Introduction

Like so many other scientific and philosophical endeavors in Ancient Greece, Zeno of Elea's paradox (or, more accurately, paradoxes at one time there were in fact up to forty recorded variations on the same theme) reflects its creator's desire to test and define the limits of human knowledge. In particular, the paradoxes were formulated as a support for the ideas of the philosopher Parmenides, who argued that the empirical evidence obtained via one's senses gives the false impression of plurality and flux, which means consequently that motion itself is an illusory concept. Accordingly the paradoxes are attempts by Zeno to prove these claims, encapsulated in his mentor's maxim that "all is one".

Some Famous Examples of the Paradoxes

For the purposes of this article I will concentrate on three of the most representative of Zeno's paradoxes, namely those dealing with Achilles and the tortoise, the dichotomy argument and the paradox of the arrow. The first two attempt to address the question of space and its relation to motion, while the third replaces the space factor with the concept of time.

Achilles and the Tortoise
The story of the Greek warrior Achilles and his footrace with a tortoise is probably the best known example of Zeno's paradox. Given the competitors and the large difference in the speeds with which each one usually moves, and with the assumption that these speeds are always constant, the tortoise is granted a generous head start. The race begins and Achilles reaches the tortoise's starting point. However, the tortoise has itself moved a certain (shorter) distance in this time. Achilles thus still remains behind it. When he does reach the tortoise's new position, the latter has since moved again and is still ahead in the race, albeit even less so than before. The situation repeats itself ad infinitum. Achilles continues to close in on the tortoise, without actually ever catching up with it. There always remains some distance, however small, between himself and his competitor.

Dichotomy
Someone wants to reach a certain point at a fixed distance from his present position. However, to get there he must first cover half of the distance. In order to do this he first has to move half of the new distance (one quarter of the original), half again of that, and so on forever. His "progress" can then be illustrated by the following sequence of numbers: , , , , 1. The various distances are thus continually cut in two, hence,

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