Achilles and the Tortoise

Zeno of Elea, a Greek philosopher born around 490 BC, challenged conventional perceptions of motion with his paradoxes, notably the Achilles and the Tortoise. These paradoxes, part of his broader argument that motion is an illusion, presented a stark contrast to everyday experiences such as runners running or arrows flying. Zeno’s work hinged on the idea that space and time are infinitely divisible, marking him as a pioneer in grappling with the complexities of infinity.

Limited information is available about Zeno’s life, with most knowledge derived from Plato’s works, written about a century later. Although none of Zeno’s original writings survive, he is said to have authored a book containing 40 paradoxes, with the Achilles and the Tortoise being the most renowned.

The paradox describes the Greek hero Achilles racing a tortoise that has a head-start. Common sense suggests that Achilles, being swifter, should eventually overtake the tortoise. Zeno, however, challenges this notion. He posits that as Achilles reaches the tortoise’s initial position, the tortoise will have advanced to a new spot. This sequence continues with the tortoise always inching forward each time Achilles arrives at the tortoise’s previous position. Consequently, Achilles can never surpass the tortoise. Zeno argued that distances and durations can be divided into an infinite number of parts, implying that completing these parts is impossible for a runner.

To illustrate, imagine the tortoise has a 10-meter head start. While it moves at 1 meter per second, Achilles runs ten times faster, at 10 meters per second. When Achilles covers the initial 10 meters, the tortoise, at a tenth of his speed, has moved an additional meter. In chasing this new one-meter gap, Achilles finds the tortoise has again moved ahead, this time by 0.1 meters. This process repeats indefinitely, with Achilles always reaching where the tortoise was, while the tortoise moves slightly ahead each time. This scenario can be represented by a geometric series: 10 + 1 + 0.1 + 0.01…

Common sense tells us Achilles should catch the tortoise in a second, but Zeno’s logic appears sound. The paradox lies in the fact that Zeno lived before the development of mathematical analysis and calculus, which later provided insights into the concept of infinity.

Aristotle countered Zeno by proposing a potential, rather than actual, infinity of divisions in distance and time. This means any division results in a finite number of parts, which a runner can feasibly complete. Aristotle’s interpretation, among others, remained one of the dominant views until the late 19th century.

The current mathematical solution agrees that Achilles’s path contains an actual infinity of parts but refutes Zeno’s claim that this is too many for a runner to complete. Utilizing calculus, a branch of mathematics developed in the 17th century that examines continuous change, it demonstrates that the sum of an infinite geometric series can be finite, explaining how Achilles overtakes the tortoise.

Zeno’s paradox of Achilles and the tortoise profoundly influenced philosophy and science, sparking debates on infinity, the structure of space and time, and the nature of mathematical concepts. It underscores the complexities in understanding motion and change, essential elements in our perception of the world.