistake. There is nothing to solve here. After all, experience shows that movement happens. What we have here is a case where a model for movement (‘dichotomy’) is simply inadequate for the job at hand. Why the model fails doesn’t have to concern us. We can simply go our merry way and produce models more adequate for the job, say the kinematics of Galileo – or those of Einstein. We side-step the paradox rather than address it.
Should ‘dichotomy’ be seen as a reductio ad absurdum? It certainly can be thought as such: after all if at least one the premises (or the logic used) is not mistaken, then movement would be impossible. However, on this view any bad model should be taken as a reductio, which I suppose they technically are, but do they warrant such seriousness?
Still, one might argue, the paradox is forced upon us. But how is it so forced? The world does not force it upon us, as the world points the exact other way – movement is after all empirically possible. Perhaps we are then logically forced to accept the paradox. Now problems multiply. Which logic forces it upon us? Can (any) logic force anything upon us? Regarding the first question, if not all of possible logics support ‘dichotomy’, then we can simply choose a logic which does not, and be done with the paradox. Recall that logic is a tool used for a purpose, and as such we should always choose the best logic for the job. If, by some miracle, all possible logics do in fact support ‘dichotomy’ then we are still left with the second question.
So, can logic force a claim about the world upon us? Simply put, no. A central failure of philosophy has been an attitude that takes all kinds of ‘ontological proofs’ and other logical deductions as showing this or that to be true (or false) about the world or its parts. Logic is simply about valid deductions. Logic can tell us things about (suitably formalized) concepts and their relations to each other, but the question of whether such concepts are applicable to anything is not a question of logic. It is an empirical or a pragmatic question. Further, one should recall what the Tortoise told Achilles after their race: declining to use logic is not a logical mistake (Carroll 1985). Compare this to a carpenter deciding to drive nails with a screwdriver instead of a hammer. She is not guilty of using a hammer wrong – after all she is not using a hammer at all. Similarly the Tortoise is not using logic wrong in declining to accept a sound conclusion, he simply declines to use logic.
So finally, Atalanta reaches her goal, and wins the race. To see this, one need not take heed of differential calculus, refined logics or, pace Siegel, physics. One simply needs to remember to use the right tools for the job. If your tools are lacking, find some others. And in cases like ‘dichotomy’ one can simply refrain from modelling and just go with the empirical evidence.
 Studying such a failure might of course help us avoid pitfalls in further attempts at modelling.
 Which may, or may not, be a well-defined set.
Carroll, Lewis (1895): What the Tortoise Said to Achilles. Mind 104 (416): 691–693.