Achilles and the tortoise
The fastest runner can never catch a tortoise. Or can he?
Zeno of Elea, 5th century BCE
The race at the temple steps
Long ago in the Greek city of Elea, the philosopher Zeno told a story to make people's heads spin. Achilles, the swiftest runner in all of Greece, agrees to race a tortoise. Being a good sport, he gives the tortoise a 100-metre head start.
The race begins. Achilles sprints to the spot where the tortoise started. But in that time the tortoise has plodded forward a little, say 10 metres. So Achilles runs those 10 metres. By then the tortoise has crawled another metre. Achilles runs that metre; the tortoise has moved 10 more centimetres.
Every time Achilles reaches where the tortoise was, the tortoise has moved on. There are infinitely many of these catch-up stages, and each one takes some time. So, said Zeno, Achilles can never pass the tortoise.
And yet anyone who has ever overtaken a slow cyclist knows that's wrong. Something in the story must be cheating.
The puzzle
Achilles must complete infinitely many catch-up stages, each taking time. How can he ever finish them all?
See it for yourself
What's really going on
The trick is that infinitely many stages can add up to a finite total. If Achilles runs at 10 metres per second and the tortoise at 1, the stages take 10 s, then 1 s, then 0.1 s, then 0.01 s… Add them all up and you get 11.111… seconds, a perfectly ordinary number. At that moment Achilles is level, at 111.1… metres, and after it he's ahead.
Mathematicians made this rigorous in the 1800s with the idea of a limit, which is the foundation of calculus. An infinite series like 10 + 1 + 0.1 + 0.01 + … can converge to a finite value. Zeno's mistake was assuming that infinitely many things must take forever.
Think about it
Zeno also argued you can never cross a room, because you must first get halfway, then half the remaining distance, and so on. Can you use the same idea to defeat that one?