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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that Carnap's formalist position has not adequately answered the problem of applicability of mathematics

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    Reasons For

    1 perspective
    Reason for
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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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    Reasons Against

    2 perspectives
    Reason against 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 against 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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