Lectures on the Philosophy of Mathematics
ISBN: 9780262362986 | Copyright 2020
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| Preface (pg. xvii) | |
| About the Author (pg. xix) | |
| Numbers (pg. 1) | |
| Numbers versus numerals (pg. 1) | |
| Number systems (pg. 2) | |
| Natural numbers (pg. 2) | |
| Integers (pg. 3) | |
| Rational numbers (pg. 3) | |
| Incommensurable numbers (pg. 4) | |
| An alternative geometric argument (pg. 5) | |
| Platonism (pg. 6) | |
| Plenitudinous platonism (pg. 7) | |
| Logicism (pg. 7) | |
| Equinumerosity (pg. 7) | |
| The Cantor-Hume principle (pg. 8) | |
| The Julius Caesar problem (pg. 10) | |
| Numbers as equinumerosity classes (pg. 10) | |
| Neologicism (pg. 11) | |
| Interpreting arithmetic (pg. 12) | |
| Numbers as equinumerosity classes (pg. 12) | |
| Numbers as sets (pg. 12) | |
| Numbers as primitives (pg. 14) | |
| Numbers as morphisms (pg. 15) | |
| Numbers as games (pg. 16) | |
| Junk theorems (pg. 18) | |
| Interpretation of theories (pg. 18) | |
| What numbers could not be (pg. 19) | |
| The epistemological problem (pg. 20) | |
| Dedekind arithmetic (pg. 21) | |
| Arithmetic categoricity (pg. 22) | |
| Mathematical induction (pg. 23) | |
| Fundamental theorem of arithmetic (pg. 24) | |
| Infinitude of primes (pg. 26) | |
| Structuralism (pg. 27) | |
| Definability versus Leibnizian structure (pg. 28) | |
| Role of identity in the formal language (pg. 29) | |
| Isomorphism orbit (pg. 30) | |
| Categoricity (pg. 31) | |
| Structuralism in mathematical practice (pg. 32) | |
| Eliminative structuralism (pg. 34) | |
| Abstract structuralism (pg. 35) | |
| What is a real number? (pg. 36) | |
| Dedekind cuts (pg. 36) | |
| Theft and honest toil (pg. 37) | |
| Cauchy real numbers (pg. 38) | |
| Real numbers as geometric continuum (pg. 38) | |
| Categoricity for the real numbers (pg. 38) | |
| Categoricity for the real continuum (pg. 40) | |
| Transcendental numbers (pg. 42) | |
| The transcendence game (pg. 42) | |
| Complex numbers (pg. 43) | |
| Platonism for complex numbers (pg. 44) | |
| Categoricity for the complex field (pg. 44) | |
| A complex challenge for structuralism? (pg. 45) | |
| Structure as reduct of rigid structure (pg. 46) | |
| Contemporary type theory (pg. 47) | |
| More numbers (pg. 48) | |
| What is a philosophy for? (pg. 48) | |
| Finally, what is a number? (pg. 49) | |
| Questions for further thought (pg. 49) | |
| Further reading (pg. 51) | |
| Credits (pg. 52) | |
| Rigor (pg. 53) | |
| Continuity (pg. 53) | |
| Informal account of continuity (pg. 53) | |
| The definition of continuity (pg. 55) | |
| The continuity game (pg. 56) | |
| Estimation in analysis (pg. 56) | |
| Limits (pg. 57) | |
| Instantaneous change (pg. 57) | |
| Infinitesimals (pg. 58) | |
| Modern definition of the derivative (pg. 59) | |
| An enlarged vocabulary of concepts (pg. 59) | |
| The least-upper-bound principle (pg. 61) | |
| Consequences of completeness (pg. 62) | |
| Continuous induction (pg. 63) | |
| Indispensability of mathematics (pg. 64) | |
| Science without numbers (pg. 65) | |
| Fictionalism (pg. 67) | |
| The theory/metatheory distinction (pg. 68) | |
| Abstraction in the function concept (pg. 68) | |
| The Devil's staircase (pg. 69) | |
| Space-filling curves (pg. 70) | |
| Conway base-13 function (pg. 71) | |
| Infinitesimals revisited (pg. 74) | |
