A naive reading of what you said suggests eg. no spaceflight is possible because no astronomical (or any...) calculation is reliable. Can you clarify?
It's analogous to talking about someone throwing a ball on a train. A person on the train can say the ball moves relative to the train, and someone outside can say the ball and train are both moving relative to the Earth, and a guy in a spaceship can say that all three are actually moving relative to the sun. But choosing one of those claims over the others is purely a matter of convenience or perspective, not of physical reality.
As with most paradoxes, it's about how a seemingly innocuous question turns out to either be underspecified to have an answer, or tricks even the mind of mathematicians to assume too much.
Here it's about what it means to have a "random chord of a circle"; several simple ways to generate random chords lead to very different results, but true randomness should not be easy distinguishable from other true randomness...
E.T.J's solution to invoke the maximum ignorance principle to require any source of random chords to be size- and translation-invariant (because those dimensions are not specified in the problem) seems elegant.
Now, comparing to orbits, or to balls on a train, the thing here is: all these different viewpoints, although very different, do lead to the same result. Even if you calculate stuff; and if I'm not mistaken, that's because of relativity (Galilean relativity should be sufficient for the train example to work, the more modern ones for the rest).
“there's no particular physical reality to the matter”
Wrong.
https://en.wikipedia.org/wiki/Ellipse#Planetary_orbits
"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.
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.
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.
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!
(from The Evolution of Physics)
Inertial reference frame, right? Otherwise you need to add in fictitious forces.
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.
The center of mass in a two body system is a nice inertial reference frame which simplifies calculations.
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.
Still very much existent.