A history of mathematical impossibility:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford, United Kingdom
Oxford University Press
[2022]
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite [267]-278. - Index |
| Physical Description: | xi, 284 Seiten Illustrationen, Diagramme, Porträts (s/w) 24,3 cm |
| ISBN: | 9780192867391 |
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Record in the Search Index
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C
ONTENTS
1.
INTRODUCTION
1.1
THE ORGANIZATION OF THE BOOK
1.
2 WHAT IS AN IMPOSSIBILITY THEOREM?
1.
3 META STATEMENTS AND MATHEMATICAL RESULTS
1.
4 WHY ARE IMPOSSIBILITY RESULTS OFTEN MISUNDERSTOOD
A
MONG AMATEUR MATHEMATICIANS?
1.5 IMPOSSIBILITY RESULTS IN MATHEMATICS AND ELSEWHERE
1.
6 A CLASSIFICATION OF MATHEMATICAL IMPOSSIBILITY RESULTS
1.7 IMPOSSIBILITY AS A CREATIVE FORCE
2
. PREHISTORY: RECORDED AND NON-RECORDED IMPOSSIBILITIES
3
. THE FIRST IMPOSSIBILITY PROOF: INCOMMENSURABILITY
3.1 THE DISCOVERY
3
.2 THE CONSEQUENCES OF THE IMPOSSIBILITY THEOREM
3.3 INCOMMENSURABLE QUANTITIES IN EUCLID'S
ELEMENTS
4
. CLASSICAL PROBLEMS OF ANTIQUITY: CONSTRUCTIONS AND
POSITIVE THEOREMS
4.1 SQUARING A CIRCLE
4.2 DOUBLING THE CUBE
4
.3 TRISECTING THE ANGLE
1
1
4
5
6
9
1
2
1
3
1
5
1
9
1
9
2
3
25
2
7
28
3
5
39
V
5
. THE CLASSICAL PROBLEMS: THE IMPOSSIBILITY QUESTION IN ANTIQUITY 42
5.1 EXISTENCE AND CONSTRUCTABILITY 42
5
.2 PAPPUS ON THE CLASSIFICATION OF GEOMETRIC PROBLEMS 44
5
.3 THE QUADRATURE OF A CIRCLE 47
5
.4 USING NON-CONSTRUCTIBLE QUANTITIES: ARCHIMEDES AND PTOLEMY 48
6
. DIORISMS: CONCLUSIONS ABOUT THE GREEKS AND MEDIEVAL ARABS 52
6.1 DIORISMS
6
.2 CONCLUSION ON IMPOSSIBILITIES IN GREEK MATHEMATICS
6.3 MEDIEVAL ARABIC CONTRIBUTIONS
7
. CUBE DUPLICATION AND ANGLE TRISECTION IN THE
SEVENTEENTH AND EIGHTEENTH CENTURIES
7.1 THE SEVENTEENTH CENTURY
7.2 DESCARTES'S ANALYTIC GEOMETRY
7
.3 DESCARTES ON THE DUPLICATION OF A CUBE AND THE
TRISECTION OF AN ANGLE
5
2
56
59
6
5
65
66
6
9
X
CONTENTS
1
4. IMPOSSIBLE INTEGRALS
1
4.1 EARLY CONSIDERATIONS
1
4.2 ABEL'S MOSTLY UNPUBLISHED RESULTS
1
4.3 JOSEPH LIOUVILLE ON INTEGRATION IN ALGEBRAIC TERMS
1
4.4 LIOUVILLE ON INTEGRATION IN FINITE TERMS . DRATURE
14.5 LIOUVILLE ON SOLUTION OF DIFFERENTIAL EQUAT10NS BY QUA
1
4.6 LATER DEVELOPMENTS
1
4.7 CONCLUDING REMARKS ON THE SITUATION C.1830
1
5. IMPOSSIBILITY OF PROVING THE PARALLEL POSTULATE
1
5.1 THE AXIOMATIC DEDUCTIVE METHOD
1
5.2 THE PARALLEL POSTULATE AND THE ATTEMPTS TO PROVE IT
1
5.3 INDIRECT PROOFS: IMPLICIT NON-EUCLIDEAN GEOMETRY
1
5.4 NON-EUCLIDEAN GEOMETRY: THE INVENTION
1
5.5 THE HELP FROM DIFFERENTIAL GEOMETRY OF SURFACES
1
5.6. CONCLUSIONS
1
6. HILBERT AND IMPOSSIBLE PROBLEMS
1
6.1 IMPOSSIBILITY AS A SOLUTION; REJECTION OF
IGNORABIMUS
1
