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    The semantics of a formal system rich enough to contain e... — Carmelics
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    The semantics of a formal system rich enough to contain elementary mathematics cannot be fully defined in terms of mathematical functions within that same system.

    Philosophy of LanguageTruth & 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.Any consistent formal system containing elementary arithmetic contains true statements that cannot be proved within the system.
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    • 2.A statement that is well-formed, meaningful, and truthful carries semantic information about the system.
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    • 3.If semantic information about a system exists that cannot be captured by provability within the system, then the system's semantics exceed what its internal mathematical functions can define.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Tarski's undefinability theorem shows truth is undefinable within a system, but this is consistent with truth being definable by a richer metalanguage that is itself mathematical.
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    • 2.The claim conflates the limits of object-language self-reference with the limits of mathematical semantics as such, since no system is required to define its own truth predicate.
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    • 3.Hartry Field's deflationary program demonstrates that semantic notions like truth can be reduced to purely formal, non-semantic primitives without invoking anything beyond mathematical structure.
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    Reason against 2 of 2
    ?
    • 1.Gödel's incompleteness theorems concern provability within a fixed axiomatic system, not the expressive capacity of mathematical functions in general.
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    • 2.A formal system's semantics can be fully specified by a mathematical model in the set-theoretic sense, as Tarski's model theory demonstrates, without any non-mathematical remainder.
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    • 3.The supporting argument equivocates between 'definable within the system' and 'definable mathematically,' and rejecting the former does not entail rejecting the latter.
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    Topics

    Philosophy of LanguageTruth & Knowledge

    Notable Defenders

    Alan Turingcontemporary
    Andrey Kolmogorovcontemporary
    Andrey KolmogorovcontemporaryKolmogorov (1965)
    Bertrand RussellcontemporaryRussell 1905
    Bertrand Russellcontemporary
    Claude ShannoncontemporaryShannon 1948; Shannon & Weaver 1949
    Claude Shannoncontemporary
    Erik VerlindecontemporaryVerlinde 2011, 2017
    Fred DretskecontemporaryDretske 1981
    Gregory ChaitincontemporaryChaitin 1987
    Gregory ChaitincontemporaryChaitin (1969)
    HamkinscontemporaryHamkins and Lewis 2000
    J. Gerard WolffcontemporaryWolff 2006
    John MilnorcontemporaryBott & Milnor (1958)
    Karl PoppercontemporaryLogik der Forschung (1934 [1977: 42])
    Karl Poppercontemporary
    Kurt Gödelcontemporary
    L.E.J. Brouwercontemporary
    Leonid LevincontemporaryLevin (1974)
    LewiscontemporaryHamkins and Lewis 2000
    Luc BovenscontemporaryBovens & Hartmann 2003
    Luciano Floridicontemporary
    Ludwig Wittgensteincontemporary
    Marcus HuttercontemporaryHutter 2005, 2007a, 2007b
    Martin DaviscontemporaryDavis 2006
    Martin HeideggercontemporaryDie Wissenschaft denkt nicht
    Michel KervairecontemporaryKervaire (1958)
    Murray Gell-ManncontemporaryGell-Mann & Lloyd 2003
    Nachum DershowitzcontemporaryDershowitz and Gurevich 2008
    Nick ChatercontemporaryChater & Vitányi 2003
    Paul VitányicontemporaryChater & Vitányi 2003
    R. L. GoodsteincontemporaryGoodstein 1957
    Raoul BottcontemporaryBott & Milnor (1958)
    Ray SolomonoffcontemporarySolomonoff 1997
    Rudolf CarnapcontemporaryCarnap 1928
    Rudolf Carnapcontemporary
    Samuel RathmannercontemporaryRathmanner & Hutter 2011
    Stephan HartmanncontemporaryBovens & Hartmann 2003
    W. V. O. QuinecontemporaryQuine (1951)
    W.V.O. Quinecontemporary
    Yuri GurevichcontemporaryDershowitz and Gurevich 2008
    Anselm of Canterburymedieval
    Ada Lovelacemodern1815–1852, first programming language for the Analytical Engine
    Alfred Vailmoderndetermined letter frequencies from Morristown, NJ newspaper, 1844
    Bertrand Russellmodern
    Charles BabbagemodernDifference Engine 1821, Analytical Engine 1834–1871
    Christian GoldbachmodernGoldbach's conjecture, 1742
    Daniel BernoullimodernHydrodynamica (1738)
    David HilbertmodernFrege-Hilbert controversy
    Edmund Husserlmodern
    Galileo Galileimodern1623
    GibbsmodernGibbs (1906)
    Gottlob FregemodernFrege 1879, 1892
    Gottlob Fregemodern
    Immanuel Kantmodern
    Immanuel Kantmodern
    Immanuel KantmodernKritik der reinen Vernunft (1781)
    James Clerk Maxwellmodern1857
    John Lockemodern
    Karl PoppermodernPopper 1934
    Ludwig Boltzmannmodern
    Ludwig Boltzmannmodern
    Max Planckmodern
    Michael StifelmodernArithmetica integra (1544)
    Ralph Hartleymodern
    René DescartesmodernImplied by the explicit reference to 'the Cartesian notion' of innate ideas
    René DescartesmodernMeditationes de Prima Philosophia; second answer to Mersenne
    René Descartesmodern
    Rudolf CarnapmodernCarnap 1945, 1950
    Rudolf Clausiusmodern1850

    Connections

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    Skepticism1 linked

    Related

    A formal system's semantics can be fully specified by a mathematical model in th...A statement that is well-formed, meaningful, and truthful carries semantic infor...Any consistent formal system containing elementary arithmetic contains true stat...Gödel's incompleteness theorems concern provability within a fixed axiomatic sys...

    Source

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    In a landmark paper in 1931 Kurt Gödel proved that any consistent formal system that contains elementary arithmetic is fundamentally incomplete in the sense that it contains true statements that cannot be proved within the system. In a philosophical context this implies that the semantics of a formal system rich enough to contain elementary mathematics cannot be defined in terms of mathematical functions within the system, i.e., there are statements that contain semantic information about the sy
    Extraction notes

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    Details

    +5 moreShow less
    Hartry Field's deflationary program demonstrates that semantic notions like trut...If semantic information about a system exists that cannot be captured by provabi...Tarski's undefinability theorem shows truth is undefinable within a system, but ...The claim conflates the limits of object-language self-reference with the limits...The supporting argument equivocates between 'definable within the system' and 'd...

    Similar

    If semantic information about a system exists that cannot be captured ...81%Any deterministic formal system can be represented in terms of element...81%Any consistent formal system containing elementary arithmetic is funda...79%Any consistent formal system that contains arithmetic as a subsystem i...79%
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit