Combining Philosophers
Ideas for B Hale / C Wright, C.I. Lewis and Geoffrey Hellman
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10 ideas
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
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Neo-logicism founds arithmetic on Hume's Principle along with second-order logic [Hale/Wright]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
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The Julius Caesar problem asks for a criterion for the concept of a 'number' [Hale/Wright]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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Structuralism is now common, studying relations, with no regard for what the objects might be [Hellman]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
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Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend]
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Statements of pure mathematics are elliptical for a sort of modal conditional [Hellman, by Chihara]
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Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman]
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Maybe mathematical objects only have structural roles, and no intrinsic nature [Hellman]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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If structures are relative, this undermines truth-value and objectivity [Hale/Wright]
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The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright]
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