5.1 Achilles and the Tortoise

Here’s one way to begin thinking about the infinitely small. Consider the paradox of ‘Achilles and the Tortoise’ which was devised by the Greek philosopher Zeno in the 5th century BC. Watch Video 3 for an introduction to this paradox, which took hundreds of years to be logically refuted.

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Transcript: Video 3 60-Second Adventures in Thought: Achilles and the Tortoise

DAVID MITCHELL

60-second Adventures in Thought. Number one: Achilles and the Tortoise.

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How could a humble tortoise beat the legendary Greek hero Achilles in a race? The Greek philosopher Zeno liked a challenge and came up with this paradox.

First, the tortoise is given a slight head start. Anyone fancying a flutter would still rush to put their money on Achilles, but Zeno pointed out that to overtake him, Achilles would first have to cover the distance to the point where the tortoise began. In that time, the tortoise would have moved, so Achilles would have to cover that distance, giving the tortoise time to amble forwards a bit more.

Logically, this would carry on forever. However small a gap between them, the tortoise would still be able to move forward while Achilles was catching up, meaning that Achilles could never overtake.

Taken to an extreme, this bizarre paradox suggests that all movement is impossible, but it did lead to the realisation that something finite can be divided an infinite number of times. This concept of an infinite series is used in finance to work out mortgage payments, which is why they take an infinite amount of time to pay off.

End transcript: Video 3 60-Second Adventures in Thought: Achilles and the Tortoise

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Video 3 60-Second Adventures in Thought: Achilles and the Tortoise

Zeno argued in this paradox that Achilles can never catch up to the tortoise in the race. Even though the distances involved are ever smaller, each time Achilles closes the gap, the tortoise has always moved on a tiny bit further. This contradicts what our senses tell us and therefore, Zeno argued, our senses are deceiving us.

It took the work of mathematicians such as Leibniz and Newton to advance our understanding of the infinitely small, and thereby resolve this paradox. What thinkers at the time of Zeno didn’t realise is that you can take something finite and divide it (mathematically, if not physically) into infinitely many parts.