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    A formalist or deflationist (following Wittgenstein's Rem... — Carmelics
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    Challenges→The Gödel sentence G_F is true (when F is consistent and the provability predicate is a Σ⁰₁-formula)

    A formalist or deflationist (following Wittgenstein's Remarks on the Foundations of Mathematics) can reject that 'true but unprovable' is coherent without a privileged intended model.

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Truth without a model is metaphysically idle; calling something 'true' merely marks it as provable within a rule-governed system.
      ?

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    • 2.Gödel's incompleteness theorems show formal systems are incomplete, but this reflects the limits of *our chosen axioms*, not existence of mind-independent mathematical facts.
      ?

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    • 3.Positing unprovable truths requires a privileged model (Platonic realm), which violates parsimony and lacks empirical access.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Mathematical statements exhibit determinate truth-values independently of proof systems; unprovability in PA doesn't make a statement indeterminate.
      ?

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    • 2.Rejecting 'true but unprovable' conflates *recognizing* a statement's truth with *epistemically accessing* it—they are distinct notions.
      ?

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    • 3.Formalism cannot explain why different formal systems produce consistent, interlocking results unless mathematical truths constrain them externally.
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    Key Terms

    Deflationist(as used in metaphysics)
    A philosopher who believes that certain big philosophical questions (especially about what exists) don't have real answers based on facts, but are more about how we choose to use words.
    Formalist(as used in philosophy of mathematics)
    A philosophical view that treats mathematics as a game played with symbols according to fixed rules, rather than as describing something real.
    Privileged intended model(in philosophy of mathematics)
    The idea that there's one special, correct way to interpret the symbols and rules of mathematics that matters more than any other interpretation.
    Remarks on the Foundations of Mathematics(the specific work being referenced)
    A book by Wittgenstein where he questions basic assumptions about what mathematics is and how it works, rather than accepting math as simply describing truths about numbers.
    Unprovable (true but unprovable)(in mathematical logic)
    A statement that is actually correct, but there's no logical chain of reasoning within a particular system that can demonstrate it's correct.
    Wittgenstein, Ludwig(as referenced in philosophy of language and philosophy of mind)
    An influential 20th-century philosopher who argued that language and meaning are shaped by how we use words with others, not by private inner experiences.
    coherent(de Finetti's usage in the context of the Dutch Book argument for probabilism)
    A subject is coherent if their unconditional degrees of belief do not permit a Dutch Book (a guaranteed loss through a combination of bets) to be made against them
    intended model(Model theory / philosophy of mathematics)
    The canonical structure that a theory is meant to describe; for ZFC, this is the proper class V with the ∈ relation

    Connections

    2 topics

    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    Formalism cannot explain why different formal systems produce consistent, interl...Gödel's incompleteness theorems show formal systems are incomplete, but this ref...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    Mathematical statements exhibit determinate truth-values independently of proof ...
    Positing unprovable truths requires a privileged model (Platonic realm), which v...
    +3 moreShow less
    Rejecting 'true but unprovable' conflates *recognizing* a statement's truth with...The Gödel sentence G_F is true (when F is consistent and the provability predica...Truth without a model is metaphysically idle; calling something 'true' merely ma...