The non-problem of Xeno’s Paradox
My son used to be so frustrated when my nieces would disrupt his Dungeons and Dragons game because they wouldn’t play by the rules in the box (rules which for two bright girls were fun for boys only). When he would complain I would tell him that we bought the game, it was now our game and no longer the manufacturer’s so we were free to change the rules.
“But those are the rules, dad,” he would complain.
“Then get used to them having all the fun,” I’d say.
Zeno’s paradox is framed on the assumption that distance resets with time. But we know this is only true from the perspective of the runner at this moment at this location. In fact there are two systems of measurement in play:
- Distance to goal by half-life (Zeno’s measurement). But even now we understand that half-life incrementalism eventually zeros out—i.e. the values involved are so negligible as to be meaningless. The difference between .0001 and 0 are so minuscule that they matter only mathematically. Which is why even a nuclear fuel rod becomes safe eventually even though it is still radioactive.
- Distance to goal by measurable units, which is how most people calculate. The distance to goal does not diminish by half, but by steps. Say the distance to goal is one hundred steps. The runner won’t diminish the distance by half in a single step, but in fifty. On the fifty-first step, the distance to goal from the point of origin is less than half.
The assumption behind Xeno’s paradox is that each step resets the distance to goal, when in fact it merely resets the distance to goal from this step forward. It does not reset the original distance to goal, and from that perspective the runner will finish the race.
As with many classical problems, the solution is presented by how the paradox is framed.One might call this not a paradox so much as a language game whose outcome is conditioned by the rules.
