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    It is not the case that The mathematical framework physicists actually use presupposes structures (delta functions, plane waves) that only become rigorous in the larger rigged Hilbert space setting.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Physicists' informal methods work without invoking rigged Hilbert spaces; their success doesn't prove these structures are actually presupposed rather than post-hoc justifications.
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    • 2.Distribution theory (Schwartz distributions) makes delta functions rigorous within standard mathematics without requiring the full apparatus of rigged Hilbert spaces.
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    • 3.The claim conflates pedagogical convenience with metaphysical necessity; physicists could adopt distribution theory straightforwardly without philosophical conclusions.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Physicists routinely use delta functions and plane waves in calculations that yield experimentally verified predictions, suggesting these tools capture real structure.
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    • 2.Standard Hilbert space cannot accommodate eigenstates of continuous spectra; rigged Hilbert spaces mathematically justify what physicists do informally.
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    • 3.The gap between physics practice and standard mathematical rigor indicates the formalism used presupposes a deeper structure than conventional mathematics provides.
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