Lectures on the philosophy of mathematics:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge, Massachusetts ; London, England
The MIT Press
[2020]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | 2103 |
| Physical Description: | xviii, 329 Seiten Illustrationen, Diagramme (teilweise farbig) |
| ISBN: | 9780262542234 0262542234 |
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Contents Preface About the Author 1 Numbers 1.1 1.2 1.3 1.4 1.5 1.6 Numbers versus numerals Number systems Natural numbers Integers Rational numbers Incommensurable numbers An alternative geometric argument Platonism Plenitudinous platonism Logicism Equinumerosity The Cantor-Hume principle The Julius Caesar problem Numbers as equinumerosity classes Neologicism Interpreting arithmetic Numbers as equinumerosity classes Numbers as sets Numbers as primitives Numbers as morphisms Numbers as games Junk theorems Interpretation of theories xvii xix 1 1 2 2 3 3 4 5 6 7 7 7 8 10 10 11 12 12 12 14 15 16 18 18
Contents 1.7 What numbers could not be The epistemological problem 19 20 1.8 Dedekind arithmetic 21 1.9 Arithmetic categoricity Mathematical induction Fundamental theorem of arithmetic 22 23 24 Infinitude of primes 1.10 Structuralism 1.11 1.12 1.13 1.14 1.15 1.16 1.17 26 27 Definability versus Leibnizian structure Role of identity in the formal language Isomorphism orbit Categoricity 28 29 30 31 Structuralism in mathematical practice Eliminative structuralism Abstract structuralism 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 Transcendental numbers The transcendence game Complex numbers Platonism for complex numbers Categoricity for the complex field A complex challenge for structuralism? Structure as reduct of rigid structure Contemporary type theory 32 34 35 36 36 37 38 38 38 40 42 42 43 44 44 45 46 47 More numbers What is a philosophy for? Finally, what is a number? Questions for further thought Further reading Credits 48 48 49 49 51 52
ix Contents 2 Rigor 53 2.1 Continuity Informal account of continuity The definition of continuity 53 53 55 The continuity game Estimation in analysis Limits Instantaneous change 56 56 57 57 2.2 Infinitesimals Modern definition of the derivative 2.3 2.4 An enlarged vocabulary of concepts The least-upper-bound principle Consequences of completeness Continuous induction 2.5 Indispensability of mathematics Science without numbers Fictionalism The theory/metatheory distinction 58 59 59 61 62 63 64 65 67 68 2.6 Abstraction in the function concept The Devil’s staircase Space-filling curves 68 69 70 Conway base-13 function 2.7 Infinitesimals revisited Nonstandard analysis and the hyperreal numbers Calculus in nonstandard analysis Classical model-construction perspective 71 74 75 76 77 Axiomatic approach 78 “The” hyperreal numbers? Radical nonstandardness perspective 78 79 Translating between nonstandard and classical perspectives Criticism of nonstandard analysis 80 81 Questions for further thought Further reading 82 84 Credits 85
Contents 3 Infinity 3.1 g7 $$ gg gg 90 3.4 Hilbert’s half-marathon Cantor's cruise ship 92 93 3.5 Uncountability 93 3.2 3.3 Cantor's original argument Mathematical cranks 3.6 Cantor on transcendental numbers Constructive versus nonconstructive arguments 3.7 On the number of subsets of a set On the number of infinities Russell on the number of propositions On the number of possible committees The diary of Tristram Shandy 3.8 3.9 3.10 3.11 3.12 4 o- Hilbert's Grand Hotel Hilbert’s bus Hilbert's train Countable sets Equinumerosity The cartographer’s paradox The Library of Babel On the number of possible books Beyond equinumerosity to the comparative size principle Place focus on reflexive preorders What is Cantor’s continuum hypothesis? Transfinite cardinals—the alephs and the beths Lewis on the number of objects and properties Zeno’s paradox Actual versus potential infinity How to count Questions for further thought Further reading Credits Geometry 4.1 Geometric constructions Contemporary approach via symmetries Collapsible compasses Constructible points and the constructible plane Constructible numbers and the number line 95 96 96 97 99 99 100 100 101 101 103 103 104 106 107 109 110 111 112 112 114 116 112 119 119 121 122 123 125
Contents Nonconstructible numbers Doubling the cube Trisecting the angle Squaring the circle Circle-squarers and angle-trisectors 4.3 Alternative tool sets Compass-only constructibility Straightedge-only constructibility Construction with a marked ruler Origami constructibility Spirograph constructibility 4.4 The ontology of geometry 4.5 The role of diagrams and figures Kant Hume on arithmetic reasoning over geometry Manders on diagrammatic proof Contemporary tools How to lie with figures Error and approximation in geometric construction Constructing a perspective chessboard 4.6 Non-Euclidean geometry Spherical geometry Elliptical geometry Hyperbolic geometry Curvature of space 4.7 Errors in Euclid? Implicit continuity assumptions The missing concept of “between" Hilbert’s geometry Tarski’s geometry 4.8 Geometry and physical space 4.9 Poincaré on the nature of geometry 4.10 Tarski on the decidability of geometry Questions for further thought Further reading Credits 4.2 126 127 127 128 128 128 129 129 130 130 131 132 133 133 134 135 135 136 137 140 142 143 145 145 146 147 147 148 149 149 149 151 15) 153 154 155
