Zeno's paradox...

Learn more about some of the mathematical, scientific and philosophical conundrums in Tom Stoppard's Jumpers.

Mathematician Helen Joyce and philosopher Tim LeBon talk about Zeno's paradoxes, infinitesimals, the limit of polygons, logical positivism, space travel, the existence of God and other themes from the play.

The paradoxes of the philosopher Zeno, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. The most famous of Zeno's arguments is the Achilles:

'The slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.'

This is usually put in the context of a race between Achilles and the Tortoise. Achilles gives the Tortoise a head start of, say 10 m, since he runs at 10 ms-1 and the Tortoise moves at only 1 ms-1. Then by the time Achilles has reached the point where the Tortoise started (T0 = 10 m), the slow but steady individual will have moved on 1 m to T1 = 11 m. When Achilles reaches T1, the labouring Tortoise will have moved on 0.1 m (to T2 = 11.1 m). When Achilles reaches T2, the Tortoise will still be ahead by 0.01 m, and so on. Each time Achilles reaches the point where the Tortoise was, the cunning reptile will always have moved a little way ahead.

This seems very peculiar. We know that Achilles should pass the Tortoise after 1.11 seconds when they have both run just over 11 m, so Achilles will win any race longer than 11.11m. But why in Zeno's argument does it seem that Achilles will never catch the tortoise…?

From Plus online maths magazine (http://plus.maths.org) - Mathematical Mysteries: Zeno's Paradoxes by Rachel Thomas

Zeno's paradox... finished on: 14 July 2003