The conceptual roots of mathematics: an essay on the philosophy of mathematics
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
London ; New York
Routledge, Taylor & Francis Group
2011
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| Ausgabe: | First issued in paperback |
| Schriftenreihe: | International library of philosophy
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xvii, 452 Seiten Illustrationen, Diagramme |
| ISBN: | 9780415513722 |
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Contents section page Chapter 1 Plato’s Philosophies of Mathematics 1.1 Meno 1 1.2 A Priori 3 1.3 Relevance 5 1.4 What Are We Talking About? 9 1.5 How Do We Know? 13 1.6 Modality 20 1.7 Cogency 22 1.8 Deduction 24 1.9 Whence the Premisses? 27 Chapter 2 Geometry 2.1 Euclid 2.2 The Fifth Postulate 2.3 Non-Euclidean Geometries 2.4 Formal and Physical Geometry 2.5 Conceptual Constraints 2.6 Which Geometry? 2.7 The Theory of Groups 2.8 Pythagorean Geometry has a Better Metric 2.9 Desargues 2.10 Conclusions 33 34 39 44 49 54 57 60 63 64 Chapter 3 Formalism 3.1 More Geomeirico 3.2 Formalising 3.3 Maximum Cogency 3.4 The Theory of Formal Systems 3.5 Meaning and Interpretation 3.6 Epistemological Formalism 3.7 The Logicist Programme 68 70 70 75 81 84 87 xiii
The Conceptual Roots of Mathematics Chapter 4 Numbers: The Cardinal Approach 4.1 Etymology 90 4.2 The Uses of Numbers 92 4.3 How Many? 95 4.4 Nought 101 4.5 Quotifiers and Quotities 102 4.6 Frege’s Extensions and Sets 106 4.7 Paradigm Sets 109 Chapter 5 Numbers: The Ordinal Approach 5.1 The Superlative Approach 5.2 Dedekind’s Successor 5.3 And So On 5.5 Grounding the Ordinals 5.5 How to Count 5.6 Ordinals and Cardinals 5.7 Conclusion 114 115 118 120 122 123 124 Chapter 6 Numbers: TheAbstract Approach 6.1 The Third Approach 6.2 Peano 6.3 Monomorphism and Non-standard Models 6.4 Immigration Control 6.5 The Fifth Postulate 6.6 Sorites Arithmetic 6.7 Dialogues 6.8 Recursive Reasoning 6.9 The Natural Numbers 126 127 132 138 143 145 146 152 154 Chapter 7 The Infinite 7.1 In Defence of Doubt 7.2 Cardinality 7.3 The Mostest 7.4 Characterization of Intuitionism 7.5 Proofs and Dialogues 7.6 Verificationist Arguments for Intuitionism 7.7 Selective Scepticism 7.8 Ultrafinitism 7.9 Lax Finitism 15? 159 165 169 1?2 181 18? 189 192 xiv
The Conceptual Roots of Mathematics 7.10 Actualising Potentiality 7.11 All 193 196 Chapter 8 The Implications of Gödel’s Theorem 8.1 Pons Asinorum 199 8.2 The Flavour of the GödelianArgument 200 8.3 Gödel Numbering 201 8.4 Translation 203 8.5 Diagonalization 206 8.6 Conditions 210 8.7 Corollaries and Consequences 212 8.8 Church’s Theorem and Turing’sTheorem 214 8.9 Gödel’s Second Theorem 216 8.10 Mechanism 218 8.11 Gödel's Theorem and Provability 219 Chapter 9 Transitive Relations 9.1 Logic 9.2 Equivalence Relations 9.3 Functions 9.4 Identity in Difference 9.5 Ordering Relations 9.6 Macrostructure and Microstructure 9.7 The Continuum 9.8 The Marriage of Equivalencewith Order 9.9 Converse Transitivity 9.10 Paradigm Partial Orderings 9.11 Lattices and Set Theory 9.12 Trees and Mereology 224 226 232 234 236 239 245 249 253 255 257 261
