What I mean is, if two people disagree about whether the Earth orbits the sun, there aren't any experiments where we'd expect them to predict different results. They're both describing the same physical reality, viewed from different frames of reference.
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.
IIRC objects experiencing acceleration (e.g. planets orbiting each other) are not subject to Galilean relativity like objects that are moving at a constant velocity (e.g. ball being thrown on a train). So when planets orbit each other, there is only a singe valid frame of reference.
The disagreement can exist because "orbits" is left under-defined. If you say A orbits B if A makes circles around B and B creates a bigger gravity field, then everyone will agree; they'll also agree on the fact if both objects have roughly equal masses, then it is true to say that both A orbits B and B orbits A.
But that is not what "there is no physical reality to the matter" means, if one chooses not to use any relative point there is still an objective distance between each object with respect to every other object and accelerations for each of those distances.
"The matter" in that quote is the matter of whether the Earth orbits the sun or vice versa. Naturally there's observable physical reality to distances and whatnot, but if you use the laws of physics to make predictions about those observables, the math works regardless of which body you assume the others revolve around.
While I think the connection with what GP wanted to convey is tenuous, Betrands paradox _is_ interesting, and even more so E.T. Jaynes' solution.
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).
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.