Zeno's Paradoxes of Motion: Achilles and the Arrow

Introduction

For more than two thousand years, mathematicians and philosophers 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 thinkers as well. To gain some additional insights into these four paradoxes, this paper reviews the relevant literature to determine who Zeno was, the content of his paradoxes, and an explanation of the mathematics and reasoning behind each. A discussion of possible solutions to the four paradoxes is followed by a summary of key findings in the conclusion.

Who Was Zeno of Elea?

Predating both 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, as one commentator notes, "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 and 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 contains only a sparse amount of Zeno's actual writings — fewer than 200 words — (Dowden, 2010), but of the four paradoxes attributed to him, the two most important are the paradox of Achilles and the tortoise and the paradox of the arrow, both of which are discussed further below.

The Paradox of Achilles and the Tortoise

Because Achilles was known to be a great runner, a race between a slow-moving tortoise and the fleet-footed Achilles should be a straightforward affair, even if Achilles provided the tortoise with a substantial head start. However, Zeno introduces some complicating factors:

1. Before Achilles can overtake the tortoise, he must first 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 little further along in the race.

4. This same logic applies to each successive stage of the race. Achilles will rush to where the tortoise was when he closed the initial gap, but the tortoise will have moved on again — a little less far each time, certainly, but some distance nonetheless. He is always getting closer, but never fully makes up the lost ground.

Intuitively, it is clear that a human runner can quickly outdistance a 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.

Conclusion

The research showed that Zeno was an ancient Greek philosopher who predated Aristotle and Plato and who propounded a series of four paradoxes — the Achilles, the Dichotomy, the Arrow, and the Moving Rows — all of which concern motion and all of which have challenged mathematicians for more than two thousand years. Of these four paradoxes, the Achilles and the Arrow were shown to be the most well-known, and both involved similar extensions of Zeno's logic as applied to moving objects.

As demonstrated above, Zeno's paradoxes do appear to defy easy solutions; however, they ignore the practical realities of motion. A single stride by Achilles will overtake the tortoise irrespective of how infinitesimally it continues to move, and the arrow in fact remains in motion from moment to moment regardless of its apparent status at any single instant. Therefore, to assert that an object in motion is "motionless" is not a paradox but is rather an oxymoron. Nonetheless, the enduring appeal of Zeno's philosophical puzzles lies in their capacity to expose the limits of intuitive reasoning about infinity, time, and space — questions that continue to engage scholars across philosophy, mathematics, and physics.