In what sense does the earth rotate around the sun? couldn't the entire universe be thought to rotate around any arbitrary point?

In the century and a half following Copernicus, when the debate around this issue was at its height, there were actually several major differences of opinion between those figures (such as Kepler, Bruno, Galileo or Descartes) whom we tend to lump together as adherents of the new astronomy. The debate between Medieval and modern astronomers is usually set up in terms of a pair of interconnected differences of opinion, over (i) whether the Earth or the Sun is stationary or moving, and (ii) whether the Earth or the Sun is at the centre of things. The Medieval view was that the Earth was stationary at the centre of the universe and the Sun revolved around it; the modern view was that the Sun was at the centre and the Earth revolved around that. Except that that's too simplistic. With regard to that notion of a centre, some people (like Bruno) were firmly committed to the notion that the universe was infinite, and they explicitly stressed that no centre could be defined in an infinite universe at all, whether for the Earth or for the Sun. Over on the other side, Nicholas of Cusa, who gets lumped together with the more traditional (so-called) geocentrists, also made exactly the same point. With regard to motion, meanwhile, Descartes (for rather complex reasons of his own, not worth exploring here) believed that both the Earth and the Sun were at rest -- indeed, that if motion in the true sense was going to be ascribed to either of them at all, it would have to be to the Sun.

So why do we classify these figures in the way we do? What did the new astronomers all agree on, to set them apart from the traditionalists? It was that, whatever they decided about the behaviour of the Earth, their account of that case would resemble the account they gave of the behaviour of the other planets, and would differ from the account they gave of the behaviour of the Sun. The Medieval astronomers thought that the Sun moved in a circular orbit around a resting Earth, and (a few epicycles notwithstanding) they thought exactly the same thing about a planet such as Mars. The Earth was the odd one out, the one that did something different from the others. The modern astronomers thought that both the Earth and Mars described the same kind of circular or elliptical orbit around the Sun. For them it was the Sun that was doing something different from the other astral bodies, even if they didn't always agree on precisely what it was doing.

Suppose we do select some arbitrary points, define a frame of reference in relation to them, and then use this as an foundation for defining the motions of bodies. To make things vivid, let's base this alternative frame of reference on the Earth itself: the Earth will then be stationary by definition, and the other astral bodies might indeed, as you suggest, turn out to be revolving around it. But the difference between the Medieval and the modern astronomers will lie in the details of precisely how those other bodies' motions will be defined under this scheme. A Medieval philosopher would be happy to say that both the Sun and Mars were revolving around this stationary Earth, and that these two orbits had the same roughly circular shape. But a modern astronomer, if he could be persuaded to adopt this frame of reference (on the basis that all such selections of frames of reference are arbitrary anyway), might then agree that, looking at things in this way, both the Sun and Mars would indeed be revolving around the Earth: but he would insist that their motions would have very different characters. The relative motion of the Sun would be (roughly) circular or elliptical, but the relative motion of Mars would be much more complex than this. It would involve two components: a shared revolution with the Sun, following the latter's own motion relative to the Earth, plus an independent motion of its own around the Sun itself.

But then something else to consider is whether such selections between conflicting frames of reference are arbitrary at all. Some figures in this early modern period (for I must confess that I know this stuff better than I do the General Theory of Relativity and matters arising from that) did indeed regard them as arbitrary, the reason being that they thought that the only available frames of reference, in relation to which places and motions might be defined, were ones based on bodies, and they did not think that any particular body had a privileged status over the rest as the one that defined the 'correct' frame of reference. But others, most notably Newton, thought that there was such a thing as an absolute frame of reference, defined not by bodies at all but rather by an eternal and immutable space in which these bodies were located. He acknowledged that we could never discover by empirical means which objects (if any) in the physical world were at absolute rest, in relation to this absolute space, but he felt that it was reasonable to conjecture that the centre of mass of the solar system might be. Notice that he did not say this of the Sun itself, because his opinion was that the Sun actually jiggled about a little bit. In Newtonian physics, it's not so much that the Earth revolves around the Sun, but more that the Earth and the Sun both revolve around the centre of mass of the two of them taken together; but, since this point is very close to the centre of the Sun alone, the effect on the Sun is very much less pronounced than the effect on the Earth. And then Newton also gave exactly the same account of the interaction between the Sun and Mars. Indeed, it was really Newton's work that signalled the end of this particular debate. The reason why people aren't still arguing about this issue isn't because anyone managed to prove beyond doubt that theirs was the correct answer to the question. Rather, Newton showed that the question itself didn't actually make much sense. The Medieval astronomers would tell one story about the Earth, and a different story about both the Sun and Mars; the modern astronomers up until Newton would tell one story about the Sun, and a different story about both the Earth and Mars. But, as far as Newton was concerned, the Earth and Mars and the Sun were all doing the same thing, all revolving around their common centre of mass. This revolution was less noticeable in some cases than in others, but nevertheless the same cause (namely, gravity) was producing the same effect in, as a matter of fact, every object in the universe.

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