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    If mathematical utterances express propositions about for... — Carmelics
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    Challenges→Curry's formalist position threatens to collapse into structuralism

    If mathematical utterances express propositions about formal systems without privileging any particular interpretation, they function as schemata implicitly generalizing over abstract structures satisfying those schemata

    Modality & PossibilityPhilosophy of Language
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    Philosophy of LanguageModality & Possibility

    Key Terms

    Formal systems(the systems Hilbert tried to use)
    A set of basic symbols and strict rules for manipulating them (like a logical language), designed to avoid ambiguity and prove statements with certainty.
    abstract structures(as used in philosophy of mathematics)
    Patterns or systems of relationships that exist only as ideas, not as physical objects—like the structure of numbers or geometric shapes.
    generalizing over

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    Related propositions within the same area of thought.
    (as used in logic and mathematics)
    Making a broad statement that applies to many different examples or cases rather than just one specific thing.
    interpretation(Formal semantics for modal nonmonotonic logic)
    A complete, consistent set of literals
    mathematical utterances(as used in philosophy of mathematics)
    Statements or expressions made using mathematical language, like equations or theorems.
    propositions(Answer to the question of what metaphysical category propositions belong to)
    Entities belonging to a sui generis metaphysical category of their own kind, not reducible to other categories
    schemata (or schema)(as used in logic and epistemology)
    A general template or pattern that can be applied to many different specific cases.

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    Truth & Knowledge3 linked

    Related

    Curry's formalist position threatens to collapse into structuralismGeneralizing over abstract structures that satisfy schemata is the core commitme...On Curry's view, mathematics becomes metamathematics — a contentful theory about...

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    The upshot is that mathematics in general becomes metamathematics, a contentful theory—Curry’s sentences express propositions with truth values—setting out the truths about what is provable from what in underlying formal systems whose interpretation, or rather interpretations, are not taken to be mathematically important. This standpoint, however, threatens to collapse into structuralism, into viewing mathematical utterances as schemata implicitly generalising over a range of (in general) abstra

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