Mathematics-The Music of Reason

by Dieudonne, Jean A.; Dales, H. G.; Dales, J. C.

• ISBN13: 9783540533467
• ISBN10: 354053346X
• Format: Hardcover
• Copyright: 1992-09-01
• Publisher: Springer Verlag

Summary

This book is of interest for students of mathematics or of neighboring subjects like physics, engineering, computer science, and also for people who have at least school level mathematics and have kept some interest in it. Also good for younger readers just reaching their final school year of mathematics.

Table of Contents

Introduction	p. 1
Mathematics and Mathematicians	p. 7
The Concept of Mathematics	p. 7
A Mathematician's Life	p. 8
The Work of Mathematicians and the Mathematical Community	p. 12
Masters and Schools	p. 14
The Nature of Mathematical Problems	p. 19
"Pure" Mathematics and "Applied" Mathematics	p. 19
Theoretical Physics and Mathematics	p. 21
Applications of Mathematics in the Classical Era	p. 21
The Utilitarian Attacks	p. 25
Fashionable Dogmas	p. 26
Conclusions	p. 28
Objects and Methods in Classical Mathematics	p. 29
The Birth of Pre-Mathematical Ideas	p. 31
The Idea of Proof	p. 33
Axioms and Definitions	p. 35
Geometry, from Euclid to Hilbert	p. 37
Numbers and Magnitudes	p. 41
The Idea of Approximation	p. 45
The Evolution of Algebra	p. 49
The Method of Coordinates	p. 51
The Concept of Limit and the Infinitesimal Calculus	p. 56
App. 1. Calculation of Ratios in Euclid's Book V	p. 64
App. 2. The Axiomatic Theory of Real Numbers	p. 65
App. 3. Approximation of the Real Roots of a Polynomial	p. 68
App. 4. Arguments by "Exhaustion"	p. 70
App. 5. Applications of Elementary Algorithms of the Integral Calculus	p. 72
Some Problems of Classical Mathematics	p. 77
Intractable Problems and Sterile Problems	p. 77
Prolific Problems	p. 82
App. 1. Prime Numbers of the Form 4k-1 or 6k-1	p. 93
App. 2. The Decomposition of [actual symbol not reproducible](s) as a Eulerian Product	p. 93
App. 3. Lagrange's Method for the Solution of ax[superscript 2] + bxy + cy[superscript 2] = n in Integers	p. 95
App. 4. Bernoulli Numbers and the Zeta Function	p. 98
New Objects and New Methods	p. 103
New Calculations	p. 105
The First Structures	p. 121
The Language of Sets and General Structures	p. 133
Isomorphisms and Classifications	p. 146
Mathematics of Our Day	p. 152
Intuition and Structures	p. 163
App. 1 The Resolution of Quartic Equations	p. 167
App. 2 Additional Remarks on Groups and on the Resolution of Algebraic Equations	p. 168
App. 3 Additional Remarks on Rings and Fields	p. 174
App. 4. Examples of Distances	p. 184
App. 5. Fourier Series	p. 192
Problems and Pseudo-Problems about "Foundations"	p. 203
Non-Eueclidean Geometries	p. 204
The Deepening of the Concept of Number	p. 213
Infinite Sets	p. 218
"Paradoxes" and their Consequences	p. 222
The Rise of Mathematical Logic	p. 228
The Concept of "Rigorous Proof"	p. 235
App. 1. Geometry on a Surface	p. 239
App. 2. Models of the Real Numbers	p. 245
App. 3. Theorems of Cantor and of his School	p. 247
Index 1. Historical Index	p. 255
Index 2. Standard Notation	p. 281
Index 3. Index of Terminology	p. 282
Table of Contents provided by Blackwell. All Rights Reserved.