Philosophy of mathematics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English German |
| Published: |
Berlin ; Boston
De Gruyter
[2018]
|
| Series: | De Gruyter STEM
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Physical Description: | XV, 457 Seiten Illustrationen 24 cm |
| ISBN: | 9783110468304 3110468301 |
Staff View
MARC
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| 240 | 1 | 0 | |a Philosophie der Mathematik |
| 245 | 1 | 0 | |a Philosophy of mathematics |c Thomas Bedürftig and Roman Murawski |
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| 264 | 4 | |c © 2018 | |
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Record in the Search Index
| _version_ | 1804179564644007936 |
|---|---|
| adam_text | CONTENTS
PREFACE* XIII
INTRODUCTION * 1
1 ON THE WAY TO THE REALS * 7
1.1 IRRATIONALITY * 7
1.2 INCOMMENSURABILITY* 9
1.3 CALCULATING WITH V2?* 13
1.4 PROCEDURE OF APPROXIMATING, NESTING OF INTERVALS AND
COMPLETENESS * 14
1.5 ON THE CONSTRUCTION OF THE REALS * 19
1.6 ON THE HANDLING OF THE INFINITE * 21
1.7 INFINITE NON-PERIODIC DECIMAL FRACTIONS * 23
2 ON THE HISTORY OF THE PHILOSOPHY OF MATHEMATICS * 27
2.1 PYTHAGORAS AND PYTHAGOREANS * 28
2.2 PLATO * 31
2.3 ARISTOTLE * 34
2.4 EUCLID
-----
38
2.5 PROCLUS DIADOCHUS * 40
2.6 NICHOLAS OF CUSA * 42
2.7 DESCARTES * 45
2.8 PASCAL * 48
2.9 LEIBNIZ * 50
2.10 KANT * 53
2.11 MILL AND THE EMPIRICAL CONCEPTIONS * 58
2.12 BOLZANO * 62
2.13 GAUSS
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65
2.14 CANTOR * 66
2.15 DEDEKIND * 70
2.16 POINCARE * 75
2.17 PEIRCE*S PRAGMATISM AND THE WORLD OF SYMBOLS * 78
2.18 HUSSERL*S PHENOMENOLOGY * 82
2.19 LOGICISM * 89
2.20 INTUITIONISM * 98
2.21 CONSTRUCTIVISM * 107
2.22 FORMALISM * 109
2.23 PHILOSOPHY OF MATHEMATICS BETWEEN 1931 AND THE END OF THE
1950S * 117
2.24 THE EVOLUTIONARY POINT OF VIEW - A NEW BASIC POSITION IN
PHILOSOPHY
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123
2.24.1 CHARACTERIZATION
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123
2.24.2 ON STUDIES OF EVOLUTION OF NUMBER CONCEPT * 127
2.24.3 CONCLUDING REMARKS * 136
2.25 PHILOSOPHY OF MATHEMATICS AFTER 1960 * 137
2.25.1 QUASI-EMPIRICAL CONCEPTIONS * 138
2.25.2 REALISM AND ANTIREALISM * 146
3 ON FUNDAMENTAL QUESTIONS OF THE PHILOSOPHY OF MATHEMATICS * 149
3.1 ON THE CONCEPT OF NUMBER * 149
3.1.1 SURVEY OF SOME VIEWS * 150
3.1.2 RESUMS * 154
3.2 INFINITIES * 159
3.2.1 ON PROBLEMS WITH THE INFINITE * 160
3.2.2 CONCEPTION OF ARISTOTLE * 163
3.2.3 IDEALISTIC APPROACH * 163
3.2.4 EMPIRICIST POINT OF VIEW * 164
3.2.5 INFINITY BY KANT * 165
3.2.6 INTUITIONISTLC INFINITY * 167
3.2.7 LOGICIST HYPOTHESIS OF THE INFINITE * 167
3.2.8 INFINITY AND THE NEW PHILOSOPHY OF MATHEMATICS * 168
3.2.9 FORMALISTIC APPROACH AND NOWADAYS TENDENCIES * 169
3.3 THE CONTINUUM AND THE INFINITELY SMALL * 170
3.3.1 GENERAL PROBLEM * 171
3.3.2 ON THE HISTORY OF THE CONTINUUM * 173
3.3.3 WHAT IS A POINT? * 184
3.3.4 ON THE HISTORY OF THE CONTINUUM - CONTINUATION * 190
3.3.5 SURVEY OF CONCEPTIONS OF THE CONTINUUM * 194
3.3.6 NOTES ON THE ARITHMETIZATION OF THE CONTINUUM * 196
3.3.7 THE END OF INFINITESIMALS AND THE REDISCOVERY OF THEM * 198
3.3.8 NONSTANDARD NUMBERS AND THE CONTINUUM * 205
3.3.9 CONSEQUENCES FOR THE CONCEPTION OF THE CONTINUUM * 207
