Combining Texts
Ideas for
'Thinking About Mathematics', 'Sets and Numbers' and 'The Metaphysics of Space-Time Substantivalism'
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9 ideas
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8764
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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 / f. Zermelo numbers
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 / 6. Mathematics as Set Theory / a. Mathematics is set theory
17825
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Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy]
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17826
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Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy]
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17828
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Numbers are properties of sets, just as lengths are properties of physical objects [Maddy]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
17827
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Sets exist where their elements are, but numbers are more like universals [Maddy]
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17830
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Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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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