The paradoxes of the philosopher Zeno, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher Parmenides. Parmenides believed in monism, that reality was a single, constant, unchanging thing that he called 'Being'. In defending this radical belief, Zeno fashioned 40 arguments to show that change (motion) and plurality are impossible.
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 (the legendary Greek warrior) 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?
4 of Zeno's famous arguments:
The Dichotomy: There is no motion, because that which is moved must arrive at the middle before it arrives at the end, and so on ad infinitum.
The Achilles: The slower 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 always be some distance ahead.
The Arrow: If everything is either at rest or moving when it occupies a space equal to itself, while the object moved is always in the instant, a moving arrow is unmoved.
The Stadium: Consider two rows of bodies, each composed of an equal number of bodies of equal size. They pass each other as they travel with equal velocity in opposite directions. Thus, half a time is equal to the whole time.