3.3.10 MEDIUM OF FREE EVOLUTION * 210
3.3.11 THE DISAPPEARANCE OF MAGNITUDES * 212
3.3.12 FINAL REMARKS * 217
3.4 ON THE PROBLEM OF APPLICATIONS OF MATHEMATICS * 222
3.4.1 ASPECTS OF THE PROBLEM * 223
3.4.2 THE PROBLEM OF APPLICABILITY IN THE HISTORICAL SETTING* 227
3.4.3 CLASSICAL POSITIONS * 233
3.4.4 NEW CONCEPTIONS * 236
3.4.5 RETROSPECT * 236
3.5 CONCLUSION------241
3.5.1 FROM NATURAL TO RATIONAL NUMBERS * 242
3.5.2 INCOMMENSURABILITY AND IRRATIONALITY * 243
3.5.3 ADJUNCTION
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245
3.5.4 ON THE LINEAR CONTINUUM * 246
3.5.5 INFINITELY SMALL QUANTITIES * 247
3.5.6 CONSTRUCTION, INFINITY, INFINITE NON-PERIODIC DECIMAL FRACTIONS *
248
3.5.7 CONCLUDING REMARKS * 249
4 SETS AND SET THEORIES * 253
4.1 PARADOXES OF THE INFINITE * 254
4.2 ON THE CONCEPT OF A SET * 256
4.2.1 COLLECTING TOGETHER VERSUS PUTTING TOGETHER* 256
4.2.2 SETS AND THE PROBLEM OF UNIVERSAL * 257
4.3 TWO SET THEORIES * 260
4.3.1 SET THEORY ACCORDING TO ZERMELO AND FRAENKEL * 261
4.3.2 VON NEUMANN, BERNAYS AND GOEDEL SET THEORY * 269
4.3.3 REMARKS * 274
4.3.4 ON MODIFICATIONS
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275
4.4 THE AXIOM OF CHOICE AND THE CONTINUUM HYPOTHESIS * 276
4.4.1 SEARCH FOR NEW AXIOMS * 281
4.4.2 FURTHER REMARKS AND QUESTIONS * 286
4.5 FINAL REMARKS * 287
5 AXIOMATIC APPROACH AND LOGIC * 293
5.1 SOME ELEMENTS OF MATHEMATICAL LOGIC * 294
5.1.1 SYNTAX * 294
5.1.2 SEMANTICS * 296
5.1.3 CALCULUS * 299
5.2 HISTORICAL REMARKS * 301
5.2.1 FROM THE HISTORY OF LOGIC * 302
5.2.2 ON THE HISTORY OF THE AXIOMATIC APPROACH * 310
5.3 LOGICAL AXIOMS AND THEORIES * 315
5.3.1 PEANO ARITHMETIC * 317
5.3.2 ON THE AXIOMS FOR REAL NUMBERS * 320
5.4 ON THE ARITHMETIC OF NATURAL NUMBERS * 323
5.4.1 SYNTACTICAL ASPECT * 324
5.4.2 SEMANTICAL ASPECT * 326
5.5 TRUTH AND PROVABILITY * 330
5.5.1 FORMAL TRUTH * 331
5.5.2 COMPLETENESS AND TRUTH * 332
5.5.3 SYNTACTIC REDUCTION OF TRUTH * 334
5.5.4 TRUTH IS UNEQUAL TO PROVABILITY * 337
5.5.5 SEARCH FOR THE WAY OUT* 339
5.5.6 CONCLUDING REMARKS * 341
5.6 FINAL CONCLUSIONS * 341
5.6.1 LOGIC AS THE BACKGROUND OF MATHEMATICS * 342
5.6.2 CONSEQUENCES FOR THE SHAPE OF MATHEMATICS * 343
6 THINKING AND CALCULATING INFINITESIMALLY - FIRST NONSTANDARD STEPS *
347
6.1 PRELIMINARY REMARK* 347
6.2 THE QUESTION ABOUT 0 .9 9 9 ...
-----
348
6.2.1 EMPIRICAL APPROACH * 348
6.2.2 THE PROBLEM
-----
349
6.2.3 ANSWER
-----
356
6.2.4 FINAL REMARKS * 361
6.3 A BIT OF INFINITESIMAL CALCULUS * 362
6.3.1 INFINITESIMAL CALCULATIONS * 363
6.3.2 CONTINUITY, DIFFERENTIAL QUOTIENT, DERIVATIVE * 365
6.4 ON THE CONSTRUCTION OF HYPERREAL NUMBERS * 374
6.4.1 HYPERNATURAL NUMBERS * 375
6.4.2 HYPERREAL NUMBERS * 376
6.4.3 HOW WILL *R BECOME A MODEL OF 1R? * 378
6.4.4 ON THE JUSTIFICATION OF THE NAIVE INFINITESIMAL CALCULUS * 379
6.5 ON THE STATUS OF NONSTANDARD NUMBERS * 380
7 RETROSPECTION * 387
7.1 SETTING OF REAL NUMBERS * 388
7.2 AXIOMATIC METHOD * 389
7.3 THE CONCEPT OF NUMBER* 390
7.4 INFINITY, AXIOM OF CHOICE AND CONTINUUM HYPOTHESIS * 392
7.5 INFINITELY SMALL AND THE CONTINUUM * 393
7.6 APPLICABILITY * 395
7.7 THEORETICAL LIMITS * 395
7.8 USAGE OF COMPUTERS * 396
7.9 WHAT IS THE PHILOSOPHY OF MATHEMATICS AND WHAT DOES IT PROVIDE? *
397
7.10 EVIDENCE AND TRANSCENDENCE * 399