6.2 HILBERT'S THIRD PROBLEM: EQUIDECOMPOSABILITY
1
6.3 HILBERT'S SEVENTH PROBLEM
1
6.4 HILBERT'S FIRST PROBLEM
1
7. HILBERT AND GODEL ON AXIOMATIZATION AND INCOMPLETENESS
1
7.1 THE AXIOMATIZATION OF MATHEMATICS
1
7.2 HILBERT'S SECOND PARIS PROBLEM
1
7.3 THE FOUNDATIONAL CRISIS
1
7.4 GODELS INCOMPLETENESS THEOREMS
1
7.5 HILBERT'S TENTH PARIS PROBLEM
17.6 CONCLUSION
1
8. FERMAT'S LAST THEOREM
1
8.1 FERMAT'S CONTRIBUTION
1
8.2 NINETEENTH-CENTURY CONTRIBUTIONS
1
8.3 THE TWENTIETH-CENTURY PROOF
1
9. IMPOSSIBILITY IN PHYSICS
1
9.1 THE IMPOSSIBILITY OF PERPETUAL MOTION MACHINES
1
9.2 TWENTIETH-CENTURY IMPOSSIBILITIES IN PHYSICS
20
. ARROW'S IMPOSSIBILITY THEOREM
20.1 THE THEORY OF VOTING
20.2 WELFARE ECONOMICS
20.3 THE IMPOSSIBILITY THEOREM
20.4 THE GIBBARD-SATTERTHWAITE THEOREM
1
68
1
68
1
70
1
71
1
73
1
77
1
78
1
79
1
81
1
82
1
84
1
87
1
90
1
94
1
97
1
99
1
99
2
02
204
205
2
10
210
212
215
216
221
223
2
25
2
25
230
233
2
38
2
38
242
2
48
2
48
251
253
256
C
ONTENTS
XI
2
1. CONCLUSION 259
2
1.1 FROM UNIMPORTANT NON-RESULTS TO REMARKABLE "SOLUTIONS" 259
21.2 FROM META-STATEMENTS TO MATHEMATICAL THEOREMS 260
21.3 DIFFERENT TYPES OF PROBLEMS AND DIFFERENT TYPES OF PROOFS 260
21.4 PURE AND APPLIED IMPOSSIBILITY THEOREMS 262
2
1.5 CONTROVERSIES 263
2
1.6 IMPOSSIBILITY AS A CREATIVE FORCE 263
R
ECOMMENDED SUPPLEMENTARY READING
265
R
EFERENCES
267
I
NDEX
279
C
ONTENTS
IX
7
.4 DESCARTES'S CONTRIBUTIONS 74
7
.5 THE EIGHTEENTH CENTURY 75
7
.6 MONTUCLA AND CONDORCET COMPARED WITH DESCARTES 79
8
. CIRCLE QUADRATURE IN THE SEVENTEENTH CENTURY 81
8
.1 "SOLUTIONS" AND POSITIVE RESULTS 81
8
.2 DESCARTES ON THE QUADRATURE OF A CIRCLE 83
8
.3 WALLIS ON THE IMPOSSIBILITY OF AN ANALYTIC QUADRATURE OF
A CIRCLE 85
8
.4 DIFFERENT QUADRATURES OF A CIRCLE 89
8
.5 GREGORY ON IMPOSSIBILITY PROOFS AND THE NEW ANALYSIS 91
8.6 GREGORY'S ARGUMENT FOR THE IMPOSSIBILITY OF THE
ALGEBRAIC INDEFINITE CIRCLE QUADRATURE 93
8
.7 HUYGENS' AND WALLIS' CRITIQUE OF GREGORY 96
8
.8 LEIBNIZ ON THE IMPOSSIBILITY OF THE INDEFINITE CIRCLE QUADRATURE 99
8.9 NEWTON'S ARGUMENT FOR THE IMPOSSIBILITY OF THE
ALGEBRAIC INDEFINITE OVAL QUADRATURE 102
8
.10 WHY PROVE IMPOSSIBILITY 105
9
. CIRCLE QUADRATURE IN THE EIGHTEENTH CENTURY 108
9
.1 JOSEPH SAURIN (1659-1737) 108
9
.2 ANONYMOUS 110
9
.3 THOMAS FANTET DE LAGNY (1660-1734) 110
9
.4 THE ENLIGHTENED OPINION 112
9
.5 D'ALERNBERT 114
9
.6 THE FRENCH ACADEMY OF SCIENCES. CONDORCET 116
9
.7 ENLIGHTENING THE AMATEURS 117
9
.8 LAMBERT AND THE IRRATIONALITY OF
RE
118
IO
. IMPOSSIBLE EQUATIONS MADE POSSIBLE: THE COMPLEX NUMBERS 121
10.1 THE EXTENSION OF THE NUMBER SYSTEM: WALLIS'S ACCOUNT 121
10.2 CARDANOS SOPHISTICATED AND USELESS NUMBERS 124
10
.3 THE UNREASONABLE USEFULNESS OF THE COMPLEX NUMBERS 127
10.4 A DIGRESSION ABOUT INFINITESIMALS 130
1
