Mathematics

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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, literally, the name "dichotomy". The sequence in question contains infinitely many numbers, representing the infinitely many steps required to travel the distance. Zeno held that nobody could perform an unlimited number of such operations, and that therefore travel itself must be impossible, because one would not be able even to start moving. Motion by single objects (as well as by multiple objects, as in the case of Achilles and the tortoise) is thus an illusion.

The Arrow
Unlike the previous paradoxes, where space is divided into ever smaller segments, in the paradox of the arrow it is time that is split up, into the familiar and seemingly clearly defined categories of past, present and future. If one thus views a given arrow in flight, one can conceivably fix its position at a certain point in time relative to others. Now one comes up against what for Zeno is an intractable problem. At any fixed point in time, the arrow has an exact location, and therefore cannot be simultaneously in motion. Motion itself only makes sense when viewed as a phenomenon in the present, not in the past or future. Time itself consists of a series of present moments in which the arrow is by Zeno's definition stationary, thereby rendering its motion impossible.

Common sense and intuition should tell one that the arguments of Zeno's paradoxes are fatally flawed, and indeed fields of knowledge such as physics and mathematics can be invoked to refute them. The relatively modern concept of a convergent geometric series, for instance, proves that at some point in space two objects will meet one another, when given the type of conditions described for the footrace. Achilles will eventually catch and overtake the tortoise, and despite the dichotomy argument one can finally reach a desired destination. As for the arrow, the very fact that it is granted different positions relative to one another in the same flight trajectory means that it has to perforce move between them. Motion is not only possible; under the imposition of these parameters it is inevitable. In short, what finally sinks the paradoxes as viable methods of reasoning is their reliance on a conception of space and time as a series of discrete, exclusive units, rather than as the fluid continua that they are now known to be.

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