Achilles and the Tortoise
Achilles, though ten times faster than a tortoise with a head start, seems logically unable to ever catch it, since by the time he reaches where the tortoise was, it has always moved a little further ahead.
History and references
Settled in 1821 by Augustin-Louis Cauchy (rigorous theory of limits and convergent series), building on Newton/Leibniz calculus.
Zeno of Elea proposed this (and several related motion paradoxes) around 450 BCE to defend Parmenides' claim that motion is illusory. It resisted a fully rigorous resolution for two millennia until the formal mathematics of convergent infinite series and limits — developed by Newton and Leibniz and made fully rigorous by Cauchy and Weierstrass in the 19th century — showed that an infinite sum of shrinking positive terms can converge to a finite value. Lange opens his 'Kinematics' chapter with this problem to establish that physical intuition about infinity is often wrong. It remains a staple of popular-physics and philosophy-of-mathematics writing.
It shows that an apparently airtight logical proof about physical motion is really a hidden error in reasoning about infinite series, not a fact about the world.
Explanations that circulated
- The naive argument treats the infinite sequence of ever-shorter chase steps as necessarily requiring an infinite amount of TIME to complete, conflating 'infinitely many steps' with 'infinite duration'.
- It ignores that the corresponding infinite geometric series of time intervals (t + t/10 + t/100 + ...) has a finite sum, so infinitely many steps can occur within a finite time — Achilles passes the tortoise at a definite, calculable moment.
References
- Zeno's Paradoxes, Stanford Encyclopedia of Philosophy (Nick Huggett), SEP, 2022
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