1. EULER AND THE BRIDGES OF KONIGSBERG
1
2. THE INSOLVABILITY OF THE QUINTIC BY RADICALS
1
2.1 EARLY RESULTS
1
2.2 PAOLO RUFFINI
1
2.3 NIELS HENRIK ABEL
13
. CONSTRUCTIONS WITH RULER AND COMPASS: THE FINAL
I
MPOSSIBILITY PROOFS
13
.1 GAUSS ON REGULAR POLYGONS
13.2 WANTZCL
1
3.3 THE QUADRATURE OFA CIRCLE
13
3
1
40
1
40
1
45
1
49
1
55
1
55
1
60
1
63 |
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| spelling | Lützen, Jesper Verfasser (DE-588)132955202 aut A history of mathematical impossibility Jesper Lützen (Department of Mathematical Sciences, University of Copenhagen, Denmark) Oxford, United Kingdom Oxford University Press [2022] © 2022 xi, 284 Seiten Illustrationen, Diagramme, Porträts (s/w) 24,3 cm txt rdacontent n rdamedia nc rdacarrier Literaturverzeichnis Seite [267]-278. - Index Geschichte gnd rswk-swf Unmöglichkeit Philosophie (DE-588)7700608-2 gnd rswk-swf Mathematisches Problem (DE-588)4114530-6 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematical analysis / History Mathematical analysis History Mathematik (DE-588)4037944-9 s Unmöglichkeit Philosophie (DE-588)7700608-2 s Mathematisches Problem (DE-588)4114530-6 s Geschichte z DE-604 Erscheint auch als Online-Ausgabe 10.1093/oso/9780192867391.001.0001 978-0-19-195947-9 978-0-19-269303-7 (DE-604)BV048690626 Digitalisierung Deutsches Museum application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034073885&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lützen, Jesper A history of mathematical impossibility Unmöglichkeit Philosophie (DE-588)7700608-2 gnd Mathematisches Problem (DE-588)4114530-6 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)7700608-2 (DE-588)4114530-6 (DE-588)4037944-9 |
| title | A history of mathematical impossibility |
| title_auth | A history of mathematical impossibility |
| title_exact_search | A history of mathematical impossibility |
| title_exact_search_txtP | A history of mathematical impossibility |
| title_full | A history of mathematical impossibility Jesper Lützen (Department of Mathematical Sciences, University of Copenhagen, Denmark) |
| title_fullStr | A history of mathematical impossibility Jesper Lützen (Department of Mathematical Sciences, University of Copenhagen, Denmark) |
| title_full_unstemmed | A history of mathematical impossibility Jesper Lützen (Department of Mathematical Sciences, University of Copenhagen, Denmark) |
| title_short | A history of mathematical impossibility |
| title_sort | a history of mathematical impossibility |
| topic | Unmöglichkeit Philosophie (DE-588)7700608-2 gnd Mathematisches Problem (DE-588)4114530-6 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Unmöglichkeit Philosophie Mathematisches Problem Mathematik |
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