Mathematical Thought From Ancient to Modern Times, Volume 3

by Kline, Morris

• ISBN13: 9780195061376
• ISBN10: 0195061373
• Format: Paperback
• Copyright: 1990-03-01
• Publisher: Oxford University Press

Summary

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 nineteenth 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 Gödel on recent mathematical study.

Author Biography

Morris Kline is Professor of Mathematics, Emeritus, at the Courant Institute of Mathematical Sciences, New York University, where he directed the Division of Electromagnetic Research for twenty years

Table of Contents

The Theory of Numbers in the Nineteenth Century	p. 813
Introduction	p. 813
The Theory of Congruences	p. 813
Algebraic Numbers	p. 818
The Ideals of Dedekind	p. 822
The Theory of Forms	p. 826
Analytic Number Theory	p. 829
The Revival of Projective Geometry	p. 834
The Renewal of Interest in Geometry	p. 834
Synthetic Euclidean Geometry	p. 837
The Revival of Synthetic Projective Geometry	p. 840
Algebraic Projective Geometry	p. 852
Higher Plane Curves and Surfaces	p. 855
Non-Euclidean Geometry	p. 861
Introduction	p. 861
The Status of Euclidean Geometry About 1800	p. 861
The Research on the Parallel Axiom	p. 863
Foreshadowings of Non-Euclidean Geometry	p. 867
The Creation of Non-Euclidean Geometry	p. 869
The Technical Content of Non-Euclidian Geometry	p. 874
The Claims of Lobatchevsky and Bolyai to Priority	p. 877
The Implications of Non-Euclidean Geometry	p. 879
The Differential Geometry of Gauss and Riemann	p. 882
Introduction	p. 882
Gauss's Differential Geometry	p. 882
Riemann's Approach to Geometry	p. 889
The Successors of Riemann	p. 896
Invariants of Differential Forms	p. 899
Projective and Metric Geometry	p. 904
Introduction	p. 904
Surfaces as Models of Non-Euclidean Geometry	p. 904
Projective and Metric Geometry	p. 906
Models and the Consistency Problem	p. 913
Geometry from the Transformation Viewpoint	p. 917
The Reality of Non-Euclidean Geometry	p. 921
Algebraic Geometry	p. 924
Background	p. 924
The Theory of Algebraic Invariants	p. 925
The Concept of Birational Transformations	p. 932
The Function-Theoretic Approach to Algebraic Geometry	p. 934
The Uniformization Problem	p. 937
The Algebraic-Geometric Approach	p. 939
The Arithmetic Approach	p. 942
The Algebraic Geometry of Surfaces	p. 943
The Instillation of Rigor in Analysis	p. 947
Introduction	p. 947
Functions and Their Properties	p. 949
The Derivative	p. 954
The Integral	p. 956
Infinite Series	p. 961
Fourier Series	p. 966
The Status of Analysis	p. 972
The Foundations of the Real and Transfinite Numbers	p. 979
Introduction	p. 979
Algebraic and Transcendental Numbers	p. 980
The Theory of Irrational Numbers	p. 982
The Theory of Rational Numbers	p. 987
Other Approaches to the Real Number System	p. 990
The Concept of an Infinite Set	p. 992
The Foundation of the Theory of Sets	p. 994
Transfinite Cardinals and Ordinals	p. 998
The Status of Set Theory by 1900	p. 1002
The Foundations of Geometry	p. 1005
The Defects in Euclid	p. 1005
Contributions to the Foundations of Projective Geometry	p. 1007
The Foundations of Euclidean Geometry	p. 1010
Some Related Foundational Work	p. 1015
Some Open Questions	p. 1017
Mathematics as of 1900	p. 1023
The Chief Features of the Nineteenth-Century Developments	p. 1023
The Axiomatic Movement	p. 1026
Mathematics as Man's Creation	p. 1028
The Loss of Truth	p. 1032
Mathematics as the Study of Arbitrary Structures	p. 1036
The Problem of Consistency	p. 1038
A Glance Ahead	p. 1039
The Theory of Functions of Real Variables	p. 1040
The Origins	p. 1040
The Stieltjes Integral	p. 1041
Early Work on Content and Measure	p. 1041
The Lebesgue Integral	p. 1044
Generalizations	p. 1050
Integral Equations	p. 1052
Introduction	p. 1052
The Beginning of a General Theory	p. 1056
The Work of Hilbert	p. 1060
The Immediate Successors of Hilbert	p. 1070
Extensions of the Theory	p. 1073
Functional Analysis	p. 1076
The Nature of Functional Analysis	p. 1076
The Theory of Functionals	p. 1077
Linear Functional Analysis	p. 1081
The Axiomatization of Hilbert Space	p. 1091
Divergent Series	p. 1096
Introduction	p. 1096
The Informal Uses of Divergent Series	p. 1098
The Formal Theory of Asymptotic Series	p. 1103
Summability	p. 1109
Tensor Analysis and Differential Geometry	p. 1122
The Origins of Tensor Analysis	p. 1122
The Notion of a Tensor	p. 1123
Covariant Differentiation	p. 1127
Parallel Displacement	p. 1130
Generalizations of Riemannian Geometry	p. 1133
The Emergence of Abstract Algebra	p. 1136
The Nineteenth-Century Background	p. 1136
Abstract Group Theory	p. 1137
The Abstract Theory of Fields	p. 1146
Rings	p. 1150
Non-Associative Algebras	p. 1153
The Range of Abstract Algebra	p. 1156
The Beginnings of Topology	p. 1158
The Nature of Topology	p. 1158
Point Set Topology	p. 1159
The Beginnings of Combinational Topology	p. 1163
The Combinational Work of Poincare	p. 1170
Combinatorial Invariants	p. 1176
Fixed Point Theorems	p. 1177
Generalizations and Extensions	p. 1179
The Foundations of Mathematics	p. 1182
Introduction	p. 1182
The Paradoxes of Set Theory	p. 1183
The Axiomatization of Set Theory	p. 1185
The Rise of Mathematical Logic	p. 1187
The Logistic School	p. 1192
The Intuitionist School	p. 1197
The Formalist School	p. 1203
Some Recent Developments	p. 1208
List of Abbreviations
Index
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