> 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!
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.