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'Thinking About Mathematics', 'fragments/reports' and 'Philosophy of Mathematics'
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12 ideas
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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8764
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Categories are the best foundation for mathematics [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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18156
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Modern axioms of geometry do not need the real numbers [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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18097
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The Peano Axioms describe a unique structure [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
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18148
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Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
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18145
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Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
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18149
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There are many criteria for the identity of numbers [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
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18143
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Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
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8762
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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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8760
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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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8761
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A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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18116
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Numbers can't be positions, if nothing decides what position a given number has [Bostock]
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18117
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Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
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