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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21 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
8763
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The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
17928
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Ordinal numbers represent order relations [Colyvan]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
18249
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Cauchy gave a formal definition of a converging sequence. [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
17923
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Intuitionists only accept a few safe infinities [Colyvan]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
17941
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Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan]
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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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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
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
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
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
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
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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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
8744
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Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
8749
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Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
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8750
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Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
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8752
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Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8753
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Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
8731
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Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
8730
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'Impredicative' definitions refer to the thing being described [Shapiro]
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