| Nonstandard analysis and the hyperreal numbers (pg. 75) | |
| Calculus in nonstandard analysis (pg. 76) | |
| Classical model-construction perspective (pg. 77) | |
| Axiomatic approach (pg. 78) | |
| ``The'' hyperreal numbers? (pg. 78) | |
| Radical nonstandardness perspective (pg. 79) | |
| Translating between nonstandard and classical perspectives (pg. 80) | |
| Criticism of nonstandard analysis (pg. 81) | |
| Questions for further thought (pg. 82) | |
| Further reading (pg. 84) | |
| Credits (pg. 85) | |
| Infinity (pg. 87) | |
| Hilbert's Grand Hotel (pg. 87) | |
| Hilbert's bus (pg. 88) | |
| Hilbert's train (pg. 88) | |
| Countable sets (pg. 89) | |
| Equinumerosity (pg. 90) | |
| Hilbert's half-marathon (pg. 92) | |
| Cantor's cruise ship (pg. 93) | |
| Uncountability (pg. 93) | |
| Cantor's original argument (pg. 95) | |
| Mathematical cranks (pg. 96) | |
| Cantor on transcendental numbers (pg. 96) | |
| Constructive versus nonconstructive arguments (pg. 97) | |
| On the number of subsets of a set (pg. 99) | |
| On the number of infinities (pg. 99) | |
| Russell on the number of propositions (pg. 100) | |
| On the number of possible committees (pg. 100) | |
| The diary of Tristram Shandy (pg. 101) | |
| The cartographer's paradox (pg. 101) | |
| The Library of Babel (pg. 103) | |
| On the number of possible books (pg. 103) | |
| Beyond equinumerosity to the comparative size principle (pg. 104) | |
| Place focus on reflexive preorders (pg. 106) | |
| What is Cantor's continuum hypothesis? (pg. 107) | |
| Transfinite cardinals—the alephs and the beths (pg. 109) | |
| Lewis on the number of objects and properties (pg. 110) | |
| Zeno's paradox (pg. 111) | |
| Actual versus potential infinity (pg. 112) | |
| How to count (pg. 112) | |
| Questions for further thought (pg. 114) | |
| Further reading (pg. 116) | |
| Credits (pg. 117) | |
| Geometry (pg. 119) | |
| Geometric constructions (pg. 119) | |
| Contemporary approach via symmetries (pg. 121) | |
| Collapsible compasses (pg. 122) | |
| Constructible points and the constructible plane (pg. 123) | |
| Constructible numbers and the number line (pg. 125) | |
| Nonconstructible numbers (pg. 126) | |
| Doubling the cube (pg. 127) | |
| Trisecting the angle (pg. 127) | |
| Squaring the circle (pg. 128) | |
| Circle-squarers and angle-trisectors (pg. 128) | |
| Alternative tool sets (pg. 128) | |
| Compass-only constructibility (pg. 129) | |
| Straightedge-only constructibility (pg. 129) | |
| Construction with a marked ruler (pg. 130) | |
| Origami constructibility (pg. 130) | |
| Spirograph constructibility (pg. 131) | |
| The ontology of geometry (pg. 132) | |
| The role of diagrams and figures (pg. 133) | |
| Kant (pg. 133) | |
| Hume on arithmetic reasoning over geometry (pg. 134) | |
| Manders on diagrammatic proof (pg. 135) | |
| Contemporary tools (pg. 135) | |
| How to lie with figures (pg. 136) | |
| Error and approximation in geometric construction (pg. 137) | |
| Constructing a perspective chessboard (pg. 140) | |
| Non-Euclidean geometry (pg. 142) | |
| Spherical geometry (pg. 143) | |
| Elliptical geometry (pg. 145) | |
| Hyperbolic geometry (pg. 145) | |
| Curvature of space (pg. 146) | |
| Errors in Euclid? (pg. 147) | |
| Implicit continuity assumptions (pg. 147) | |
| The missing concept of ``between'' (pg. 148) | |
| Hilbert's geometry (pg. 149) | |
