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    Paraphrase strategies (Field, Hellman) can restate all tr... — Carmelics
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    Challenges→Extensional sentences containing numerical singular terms, if true, are ontologically committed to numbers.

    Paraphrase strategies (Field, Hellman) can restate all true arithmetic claims using second-order logic over physical regions, eliminating numerical singular terms entirely.

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    Reasons For

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    Reason for
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    • 1.Second-order quantification over physical regions can express cardinality and structure without positing abstract numbers as ontological commitments.
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    • 2.Paraphrase strategies preserve mathematical truth while achieving nominalist goals, avoiding Platonism without sacrificing arithmetic validity.
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    • 3.Field's approach demonstrates that numerical language functions pragmatically within physical domains without requiring numbers as fundamental entities.
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    Reasons Against

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    Reason against
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    • 1.Second-order logic itself invokes quantification over sets/properties, which is ontologically costly and arguably more problematic than numbers themselves.
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    • 2.Paraphrases become increasingly complex and artificial for higher mathematics, raising questions about whether they genuinely restate or merely approximate claims.
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    • 3.The mapping between physical regions and arithmetic structures requires independent mathematical justification, undermining nominalist reduction claims.
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    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Extensional sentences containing numerical singular terms, if true, are ontologi...Field's approach demonstrates that numerical language functions pragmatically wi...Paraphrase strategies preserve mathematical truth while achieving nominalist goa...Paraphrases become increasingly complex and artificial for higher mathematics, r...
    +3 moreShow less
    Second-order logic itself invokes quantification over sets/properties, which is ...Second-order quantification over physical regions can express cardinality and st...The mapping between physical regions and arithmetic structures requires independ...

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