Achilles, a tortoise and Zeno

By Gaurav Pundir

Imagine a person named Achilles is standing on a road with a tortoise a few metres ahead of him. Let both of them start running at the same time. For Achilles to overtake the tortoise, he must first cover the distance between them. But during the time it takes him to do so, the tortoise has moved ahead by some finite distance. Now, Achilles again covers the distance between him and the tortoise, but the tortoise has also moved during this time. This goes on forever, implying that there is always a distance remaining between Achilles and the tortoise. Hence, Achilles cannot overtake even a tortoise.
If this sounds absurd to you, you are not alone. This is known as the β€˜Achilles and the tortoise’ paradox and was put forth by Zeno of Elea, a 5th century BCE Greek philosopher. In another paradox, known as the β€˜Dichotomy paradox’, he claimed that motion is impossible because to cover a certain distance d you have to cover half of it first (d/2). To cover this half, you again have to cover half of it (d/4), resulting in an infinite number of tasks.

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Upon hearing this paradox, another ancient Greek philosopher named Diogenes the Cynic said nothing but simply stood up and walked away to demonstrate the obviously false nature of the argument. Aristotle attempted to resolve the paradox by saying that as the distance between Achilles and the tortoise gets smaller, the time taken to cover those distances also becomes smaller. Thus, the time it takes in total for Achilles to overtake the tortoise is finite.
Modern mathematics can provide a solution to these paradoxes by using something called a β€˜convergent geometric series’. Nonetheless, Zeno’s paradoxes were quite good at messing with people’s intuition about time and space for a long time, teaching us that thinking about even simple things differently can lead to interesting observations!

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