Zeno's Paradox of Achilles and the Tortoise is a thought experiment that raises questions about the nature of motion and the concept of infinity. It was proposed by the ancient Greek philosopher Zeno of Elea.

The paradox goes as follows: Achilles, a swift runner, challenges a tortoise to a race. However, to make it fair, he gives the tortoise a head start. Let's say the tortoise starts 10 meters ahead of Achilles. According to Zeno, Achilles can never overtake the tortoise and win the race.

Zeno argues this by dividing the race into an infinite number of smaller distances. First, Achilles must reach the point where the tortoise started, but by the time he gets there, the tortoise has moved a small distance ahead. Achilles then needs to reach this new point, but again, the tortoise has moved slightly forward. This process continues infinitely, so Achilles can never catch up to the tortoise.

However, Zeno's paradox is based on a flawed understanding of the mathematical concept of infinity. In reality, we know that the sum of an infinite geometric series can have a finite value. In this case, the sum of the distances Achilles needs to cover becomes a convergent series, meaning it eventually reaches a limit.

To illustrate this, let's assign some numbers to the distances involved. Suppose the tortoise starts 10 meters ahead and moves at a constant speed of 1 meter per minute. Achilles, on the other hand, runs at a speed of 10 meters per minute. In the first minute, the tortoise moves 1 meter, but Achilles covers 10 meters, so he is now 9 meters behind. In the next minute, the tortoise moves 1 more meter, while Achilles covers 10 meters again, now being 8 meters behind. This pattern continues:

• After 2 minutes: Tortoise: 2 meters, Achilles: 20 meters, 18 meters behind.
• After 3 minutes: Tortoise: 3 meters, Achilles: 30 meters, 27 meters behind.
• After 4 minutes: Tortoise: 4 meters, Achilles: 40 meters, 36 meters behind.

We can observe that Achilles is closing the gap between them at a constant rate of 9 meters per minute. Therefore, we can calculate the exact time it takes for Achilles to overtake the tortoise by solving the equation: 10 meters / 9 meters per minute = 1.111 minutes. This demonstrates that Achilles will eventually surpass the tortoise and win the race.

Zeno's Paradox of Achilles and the Tortoise highlights the philosophical concept of potential infinity versus actual infinity. While an infinite number of distances can be divided, they can still be summed up to reach a finite value. Thus, the paradox is resolved by understanding the mathematical properties of infinity and the concept of limits.

References:

1. Stanford Encyclopedia of Philosophy. (2021). Zeno's Paradoxes. Retrieved from https://plato.stanford.edu/entries/paradox-zeno/
2. Dainton, B. (2017). Time and Space. Routledge.