Combining Texts
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'Thinking About Mathematics', 'An Analysis of Knowledge and Valuation' and 'Introduction to the Philosophy of Mathematics'
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9 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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17922
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Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
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17936
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Transfinite induction moves from all cases, up to the limit ordinal [Colyvan]
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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 / 6. Mathematics as Set Theory / a. Mathematics is set theory
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17940
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Most mathematical proofs are using set theory, but without saying so [Colyvan]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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17931
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Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan]
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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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17932
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If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan]
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