Combining Philosophers
Ideas for B Hale / C Wright, George Boolos and Richard Fitzralph
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14 ideas
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
10491
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Infinite natural numbers is as obvious as infinite sentences in English [Boolos]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
10483
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Mathematics and science do not require very high orders of infinity [Boolos]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10833
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Many concepts can only be expressed by second-order logic [Boolos]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10624
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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
8784
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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
8787
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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
10629
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If structures are relative, this undermines truth-value and objectivity [Hale/Wright]
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10628
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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 / 1. Mathematical Platonism / a. For mathematical platonism
10490
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Mathematics isn't surprising, given that we experience many objects as abstract [Boolos]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
8788
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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
10622
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The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright]
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8783
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Logicism might also be revived with a quantificational approach, or an abstraction-free approach [Hale/Wright]
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12225
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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
12224
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Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright]
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