Very interesting read. I've been thinking about it for a while, and although the explanation in the article makes sense to me, I wonder if the following is equivalent:
Consider a spherical surface S. We can construct a mapping from the point of S to the points in a finite and limited subset of R^2 (say, the square [0, 1] X [0, 1]). One way to do that is to map each point of "latitude" ρ and "longitude" θ (in degrees), to the point ((ρ+90)/180, θ/360).
Now, let's consider the rectangle [0, .5] x [0, 1]. There is obviously a 1:1 mapping from this to the original square (just halve or double one of the coordinates to change from one to the other). In an intuitive sense, there are "as many" points in half the square as in the whole square, a bit like there are as many natural numbers as there are even numbers. (This is one definition of infinite set, if I remember correctly).
Now this means that we can remap all the points from half the square to a complete sphere, and the points from the other half square to a "new" complete sphere, in fact mapping points from the original sphere S to two new identical spheres.
Is this reasoning correct, or am I missing anything (probably something obvious...)?
i'm not a mathematician either, but i think the mistake you're making is in neglecting the difference between "subset A of the sphere has as many points as the entire sphere" and "here's a way to cut a sphere into subsets, and reassemble the subsets into two spheres". intuitively, your mapping scheme involves stretching, shearing and discontinuous transforms on the subsets before you get two spheres out of it - it is pretty obvious from the mathematics of infinite sets that you can always do this, so there's no real "paradox" involved.