| Tarski's geometry (pg. 149) | |
| Geometry and physical space (pg. 149) | |
| Poincaré on the nature of geometry (pg. 151) | |
| Tarski on the decidability of geometry (pg. 151) | |
| Questions for further thought (pg. 153) | |
| Further reading (pg. 154) | |
| Credits (pg. 155) | |
| Proof (pg. 157) | |
| Syntax-semantics distinction (pg. 157) | |
| Use/mention (pg. 158) | |
| What is proof? (pg. 159) | |
| Proof as dialogue (pg. 160) | |
| Wittgenstein (pg. 161) | |
| Thurston (pg. 161) | |
| Formalization and mathematical error (pg. 162) | |
| Formalization as a sharpening of mathematical ideas (pg. 163) | |
| Mathematics does not take place in a formal language (pg. 163) | |
| Voevodsky (pg. 165) | |
| Proofs without words (pg. 165) | |
| How to lie with figures (pg. 166) | |
| Hard arguments versus soft (pg. 166) | |
| Moral mathematical truth (pg. 167) | |
| Formal proof and proof theory (pg. 169) | |
| Soundness (pg. 170) | |
| Completeness (pg. 170) | |
| Compactness (pg. 171) | |
| Verifiability (pg. 172) | |
| Sound and verifiable, yet incomplete (pg. 172) | |
| Complete and verifiable, yet unsound (pg. 173) | |
| Sound and complete, yet unverifiable (pg. 173) | |
| The empty structure (pg. 174) | |
| Formal deduction examples (pg. 175) | |
| The value of formal deduction (pg. 176) | |
| Automated theorem proving and proof verification (pg. 177) | |
| Four-color theorem (pg. 177) | |
| Choice of formal system (pg. 178) | |
| Completeness theorem (pg. 180) | |
| Nonclassical logics (pg. 182) | |
| Classical versus intuitionistic validity (pg. 183) | |
| Informal versus formal use of ``constructive'' (pg. 185) | |
| Epistemological intrusion into ontology (pg. 186) | |
| No unbridgeable chasm (pg. 186) | |
| Logical pluralism (pg. 187) | |
| Classical and intuitionistic realms (pg. 187) | |
| Conclusion (pg. 188) | |
| Questions for further thought (pg. 188) | |
| Further reading (pg. 190) | |
| Credits (pg. 191) | |
| Computability (pg. 193) | |
| Primitive recursion (pg. 194) | |
| Implementing logic in primitive recursion (pg. 195) | |
| Diagonalizing out of primitive recursion (pg. 197) | |
| The Ackermann function (pg. 198) | |
| Turing on computability (pg. 200) | |
| Turing machines (pg. 201) | |
| Partiality is inherent in computability (pg. 202) | |
| Examples of Turing-machine programs (pg. 202) | |
| Decidability versus enumerability (pg. 203) | |
| Universal computer (pg. 204) | |
| ``Stronger'' Turing machines (pg. 205) | |
| Other models of computatibility (pg. 206) | |
| Computational power: Hierarchy or threshold? (pg. 207) | |
| The hierarchy vision (pg. 207) | |
| The threshold vision (pg. 207) | |
| Which vision is correct? (pg. 207) | |
| Church-Turing thesis (pg. 208) | |
| Computation in the physical universe (pg. 209) | |
| Undecidability (pg. 210) | |
| The halting problem (pg. 210) | |
| Other undecidable problems (pg. 211) | |
| The tiling problem (pg. 211) | |
| Computable decidability versus enumerability (pg. 212) | |
| Computable numbers (pg. 212) | |
| Oracle computation and the Turing degrees (pg. 214) | |
| Complexity theory (pg. 215) | |
| Feasibility as polynomial-time computation (pg. 217) | |
| Worst-case versus average-case complexity (pg. 217) | |
| The black-hole phenomenon (pg. 218) | |
| Decidability versus verifiability (pg. 219) | |
