Zeno of Elea was a student of Parmenides and the author of a set of arguments designed to show that motion and plurality generate contradictions.

The purpose

Parmenides had argued that being is one and unchanging, which contradicts everything anyone observes. Zeno's method was not to defend that conclusion directly but to attack its denial: if you believe in many things that move, here is what follows.

Aristotle credits him with inventing dialectic, meaning argument that proceeds by taking an opponent's premise and deriving an absurdity from it.

The paradoxes

The dichotomy. To cross a room you must first reach the halfway point, and before that its halfway point, without end. The journey requires completing infinitely many tasks, which seems impossible, so motion never begins.

Achilles and the tortoise. Achilles gives the tortoise a head start. By the time he reaches where it was, it has moved a little; by the time he covers that, it has moved again. He never passes it.

The arrow. At any instant a flying arrow occupies a space exactly its own size, and so is at rest. Time is composed of instants, so the arrow is at rest throughout its flight.

What answers them

The first two are dissolved by the mathematics of convergent series: an infinite number of intervals can sum to a finite distance, so the task is not infinite in the sense that matters. That machinery arrived two thousand years later.

The arrow is harder and remains interesting, because it concerns whether motion can be defined at an instant at all. The modern treatment defines velocity as a limit rather than as something an instant possesses.

Why they lasted

Because each one is valid on its premises. Refuting a Zeno paradox means identifying which premise about space, time or infinity fails, and doing that took the development of analysis. They are the earliest arguments in philosophy whose resolution required new mathematics rather than better reasoning. See causation.