A Greek philosopher 'proved' that the fastest runner can never catch a tortoise.
Give a tortoise a head start, said Zeno of Elea, and Achilles must first reach where it was. By then it has crept ahead, so he must reach that point too, and so on forever. The argument sounds absurd, yet philosophers and mathematicians have wrestled with it for 2,400 years.
Zeno of Elea lived in the fifth century BC and was a follower of Parmenides, who taught that reality is a single, unchanging whole and that motion and change are illusions. Zeno's paradoxes were designed to defend that view by showing that ordinary beliefs about motion lead to contradictions. His own writings are lost; the arguments reach us through Plato, Aristotle and the later commentator Simplicius.
The three most famous target motion. In the Dichotomy, to cross a path you must first cover half of it, then half of what remains, and so on without end, so the journey seems impossible to finish or even to begin. Achilles and the tortoise applies the same logic to a chase. The Arrow slices time instead of space: at any single instant a flying arrow occupies one position and is not moving, so if time is made only of instants, when does it ever move?
Popular retellings often say Zeno believed an infinite sum must be infinite. The ancient sources don't show that. His worry, as Simplicius reports it, was how anyone could complete an infinite number of tasks in a finite time. Aristotle replied that the shrinking distances take shrinking amounts of time, and that time is not built from indivisible 'nows'. Diogenes the Cynic is said to have answered by simply standing up and walking.
In the 19th century, Cauchy and Weierstrass gave calculus a rigorous theory of limits, and many mathematicians consider the puzzles solved: an infinite series of ever-smaller steps can add up to a finite distance and time. Some philosophers disagree, arguing that the maths tells you where Achilles overtakes the tortoise without explaining how an infinite sequence of acts can actually be completed. Zeno may also have given us one of the earliest examples of proof by contradiction.
Source: Wikipedia — Zeno's paradoxes · Text summarised from Wikipedia (CC BY-SA 4.0)
