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    Goldbach's Conjecture (GC) is not a mathematical proposition — Carmelics
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    Goldbach's Conjecture (GC) is not a mathematical proposition

    Philosophy of LanguageTruth & Knowledge
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    2 reasons for
    1 reason against

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.For Wittgenstein, mathematical meaning is constituted by proof techniques, not by the mere syntactic form of a proposition.
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    • 2.GC has no proof technique: no known method determines its truth, making it semantically idle rather than genuinely mathematical.
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    • 3.A string of symbols that plays no role in mathematical practice—yielding no calculations, proofs, or decisions—lacks the use that constitutes meaning.
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    Reason for 2 of 2
    ?
    • 1.Hilbert's formalist program implicitly concedes that undecidable statements require new axioms or methods before they can be treated as genuine mathematical claims.
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    • 2.Gödel's incompleteness results show that statements like GC may be undecidable within any consistent formal system, meaning their 'truth' is system-relative rather than intrinsic.
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    • 3.A claim whose truth-value is permanently inaccessible within any specifiable formal system cannot function as a proposition in the normative sense mathematics requires.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.A meaningful mathematical proposition requires a known method of algorithmic decision
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    • 2.We do not know how to decide GC algorithmically
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    Topics

    Philosophy of LanguageTruth & Knowledge

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    Skepticism1 linked

    Related

    A claim whose truth-value is permanently inaccessible within any specifiable for...A meaningful mathematical proposition requires a known method of algorithmic dec...A string of symbols that plays no role in mathematical practice—yielding no calc...For Wittgenstein, mathematical meaning is constituted by proof techniques, not b...
    +4 moreShow less
    GC has no proof technique: no known method determines its truth, making it seman...Gödel's incompleteness results show that statements like GC may be undecidable w...Hilbert's formalist program implicitly concedes that undecidable statements requ...We do not know how to decide GC algorithmically

    Similar

    An expression that is not decidable in some actual calculus is not a m...83%An expression lacking a place in any actual calculus cannot qualify as...82%Church's Thesis is an empirically supported thesis, not a proven mathe...81%Sense does not exist outside the proposition that expresses it.81%

    Source

    AI-extracted1/3 agreementValid
    SEP: wittgenstein-mathematics
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    Together, Wittgenstein’s finitism and his criterion of algorithmic decidability shed considerable light on his highly controversial remarks about putatively meaningful conjectures such as FLT and GC. GC is not a mathematical proposition because we do not know how to decide it, and if someone like G. H. Hardy says that he ‘believes’ GC is true (PG 381; LFM 123; PI §578), we must answer that s/he only “has a hunch about the possibilities of extension of the present system” (LFM 139)—that one can o
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    Details

    Type
    claim
    Perspectives
    3 (2 for, 1 against)
    Edits
    1 edit