My apologies for yet another technical post that's just notes-to-self.
Quantum Mechanics models the world using a Hilbert space. I wonder what we can say about just how much of the structure of the model is meant to be found in what is modeled. In contemporary mathematics, I guess ultimately any Hilbert space will be a very complex construction out of the empty set. Yet it seems absurd to think that the low-level details of the set-theoretic implementation (say, different ways of constructing the natural numbers out of the empty set) would reflect differences in the world. There are way too many ways to implement these details.
But there will also be differences at higher levels. For instance, there will be cases where the Hilbert space is L2(X), "the space of square-integrable functions" on some set X. I put that in scare quotes, because that's not what L2 is, despite often being described so. Rather, it's the space of equivalence classes of square-integrable functions, where two functions are equivalent provided that the set of points where they differ has measure zero. So now we have a question about the model and the modeled. You could think that different members of an equivalence class correspond to different empirically indistinguishable physical states, and the physics simply makes no prediction as to which of the indistinguishable states is exemplified when. Or you could think that each equivalence class corresponds to a single possible physical state. The latter makes for a theory that is simpler and yet seems to give less understanding. It is simpler because it doesn't posit unexplained differences between states. But it seems to give less understanding, because it means that the wavefunction can no longer be seen as an assignment of values to different points in phase space, but rather a more mysterious kind of entity—one modeled as an equivalence class of such assignments.
There may be a third option: There is a privileged member of each equivalence class, and only the privileged member can be physically actualized. This would give us the best of both worlds. We would have a field over phase-space, and no extra indistinguishable physical possibilities. The lack of linear liftings on L2[0,1] makes it a bit harder to realize this hope than one might have wished, but maybe there is still some hope.