The Conceptual Roots of Mathematics Chapter 10 Prototopology 10.1 Togetherness 10.2 Axiomatic Approaches 10.3 Whitehead’s Programme 10.4 10.5 10.6 10.7 10.8 Failure Pointed and Linear Hopes Alternatives Mooreology More Mereology Chapter 11 Magnitude and Measure Quantum? and Quot? 11.1 11.2 The Point of Measuring 11.3 Equivalence Structures 11.4 11.5 11.6 11.7 271 272 274 277 279 282 283 285 291 292 294 298 Addition Rules Limits and Zero Zeno Measures and Numbers Chapter 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12 Down With Set Theory Multitudes and Magnitudes Russell’s Paradox Responses to Paradox Formal Approaches The Skolem Paradox The Axiom of Choice The Continuum Hypotheses Axioms and Existence The Axiom of Extensionality Conclusion Chapter 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13 Chastened Logicism? Logicism What is Logic? Boolean Plus Iterated Modalities Completeness Paradox Second-order Logic xvi 301 302 306 311 314 315 319 322 323 328 330 333 337 340 342 343 348 353 356 358
The Conceptual Roots of Mathematics Analytic and A Priori Truth 361 Chapter 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14 Mathematical Knowledge Synthetic A Priori? Not Seeing But Doing Pattern Recognition Lakatos Cogency and Dialogue On Behalf of the Fool Hilbert The Bed Theory of Truth Mathematical Knowledge 363 366 368 372 377 379 383 388 394 Chapter 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15 Realism Revisited Existence and Reality Self-Subsistent Objects Meaning and Impredicativity Bivalence and Determinacy Competing Truths Contingency and Structure Coherence and Depth Laws of the Laws of Nature Chastened Isms 399 403 407 413 415 418 422 424 427 13.8 Envoi 433 Summaries 435 Index 442 xvii |
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| spelling | Lucas, John R. 1929-2020 Verfasser (DE-588)124931537 aut The conceptual roots of mathematics an essay on the philosophy of mathematics J.R. Lucas First issued in paperback London ; New York Routledge, Taylor & Francis Group 2011 xvii, 452 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier International library of philosophy Filosofische aspecten gtt Grondslagen gtt Mathématiques - Philosophie Wiskunde gtt Mathematik Philosophie Mathematics Philosophy Mathematik (DE-588)4037944-9 gnd rswk-swf Philosophie (DE-588)4045791-6 gnd rswk-swf Mathematik (DE-588)4037944-9 s Philosophie (DE-588)4045791-6 s DE-604 Äquivalent Druck-Ausgabe, Hardcover 978-0-415-20738-6 Erscheint auch als Online-Ausgabe 978-0-203-02842-1 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=034966834&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lucas, John R. 1929-2020 The conceptual roots of mathematics an essay on the philosophy of mathematics Filosofische aspecten gtt Grondslagen gtt Mathématiques - Philosophie Wiskunde gtt Mathematik Philosophie Mathematics Philosophy Mathematik (DE-588)4037944-9 gnd Philosophie (DE-588)4045791-6 gnd |
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| title | The conceptual roots of mathematics an essay on the philosophy of mathematics |
| title_auth | The conceptual roots of mathematics an essay on the philosophy of mathematics |
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| title_exact_search_txtP | The conceptual roots of mathematics an essay on the philosophy of mathematics |
| title_full | The conceptual roots of mathematics an essay on the philosophy of mathematics J.R. Lucas |
| title_fullStr | The conceptual roots of mathematics an essay on the philosophy of mathematics J.R. Lucas |
| title_full_unstemmed | The conceptual roots of mathematics an essay on the philosophy of mathematics J.R. Lucas |
| title_short | The conceptual roots of mathematics |
| title_sort | the conceptual roots of mathematics an essay on the philosophy of mathematics |
| title_sub | an essay on the philosophy of mathematics |
| topic | Filosofische aspecten gtt Grondslagen gtt Mathématiques - Philosophie Wiskunde gtt Mathematik Philosophie Mathematics Philosophy Mathematik (DE-588)4037944-9 gnd Philosophie (DE-588)4045791-6 gnd |
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