Inverted Zeno's Paradox

The conventional “solution” to Zeno’s Paradox should be turned upside down. Instead of saying: the arrow moves a finite distance across infinitessimal subdivisions of space (calculus’s answer), the arrow does actually move an infinite distance across finite quanta of spacetime (physics). Since arrows are physical, we should prefer the physical (not mathematical) solution.

Zeno shot an arrow at a target 100 feet away. It hit the target 1 second later. How fast did the arrow move? 100 feet per second seems reasonable.

Now move the target closer — half the distance. He shoots a new arrow, and it hits the target in 0.5 seconds. Moving 50 feet in 0.5 seconds is still 100 feet per second.

Repeat the process — halving the distance, shooting the arrow — and each shot moves the same speed.

So far so good. But now let’s get philosophical. When the original arrow hit the original target 100 feet away, that arrow did not hit the target until after it travelled half the distance. But it didn’t travel half the distance until after it travelled a quarter of the distance. Same goes for 1/8, 1/16, 1/32 … ad infinitum. Infinitum you say? So the arrow travelled an infinite distance? What went wrong?

Newton comes to the rescue with his notion of “limits” and “infinitessimals.” We learn these rules for working with limits and sums of infinite series in “calculus.” And they work. But do they explain Zeno’s paradox or explain it away?

They explain it away. Calculus just says: don’t worry about it. Nothing to see here. Yes, the path of the arrow can be chopped up into an infinite series — but the “pixels” of this infinite series are infinitely small when you chop up 100 feet an infinite number of times. And so they add up to 100 feet.

The familiarity of the equations and the fact that arrows do hit their targets has rubbed out the cramp of the paradox.

But the comfort of tidy math equations is a false comfort. Calculus is about numbers, while arrows are naturally occuring things.

Do naturally occuring things actually move across an infinity of infinitelly sliced up magnitudes? Well, not in any sense that we could observe. The Planck-scale pixelization puts a non-infinitessimal limit on how far we can push a purely calculus-oriented limit. The slicing of the distances the arrow travels bottoms out at a hard quantity, tinier than which there is no meaningful difference to discern.

How to reconcile that calculus models the arrow with infinitessimals that have no natural (real) meaning? There is another way to explain it away - but one which is more natural than abstract calculus. The arrow is actually moving an infinite distance along finite pixels, not a finite distance summed from inifinitely small pixels.

It is moving an infinite distance because it is moving at the speed of light. This is the deep lesson of the “four-velocity vector.” The velocity of the arrow is space divided by time. If you sub-divide the space it travels down to the Planck length, it is not moving at all, and the time-vector goes to c itself. And something moving at the speed of c is moving infinitely fast, because it has already arrived at its destination. There is no space for it to traverse.