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    Turing's thesis is not susceptible to mathematical proof — Carmelics
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    Turing's thesis is not susceptible to mathematical proof

    SkepticismTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Turing did not consider argument I to be a mathematical demonstration of his thesis
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    • 2.Turing did not consider argument II to be a mathematical demonstration of his thesis
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    • 3.Turing asserted that all arguments which can be given for the thesis share this non-demonstrative character
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Church's theorem and Gödel's incompleteness results demonstrate that formal unprovability claims are themselves susceptible to rigorous mathematical proof.
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    • 2.If the boundaries of mechanical computability can be precisely formalized, then whether a thesis accurately characterizes those boundaries becomes a mathematical question.
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    • 3.Wilfried Sieg's work on 'mechanical procedures' shows that Turing's own structural analysis of computation admits formalization sufficient for proof-theoretic treatment.
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    Reason against 2 of 2
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    • 1.Turing's own unprovability assertion is itself a metamathematical claim, and metamathematical claims can be given mathematical proofs, as Gödel demonstrated in 1931.
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    • 2.The inference from 'Turing did not attempt a mathematical proof' to 'no mathematical proof is possible' commits an argument from ignorance that undermines the claim's scope.
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    Topics

    SkepticismTruth & Knowledge

    Notable Defenders

    Abramsoncontemporary
    Alan TuringcontemporaryTuring in Copeland 2004b: 590
    Allen Newellcontemporary
    Allen Newellcontemporary
    Andrey MarkovcontemporaryMarkov 1960
    Daniel DennettcontemporaryDennett 1991
    Emil PostcontemporaryPost 1943, 1946
    Haskell CurrycontemporaryCurry 1929, 1930, 1932
    Howard SturgiscontemporaryShepherdson and Sturgis 1963
    John ShepherdsoncontemporaryShepherdson and Sturgis 1963
    Kurt Gödelcontemporary
    Kurt GödelcontemporaryGödel 1936
    Martin DaviscontemporaryDavis 1958: 70
    Moses SchönfinkelcontemporarySchönfinkel 1924
    Nachum DershowitzcontemporaryDershowitz and Gurevich 2008
    Oron ShagrircontemporaryShagrir 2006
    Patricia ChurchlandcontemporaryChurchland and Churchland 1990
    Patricia ChurchlandcontemporaryChurchland and Churchland 1983: 6
    Paul ChurchlandcontemporaryChurchland and Churchland 1990
    Paul ChurchlandcontemporaryChurchland and Churchland 1983: 6
    Richard GregorycontemporaryGregory 1987
    Richard GregorycontemporaryGregory 1987
    Robin GandycontemporaryGandy 1980
    Saul KripkecontemporaryKripke 2013: 80
    Stephen Cole KleenecontemporaryKleene 1952
    Wilfried SiegcontemporarySieg 2002, 2008
    Yuri GurevichcontemporaryDershowitz and Gurevich 2008
    Alan TuringmodernTuring 1936, Section 9 of "On Computable Numbers"
    Alan TuringmodernTuring 1936
    Alan TuringmodernTuring's thesis referenced in the passage
    Alan Turingmodern
    Alan TuringmodernTuring's proof of uncomputable real numbers
    Alan Turingmodern
    Alan Turingmodern
    Alan Turingmodern
    Alan TuringmodernTuring's thesis; proved computable functions coincide with lambda-definable/recursive functions
    Alonzo Churchmodern
    Alonzo Churchmodern
    Alonzo ChurchmodernChurch's thesis; proved lambda-definable functions coincide with recursive functions
    Georg CantormodernCantor 1874
    Stephen KleenemodernContributed to results establishing equivalence of lambda-definable and recursive functions

    Related

    Church's theorem and Gödel's incompleteness results demonstrate that formal unpr...If the boundaries of mechanical computability can be precisely formalized, then ...The inference from 'Turing did not attempt a mathematical proof' to 'no mathemat...Turing asserted that all arguments which can be given for the thesis share this ...
    +4 moreShow less
    Turing did not consider argument I to be a mathematical demonstration of his the...Turing did not consider argument II to be a mathematical demonstration of his th...

    Source

    AI-extracted1/3 agreementValid
    SEP: church-turing
    View source passageHide passage
    Turing’s own view of the status of his thesis is very different from that expressed by Kripke, Sieg, and Dershowitz and Gurevich. According to Turing, his thesis is not susceptible to mathematical proof. He did not consider either argument I or argument II to be a mathematical demonstration of his thesis: he asserted that I and II, and indeed “[a]ll arguments which can be given” for the thesis, are
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Turing's own unprovability assertion is itself a metamathematical claim, and met...
    Wilfried Sieg's work on 'mechanical procedures' shows that Turing's own structur...
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit