# Zeno of Elea

Zeno of Elea (c. 490 – 430 BCE) was the devoted pupil of Parmenides and, in
Plato's telling, the man who invented dialectic. He wrote one book, a set of
some forty arguments, and every one of them was built to defend his teacher.
Parmenides had said that what is, is one and unmoving; people laughed at the
conclusion; Zeno set out to show that the ordinary belief in many moving things
led to worse absurdities still.

## The method

Zeno's arguments share a shape. Take an assumption the opponent holds, follow
it honestly, and arrive at two conclusions that cannot both be true. He did not
argue *for* the One directly; he argued *against* the Many, and let the
contradiction do the work. This is the reductio ad absurdum, and he used it
with such control that Aristotle named him the father of the technique.

## The paradoxes of motion

Four of his arguments survive in Aristotle's summary, and they are the ones
everyone remembers.

- **The Dichotomy.** To cross a room you must first reach the halfway point,
  then the half of what remains, and so on. There are infinitely many points to
  pass, so you never begin.
- **Achilles and the tortoise.** The fastest runner can never overtake the
  slowest if the slow one starts ahead, because whenever Achilles reaches where
  the tortoise was, the tortoise has moved on.
- **The Arrow.** At each instant an arrow in flight occupies a space equal to
  itself, and is therefore at rest. Time is made of instants, so the arrow never
  moves.
- **The Stadium.** Rows of bodies passing one another at equal speed seem to
  take both a whole and a half of the same time, so half a time equals a whole.

> That which is in motion moves neither in the place where it is nor in the
> place where it is not.

## What the arguments showed

Zeno was not confused about whether runners overtake tortoises. He was showing
that the concepts of space, time and infinity that his listeners used without
thought could not be put together consistently. The mathematics needed to
answer him, a workable theory of infinite series and of the continuum, took
more than two thousand years to arrive, and some philosophers hold that the
Arrow has still not been fully answered.

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*Stoa Press · Philosophy Essays · Markdown edition*