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    Set theory is unnecessary for mathematics. — Carmelics
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    Set theory is unnecessary for mathematics.

    SkepticismTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    2 reasons for
    1 reason against

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Mathematical practice across analysis, geometry, and number theory proceeded rigorously for centuries without set-theoretic foundations.
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    • 2.Mathematicians like Euler, Gauss, and Cauchy established lasting results using proof techniques that carry no essential set-theoretic commitments.
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    • 3.If a foundational framework is genuinely necessary, its absence would have generated systematic failures in pre-Cantorian mathematics, yet none occurred.
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    Reason for 2 of 2
    ?
    • 1.Predicativist programs (Weyl, Feferman) reconstruct virtually all scientifically applicable mathematics without impredicative set theory.
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    • 2.Feferman demonstrated that the mathematics needed for physical science is fully formalizable in predicative systems far weaker than ZFC.
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    • 3.If set theory were necessary, no adequate substitute could exist, but the existence of predicativist alternatives refutes that necessity claim.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Set theory was invented to provide mathematics with a foundation.
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    • 2.Mathematics is a motley of techniques of proof and does not require a foundation.
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    • 3.Mathematics cannot be given a self-evident foundation.
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    Related

    Feferman demonstrated that the mathematics needed for physical science is fully ...If a foundational framework is genuinely necessary, its absence would have gener...If set theory were necessary, no adequate substitute could exist, but the existe...Mathematical practice across analysis, geometry, and number theory proceeded rig...
    +5 moreShow less
    Mathematicians like Euler, Gauss, and Cauchy established lasting results using p...Mathematics cannot be given a self-evident foundation.Mathematics is a motley of techniques of proof and does not require a foundation...Predicativist programs (Weyl, Feferman) reconstruct virtually all scientifically...Set theory was invented to provide mathematics with a foundation.

    Similar

    Set theory was invented to provide mathematics with a foundation.79%Category theory has been proposed as an alternative foundation for mat...78%It is not clear how scientific theories could be expressed without mat...77%Set theory underlies all branches of mathematics.77%

    Source

    AI-extracted1/3 agreementValid
    SEP: wittgenstein-mathematics
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    Largely a product of his anti-foundationalism and his criticism of the extension-intension conflation, Wittgenstein’s later critique of set theory is highly consonant with his intermediate critique (PR §§109, 168; PG 334, 369, 469; LFM 172, 224, 229; and RFM III, §43, 46, 85, 90; VII, §16). Given that mathematics is a “MOTLEY of techniques of proof” (RFM III, §46), it does not require a foundation (RFM VII, §16) and it cannot be given a self-evident foundation (PR §160; WVC 34 & 62; RFM IV,
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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