Rowe / McCleary Ideas and Their Reception

Proceedings of the Symposium on the History of Modern Mathematics, Vassar College, Poughkeepsie, New York, June 20-24, 1989
1. Auflage 2014
ISBN: 978-1-4832-6621-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 470 Seiten, Web PDF

The History of Modern Mathematics, Volume I: Ideas and their Reception documents the proceedings of the Symposium on the History of Modern Mathematics held at Vassar College in Poughkeepsie, New York on June 20-24, 1989. This book is concerned with the emergence and reception of major ideas in fields that range from foundations and set theory, algebra and invariant theory, and number theory to differential geometry, projective and algebraic geometry, line geometry, and transformation groups. Other topics include the theory of reception for the history of mathematics and British synthetic vs. French analytic styles of algebra in the early American Republic. The early geometrical works of Sophus Lie and Felix Klein, background to Gergonne's treatment of duality, and algebraic geometry in the late 19th century are also elaborated. This volume is intended for students and researchers interested in developments in pure mathematics.

Autoren/Hrsg.

Rowe, David E.

Mccleary, John

Inhaltsverzeichnis

1;Front Cover;1
2;Ideas and their Reception;4
3;Copyright Page;5
4;Table of Contents;6
5;Contents for Volume II;8
6;Contributors List;10
7;Preface;12
8;Part I: The Context of Reception;18
8.1;Chapter 1. A Theory of Reception for the History of Mathematics;20
8.1.1;1. INTRODUCTION;20
8.1.2;2. POINTS OF VIEW;21
8.1.3;3. RECEPTION;23
8.1.4;4. THE CONTEXT OF MATHEMATICS;25
8.1.5;5. THE RECEPTION OF TOPOLOGY;26
8.1.6;6. SUMMARY;29
8.1.7;NOTES;30
8.1.8;BIBLIOGRAPHY;30
8.2;Chapter 2. Riemann's Habilitationsvortrag And the Synthetic A Priori Status of Geometry;34
8.2.1;CONCLUSION;53
8.2.2;NOTES;54
9;Part II: Foundations of Mathematics;64
9.1;Chapter 3. Cantor's Views on the Foundations of Mathematics;66
9.1.1;ACKNOWLEDGEMENTS;79
9.1.2;ARCHIVAL SOURCES;79
9.1.3;REFERENCES;80
9.2;Chapter 4. Kronecker's Views on the Foundations of Mathematics;84
9.2.1;NOTES;92
9.3;Chapter 5. Towards A History of Cantor's Continuum Problem;96
9.3.1;1. INTRODUCTION;96
9.3.2;2. THE ORIGINS OF THE CONTINUUM HYPOTHESIS;99
9.3.3;3. DERIVED SETS: AN INTERLUDE;100
9.3.4;4. MITTAG-LEFFLER AND INFINITE ORDINALS;101
9.3.5;6. DEDEKIND AND THE EQUIVALENCE THEOREM;103
9.3.6;7. CANTOR'S "GRUNDLAGEN";104
9.3.7;8. BENDIXSON AND PERFECT SETS;106
9.3.8;9. THE CONTINUUM HYPOTHESIS AND CLOSED SETS;107
9.3.9;10. THE NEGATION OF THE CONTINUUM HYPOTHESIS;108
9.3.10;11. TANNERY'S ATTEMPTED PROOF;110
9.3.11;12. MITTAG-LEFFLER AND THE EARLY RECEPTION OF SET THEORY;111
9.3.12;13. THE EARLY RECEPTION OF CH IN GERMANY AND FRANCE;114
9.3.13;14. CANTOR'S LAST WORK;116
9.3.14;15. RUSSELL'S CHANGING PERSPECTIVE;117
9.3.15;16. C.S. PEIRCE AND THE BETH HIERARCHY;118
9.3.16;17. BOREL;121
9.3.17;18. SCHOENFLIES AND HlLBERT;121
9.3.18;19. BERNSTEIN, HAUSDORFF, AND LEVI;122
9.3.19;20. KÖNIG'S ATTEMPTED REFUTATION;124
9.3.20;21. BERNSTEIN'S ATTEMPTED PROOF;125
