🐢 ∞
Achilles and the Tortoise
Zeno of Elea · c. 5th century BCE
The puzzle
Achilles gives the tortoise a head start. By the time he reaches where it was, it has moved a little. By the time he reaches that point, it has moved again. He never overtakes.
Note
The paradox isn’t that the math is hard. Aristotle pointed out that the geometric series 1 + ½ + ¼ + … sums to 2. The paradox is that the analytic operation, taking the sum of infinitely many terms, doesn’t obviously match what motion does. Modern measure theory and the Cauchy definition of the limit absorb the move easily, but it isn’t entirely clear whether they dissolve the paradox or just rename it. Performing infinitely many sub-actions in finite time is what motion seems to require — and what no human-scale procedure can demonstrate. Zeno’s instinct that motion is metaphysically suspicious has held up better than most ancient instincts.