Greece & Rome · c. 490–430 BCE
Zeno of Elea
Parmenides’ pupil, famous for paradoxes of motion and plurality
Defended Parmenides by showing that common-sense belief in motion and plurality leads to absurdities, as in Achilles and the Tortoise and the flying Arrow. His method, reductio ad absurdum, became a basic tool of philosophy.
Life and thought
Zeno of Elea was Parmenides’ younger associate; Plato describes him at about forty, visiting Athens with his master. He wrote a book of arguments, now lost, and ancient reports say he died bravely resisting a tyrant. Our knowledge of his paradoxes comes mostly from Aristotle’s Physics and its commentators.
His method was to defend Parmenides indirectly: assume what common sense says, that there are many things and that they move, and show it leads to absurdity. To cross a stadium you must first reach the halfway point, then half of the rest, and so on without end, so you can never finish. Swift Achilles never catches the tortoise, since each time he reaches where it was, it has moved on. A flying arrow occupies a space equal to itself at every instant, so it is always at rest. These puzzles about infinity, space and time were not well answered until the mathematics of limits in the 19th century, and some philosophers think they still are not.