What they mean is that you may take whatever point in space to be your immobile frame of reference and describe all motion with respect to that point. It happens to be convenient to take the center of the Sun as such a point when doing calculations relevant to the solar system but it's not more 'valid' than literally any other point in the universe. You just have to change the equations a bit.
No, they said "there is no physical reality to the matter", where, judging by context, "the matter" is the question of whether the Earth revolves around the Sun.
The conclusion that there is no physical reality to this is based on the what @aluren mentioned - you can take any immutable point of reference and describe the movements of the others from it using the same laws of physics, thus "the matter" becomes one of philosophy and choice of point of reference rather than physics itself.
You may disagree with this point but I don't think anyone disputed the existence of a physical reality, they simply moved one particular question out of its domain.
Changing the point of reference will change your mathematical model, sure, but it will still describe a movement around a barycenter, and the statement remains true whatever your philosophy. Numerous ways to model a phenomenon don't deny its existence.
Again: not denying the existence of the phenomenon, only removing it from the realm of physical reality to that of philosophy and mathematical point of reference.
The laws of motion are the same in every reference frame, which means that statements like "X revolves around Y and not vice versa" are not "physically real" in the same way as "X and Y attract each other gravitationally" is, because they select one reference frame as privileged.
Rotating around a point still applies in any frame of reference. Spin a top and the axes of rotation is independent of your frame of reference. Pick an atom on that top and it’s rotating around a specific point in any frame of reference.
This goes back to the ambiguity of the definition of orbit. If I have a system with two points involved in some sort of rotational motion, then there isn't a "best choice" for which point should be considered the origin of my coordinate system.
If you mean exactly 2 abstract points in Euclidean geometry then sure. 3D objects are not points which is the first issue. Relatively also cares about accelerating or rotating reference frames which applies to both point masses.
The center of mass in a two body system is a nice inertial reference frame which simplifies calculations.
The "physical reality" of describing motion is transformations relative to a frame of reference. The origin (eg where you put 0,0,0) of that frame is a mathematical choice; it's not a real physical thing.
That the formulas can be calculated relative to any reference point doesn’t change the fact that the physical reality is ellipsoid movement of planets around the Sun.
Even as you insist on choosing a privileged reference frame for, say, aesthetic reasons, it's still not all the planets moving around the Sun. Did you not read the article? It's all the planets and the Sun moving around the barycenter. If you try to think through what your words even mean, you'll realize that they don't even mean anything, much less describe a physical reality.
"In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun [approximately] at one focus, in his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation.
More generally, in the gravitational two-body problem, if the two bodies are bound to each other (that is, the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. The orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus."
The existence of barycenter it a physical reality. The elliptic movement is a physical reality. Compared to all other distances, the barycenter is for all the motions of all the planets effectively Sun. Even it it is approximation, it's a good approximation. Even if it moves and is sometimes a little outside of the Sun surface, it's still compared to the other distances practically there.
Sun.
It was known even three centuries ago, but now some "philosophers" think that they "relativize" that away. No. It's there.
> The existence of barycenter it a physical reality.
A barycenter is a mathematically defined point that we choose because it makes equations simpler. There's nothing physically real (i.e. directly observable) at the location.
Consider: if you fire a cannonball it will follow a parabolic path, and that parabola will have a focus, right? But that doesn't mean there's anything physically real about the focus of the parabola - it's a mathematically abstraction we invent because it's useful for describing the cannonball.
> some "philosophers" think that they "relativize" that away.
The relativity we're talking about comes from Einstein, not philosophers. Deciding whether the Earth revolves around the sun or vice-versa ultimately boils down to choosing a frame of reference, and one of the grand results of relativity is that there really is no absolute frame of reference we can measure from.
> Consider: if you fire a cannonball it will follow a parabolic path, and that parabola will have a focus, right? But that doesn't mean there's anything physically real about the focus of the parabola
The parabolic path is physically real, it will not go any other way.
The same is with the paths the planets make: their form is real, they don't take any other.
If you try to draw these paths on scale, you have to draw the ellipse (which for many planets is hard to distinguish from circle, the measurements had to become precise to learn that). Also, you have to place the Sun directly in one of the foci of the ellipse.
All of these steps you have to do to make a picture that corresponds to what is measured and observed -- to make a picture reflect the physical reality.
You obviously agree that much. So when you then write "whether the Earth revolves around the sun. ... I don't think there's any valid answer besides "you can define your terms and points of reference such that it does or doesn't; there's no particular physical reality to the matter"."
You are confused with the fact that the calculations can be done using different frames of reference to make a statement that "there's no particular physical reality to the matter."
The particular reality is the ellipsoid paths and the Sun in one of the foci, and whichever calculations you do, you have to reconstruct that physical reality. These shapes and that the Sun is actually in one of the foci is what you can't "relativize away."
> Deciding whether the Earth revolves around the sunb or vice-versa ultimately boils down to choosing a frame of reference
It doesn't. The Sun is in the focus of the ellipses made by planets around it. The opposite doesn't hold.
> there really is no absolute frame of reference we can measure from.
And that has nothing to do with the fact that the Sun is in the focus of the elliptic path of each planet and not the vice versa. The planets do make that path around the Sun, what can be simplified to "they rotate around the Sun." If you ever tried to draw it it would be obvious to you too.
So you were just confused after taking that "Philosophy of Physics" class.
> The Sun is in the focus of the ellipses made by planets around it. The opposite doesn't hold.
Certainly the opposite holds - if you draw the Sun's orbit from a planet's frame of reference, you'll draw an elliptical path with the planet at one focus. That's the whole point here, that neither drawing describes observable reality any better than the other.
I don't think I'm making any headway here so I'll defer to Einstein:
> Can we formulate physical laws so that they are valid for all CS [coordinate systems] ... ? If this can be done ... the struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either CS could be used with equal justification. The two sentences, 'the sun is at rest and the Earth moves', or 'the sun moves and the Earth is at rest', would simply mean two different conventions concerning two different CS. Could we build a real relativistic physics valid in all CS; a physics in which there would be no place for absolute, but only for relative, motion? This is indeed possible!
