Measure theory and probability theory was developed to describe stochasticity and uncertainty, but they formalize it in many-worlds terms, closely analogous to how the wavefunction is formalized in quantum mechanics. If one takes the wavefunction formalism literally to the point of believing that quantum mechanics must have many worlds, it seems natural to take the probability distribution formalism equally literally to the point of believing that probability must have many worlds too. Or well, you can have a hidden variables theory of probability too, but the point is it seems like you would have to abandon True Stochasticity.
True Stochasticity vs probability distributions provides a non-quantum example of the non-native embedding, so if you accept the existence of True Stochasticity as distinct from many worlds of simultaneous possibility or ignorance of hidden variables, then that provides a way to understand my objection. Otherwise, I don’t yet know a way to explain it, and am not sure one exists.
As for the case of how a new branch of math could describe wavefunctions more natively, there’s a tradeoff where you can put in a ton of work and philosophy to make a field of math that describes an object completely natively, but it doesn’t actually help the day-to-day work of a mathematician, and it often restricts the tools you can work with (e.g. no excluded middle and no axiom of choice), so people usually don’t. Instead they develop their branch of math within classical math with some informal shortcuts.
Measure theory and probability theory was developed to describe stochasticity and uncertainty, but they formalize it in many-worlds terms, closely analogous to how the wavefunction is formalized in quantum mechanics. If one takes the wavefunction formalism literally to the point of believing that quantum mechanics must have many worlds, it seems natural to take the probability distribution formalism equally literally to the point of believing that probability must have many worlds too. Or well, you can have a hidden variables theory of probability too, but the point is it seems like you would have to abandon True Stochasticity.
True Stochasticity vs probability distributions provides a non-quantum example of the non-native embedding, so if you accept the existence of True Stochasticity as distinct from many worlds of simultaneous possibility or ignorance of hidden variables, then that provides a way to understand my objection. Otherwise, I don’t yet know a way to explain it, and am not sure one exists.
As for the case of how a new branch of math could describe wavefunctions more natively, there’s a tradeoff where you can put in a ton of work and philosophy to make a field of math that describes an object completely natively, but it doesn’t actually help the day-to-day work of a mathematician, and it often restricts the tools you can work with (e.g. no excluded middle and no axiom of choice), so people usually don’t. Instead they develop their branch of math within classical math with some informal shortcuts.