Mathematical Thought From Ancient to Modern Times, Volume 3

Name: Mathematical Thought From Ancient to Modern Times, Volume 3 | Volume 1
Brand: OUP USA
Price: 30.19 EUR
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Volume 1

Morris Kline(Autor*in)

OUP USA (Verlag)

1. Auflage

Erschienen am 1. März 1990

439 Seiten

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978-0-19-977048-9 (ISBN)

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Inhalt

Cover
Contents
34. The Theory of Numbers in the Nineteenth Century
1. Introduction
2. The Theory of Congruences
3. Algebraic Numbers
4. The Ideals of Dedekind
5. The Theory of Forms
6. Analytic Number Theory
35. The Revival of Projective Geometry
1. The Renewal of Interest in Geometry
2. Synthetic Euclidean Geometry
3. The Revival of Synthetic Projective Geometry
4. Algebraic Projective Geometry
5. Higher Plane Curves and Surfaces
36. Non-Euclidean Geometry
1. Introduction
2. The Status of Euclidean Geometry About 1800
3. The Research on the Parallel Axiom
4. Foreshadowings of Non-Euclidean Geometry
5. The Creation of Non-Euclidean Geometry
6. The Technical Content of Non-Euclidian Geometry
7. The Claims of Lobatchevsky and Bolyai to Priority
8. The Implications of Non-Euclidean Geometry
37. The Differential Geometry of Gauss and Riemann
1. Introduction
2. Gauss's Differential Geometry
3. Riemann's Approach to Geometry
4. The Successors of Riemann
5. Invariants of Differential Forms
38. Projective and Metric Geometry
1. Introduction
2. Surfaces as Models of Non-Euclidean Geometry
3. Projective and Metric Geometry
4. Models and the Consistency Problem
5. Geometry from the Transformation Viewpoint
6. The Reality of Non-Euclidean Geometry
39. Algebraic Geometry
1. Background
2. The Theory of Algebraic Invariants
3. The Concept of Birational Transformations
4. The Function-Theoretic Approach to Algebraic Geometry
5. The Uniformization Problem
6. The Algebraic-Geometric Approach
7. The Arithmetic Approach
8. The Algebraic Geometry of Surfaces
40. The Instillation of Rigor in Analysis
1. Introduction
2. Functions and Their Properties
3. The Derivative
4. The Integral
5. Infinite Series
6. Fourier Series
7. The Status of Analysis
41. The Foundations of the Real and Transfinite Numbers
1. Introduction
2. Algebraic and Transcendental Numbers
3. The Theory of Irrational Numbers
4. The Theory of Rational Numbers
5. Other Approaches to the Real Number System
6. The Concept of an Infinite Set
7. The Foundation of the Theory of Sets
8. Transfinite Cardinals and Ordinals
9. The Status of Set Theory by 1900
42. The Foundations of Geometry
1. The Defects in Euclid
2. Contributions to the Foundations of Projective Geometry
3. The Foundations of Euclidean Geometry
4. Some Related Foundational Work
5. Some Open Questions
43. Mathematics as of 1900
1. The Chief Features of the Nineteenth-Century Developments
2. The Axiomatic Movement
3. Mathematics as Man's Creation
4. The Loss of Truth
5. Mathematics as the Study of Arbitrary Structures
6. The Problem of Consistency
7. A Glance Ahead
44. The Theory of Functions of Real Variables
1. The Origins
2. The Stieltjes Integral
3. Early Work on Content and Measure
4. The Lebesgue Integral
5. Generalizations
45. Integral Equations
1. Introduction
2. The Beginning of a General Theory
3. The Work of Hilbert
4. The Immediate Successors of Hilbert
5. Extensions of the Theory
46. Functional Analysis
1. The Nature of Functional Analysis
2. The Theory of Functionals
3. Linear Functional Analysis
4. The Axiomatization of Hilbert Space
47. Divergent Series
1. Introduction
2. The Informal Uses of Divergent Series
3. The Formal Theory of Asymptotic Series
4. Summability
48. Tensor Analysis and Differential Geometry
1. The Origins of Tensor Analysis
2. The Notion of a Tensor
3. Covariant Differentiation
4. Parallel Displacement
5. Generalizations of Riemannian Geometry
49. The Emergence of Abstract Algebra
1. The Nineteenth-Century Background
2. Abstract Group Theory
3. The Abstract Theory of Fields
4. Rings
5. Non-Associative Algebras
6. The Range of Abstract Algebra
50. The Beginnings of Topology
1. The Nature of Topology
2. Point Set Topology
3. The Beginnings of Combinational Topology
4. The Combinational Work of Poincaré
5. Combinatorial Invariants
6. Fixed Point Theorems
7. Generalizations and Extensions
51. The Foundations of Mathematics
1. Introduction
2. The Paradoxes of Set Theory
3. The Axiomatization of Set Theory
4. The Rise of Mathematical Logic
5. The Logistic School
6. The Intuitionist School
7. The Formalist School
8. Some Recent Developments
List of Abbreviations
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

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