Limits…Achilles, the tortoise, and flawed logic

Zeno of Elea (c. 450 BCE) is credited with creating several famous paradoxes, the best known of which is that of Achilles and the tortoise.

The anecdote goes that the tortoise challenged Achilles to a race, claiming he would win as long as Achilles gave him a small head start.  His argument was that, whatever head start the tortoise was given (e.g. 10 metres), Achilles would first need to cover half the distance (5 metres). Then he would need to cover half the remaining distance. Then he would need to cover half of the new remaining distance…and so on ad infinitum. The consequence is that Achilles would never catch up with the tortoise.

What this actually does is to make all motion impossible: before I can cover half the distance I must cover half of half the distance, and before I can do that I must cover half of half of half of the distance, and so on, so that in reality I can never move any distance at all, because doing so involves moving an infinite number of small intermediate distances first.

Now, since motion obviously is possible, the question arises, what is the “flaw in the logic?”

The solution is as follows: suppose I needed to cover a distance of 1 metre. I could see that 1 metre as the sum of an infinite number of small distances:

1 = 1/2 + 1/4 + 1/8 +1/16 +1/32…

…which looks a lot like an infinite series which gives a finite sum, and actually makes sense intuitively as well: if I can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give me back the finite distance I started with!

So back to Achilles: it would take him some fixed time to cross half the distance of the head start, say 2 seconds. To cross half the remaining distance would take half as long — only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second. And so on. And once he had covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 4 seconds, at which point he would have overtaken the tortoise.

It would seem that slow and steady does not win the race after all!

Read more Lighter Side articles here.

Simone O'Connor
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