Comment 25 by wtfbits :

Here's an excerpt from an article which explains why the heliocentric view is not any more valid than the geocentric one.

A famous example of different pictures of reality is the model introduced around A.D. 150 by Ptolemy (ca. 85–ca. 165) to describe the motion of the celestial bodies. Ptolemy published his work in a treatise explaining reasons for thinking that the earth is spherical, motionless, positioned at the center of the universe, and negligibly small in comparison to the distance of the heavens.

This model seemed natural because we don't feel the earth under our feet moving (except in earthquakes or moments of passion). Ptolemy's model of the cosmos was adopted by the Catholic Church and held as official doctrine for fourteen hundred years. It was not until 1543 that an alternative model was put forward by Copernicus. So which is real? Although it is not uncommon for people to say Copernicus proved Ptolemy wrong, that is not true. As in the case of the goldfish, one can use either picture as a model of the universe. The real advantage of the Copernican system is that the mathematics is much simpler in the frame of reference in which the sun is at rest.

Link to full article

This article seems to make a different argument then what you seemed to be saying in your initial post. I took your first post to mean that from a relativistic perspective taking the earth as a fixed frame of reference it made sense to say the Sun revolves around the earth.

This article seems to say that the Ptolemeic view of the universe (a completely different model than the relativistic view which starts with the concept of gravity as defined by Newton) is really as good a model for the solar system as Einstein/Newton although not quite as elegant. I think that is clearly false.

For one thing the Ptolemeic view had errors in it. It worked up to a point because the instruments of the time were mostly incapable of detecting the errors but now it would be seen to be highly innacurate compared to the relativistic view. For another thing the Ptolemeic view really has no definition for the underlying force (gravity) that keeps the planets in their orbits. The relativistic view not only has that but the same model of gravity works amazingly well for all sorts of terrestrial behavior such as how bridges stay up and artillery shells come down. The theory of gravity as defined by Newton and then enhanced by Einstein also describes all sorts of atstronomical phenomenon: galexies, black holes, quasars,... that Ptolemy couldn't even begin to describe.