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Lectures on the philosophy of mathematics
- Author / Creator
- Available as
- Online
Details
- Creator
- Joel David Hamkins
- Format
- Books
- Language
- English
- Contributors
- Publication
-
- [Place of publication not identified] : The MIT Press, 2021
- Physical Details
-
- 1 online resource
- data file
- ISBNs
- 9780262362658, 0262362651, 0262542234, 9780262542234, 0262542234, 9780262542234
- OCLC
- on1162510217
- Intro -- Title Page -- Copyright -- Dedication -- Table of Contents -- Preface -- About the Author -- 1. Numbers -- 1.1. Numbers versus numerals -- 1.2. Number systems -- Natural numbers -- Integers -- Rational numbers -- 1.3. Incommensurable numbers -- An alternative geometric argument -- 1.4. Platonism -- Plenitudinous platonism -- 1.5. Logicism -- Equinumerosity -- The Cantor-Hume principle -- The Julius Caesar problem -- Numbers as equinumerosity classes -- Neologicism -- 1.6. Interpreting arithmetic -- Numbers as equinumerosity classes -- Numbers as sets -- Numbers as primitives
- Numbers as morphisms -- Numbers as games -- Junk theorems -- Interpretation of theories -- 1.7. What numbers could not be -- The epistemological problem -- 1.8. Dedekind arithmetic -- Arithmetic categoricity -- 1.9. Mathematical induction -- Fundamental theorem of arithmetic -- Infinitude of primes -- 1.10. Structuralism -- Definability versus Leibnizian structure -- Role of identity in the formal language -- Isomorphism orbit -- Categoricity -- Structuralism in mathematical practice -- Eliminative structuralism -- Abstract structuralism -- 1.11. What is a real number? -- Dedekind cuts
- Theft and honest toil -- Cauchy real numbers -- Real numbers as geometric continuum -- Categoricity for the real numbers -- Categoricity for the real continuum -- 1.12. Transcendental numbers -- The transcendence game -- 1.13. Complex numbers -- Platonism for complex numbers -- Categoricity for the complex field -- A complex challenge for structuralism? -- Structure as reduct of rigid structure -- 1.14. Contemporary type theory -- 1.15. More numbers -- 1.16. What is a philosophy for? -- 1.17. Finally, what is a number? -- Questions for further thought -- Further reading -- Credits -- 2. Rigor
- 2.1. Continuity -- Informal account of continuity -- The definition of continuity -- The continuity game -- Estimation in analysis -- Limits -- 2.2. Instantaneous change -- Infinitesimals -- Modern definition of the derivative -- 2.3. An enlarged vocabulary of concepts -- 2.4. The least-upper-bound principle -- Consequences of completeness -- Continuous induction -- 2.5. Indispensability of mathematics -- Science without numbers -- Fictionalism -- The theory/metatheory distinction -- 2.6. Abstraction in the function concept -- The Devil's staircase -- Space-filling curves
- Conway base-13 function -- 2.7. Infinitesimals revisited -- Nonstandard analysis and the hyperreal numbers -- Calculus in nonstandard analysis -- Classical model-construction perspective -- Axiomatic approach -- "The" hyperreal numbers? -- Radical nonstandardness perspective -- Translating between nonstandard and classical perspectives -- Criticism of nonstandard analysis -- Questions for further thought -- Further reading -- Credits -- 3. Infinity -- 3.1. Hilbert's Grand Hotel -- Hilbert's bus -- Hilbert's train -- 3.2. Countable sets -- 3.3. Equinumerosity -- 3.4. Hilbert's half-marathon