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    Carnap's formalist position has not adequately answered t... — Carmelics
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    Carnap's formalist position has not adequately answered the problem of applicability of mathematics

    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.Carnap's analytic-synthetic distinction, which underpins his formalism, was fatally undermined by Quine's 'Two Dogmas of Empiricism', removing the boundary between formal and empirical content.
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    • 2.Without a stable analytic-synthetic distinction, Carnap cannot coherently designate mathematical calculi as purely formal while simultaneously explaining their indispensable role in confirmed empirical theories.
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    • 3.Quine's confirmational holism entails that mathematical statements share empirical content with the physical hypotheses they help confirm, directly contradicting Carnap's formalist insulation of mathematics.
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    Reason for 2 of 2
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    • 1.Frege's applicability constraint holds that any adequate philosophy of mathematics must explain why arithmetic applies to empirical objects without treating numbers as mere uninterpreted symbols.
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    • 2.Carnap's syntactic frameworks, as purely formal rule-governed calculi, cannot themselves specify which physical domain a given mathematical structure maps onto, leaving the correspondence relation unexplained.
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    • 3.The semantic bridge between a formal calculus and its physical application requires a substantive model-theoretic account that Carnap's deflationary, syntax-first approach is structurally prohibited from supplying.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.A conservative extension result must be a contentful truth, not merely a derivable string of symbols, to provide reassurance that a given calculus will be pragmatically useful in applications like bridge or computer design
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    • 2.Carnap's formalist position struggles to account for what constitutes the truth of such results if they are granted to be contentful
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    Related

    A conservative extension result must be a contentful truth, not merely a derivab...Carnap's analytic-synthetic distinction, which underpins his formalism, was fata...Carnap's formalist position struggles to account for what constitutes the truth ...Carnap's syntactic frameworks, as purely formal rule-governed calculi, cannot th...
    +4 moreShow less
    Frege's applicability constraint holds that any adequate philosophy of mathemati...Quine's confirmational holism entails that mathematical statements share empiric...The semantic bridge between a formal calculus and its physical application requi...Without a stable analytic-synthetic distinction, Carnap cannot coherently design...

    Similar

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    Few nowadays look for Cartesian certitude in mathematics, so Carnap’s position here may seem reasonable. It is not so clear, however, that he has answered the problem of applicability. Even if a conservative extension result can only be given in a more powerful system, we need the result to be a contentful truth, not just a string of symbols we can derive from some system, if we are to have reassurance that a particular calculus we are about to use in designing bridges or computers will be pragm
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    Details

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