xii 5 Contents Proof 157 5.1 Syntax-semantics distinction Use/mention 157 5.2 What is proof? 159 160 161 Proof as dialogue Wittgenstein Thurston Formalization and mathematical error Formalization as a sharpening of mathematical ideas Mathematics does not take place in a formal language Voevodsky Proofs without words How to lie with figures Hard arguments versus soft Moral mathematical truth 5.3 Formal proof and proof theory Soundness Completeness Compactness Verifiability Sound and verifiable, yet incomplete Complete and verifiable, yet unsound Sound and complete, yet unverifiable The empty structure Formal deduction examples 5.4 5.5 5.6 The value of formal deduction Automated theorem proving and proof verification Four-color theorem Choice of formal system Completeness theorem Nonclassical logics Classical versus intuitionistic validity Informal versus formal use of “constructive” Epistemological intrusion into ontology No unbridgeable chasm Logical pluralism Classical and intuitionistic realms 5.7 Conclusion 158 161 162 163 163 165 165 166 166 167 169 170 170 171 172 172 173 173 174 175 176 177 177 178 180 182 183 185 186 186 187 187 188
xiii Conteurs Questions for further thought Further reading Credits 6 1 BK 190 ) 91 Computability [93 6.1 194 195 197 198 200 201 202 202 203 204 205 206 207 207 207 207 208 209 210 210 211 211 212 212 214 215 217 217 218 219 219 220 221 221 224 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Primitive recursion Implementing logic in primitive recursion Diagonalizing out of primitive recursion The Ackermann function Turing on computability Turing machines Partiality is inherent in computability Examples of Turing-machine programs Decidability versus enumerability Universal computer “Stronger” Turing machines Other models of computatibility Computational power: Hierarchy or threshold? The hierarchy vision The threshold vision Which vision is correct? Church-Turing thesis Computation in the physical universe Undecidability The halting problem Other undecidable problems The tiling problem Computable decidability versus enumerability Computable numbers Oracle computation and the Turing degrees Complexity theory Feasibility as polynomial-time computation Worst-case versus average-case complexity The black-hole phenomenon Decidability versus verifiability Nondeterministic computation P versus NP Computational resiliency Questions for further thought Further readins
xiv 7 Contents Incompleteness 225 7.1 226 226 228 228 228 229 229 230 232 233 235 238 238 240 240 241 242 243 244 244 247 250 251 252 253 The Hilbert program Formalism Life in the world imagined by Hilbert The alternative 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 8 Which vision is correct? The first incompleteness theorem The first incompleteness theorem, via computability The Entscheidungsproblem Incompleteness, via diophantine equations Arithmetization First incompleteness theorem, via Gödel sentence Second incompleteness theorem Löb proof conditions Provability logic Gödel-Rosser incompleteness theorem Tarski’s theorem on the nondefinability of truth Feferman theories Ubiquity of independence Tower of consistency strength Reverse mathematics Goodstein’s theorem Lob’s theorem Two kinds of undecidability Questions for further thought Further reading Set Theory 8.1 Cantor-Bendixson theorem 8.2 Set theory as a foundation of mathematics 8.3 General comprehension principle Frege’s Basic Law V 8.4 8.5 Cumulative hierarchy Separation axiom Ill-founded hierarchies 8.6 Extensionality Other axioms Impredicativity 255 256 258 261 262 265 267 268 269 270 271
XV Contents 8.7 8.8 8.9 8.10 8.11 8.12 8.13 Replacementaxiom The number of infinities The axiom of choice and the well-order theorem Paradoxical consequences of AC Paradox without AC Solovay’s dream world for analysis Large cardinals Strong limit cardinals Regular cardinals Aleph-fixed-point cardinals Inaccessible and hyperinaccessible cardinals Linearity of the large cardinal hierarchy Large cardinals consequences down low Continuum hypothesis Pervasive independence phenomenon Universe view Categoricity and rigidity of the set-theoretic universe Criterion for new axioms Intrinsic justification Extrinsic justification What is an axiom? Does mathematics need new axioms? Absolutely undecidable questions Strong versus weak foundations Shelah Feferman 8.14 Multiverse view Dream solution of the continuum hypothesis Analogy with geometry Pluralism as set-theoretic skepticism? Plural platonism Theory/metatheory interaction in set theory Summary Questions for further thought Further reading Credits 271 272 274 276 276 277 278 279 280 280 281 283 284 284 285 286 286 288 289 289 290 292 292 293 294 294 295 295 296 296 297 297 298 298 300 301
Contents Bibliography Notation Index Subject Index 303 315 319 |
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| spelling | Hamkins, Joel David Verfasser (DE-588)1225063485 aut Lectures on the philosophy of mathematics Joel David Hamkins Cambridge, Massachusetts ; London, England The MIT Press [2020] © 2020 xviii, 329 Seiten Illustrationen, Diagramme (teilweise farbig) txt rdacontent n rdamedia nc rdacarrier 2103 Logik (DE-588)4036202-4 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Philosophie (DE-588)4045791-6 gnd rswk-swf Mathematics / Philosophy (DE-588)4123623-3 Lehrbuch gnd-content Mathematik (DE-588)4037944-9 s Logik (DE-588)4036202-4 s Philosophie (DE-588)4045791-6 s DE-604 Erscheint auch als Online-Ausgabe 978-0-262-36265-8 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032565175&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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