Turning points in the history of mathematics:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Springer
[2015]
|
| Schriftenreihe: | Compact Textbooks in Mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | ix, 109 Seiten Illustrationen (teilweise farbig) |
| ISBN: | 9781493932634 |
| ISSN: | 2296-4568 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804176248400773120 |
|---|---|
| adam_text | Titel: Turning Points in the History of Mathematics
Autor: Grant, Hardy
Jahr: 2015
Contents
1 Axiomatics-Euclid s and Hilbert s: From Material to Formal........ 1
1.1 Euclid s Elements........................................................... 1
1.2 Hilbert s Foundations ofGeometry.......................................... 3
1.3 The Modern Axiomatic Method ............................................ 5
1.4 Ancient vs. Modern Axiomatics............................................. 6
Problems and Projects..................................................... 7
References................................................................. 7
Further Reading............................................................ 8
2 Solution by Radicals of theCubic: From Equations to Groups
and from Real to Complex Numbers................................... 9
2.1 Introduction............................................................... 9
2.2 Cubic and Quartic Equations............................................... 10
2.3 Beyond the Quartic: Lagrange.............................................. 11
2.4 Ruffini, Abel, Galois........................................................ 13
2.5 Complex Numbers: Birth................................................... 13
2.6 Growth..................................................................... 15
2.7 Maturity.................................................................... 16
Problems and Projects..................................................... 17
References................................................................. 8
Further Reading............................................................ 18
3 Analytic Geometry: From the Marriage of Two Fields to the
Birth of a Third........................................................... 19
3.1 Introduction............................................................... 19
3.2 Descartes.................................................................. 19
3.3 Fermat..................................................................... 21
3.4 Descartes and Fermat s Works from a Modern Perspective................. 22
3.5 The Significance of Analytic Geometry..................................... 23
Problems and Projects..................................................... 24
References................................................................. 25
Further Reading............................................................ 25
4 Probability: From Games of Chance to an Abstract Theory.......... 27
4.1 The Pascal-Fermat Correspondence........................................ 27
4.2 Huygens:The First Book on Probability..................................... 29
4.3 Jakob Bernoulli s Ars Conjectandi (The Art of Conjecturing).................. 30
4.4 De Moivre s The Doctrine of Chances........................................ 32
4.5 Laplace s Thäorie Analytique des Probabilitäs................................ 32
4.6 Philosophy of Probability.................................................. 33
4.7 Probability as an Axiomatic Theory......................................... 33
4.8 Conclusion................................................................. 34
Problems and Projects ..................................................... 35
References................................................................. 35
Further Reading............................................................ 3S
Contents
5 Calculus: From Tangents and Areas to Derivatives
and Integrals............................................................. 37
5.1 Introduction ............................................................... 37
5.2 Seventeenth-Century Predecessors of Newton and Leibniz................. 37
5.3 Newton and Leibniz: The Inventors of Calculus............................. 40
5.4 The Eighteenth Century: Euler.............................................. 43
5.5 A Look Ahead: Foundations................................................ 45
Problems and Projects..................................................... 46
References................................................................. 46
Further Reading............................................................ 47
6 Gaussian Integers: From Arithmetic to Arithmetics.................. 49
6.1 Introduction............................................................... 49
6.2 Ancient Times.............................................................. 49
6.3 Fermat..................................................................... 49
6.4 Euler and the Bachet Equation x2 + 2 = y3................................... 51
6.5 Reciprocity Laws, Fermat s Last Theorem, Factorization of Ideals........... 51
6.6 Conclusion................................................................. 55
Problems and Projects..................................................... 55
References................................................................. 56
Further Reading............................................................ 56
7 Noneuclidean Geometry: From One Geometry to Many............. 57
7.1 Introduction............................................................... 57
7.2 Euclidean Geometry........................................................ 57
7.3 Attempts to Prove the Fifth Postulate...................................... 58
7.4 The Discovery (Invention) of Noneuclidean Geometry...................... 60
7.5 Some Implications of the Creation of Noneuclidean Geometry............. 62
Problems and Projects..................................................... 65
