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.

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.