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Achilles and the Turtle: an effort at explicit statement




Let us take this from the top. 

Just think of two creatures: one notoriously swift of foot, the other not so. 

Then think that the slower creature has a short head-start in a race.  Of course the former will catch up with and pass the later.  That is inherent in what we mean by such notions as slow and fast. 

But ... what if we assume with ancient geometers that space, and so every possible distance, is infinitely divisible?  Does that throw a wrench into things? 

Fast creature (Achilles) starts ten meters behind slow creature (Turtle). Achilles moves ten times as fast.  Within some period of time (we will call it a nonce), Achilles has advanced those ten meters.

Ah, but Turtle has now advanced one meter, and so is still ahead. 

Achilles advances THAT distance in one-tenth of a nonce.  Yet he still has not caught up, for the Turtle has by now advanced one-hundredth of a meter. 

And so forth, depressingly, on and on. It is impossible for Achilles ever to catch up, because one can continue these imaginings forever, given the presumption of infinite divisibility.  

Or, at least, that is the argument. And it set a challenge that nobody really met until the invention of calculus as a branch of mathematics two millennia later. 

One key element in the invention of calculus is the idea that an infinite series of decreasing quantities (or, say, distances) can have a finite sum. The generic name for such a case is a converging series.  

The motions of Achilles and the turtle converge ... literally.  Given the speed and starting points of each of these characters mathematicians can tell you exactly when and where Achilles catches up with the reptile. 

t=∑n=1∞t

By quite similar reasoning they can also explain, for example, the mathematics of continuously compounded interest, in which the number of compoundings "approaches infinity" at a limit. Figuring that out requires an irrational number a little bit above 2.718, known as e or Euler's number.    

Anyway: is anything left of this ancient paradox once we grasp the basic idea of converging series'? That question is itself the object of a 21st century philosophical debate.  Some "mathematical deflationists" think Zeno's paradox has become uninteresting.  Others, though, say that calculus assumes convergence and quantifies it without really explaining it on the ontological level Zeno was demanding 

 

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