# Zeno's Paradoxes

## Contents

## Paradoxes

**Zeno's paradoxes** are a set of antinomies devised by Zeno of Elea to demonstrate that the mathematical understanding of motion and space had conceptual gaps.

These paradoxes were preserved in the writings of Aristotle. Zeno's originals have not survived to the present day.

### Achilles Paradox

Achilles and a tortoise are to have a race.

Achilles (not surprisingly) runs considerably faster than the tortoise. So, to make it marginally more fair, he gives the tortoise a head start.

But it is apparent that Achilles can not actually catch up with the tortoise.

Suppose he gives the tortoise a headstart of $x_0$.

By the time he has got to $x_0$, the tortoise has moved on, to $x_1$, say.

But by the time Achilles has reached $x_1$, the tortoise has moved on, to $x_2$, say.

You can continue this indefinitely.

### The Dichotomy Paradox

Motion is impossible.

Suppose $B$ is a body which is to move from $P$ to $Q$.

Before $B$ can reach $Q$, it must first move halfway to $Q$.

But before it can reach the midpoint between $P$ and $Q$, it must first move halfway to that midpoint.

And before it can reach the point half way to the midpoint, it has to move halfway to that point too.

So on, indefinitely.

It is therefore impossible for motion to happen, because you cannot even get started. Every time you try to reach a point, you have to reach the point half way before it first.

### The Arrow Paradox

At any particular instant, a moving arrow is either at rest or in motion.

If the instant is indivisible, the arrow cannot move, because if it did, the instant would immediately be divided.

But time is made up of instants.

As the arrow cannot move in any one instant, it cannot move in time.

Hence the arrow remains at rest.

### The Stadium Paradox

Consider three rows of bodies:

- $\begin{array} {ccccc} \text {(A)} & 0 & 0 & 0 & 0 \\ \text {(B)} & 0 & 0 & 0 & 0 \\ \text {(C)} & 0 & 0 & 0 & 0 \\ \end{array}$

Let row $\text {(A)}$ be at rest, while row $\text {(B)}$ and row $\text {(C)}$ are travelling at the same speed in opposite directions.

- $\begin{array} {ccccccc} \text {(A)} & & 0 & 0 & 0 & 0 & \\ \text {(B)} & 0 & 0 & 0 & 0 & & \\ \text {(C)} & & & 0 & 0 & 0 & 0 \\ \end{array}$

By the time they are all in the same part of the course, $\text {(B)}$ will have passed twice as many of the bodies in $\text {(C)}$ as $\text {(A)}$ has.

Therefore the time it takes to pass $\text {(A)}$ is twice as long as it takes to pass $\text {(C)}$.

But the time which $\text {(B)}$ and $\text {(C)}$ take to reach the position of $\text {(A)}$ is the same.

Therefore double the time is equal to half the time.

## Source of Name

This entry was named for Zeno of Elea.

## Historical Note

Zeno of Elea presented these paradoxes to the school at Athens when he visited with his patron Parmenides.

These paradoxes have been said to have shocked the philosophers there out of their complacency, and raised questions that none of them could satisfactorily answer.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Zeno's paradoxes** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Zeno's paradoxes** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Zeno's paradoxes** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Zeno of Elea**(5th century bc)