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    Kreisel's squeezing argument demonstrates that informal p... — Carmelics
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    Challenges→When logic is understood proof-theoretically rather than model-theoretically, the validity problem coincides with derivability

    Kreisel's squeezing argument demonstrates that informal provability and formal derivability can come apart even when both fall under a proof-theoretic framework.

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    Reasons For

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    Reason for
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    • 1.Informal provability captures mathematical intuition about what 'ought' to follow from axioms, independent of formal system constraints.
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    • 2.Gödel's incompleteness theorems show formal systems have limits; informal reasoning can recognize truths formal systems cannot derive.
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    • 3.Mathematicians historically accepted statements as 'proven' before formal axiomatization existed, suggesting informal proof is conceptually prior.
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    Reasons Against

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    • 1.Without explicit criteria for informal provability, the claimed gap becomes unfalsifiable—informal intuition is too vague to meaningfully separate.
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    • 2.Any informal proof Kreisel identifies can be formalized in a stronger system, suggesting the gap is merely notational rather than substantive.
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    • 3.If informal provability requires appeal to metatheoretic reasoning, it's still formal reasoning by another name, not genuinely distinct.
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    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Any informal proof Kreisel identifies can be formalized in a stronger system, su...Gödel's incompleteness theorems show formal systems have limits; informal reason...If informal provability requires appeal to metatheoretic reasoning, it's still f...Informal provability captures mathematical intuition about what 'ought' to follo...
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    Mathematicians historically accepted statements as 'proven' before formal axioma...When logic is understood proof-theoretically rather than model-theoretically, th...Without explicit criteria for informal provability, the claimed gap becomes unfa...

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