References................................................................. 66
Further Reading............................................................ 66
8 Hypercomplex Numbers: From Algebra toAlgebras................. 67
8.1 Introduction............................................................... 67
8.2 Hamilton and Complex Numbers........................................... 68
8.3 The Quaternions........................................................... 69
8.4 Beyond the Quaternions................................................... 70
Problems and Projects..................................................... 71
References................................................................. 72
Further Reading............................................................ 73
9 The Infinite: From Potential to Actual.................................. 75
9.1 The Greeks................................................................. 75
9.2 Before Cantor.............................................................. 76
9.3 Cantor...................................................................... 77
9.4 Paradoxes Lost............................................................. 78
9.5 Denumerable (Countable) Infinity.......................................... 78
Contents
9.6 Paradoxes Regained....................................................... 79
9.7 Arithmetic................................................................. 80
9.8 Two Major Problems...................................................... 80
9.9 Conclusion................................................................ 81
Problems and Projects.................................................... 82
References................................................................ 83
Further Reading........................................................... 83
10 Philosophy of Mathematics: From Hubert to Gödel................. 85
10.1 Introduction.............................................................. 85
10.2 Logicism.................................................................. 86
10.3 Formalism................................................................. 87
10.4 Gödel s Incompleteness Theorems........................................ 88
10.5 Mathematics and Faith.................................................... 89
10.6 Intuitionism............................................................... 89
10.7 Nonconstructive Proofs................................................... 90
10.8 Conclusion................................................................ 91
Problems and Projects.................................................... 92
References................................................................ 92
Further Reading........................................................... 93
11 Some Further Turning Points.......................................... 95
11.1 Notation: From Rhetorical to Symbolic.................................... 95
11.2 Space Dimensions: From 3 to n (n 3)...................................... 96
11.3 Pathological Functions: From Calculus to Analysis......................... 97
11.4 The Nature of Proof: From Axiom-Based to Computer-Assisted............ 99
11.5 Experimental Mathematics: From Humans to Machines................... 100
References................................................................ 101
Further Reading........................................................... 102
Index..................................................................... 105
|
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| illustrated | Illustrated |
| indexdate | 2024-07-10T07:29:10Z |
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| isbn | 9781493932634 |
| issn | 2296-4568 |
| language | English |
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| spelling | Grant, Hardy Verfasser (DE-588)1104858924 aut Turning points in the history of mathematics Hardy Grant, Israel Kleiner New York Springer [2015] © 2015 ix, 109 Seiten Illustrationen (teilweise farbig) txt rdacontent n rdamedia nc rdacarrier Compact Textbooks in Mathematics 2296-4568 Mathematics Algebra Geometry History Social sciences Mathematics / Study and teaching History of Mathematical Sciences Mathematics Education Mathematics in the Humanities and Social Sciences Geschichte Mathematik Sozialwissenschaften Kleiner, Israel Verfasser (DE-588)133618234 aut Erscheint auch als Online-Ausgabe 978-1-4939-3264-1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028987004&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Grant, Hardy Kleiner, Israel Turning points in the history of mathematics Mathematics Algebra Geometry History Social sciences Mathematics / Study and teaching History of Mathematical Sciences Mathematics Education Mathematics in the Humanities and Social Sciences Geschichte Mathematik Sozialwissenschaften |
| title | Turning points in the history of mathematics |
| title_auth | Turning points in the history of mathematics |
| title_exact_search | Turning points in the history of mathematics |
| title_full | Turning points in the history of mathematics Hardy Grant, Israel Kleiner |
| title_fullStr | Turning points in the history of mathematics Hardy Grant, Israel Kleiner |
| title_full_unstemmed | Turning points in the history of mathematics Hardy Grant, Israel Kleiner |
| title_short | Turning points in the history of mathematics |
| title_sort | turning points in the history of mathematics |
| topic | Mathematics Algebra Geometry History Social sciences Mathematics / Study and teaching History of Mathematical Sciences Mathematics Education Mathematics in the Humanities and Social Sciences Geschichte Mathematik Sozialwissenschaften |
| topic_facet | Mathematics Algebra Geometry History Social sciences Mathematics / Study and teaching History of Mathematical Sciences Mathematics Education Mathematics in the Humanities and Social Sciences Geschichte Mathematik Sozialwissenschaften |
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