The Ross-Littlewood Paradox

This kind of thing ought to alert people to the difference between a mathematical formalism and reality; instead, it seems to lead them to claim that reality is paradoxical! Or maybe that's not what they really mean, but that's the way they talk, e.g.:

In the Ross-Littlewood supertask, each step (starting at some time before noon) is performed in half the time as the previous step: "This guarantees that a countably infinite number of steps is performed by noon."

Sure, within the formalism it "guarantees" that. But it's not as though any process could really be done in this fashion! So our formalism has assumed something impossible: at that point, of course it will lead to paradoxes: if we assume A = !A, anything at all follows.

And the whole rest of the "problem" arises simply because we've started out by assuming an impossibility. We put 10 balls in an urn each "turn," and take out 1: how many are left at "the end" of our infinite process? Of course, we can't ever "end" an infinite process, even if we can devise a formalism that says we can. So we get contradictory answers of "zero" and "an infinite number": the whole problem is nonsensical if looked at as "What would happen in reality?" because the entire setup can't happen in reality.

As Berkeley noted long ago about infinitesimals, if they help you to calculate the position of some moving object, fantastic! But one should not take the formalisms that are used in such calculations as realities:

"And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?"

There is nothing wrong with calculating the "limit" of an infinite series according to the rules of calculus and seeing what that gets you; when the results are useful, great! But those rules are just a convention: we could have used Cesàro summation instead, or some other convention altogether, and we would get quite different results.

We should judge the applicability of our formalisms by reality, and not reality by a formalism we happen to have found useful.


  1. This is like the theorem everyone proves in first year calculus, that a convergent but not absolutely convergent series can be rearranged to sum to *anything*.


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