Most of classical physics needs classical (not constructionist) mathematics, because it deals with emergent layers of description (like classical mechanics or GR) that are often not computable. But foundational physics must break with this and be constructionist, I think.
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Can't superposition of attributes of a variable be measuring its complementary variable in a sharp attribute with a relabeled measurement apparatus labeling the corresponding mixed states of substrates with the other variable?
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The weird thing is that QM treats superpositional states as an information surplus that the universe hides away in a complex valued Hilbert space, when they might actually be an information deficit, e.g. due to a particle being insufficiently constrained to project it into 3space
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