PZ - some comments. One this is pretty old. Two, where I think he goes of the rails is with his distinction of mathematical from real in this case. Motion is relative. Of course you can come up with an internally consistent geocentric model or representation of planetary motion. It's just a helluva lot more complicated, a Rube Goldberg construct in comparison to a heliocentric model. Using it to forecast the future positions of planets, asteroids, or comets (or, more generally, their motion), will be much harder to calculate with that model than with the geocentric one.
I think that trying to forecast the location of objects outside the solar system will also be more difficult; I'm not an astro-physicist, but I suspect that most of these forecasts are relative to the solar system's center of mass, which I would guess is more closely approximated by the sun's center than by the earth's. I would guess that if we want to track the motion of such an object, using a heliocentric frame of reference is simpler because the relative motion of the earth and sun is filtered out of the calculation in advance. Using a geocentric model implicitly retains that motion while doing the calculation. So it is mostly a matter of simplicity - for doing astronomical work - that we use the heliocentric system. It also simplifies understanding of the seasons, since we can talk about the tilt of the earth and how the incidence of solar rays changes over the course of the year as the earth revoloves around the sun.
But I would be surprised if someone could not come up with an internally consistent model that matches all the predictions of a heliocentric model, and does not contradict what we know about gravity. It would probably be an amusing intellectual exercise for those so inclined, too.