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'Thinking About Mathematics', 'Symbolic Logic (with Langford)' and 'Grundgesetze der Arithmetik 1 (Basic Laws)'
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7 ideas
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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We can't prove everything, but we can spell out the unproved, so that foundations are clear [Frege]
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Categories are the best foundation for mathematics [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
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Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions [Frege, by Hale/Wright]
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Frege ignored Cantor's warning that a cardinal set is not just a concept-extension [Tait on Frege]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
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Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
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A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
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