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Mathematics - The Music of Reason - Jean A. Dieudonne

Mathematics - The Music of Reason

By: Jean A. Dieudonne, H. G. Dales (Translator), J.C. Dales (Translator)


Published: 20th July 1998
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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.

Introductionp. 1
Mathematics and Mathematiciansp. 7
The Concept of Mathematicsp. 7
A Mathematician's Lifep. 8
The Work of Mathematicians and the Mathematical Communityp. 12
Masters and Schoolsp. 14
The Nature of Mathematical Problemsp. 19
"Pure" Mathematics and "Applied" Mathematicsp. 19
Theoretical Physics and Mathematicsp. 21
Applications of Mathematics in the Classical Erap. 21
The Utilitarian Attacksp. 25
Fashionable Dogmasp. 26
Conclusionsp. 28
Objects and Methods in Classical Mathematicsp. 29
The Birth of Pre-Mathematical Ideasp. 31
The Idea of Proofp. 33
Axioms and Definitionsp. 35
Geometry, from Euclid to Hilbertp. 37
Numbers and Magnitudesp. 41
The Idea of Approximationp. 45
The Evolution of Algebrap. 49
The Method of Coordinatesp. 51
The Concept of Limit and the Infinitesimal Calculusp. 56
App. 1. Calculation of Ratios in Euclid's Book Vp. 64
App. 2. The Axiomatic Theory of Real Numbersp. 65
App. 3. Approximation of the Real Roots of a Polynomialp. 68
App. 4. Arguments by "Exhaustion"p. 70
App. 5. Applications of Elementary Algorithms of the Integral Calculusp. 72
Some Problems of Classical Mathematicsp. 77
Intractable Problems and Sterile Problemsp. 77
Prolific Problemsp. 82
App. 1. Prime Numbers of the Form 4k-1 or 6k-1p. 93
App. 2. The Decomposition of [actual symbol not reproducible](s) as a Eulerian Productp. 93
App. 3. Lagrange's Method for the Solution of ax[superscript 2] + bxy + cy[superscript 2] = n in Integersp. 95
App. 4. Bernoulli Numbers and the Zeta Functionp. 98
New Objects and New Methodsp. 103
New Calculationsp. 105
The First Structuresp. 121
The Language of Sets and General Structuresp. 133
Isomorphisms and Classificationsp. 146
Mathematics of Our Dayp. 152
Intuition and Structuresp. 163
App. 1 The Resolution of Quartic Equationsp. 167
App. 2 Additional Remarks on Groups and on the Resolution of Algebraic Equationsp. 168
App. 3 Additional Remarks on Rings and Fieldsp. 174
App. 4. Examples of Distancesp. 184
App. 5. Fourier Seriesp. 192
Problems and Pseudo-Problems about "Foundations"p. 203
Non-Eueclidean Geometriesp. 204
The Deepening of the Concept of Numberp. 213
Infinite Setsp. 218
"Paradoxes" and their Consequencesp. 222
The Rise of Mathematical Logicp. 228
The Concept of "Rigorous Proof"p. 235
App. 1. Geometry on a Surfacep. 239
App. 2. Models of the Real Numbersp. 245
App. 3. Theorems of Cantor and of his Schoolp. 247
Index 1. Historical Indexp. 255
Index 2. Standard Notationp. 281
Index 3. Index of Terminologyp. 282
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540533467
ISBN-10: 354053346X
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 287
Published: 20th July 1998
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 0.61