BIOGRAPHIES * 403
BIBLIOGRAPHY * 421
INDEX OF NAMES * 437
INDEX OF SYMBOLS * 443
INDEX OF SUBJECTS * 445
|
| any_adam_object | 1 |
| author | Bedürftig, Thomas 1943- Murawski, Roman 1949- |
| author_GND | (DE-588)108217876 (DE-588)103247831 |
| author_facet | Bedürftig, Thomas 1943- Murawski, Roman 1949- |
| author_role | aut aut |
| author_sort | Bedürftig, Thomas 1943- |
| author_variant | t b tb r m rm |
| building | Verbundindex |
| bvnumber | BV045573991 |
| classification_rvk | CC 3700 SG 700 |
| ctrlnum | (OCoLC)1040168812 (DE-599)DNB1160531811 |
| dewey-full | 510.1 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510.1 |
| dewey-search | 510.1 |
| dewey-sort | 3510.1 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Philosophie |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-10T08:21:52Z |
| institution | BVB |
| isbn | 9783110468304 3110468301 |
| language | English German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030957565 |
| oclc_num | 1040168812 |
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| owner | DE-83 DE-11 DE-739 |
| owner_facet | DE-83 DE-11 DE-739 |
| physical | XV, 457 Seiten Illustrationen 24 cm |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | De Gruyter |
| record_format | marc |
| series2 | De Gruyter STEM |
| spelling | Bedürftig, Thomas 1943- Verfasser (DE-588)108217876 aut Philosophie der Mathematik Philosophy of mathematics Thomas Bedürftig and Roman Murawski Berlin ; Boston De Gruyter [2018] © 2018 XV, 457 Seiten Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier De Gruyter STEM Philosophie (DE-588)4045791-6 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematik (DE-588)4037944-9 s Philosophie (DE-588)4045791-6 s DE-604 Murawski, Roman 1949- Verfasser (DE-588)103247831 aut Erscheint auch als Online-Ausgabe, PDF 978-3-11-046833-5 Erscheint auch als Online-Ausgabe, EPUB 978-3-11-047077-2 B:DE-101 application/pdf http://d-nb.info/1160531811/04 Inhaltsverzeichnis DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030957565&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bedürftig, Thomas 1943- Murawski, Roman 1949- Philosophy of mathematics Philosophie (DE-588)4045791-6 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4045791-6 (DE-588)4037944-9 |
| title | Philosophy of mathematics |
| title_alt | Philosophie der Mathematik |
| title_auth | Philosophy of mathematics |
| title_exact_search | Philosophy of mathematics |
| title_full | Philosophy of mathematics Thomas Bedürftig and Roman Murawski |
| title_fullStr | Philosophy of mathematics Thomas Bedürftig and Roman Murawski |
| title_full_unstemmed | Philosophy of mathematics Thomas Bedürftig and Roman Murawski |
| title_short | Philosophy of mathematics |
| title_sort | philosophy of mathematics |
| topic | Philosophie (DE-588)4045791-6 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Philosophie Mathematik |
| url | http://d-nb.info/1160531811/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030957565&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bedurftigthomas philosophiedermathematik AT murawskiroman philosophiedermathematik AT bedurftigthomas philosophyofmathematics AT murawskiroman philosophyofmathematics |
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