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Mathematical Thought from Ancient to Modern Times : Volume 3 - Morris Kline

Mathematical Thought from Ancient to Modern Times

Volume 3

Paperback

Published: 1st March 1990
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This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the discipline's origins in Babylon and Egypt, the creation of geometry and trigonometry by the Greeks, and the role of mathematics in the medieval and early modern periods. Volume 2 focuses on calculus, the rise of analysis in the 19th century, and the number theories of Dedekind and Dirichlet. The concluding volume covers the revival of projective geometry, the emergence of abstract algebra, the beginnings of topology, and the influence of Godel on recent mathematical study.

"I have always had great regard for this book as the one which relates the development of modern mathematical ideas in a readable fashion."--Michael O'Reilly, University of Minnesota "I have always had great regard for this book as the one which relates the development of modern mathematical ideas in a readable fashion."--Michael O'Reilly, University of Minnesota

The Theory of Numbers in the Nineteenth Centuryp. 813
Introductionp. 813
The Theory of Congruencesp. 813
Algebraic Numbersp. 818
The Ideals of Dedekindp. 822
The Theory of Formsp. 826
Analytic Number Theoryp. 829
The Revival of Projective Geometryp. 834
The Renewal of Interest in Geometryp. 834
Synthetic Euclidean Geometryp. 837
The Revival of Synthetic Projective Geometryp. 840
Algebraic Projective Geometryp. 852
Higher Plane Curves and Surfacesp. 855
Non-Euclidean Geometryp. 861
Introductionp. 861
The Status of Euclidean Geometry About 1800p. 861
The Research on the Parallel Axiomp. 863
Foreshadowings of Non-Euclidean Geometryp. 867
The Creation of Non-Euclidean Geometryp. 869
The Technical Content of Non-Euclidian Geometryp. 874
The Claims of Lobatchevsky and Bolyai to Priorityp. 877
The Implications of Non-Euclidean Geometryp. 879
The Differential Geometry of Gauss and Riemannp. 882
Introductionp. 882
Gauss's Differential Geometryp. 882
Riemann's Approach to Geometryp. 889
The Successors of Riemannp. 896
Invariants of Differential Formsp. 899
Projective and Metric Geometryp. 904
Introductionp. 904
Surfaces as Models of Non-Euclidean Geometryp. 904
Projective and Metric Geometryp. 906
Models and the Consistency Problemp. 913
Geometry from the Transformation Viewpointp. 917
The Reality of Non-Euclidean Geometryp. 921
Algebraic Geometryp. 924
Backgroundp. 924
The Theory of Algebraic Invariantsp. 925
The Concept of Birational Transformationsp. 932
The Function-Theoretic Approach to Algebraic Geometryp. 934
The Uniformization Problemp. 937
The Algebraic-Geometric Approachp. 939
The Arithmetic Approachp. 942
The Algebraic Geometry of Surfacesp. 943
The Instillation of Rigor in Analysisp. 947
Introductionp. 947
Functions and Their Propertiesp. 949
The Derivativep. 954
The Integralp. 956
Infinite Seriesp. 961
Fourier Seriesp. 966
The Status of Analysisp. 972
The Foundations of the Real and Transfinite Numbersp. 979
Introductionp. 979
Algebraic and Transcendental Numbersp. 980
The Theory of Irrational Numbersp. 982
The Theory of Rational Numbersp. 987
Other Approaches to the Real Number Systemp. 990
The Concept of an Infinite Setp. 992
The Foundation of the Theory of Setsp. 994
Transfinite Cardinals and Ordinalsp. 998
The Status of Set Theory by 1900p. 1002
The Foundations of Geometryp. 1005
The Defects in Euclidp. 1005
Contributions to the Foundations of Projective Geometryp. 1007
The Foundations of Euclidean Geometryp. 1010
Some Related Foundational Workp. 1015
Some Open Questionsp. 1017
Mathematics as of 1900p. 1023
The Chief Features of the Nineteenth-Century Developmentsp. 1023
The Axiomatic Movementp. 1026
Mathematics as Man's Creationp. 1028
The Loss of Truthp. 1032
Mathematics as the Study of Arbitrary Structuresp. 1036
The Problem of Consistencyp. 1038
A Glance Aheadp. 1039
The Theory of Functions of Real Variablesp. 1040
The Originsp. 1040
The Stieltjes Integralp. 1041
Early Work on Content and Measurep. 1041
The Lebesgue Integralp. 1044
Generalizationsp. 1050
Integral Equationsp. 1052
Introductionp. 1052
The Beginning of a General Theoryp. 1056
The Work of Hilbertp. 1060
The Immediate Successors of Hilbertp. 1070
Extensions of the Theoryp. 1073
Functional Analysisp. 1076
The Nature of Functional Analysisp. 1076
The Theory of Functionalsp. 1077
Linear Functional Analysisp. 1081
The Axiomatization of Hilbert Spacep. 1091
Divergent Seriesp. 1096
Introductionp. 1096
The Informal Uses of Divergent Seriesp. 1098
The Formal Theory of Asymptotic Seriesp. 1103
Summabilityp. 1109
Tensor Analysis and Differential Geometryp. 1122
The Origins of Tensor Analysisp. 1122
The Notion of a Tensorp. 1123
Covariant Differentiationp. 1127
Parallel Displacementp. 1130
Generalizations of Riemannian Geometryp. 1133
The Emergence of Abstract Algebrap. 1136
The Nineteenth-Century Backgroundp. 1136
Abstract Group Theoryp. 1137
The Abstract Theory of Fieldsp. 1146
Ringsp. 1150
Non-Associative Algebrasp. 1153
The Range of Abstract Algebrap. 1156
The Beginnings of Topologyp. 1158
The Nature of Topologyp. 1158
Point Set Topologyp. 1159
The Beginnings of Combinational Topologyp. 1163
The Combinational Work of Poincarep. 1170
Combinatorial Invariantsp. 1176
Fixed Point Theoremsp. 1177
Generalizations and Extensionsp. 1179
The Foundations of Mathematicsp. 1182
Introductionp. 1182
The Paradoxes of Set Theoryp. 1183
The Axiomatization of Set Theoryp. 1185
The Rise of Mathematical Logicp. 1187
The Logistic Schoolp. 1192
The Intuitionist Schoolp. 1197
The Formalist Schoolp. 1203
Some Recent Developmentsp. 1208
List of Abbreviations
Index
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780195061376
ISBN-10: 0195061373
Series: Mathematical Thought from Ancient to Modern Times : Book 3
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 1211
Published: 1st March 1990
Publisher: Oxford University Press Inc
Country of Publication: US
Dimensions (cm): 23.1 x 15.2  x 2.2
Weight (kg): 0.63