Dichotomy Paradox

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.

Resolution

$\ds \sum_{n \mathop \ge 1} 2^{-n} = 1$

$\blacksquare$

Historical Note

The Dichotomy Paradox is one of Zeno's Paradoxes, as famously raised by Zeno of Elea.

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): Zeno's paradoxes
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): dichotomy paradox
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Zeno of Elea (5th century bc)
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Zeno of Elea (5th century bc)