Zeno of Elea argued that swift Achilles could never overtake a tortoise
Give a tortoise a head start and fleet-footed Achilles should pass it in seconds. Yet each time he reaches where it was, it has crept a little further on. Zeno of Elea used puzzles like this to argue that motion is an illusion, and philosophers today still cannot agree on exactly where his logic goes wrong.
Zeno was born around 490 BC in Elea, a Greek city in southern Italy, and studied under Parmenides, who taught that all reality is one single, indivisible thing. Plato's dialogue Parmenides shows him at about 40, looking back on a zealous youth spent defending his teacher. His only known book, probably from the 460s BC, is lost; Plato says it was stolen and circulated without his consent. What survives comes secondhand through Plato, Aristotle and the 6th-century commentator Simplicius of Cilicia.
Where earlier thinkers described their picture of the world, Zeno built explicit arguments meant for debate, which led Aristotle to call him the inventor of dialectic. His method was to take a rival view and show that it collapses into absurdity. Against plurality, Proclus credited him with forty arguments. If things have size they can be halved forever, so nothing has a finite size; if they have none, no amount of them adds up to anything.
Aristotle singled out four paradoxes of motion. In the dichotomy, a runner must cross half a distance, then half of what remains, and so on without end. Achilles and the tortoise is a variant of the same idea. The arrow is motionless at any instant, since it occupies just one place. The moving rows suggest a stretch of time can be both halved and doubled. Aristotle answered by separating actual infinity from potential infinity; Plato suspected Zeno himself did not take the arguments seriously.
The atomists responded by proposing a limit to division, the atom. Modern tools such as limits and set theory challenge Zeno's reasoning, and Bertrand Russell credited the paradoxes with enabling the work of Karl Weierstrass. Physicists call the way observation can freeze a quantum system the Quantum Zeno effect, after the arrow. Zeno reportedly died around 430 BC after a failed plot against a tyrant.
Source: Zeno of Elea