| Nondeterministic computation (pg. 219) | |
| P versus NP (pg. 220) | |
| Computational resiliency (pg. 221) | |
| Questions for further thought (pg. 221) | |
| Further reading (pg. 224) | |
| Incompleteness (pg. 225) | |
| The Hilbert program (pg. 226) | |
| Formalism (pg. 226) | |
| Life in the world imagined by Hilbert (pg. 228) | |
| The alternative (pg. 228) | |
| Which vision is correct? (pg. 228) | |
| The first incompleteness theorem (pg. 229) | |
| The first incompleteness theorem, via computability (pg. 229) | |
| The Entscheidungsproblem (pg. 230) | |
| Incompleteness, via diophantine equations (pg. 232) | |
| Arithmetization (pg. 233) | |
| First incompleteness theorem, via Gödel sentence (pg. 235) | |
| Second incompleteness theorem (pg. 238) | |
| Löb proof conditions (pg. 238) | |
| Provability logic (pg. 240) | |
| Gödel-Rosser incompleteness theorem (pg. 240) | |
| Tarski's theorem on the nondefinability of truth (pg. 241) | |
| Feferman theories (pg. 242) | |
| Ubiquity of independence (pg. 243) | |
| Tower of consistency strength (pg. 244) | |
| Reverse mathematics (pg. 244) | |
| Goodstein's theorem (pg. 247) | |
| Löb's theorem (pg. 250) | |
| Two kinds of undecidability (pg. 251) | |
| Questions for further thought (pg. 252) | |
| Further reading (pg. 253) | |
| Set Theory (pg. 255) | |
| Cantor-Bendixson theorem (pg. 256) | |
| Set theory as a foundation of mathematics (pg. 258) | |
| General comprehension principle (pg. 261) | |
| Frege's Basic Law V (pg. 262) | |
| Cumulative hierarchy (pg. 265) | |
| Separation axiom (pg. 267) | |
| Ill-founded hierarchies (pg. 268) | |
| Impredicativity (pg. 269) | |
| Extensionality (pg. 270) | |
| Other axioms (pg. 271) | |
| Replacement axiom (pg. 271) | |
| The number of infinities (pg. 272) | |
| The axiom of choice and the well-order theorem (pg. 274) | |
| Paradoxical consequences of AC (pg. 276) | |
| Paradox without AC (pg. 276) | |
| Solovay's dream world for analysis (pg. 277) | |
| Large cardinals (pg. 278) | |
| Strong limit cardinals (pg. 279) | |
| Regular cardinals (pg. 280) | |
| Aleph-fixed-point cardinals (pg. 280) | |
| Inaccessible and hyperinaccessible cardinals (pg. 281) | |
| Linearity of the large cardinal hierarchy (pg. 283) | |
| Large cardinals consequences down low (pg. 284) | |
| Continuum hypothesis (pg. 284) | |
| Pervasive independence phenomenon (pg. 285) | |
| Universe view (pg. 286) | |
| Categoricity and rigidity of the set-theoretic universe (pg. 286) | |
| Criterion for new axioms (pg. 288) | |
| Intrinsic justification (pg. 289) | |
| Extrinsic justification (pg. 289) | |
| What is an axiom? (pg. 290) | |
| Does mathematics need new axioms? (pg. 292) | |
| Absolutely undecidable questions (pg. 292) | |
| Strong versus weak foundations (pg. 293) | |
| Shelah (pg. 294) | |
| Feferman (pg. 294) | |
| Multiverse view (pg. 295) | |
| Dream solution of the continuum hypothesis (pg. 295) | |
| Analogy with geometry (pg. 296) | |
| Pluralism as set-theoretic skepticism? (pg. 296) | |
| Plural platonism (pg. 297) | |
| Theory/metatheory interaction in set theory (pg. 297) | |
| Summary (pg. 298) | |
| Questions for further thought (pg. 298) | |
| Further reading (pg. 300) | |
| Credits (pg. 301) | |
| Bibliography (pg. 303) | |
| Notation Index (pg. 315) | |
| Subject Index (pg. 319) | |
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