[lutusp]

> ... there exist certain models which are mathematically equivalent but which describe contradictory states of being. A reasonably good example of this is heliocentrism vs. geocentrism: classical mechanics allows you to say "the Earth is at the center of the Solar System, there are gravitational, Coriolis and centrifugal forces around it affecting all of the stuff in space", but it also allows you to say "The Sun-Jupiter barycentre is at the center of the Solar System, and the only force we need is gravity." There is no experiment which can distinguish between those two; they are mathematically equivalent.

How do heliocentrism and geocentrism represent "contradictory states of being"? They are trivially related to one another, and are mathematically equivalent as you point out. They represent a simple example of geometric relativity.

Consider a gravitational slingshot maneuver, a way to harvest some of a planet's orbital momentum to accelerate a passing spacecraft. If you make the planet the frame of reference, or the sun, or the spacecraft, the math and physics come out the same. No "contradictory states of being".

> A slightly better example comes from quantum mechanics. In the "Schrodinger picture" there is a "wavefunction of the universe" which changes from moment to moment, while the definitions of space and momentum stay the same. In the "Heisenberg picture" the wavefunction stays the same while the definitions of space and momentum change.

No,. this isn't really a "better example" -- Dirac demonstrated the mathematical equivalence of Schrodinger's wave mechanics and Heisenberg's matrix mechanics. Again, the difference is only apparent and superficial.

[drostie]

What's different in both cases is ontology, which is a fancy way of saying how things actually are. It may help to think of how you would program a computer to simulate either universe, or for that matter it may help to reduce the computation to something much simpler, like the following two functions:

    def odd_sum(n):
        return sum(range(1, 2*int(n), 2))

    def square(n):
        m = int(n)
        return 0 if m <= 0 else m ** 2

Those two functions are mathematically indistinguishable if all you are doing is putting in various values. There happens to be a vast mathematical identity that the sum of the first n odd numbers is the n'th square number, allowing for that indistinguishability. But to claim that they're exactly the same realization of this function is obviously mistaken.

Similarly, there is a qualitative difference between a wavefunction of the universe which changes and a wavefunction of the universe which does not change. It does not matter that the two positions can be made mathematically equivalent; one is X and one is not-X, and science is permanently incapable of figuring out which it really is.