Zeno Says You Never Arrive
What if it takes us forever to get there? Consider the “dichotomy paradox” posed by the Greek philosopher Zeno. He observed that to walk across a thing—for instance, Greenland (which belongs to Denmark)—first you need to walk across half of it. Then you need to walk across half of what’s left, then half of what’s left after that, and so forth. Since there is always another half to go, you’ll never reach the other end. How would you solve this and his other paradoxes of motion?
Achilles, the fastest hero in Greece, races a tortoise and gives it a head start. To catch up he must first reach where it began — but by then it has crept a little farther; and again; and again, forever. Zeno claimed this proves the swift can never overtake the slow. And yet, every day, they do. Where's the trick?
Key concepts
- The Dichotomy Paradox
- Zeno's argument that motion is impossible because covering any distance means first covering half, then half the rest, forever — infinitely many steps, so (he claims) you never finish.
- Achilles And The Tortoise
- The same logic dramatized: give a tortoise a head start and the faster Achilles supposedly can never overtake it, since each time he reaches where it was, it has crawled further.
- The Arrow Paradox
- At any frozen instant a flying arrow occupies one position and isn't moving — so if time is made of instants and the arrow is still in each, how does it ever move?
- Convergent Infinite Series
- The mathematical key: infinitely many shrinking pieces can sum to a finite total — 1/2 + 1/4 + 1/8 + … equals exactly 1, so endless steps cross a finite distance.
What to know
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01
The paradox tricks you by confusing 'infinitely many' with 'infinitely much' — there are infinite halves, but they shrink so fast their total is finite, so an endless number of steps needs no endless distance or time.
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