First, with reference to your initial post, presumably all proof requires presuppositions of some sort, and these presuppositions are often philosophical.
Secondly, geocentricism itself rests obviously on a range of philosophical assumptions derived from ancient Greek philosophy. So I don't think we can get very far with the dichotomy between proof vs. philosophical presupposition.
The functioning of any geocentric model may presuppose Greek Philosophy, the proof not so. Causality and implication are opposite directions in reasoning.
Third, you use the phrase "prove conclusively." (cf. Mr. Y’s use of "know conclusively"). One frequently finds criticisms ofscientific theories amongst creationists to the effect that such and such a scientific theory has not been proven "conclusively."
This observation also undercuts a widely held argument against the earth orbiting the sun, namely that the moon would not beable to keep up with the earth in its revolution. Galileo showed that Jupiter's moons had no problem with this. Hence, it should not be a problem for the earth.
4) Geocentric cosmology requires the earth to be stationary. But we know now, in a way neither Copernicus nor any of the early modern scientists could have, that it rotates on its axis. ...
a) ... In addition to the 1851 Foucault pendulum experiment, there is ...
b) ... the repeated evidence drawn from the orbits of artificial satellites. Suppose you launch a satellite from Cape Kennedy. It goes 100 miles above the earth moving Southeast, at an angle of 30 degrees to the equator. Once the satellite is launched, the plane of its orbit stays relatively fixed. There is no significant force exerted on the satellite to alter this aspect of its orbit. Now, if the earth is not rotating, the satellite should pass over Cape Kennedy once in every orbit. But it does not. It passes over Alabama after the first orbit, Louisiana at the end of its third, and so on further west at the completion of each successive orbit. This can only happen if the earth is rotating. Hence, geocentricism cannot be true or it must modify its claim about the stationary nature of the earth, which would run contrary to both Ptolemaic and Tychonic geocentricism.
(5) Johannes Kepler's laws of planetary motion (later fine-tuned in Newtonian mechanics) provide a simple explanation for why the planets are in the positions they are at each day of the year on a heliocentric model. Ockam's razor or the principle of economy says that we should not multiple entities or laws beyond necessity. Kepler's "elliptical orbit" did everything and more than Ptolemy's intricate system of cycles and epicycles. It ought to be preferred solely on this basis. And the mathematics is simpler than Tychonic geocentricism. But of course Kepler's position has a distinct advantage elsewhere, namely that it explains *why* the planets move as they do. Brahe's system, even if physically possible, leaves rather inexplicable why it is that the planets have such orbits. What physical laws would explain such motions? None. This brings us to the next point.
(6) Newtonian physics implies that a smaller mass object (earth) cannot be the center of orbit for a larger mass object (sun), much less the sun plus the rest of the planets. Objects will orbit around a common center of mass. Where one object has significantly more mass than another, the center of mass lies to the center of the object with greater mass. Hence, it certainly looks as if geocentricism is inconsistent with Newtonian physics.
7 ... Stellar parallax is quite important. The 16th century astronomer Tycho Brahe objected to heliocentric cosmology on precisely this basis. He said we should expect the parallax phenomenon if heliocentricism is true, but we don't observe it. Conversely, if geocentricism is true, we should not expect this. He then noted that his detailed observations produced no evidence of stellar parallax.
(8) A final bit of evidence in support of heliocentricism relates to the existence and orbits of extra-solar planets. Astronomers continue to build an impressive case for the existence of planets outside our own solar system but that orbit around their own star. The current number is around 63 (including the 11 most recent candidates discovered last month). The number and details of such planets is likely to increase dramatically in the next 10 to 20 years with the advancement and employment of new telescopic technology. These discoveries, pending further verification, support the idea that heliocentric systems exist elsewhere in the cosmos. Such discoveries provide additional weight to the view that our system is heliocentric.