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Zeno's Paradox: Achilles and the Tortoise

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I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications and fun mathematical facts.

The paradox of Achilles and the Tortoise is one of many paradoxes supposedly created by the Greek philosopher Zeno of Elea as part of his argument that motion is impossible and is merely an illusion. But what is the paradox and who was Zeno?

Zeno of Elea Shows Youths the Door to Truth and False by Pellegrino Tibaldi

Zeno of Elea Shows Youths the Door to Truth and False by Pellegrino Tibaldi

Zeno of Elea (c. 490 BC)

Zeno of Elea was a Greek philosopher born around 490 BC. There are few sources available on his life and most of what we know comes from the works of Plato approximately a century later. None of Zeno’s original works survive, but we are told that he wrote a book featuring 40 paradoxes, of which Achilles and the Tortoise is perhaps the most famous. It is from the work of Aristotle that we know of a handful of these paradoxes.

Achilles and the Tortoise: The Story

Achilles, the ancient Greek hero, is chasing after a tortoise. Achilles is much faster than the tortoise and so everyday logic suggests that he must eventually catch up with it. According to Zeno, this is not the case.

He argued that on his way to catching the tortoise, Achilles must first reach the tortoise’s starting point. By the time he has reached this however, the tortoise will also have moved further forward to a new point. By the time Achilles reaches this new point, the tortoise will have moved a bit further forward and so on. Every time Achilles reaches the point where the tortoise was, the tortoise has moved forwards to a new point, hence Achilles will never catch up.

We can attach numbers to this to make it easier to see. Suppose the tortoise starts 0.9m ahead and is travelling at 0.1 m/s, while Achilles is running at 1 m/s. Our everyday logic tells us that after one second has passed, the tortoise will have travelled 0.1m and will now be 0.9m + 0.1m = 1m ahead of Achilles' starting point, while Achilles will have travelled 1m and so will have caught up with the tortoise.

Zeno looks at this differently. In the time it takes Achilles to run the 0.9m to where the tortoise started, the tortoise will have travelled 0.09m. Achilles now runs this 0.09m, but the tortoise has travelled a further 0.009m. Achilles runs this only to find the tortoise has moved another 0.0009m ahead and so on for infinity. Zeno argued that because Achilles has an infinite number of finite catch-ups to make, he can never catch the tortoise. He uses this apparent paradox, among others, to argue that motion is impossible and is simply an illusion.

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Achilles Chasing After the Tortoise

Achilles Chasing After the Tortoise

The Problem With Zeno's Paradox

We can see from everyday common sense that Achilles will catch the tortoise after 1 second, but logically Zeno’s suggestion seems to make sense. So what’s the catch? One problem for Zeno is that he lived in a time before mathematical analysis and calculus, where even the greatest minds had a limited understanding of the concept of infinity.

Our sequence above has Achilles running to the points 0.9m, 0.99m, 0.999m, 0.9999m… and seemingly never reaching the point 1m (which we calculated earlier to be the real point where he will catch the tortoise). Modern mathematics however tells us that the limit of this sequence is 1, hence Achilles will catch the tortoise.

Zeno's Paradox on the DoingMaths YouTube Channel

A Similar Paradox Involving Infinity

Similar paradoxes can be solved in similar ways. For example, another version of the problem is the issue of crossing a room. To cross a room I must first cross half of the distance. Then I must cross half of the remaining distance, then half of this and so on. Thinking of it this way, it appears that I will only ever be able to cross half of what is remaining and so never actually reach the opposite side.

But if we think of this as a sequence, we get a very different answer. Once we have done the first 1/2, half of the remaining half is 1/4, and then half of the remaining quarter is 1/8 and so on. We therefore get the sequence 1/2 + 1/4 + 1/8 + 1/16 + … which again sums up to one. So an infinite collection of finite distances can still equal a finite distance, in this case the width of the room.

Bibliography and Further Reading

© 2021 David

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