Sophism #5: Achilles and the Tortoise

Zeno's paradoxes are among the most ancient and famous sophisms known to date. They are thought to be proposed (or collected) by the ancient greek philosopher Zeno in the V century BC. At least nine known Zeno's paradoxes survived to this day, three of them have gained international fame. Perhaps the cutest of all Zeno's paradoxes is the story of Achilles and the Tortoise:

Achilles, a legendary Greek hero, is having a footrace with a tortoise. Believing he is much quicker than a tortoise and will easily outrun her, Achilles gives the tortoise a head start of 100m. At the sound of the whistle, the two racers begin their sprint.

As Achilles started off 100m behind the tortoise, it will take him some time to reach the point where the tortoise started the race. During this time the tortoise will crawl some distance forward (say, 1m), hence as Achilles completes the first 100m of his run, he will still be behind the tortoise by a meter. He will then have to run this additional meter, but at the same time the tortoise will crawl some distance forward, staying in front of Achilles again.

Thus, every time Achilles reaches the point where the tortoise was previously, the tortoise moves a bit forward and always stays ahead. Consequently, Achilles will never outrun a tortoise!

The paradox touches on the rather deep mathematical concepts of infinity and continuity. In fact, it presented a real hurdle for the mathematicians and philosophers of the past and was only cleanly overcome with the birth of the modern calculus in the 19th century. The story itself might have been an indirect inspiration for the ancient fable of the Tortoise and the Hare, which, in turn, has reached many corners of the modern culture.

As this paradox is so famous, it makes no sense to "hide the spoilers" about it. Instead, let me point you to a great explanation (of both this and the next sophism) by Brady "Numberphile" Haran. Before you watch the explanations, though, if you haven't yet heard of this paradox before, do take some time to try and figure it out yourself to properly appreciate its complexity.

For other sophisms check out this list.