Combining Philosophers
Ideas for B Hale / C Wright, Richard Breheny and Ernest Nagel
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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 / 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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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
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Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology [Hale/Wright]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
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The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright]
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Logicism might also be revived with a quantificational approach, or an abstraction-free approach [Hale/Wright]
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Neo-Fregeanism might be better with truth-makers, rather than quantifier commitment [Hale/Wright]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright]
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