Relevant to our recent discussion of the continuum:
"The rate of decay of a radioactive substance -- in the amount of radiation it emits -- is at every moment proportional to its mass m: dm/dt = -am. The solution of this differential equation is m = m0e-at, where m0 is the initial mass of the substance (the mass at t = 0). We see from this solution that m will gradually approach 0 but never reach it -- the substance will never completely disintegrate." -- e: The Story of a Number, p. 103
Any Maor Dude with Half a Pile of Uranium
Here we get a clear glimpse into the problem of mistaking a formalism for reality. When he wrote this, Maor seems to have forgotten that this differential equation is just a model for radioactive decay, and not radioactive decay itself.
Because the model implies that the amount of radioactive material present at any time changes along a continuum, and that it changes exactly according to this equation. (It could not do the latter unless it also did the former.)
But, of course, the amount of uranium or plutonium or whatever present in some mass does not change continually. We cannot have 1.54957623 atoms of uranium in a rock: we can only have two atoms, or one atom. And once radioactive decay reduces the amount left to one atom, even that last atom will sooner or later decay, and we will have zero atoms of uranium left. And so, contra Maor, all of the uranium will eventually disappear.
Since we are typically dealing with a vast number of uranium atoms in any radioactive sample, modeling the decay process as though it were a continuous function is a useful fiction. But if we mistake the model for reality, we reach erroneous conclusions, such as "there will always be some uranium left in the sample."
I suggest that similarly, modeling space as if it were a continuum is a useful fiction. But if we mistake the model for reality... I leave the rest as an exercise for the reader.
Posted by Gene Callahan