The Infinite Chase - Zeno's Paradoxes and the Nature of Motion
Zeno of Elea shows Youths the Doors to Truth and False. Source: Singinglemon / Public Domain – Via

Zeno’s Paradoxes…Imagine a sunny day in ancient Greece, where the legendary hero Achilles is preparing for a race. His opponent? A humble tortoise. To make things fair, Achilles grants the tortoise a head start. The race begins, and Achilles, with his unmatched speed, starts closing the gap between them. But here’s where things get interesting: every time Achilles reaches the point where the tortoise was, the tortoise has moved a little further ahead. This scenario repeats infinitely, leading to the perplexing conclusion that Achilles can never overtake the tortoise. Welcome to one of Zeno’s paradoxes.

Zeno of Elea was a Greek philosopher who lived around 490-430 BCE. He is best known for his paradoxes that challenge our understanding of motion and infinity. The paradox involving Achilles and the tortoise is perhaps his most famous, but it’s just one of several thought experiments he devised to illustrate his ideas.

At first glance, Zeno’s paradox seems absurd. After all, we know from everyday experience that faster runners overtake slower ones all the time. So what’s going on here? The key lies in how we think about motion and infinity.

In the race between Achilles and the tortoise, Zeno asks us to consider an infinite series of steps. First, Achilles must reach the point where the tortoise started. By then, the tortoise has moved a bit further ahead. Next, Achilles must reach this new point, but again, the tortoise has advanced slightly. This process continues indefinitely. Each step involves covering a smaller distance than before, but there are infinitely many steps to take.

This leads us to a fascinating concept in mathematics: an infinite series. An infinite series is a sum of infinitely many terms. In this case, each term represents a segment of the race that Achilles must cover to catch up with the tortoise. The paradox arises because it seems like adding up an infinite number of terms should result in an infinite distance or time.

However, mathematics provides us with tools to handle such situations. The concept of limits allows us to make sense of infinite series in a rigorous way. When we sum an infinite series where each term gets progressively smaller, we can sometimes find that it converges to a finite value.

Let’s break it down with some numbers for clarity. Suppose Achilles runs ten times faster than the tortoise and gives it a 10-meter head start. By the time Achilles reaches that 10-meter mark, the tortoise has moved 1 meter further (since it’s ten times slower). When Achilles covers this additional meter, the tortoise moves another 0.1 meters ahead, and so on.

Mathematically speaking, we can represent this as an infinite series: 10 + 1 + 0.1 + 0.01 + … Each term gets smaller by a factor of ten compared to its predecessor. Despite having infinitely many terms, this series converges to a finite sum due to its decreasing nature.

In fact, using calculus and limits, we can show that this sum equals approximately 11.111… meters—a finite distance! So while Zeno’s paradox highlights an intriguing aspect of infinity and motion, modern mathematics resolves it by demonstrating that Achilles will indeed overtake the tortoise after running just over 11 meters.

Zeno’s paradoxes don’t stop at Achilles and the tortoise; he proposed several others that similarly challenge our intuition about motion and infinity. For instance, there’s The Dichotomy Paradox, which argues that before reaching any destination, one must first get halfway there—and before reaching halfway there, one must get halfway to halfway there—ad infinitum.

These paradoxes serve as more than mere intellectual puzzles; they force us to confront deep questions about reality and our understanding of it. They reveal how our intuitions can sometimes lead us astray when dealing with concepts like infinity.

In ancient times, Zeno’s paradoxes were used as arguments against the belief in continuous motion—a belief held by philosophers like Parmenides who argued that change and motion were illusions. Today, they are valuable tools for teaching mathematical concepts such as limits and convergence.

Moreover, Zeno’s work underscores an essential aspect of scientific inquiry: questioning assumptions and exploring counterintuitive ideas can lead to profound insights about nature and reality.

So next time you find yourself pondering how something as simple as a race between a hero and a tortoise could lead to such mind-bending conclusions about infinity and motion—remember Zeno’s paradoxes are more than just ancient riddles; they are gateways into deeper understanding of mathematics and philosophy alike.


Don Leith

By Don Leith

Retired from the real world. A love of research left over from my days on the debate team in college long ago led me to work on this website. Granted, not all these stories are "fun" or even "trivial" But they all are either weird, unusual or even extraordinary. Working on this website is "fun" in any case. Hope you enjoy it!