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    Wittgenstein's rule-following considerations suggest that... — Carmelics
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    Challenges→The empty sequent is not provable

    Wittgenstein's rule-following considerations suggest that what counts as a 'valid axiom' depends on communal practice, not intrinsic formal properties.

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    1 reason for
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    Reasons For

    1 perspective
    Reason for
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    • 1.Rule-following requires interpretation, and interpretations are only constrained by what a community recognizes as correct applications.
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    • 2.Axioms in non-Euclidean geometry were rejected then accepted, showing validity shifts with communal mathematical practice, not formal properties.
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    • 3.No formal system can bootstrap itself into justification without external appeal to shared understanding and agreement.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Some axioms (e.g., non-contradiction) are universally adopted across radically different communities, suggesting intrinsic formal properties matter.
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    • 2.A community can be *wrong* about axioms; validity cannot reduce to consensus if consensus is answerable to mathematical reality.
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    • 3.Wittgenstein's remarks on rules show how practice constrains interpretation, but don't entail that validity is *constituted by* rather than *discovered through* practice.
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    Proof of definition segments1 linkedTruth & Knowledge1 linked

    Related

    A community can be *wrong* about axioms; validity cannot reduce to consensus if ...Axioms in non-Euclidean geometry were rejected then accepted, showing validity s...No formal system can bootstrap itself into justification without external appeal...Rule-following requires interpretation, and interpretations are only constrained...
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    Some axioms (e.g., non-contradiction) are universally adopted across radically d...The empty sequent is not provableWittgenstein's remarks on rules show how practice constrains interpretation, but...

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    2 (1 for, 1 against)
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