9.3.21;22. HAUSDORFF AND ORDER-TYPES;125
9.3.22;APPENDIX;127
9.3.23;NOTES;130
10;Part III: National Styles in Algebra;140
10.1;Chapter 6. British Synthetic vs. French Analytic Styles of Algebra In the Early American Republic;142
10.1.1;JOHN FARRAR'S TRANSLATIONS OF FOREIGN ALGEBRAS;146
10.1.2;CHARLES DAVIES AND BOURDON'S ELEMENTS OF ALGEBRA;154
10.1.3;BENJAMIN PEIRCE'S ELEMENTARY TREATISE ON ALGEBRA;160
10.1.4;CONCLUSION;162
10.2;Chapter 7. Toward a History of Nineteenth-Century Invariant Theory;174
10.2.1;INVARIANT THEORY AND THE BRITISH APPROACH;175
10.2.2;THE GERMAN APPROACH TO INVARIANT THEORY;187
10.2.3;A BRITISH PROOF OF GORDAN'S THEOREM;197
10.2.4;CONCLUSIONS;202
10.2.5;NOTES;205
11;Part IV: Geometry and the Emergence Of Transformation Groups;224
11.1;Chapter 8. The Early Geometrical Works of Sophus Lie and Felix Klein;226
11.1.1;INTRODUCTION;226
11.1.2;1. LINE GEOMETRY CIRCA 1870;229
11.1.3;2. KLEIN'S FIRST PUBLICATIONS;234
11.1.4;3. LIE'S THEORY OF THE IMAGINARY;241
11.1.5;4. LIE AND KLEIN IN BERLIN;244
11.1.6;5. LIE'S WORK ON TETRAHEDRAL COMPLEXES;247
11.1.7;6. LIE'S LINE-TO-SPHERE TRANSFORMATION;256
11.1.8;7. THE ROLE OF "ÜBERTRAGUNGSPRINZIPIEN" IN KLEIN'S THOUGHT;269
11.1.9;8. CONCLUDING REMARKS;281
11.1.10;NOTES;283
11.2;Chapter 9. Line Geometry, Differential Equations And the Birth of Lie's Theory of Groups;292
11.2.1;1. THE GEOMETRY OF TETRAHEDRAL COMPLEXES;294
11.2.2;2. THE SPHERE MAPPING;312
11.2.3;3. THE WINTER OF 1873-74;326
11.2.4;NOTES;331
12;Part V: Projective and Algebraic Geometry;346
12.1;Chapter 10. The Background to Gergonne's Treatment of Duality: Spherical Trigonometry in the late 18th Century;348
12.1.1;I. GERGONNE'S APPROACH TO DUALITY;349
12.1.2;II. DUALITY IN SPHERICAL TRIGONOMETRY;351
12.1.3;III. ANALYTICAL TREATMENTS OF SPHERICAL TRIGONOMETRY;355
12.1.4;IV. EULER'S PAPER OF 1779;357
12.1.5;V. SUBSEQUENT DEVELOPMENTS IN THESE TWO DIRECTIONS;361
12.1.6;VI. WHAT WAS NEW IN GERGONNE'S APPROACH TO DUALITY?;363
12.1.7;NOTES;365
12.1.8;BIBLIOGRAPHY;373
12.2;Chapter 11. Algebraic Geometry in the late Nineteenth Century;378
12.2.1;INTRODUCTION;378
12.2.2;ABEL, CLEBSCH, AND RIEMANN;380
12.2.3;CREMONA TRANSFORMATIONS;387
12.2.4;THE ITALIAN SCHOOL;391
12.2.5;ITALIAN CONTRIBUTIONS TO ALGEBRAIC CURVE THEORY;393
12.2.6;ALGEBRAIC SURFACE THEORY;396
12.2.7;CONCLUDING REMARKS;399
12.2.8;BIBLIOGRAPHY;400
13;Part VI: Abel's Theorem;404
13.1;Chapter 12. Abel's Theorem;406
13.1.1;INTRODUCTION;406
13.1.2;1. EARLY ALGEBRAIC INTEGRALS;407
13.1.3;2. FAGNANO AND EULER;410
13.1.4;3. THE DISCOVERY OF ABEL'S THEOREM;417
13.1.5;4. THE CONTENT OF ABEL'S THEOREM;428
13.1.6;BIBLIOGRAPHY;436
14;Part VII: Number Theory;440
14.1;Chapter 13. Heinrich Weber and the Emergence of Class Field Theory;442
14.1.1;1. HISTORICAL CONCEPTS;442
14.1.2;2. EULER'S .-FUNCTION;445
14.1.3;3. DIRICHLET'S L-SERIES;448
14.1.4;4. DEDEKIND's .-FUNCTION;451
14.1.5;5. WEBER'S COMPLEX MULTIPLICATION, L-SERIES AND CLASS FIELD THEORY;454
14.1.6;6. DEDEKIND'S CONTRIBUTION TO CLASS FIELD THEORY;462
14.1.7;7. CHRONOLOGICAL TABLE TO HEINRICH WEBER;463
14.1.8;BIBLIOGRAPHY;464
15;Notes on the Contributors;468