## Wednesday, January 11, 2012

### Zeno’s Arrow Fallacy

Zeno’s Arrow Fallacy

Consider Zeno's Paradox of the Arrow: the arrow does not move within any instant during its flight; so when does it move?

To this I reply: we measure motion not in meters, but in meters per second. Within the instant the arrow moves zero meters, but this is in zero seconds; so the speed of the arrow is zero divided by zero, the indefinite ratio. Within the instant the arrow moves at an indefinite rate. It moves at any speed.

For proof of this, consider this experiment; we fire Zeno's Arrow past another arrow, which we release the moment Zeno's Arrow passes. At that instant we take our flash photo. At the instant of the flash, the two arrows are side-by-side; one is moving rapidly to the side and the other is slowing falling down; otherwise they are indistinguishable. In the instant, speed is indeterminate. You'd need another flash picture a millisecond later to tell their velocities.

Zeno argued that the arrow does not move in any instant, and so does not move. His mistake is confusing 0/1 = zero speed with 0/0 = no known speed. In each instant the arrow's speed is indeterminate; we learn its speed when we compare instants. So the arrow's speed isn't zero at every instant of its flight; it's unknown; a mystery that clears up when we compare two instants.

So Zeno's Arrow Fallacy is the misleading equation:
0/0   =   0/1
Actually 0/0 equals every number equally. It's as much 0 as it is 1 or 17 or 42 or 1/137. The 0/0 = 0 equation is misleadingly specific.

My students consistently make this mistake whenever I mention division by zero. They get 0/1 mixed with 1/0 and with 0/0. I have been driven to invent these nicknames:

0/1   =   POOF    ;   for vanishing
1/0   =   BOOM   ;   for exploding
0/0   =   GOOP   ;   for vagueness

I will give Zeno this much: this resolution of the Arrow requires that some physical quantities are indefinite.