The Paradox of Achilles and the Tortoise

Being a massive Star Trek nerd I am regularly exposed to time paradoxes. The writers seemed to throw in at least one or two time travel episodes per series and as sure as eggs is eggs, there would be a warning about protecting the time continuum. Time travel and the paradoxes they throw up are entertaining but they can’t really happen. Asides from time travel being highly unlikely there’s causality to deal with and so the loop of a time paradox can never be created nor broken.

There are other kinds of paradoxes, though, and these are also good because they nearly always provide an opportunity to learn something new. Often times the solution is simply that the problem is being tackled in the wrong way. For example, you might ruthlessly apply logic to the Potato Paradox yet come up with an absurd, but logically consistent, answer. Or a paradox might merely highlight a gap in our scientific understanding that we ultimately fill some time later, like in Olber’s Paradox or Maxwell’s Demon.

One of my favourite paradoxes is that of Achilles and the Tortoise. It goes like this…

Achilles, being the robust fellow he is, challenges a tortoise to a race. Being a fair kind of demigod he gives the tortoise a head start of 100 metres. The race begins and both demigod and beast set off at a furious pace. Now, as you may expect, Achilles has the edge over the reptile and soon covers the 100 m; however, in the time it took him to get there the tortoise has covered a further 10 m. Undeterred, Achilles ploughs on. He soon covers the 10 m but in that time the tortoise has gone another 1 m. Whenever Achilles covers the intervening distance the tortoise has always moved on a little bit further; the lead always decreases but it never becomes zero.

Obviously, this is patent nonsense. In real life Achilles would have overtaken the shelled slowcoach quicker than you could say Helen of Troy. So, what’s going on?

For those of you believe that Spock, Data, Tuvok and Seven of Nine were the best characters in Star Trek and to be emulated in all regards, and that certainly includes your humble correspondent, it may come as a bit of a shock to learn that logic does not always lead you to the correct answer.

In our paradox there is always a smaller distance that Achilles is forced to traverse, there is an infinity of them. I don’t know if you have ever gone for a walk that is infinity long before but it’s exhausting. I’d go so far to say it is impossible to walk infinity far. The problem is that we have brought mathematics to a non-mathematical situation and therefore the infinite steps argument is not a valid one here.

Another way to get around it is provided by the fact that mathematicians are always inventing new maths to solve previously unsolvable problems. In the time since Achilles was in school some clever person realised that the sum of an infinite geometric series converges absolutely. That is to say:

1/2 + 1/4 + 1/8 + 1/16 and so on = 1

And even a tortoise knows that going for a walk 1 long is a lot easier than going for a walk infinity long.

So, there you have it, like all paradoxes you just need to look at it a slightly different way and the paradox disappears. Which means, paradoxically, that there is no such thing as a paradox.

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