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Ideas for 'Thinking About Mathematics', 'fragments/reports' 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	The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers

17928

Ordinal numbers represent order relations [Colyvan]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy

18249

Cauchy gave a formal definition of a converging sequence. [Shapiro]

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite

17923

Intuitionists only accept a few safe infinities [Colyvan]

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility

17941

Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan]

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics

8764	Categories are the best foundation for mathematics [Shapiro]

17922

Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan]

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction

17936

Transfinite induction moves from all cases, up to the limit ordinal [Colyvan]

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers

8762	Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

17940

Most mathematical proofs are using set theory, but without saying so [Colyvan]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

17931

Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan]

8760	Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]

8761	A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique

17932

If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan]

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

8744	Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]

6. Mathematics / C. Sources of Mathematics / 7. Formalism

8749	Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]

8750	Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]

8752	Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism

8753	Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism

8731	Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism

8730	'Impredicative' definitions refer to the thing being described [Shapiro]