Zeno\'s Paradoxes for More Than

ZENO'S PARADOXES

For more than two thousand years, mathematicians have been challenged and perplexed by Zeno's four paradoxes concerning motion. As stated, the paradoxes presented by Zeno do in fact appear to defy easy resolution, and it is not surprising that these mathematical conundrums continue to fascinate modern mathematicians as well. To gain some additional insights concerning these four paradoxes, this paper provides a review of the relevant literature to determine who Zeno was, the content of his paradoxes and an explanation of the mathematics and the reasoning behind each. A discussion concerning the solutions to these four paradoxes is followed by a summary of the research and important findings in the conclusion.

Review and Analysis

Predating Aristotle and Plato, Zeno was an ancient Greek philosopher. According to one historian, "Like many of the ancient Greeks, Zeno of Elea (a small Greek colony seventy miles from Naples) was perplexed by apparent 'contradictions' in the way we understand the world" (Cohen, 2002, p. 137). Today, Zeno is remembered primarily for his four paradoxes, all of which concern movement (Goodkin, 1991). The four paradoxes are (a) the Achilles, (b) the Dichotomy, (c) the Arrow and (d) the Moving Rows (the Dichotomy and the Moving Rows are both alternatively known as "the Stadium") (Dowden, 2010). Not surprisingly, these challenging paradoxes have attracted a great deal of interest over the centuries because ". . . Zeno's paradoxes seem to deny the reality of movement and to suggest that the universe is static and unchanging.

In fact Zeno's paradoxes of motion have been refuted by philosophers for over two thousand years now, with the present crop of scientific/mathematical expert explanations no exception. However, they have survived because actually, no one can explain them away" (Cohen, 2001, p. 137). Unfortunately, the historical record only contains a sparse amount of Zeno's actual writings (less than 200 words) (Dowden, 2010), but of the four paradoxes that are attributed to Zeno, the two most important are the paradox of Achilles and the tortoise and the paradox of the arrow which discussed further below.

Paradox of Achilles. Because Achilles was known to be a great runner, a race between a slow-moving tortoise and the fleet-of-foot Achilles should be a straightforward affair, even if Achilles provided the tortoise with a hefty head start; however, Zeno introduces some complicating factors:

1. Before Achilles can overtake the tortoise, he must catch up with it.

2. No matter how fast Achilles is, and he is fast, it will take him some time to reach the place where the tortoise started from.

3. No matter how slow the tortoise is, and it is slow, it will during this time have moved at least a bit further on in the race.

4. This will apply for the next bit of the race too -- Achilles will rush to where the tortoise got to whilst he was making up the tortoise's head start, and the tortoise will move on again. A bit less far this time, certainly, but a bit none the less. He is always getting closer, but never makes up the lost ground completely.

Intuitively at least, it is clear that the human runner can quickly outdistance the tortoise (Papa-Grimaldi, 1996), but Zeno's paradox seems to defy this capability. In this regard, Cohen asks, "Achilles will certainly, with his celebrated speed, soon get very close behind the tortoise -- but why can't he, logically speaking at least, ever overtake the reptilian competitor?" (2002, p. 38). A similar paradox is presented by Zeno in "The Arrow," discussed further below.

Paradox of the Arrow. According to Dowden, this paradox concerns "a moving arrow [that] must occupy a space equal to itself at any moment. That is, at any moment it is at the place where it is. But places do not move. So, if at each moment, the arrow is occupying a space equal to itself, then the arrow is not moving at that moment because it has no time in which to move; it is simply there at the place" (2010, para. 3). This paradox suggests that anything that is moving is not actually moving, but is rather suspended in a static state from moment to moment. In this regard, Cohen asks, "What can we say about the arrow in flight? At each instant, Zeno says, it is motionless, since it would have time to move, that is, to occupy at least two successive positions, only if it were accorded at least two instants. At any given moment, it is thus at rest at a given point. Motionless at each point of its path, it is motionless the entire time that it is moving" (2002, p. 38).