Jean Dieudonne

Mathematics -The Music of Reason

Translated by H. G. and J. C. Dales With 41 Figures

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Contents

Introduction 1

I Mathematics and Mathematicians 7

1. The Concept of Mathematics 7 2. A Mathematician's Life 8 3. The Work of Mathematicians and the

Mathematical Community 12 4. Masters and Schools 14

II The Nature of Mathematical Problems 19

1. "Pure" Mathematics and "Applied" Mathematics 19 2. Theoretical Physics and Mathematics 21 3. Applications of Mathematics in the Classical Era 21 4. The Utilitarian Attacks 25 5. Fashionable Dogmas 26 6. Conclusions 28

III Objects and Methods in Classical Mathematics 29

1. The Birth of Pre-Mathematical Ideas 31 2. The Idea of Proof 33 3. Axioms and Defmitions 35 4. Geometry, from Euclid to Hubert 37 5. Numbers and Magnitudes 41 6. The Idea of Approximation 45 7. The Evolution of Algebra 49 8. The Method of Coordinates 51 9. The Concept of Limit and the Infinitesimal Calculus 56

Appendix

1. Calculation of Ratios in Euclid's Book V 64 2. The Axiomatic Theory of Real Numbers 65 3. Approximation of the Real Roots of a Polynomial 68 4. Arguments by "Exhaustion" 70 5. Applications of Elementary Algorithms of the

Integral Calculus 72

VIII Contents

I V S o m e P r o b l e m s of Classical M a t h e m a t i c s 77

1. Intractable Problems and Sterile Problems 77 A. Perfect Numbers 77 B. Fermat Numbers 78 C. The Four-Colour Problem 79 D. Problems of Elementary Geometry 80

2. Prolific Problems 82 A. The Sums of Squares 82 B. The Properties of Prime Numbers 86 C. The Beginnings of Algebraic Geometry 91

A p p e n d i x 93

1. P r ime Numbers of the Form 4k-l or 6k-l 93 2. The Decomposition of £(s) as a Eulerian Product 93 3. Lagrange's Method for the Solution of

ax2 + bxy + cy2 = n in Integers 95 4. Bernoulli Numbers and the Zeta Function 98

V N e w O b j e c t s a n d N e w M e t h o d s 103

1. New Calculations 105 A. Complex numbers 105 B. Vectors 108 C. Algebraic Calculation on Functions 111 D. Permuta t ions and Substitutions 113 E. Displacements and Afßnities 118 F. Calculation of Congruences of Natural Numbers 119 G. The Calculation of Classes of Quadratic Forms 120

2. The First Structures <*. 121 A. The Principal Properties of the

Laws of Composition 121 B. Groups of Transformations 124 C. "Abstract" Groups 128 D. Quaternions and Algebras 129

3. The Language of Sets and General Structures 133 A. The Concept of Set 133 B. The Language of Sets 134 C. Algebraic Structures 136 D. Order Structures 139 E. Metrie Spaces and Topological Concepts 141 F. Superposition and Dissociation of Structures 142

4. Isomorphisms and Classification 146 A. Isomorphisms 146 B. Problems of Classification 149 C. The Invention of Functors and of Structures 150

Contents IX

5. Mathematics of Our Day 152 A. A Panorama of Mathematics 152 B. Specialists and Generalists 161 C. The Evolution of Mathematical Theories 161

6. Intuition and Structures 163

A p p e n d i x 167

1. The Resolution of Quartic Equations 167 2. Additional Remarks on Groups and on the

Resolution of Algebraic Equations 168 A. The Symmetrie Group S n 168 B. The Galois Group of an Equation 168 C. Galois Groups and Groups of Automorphisms 169 D. Normal Subgroups and Simple Groups 170 E. Rotations of the Cube 172

3. Additional Remarks on Rings and Fields 174 A. Congruences modulo a Prime Number 174 B. The Ring Z[i] of Gaussian Integers 176 C. Congruences modulo a Polynomial 179 D. The/Field of Algebraic Functions 181 E. Remarks on Ordered Fields 183

4. Examples of Distances 184 A. Distances on the Space of Continuous Functions 184 B. Prehilbert Spaces 187 C. Hubert Spaces 190 D. p-adic Distances 191

5. Fourier Series 192 A. Trigonometrie Series and Fourier Coefficients 192 B. The Convergence of Fourier Series 195 C. Fourier Series of Bernoulli Polynomials 199 D. Cantor 's Problems 200

V I P r o b l e m s and P s e u d o - P r o b l e m s a b o u t "Foundat ions" . . . 203

1. Non-Euclidean Geometries 204 A. The Parallel Postulate 204 B. Geometry on a Surface 207 C. Models of Non-Euclidean Geometry 208

2. The Deepening of the Concept of Number 213 A. Irrational Numbers 213 B. Monsters 215 C. The Axiomatization of Arithmetic 216

3. Infinite Sets 218 A. Infinite Sets and Natural Numbers 218 B. The Comparison of Infinite Sets 220

X Contents

4. "Paradoxes" and their Consequences 222 A. Existence and Constructions 222 B. The Mixed Fortunes of the Concept of Set

and the Axiom of Choice 224 C. Paradoxes and Formalization 226

5. The Rise of Mathematical Logic 228 A. The Formalization of Logic 228 B. Metamathemat ics 229 C. The Triumphs of Mathematical Logic 230 D. Mathematicians ' Reactions 233 E. Relations between Mathematics and Logic 234

6. The Concept of "Rigorous Proof" 235

A p p e n d i x 239

1. Geometry on a Surface 239 A. Skew curves 239 B. Curves on a Surface 239 C. Poincare's Half-Plane 241

2. Models of the Real Numbers 245 A. The Theory of Rational Numbers 245 B. The Model of Dedekind (Simplified Account) 246 C. The Model of Meray-Cantor (Simplified Account) 246

3. Theorems of Cantor and of his School 247 A. The Set of Real Numbers is not Countable 247 B. The Order Relation among Cardinais 248

C. The Equipotence of IR and IR2 = 1R x IR y 250 D. The Cardinal of the Set of Subsets 253

I n d e x 255

1. Historical Index 255 2. Standard Notation 281 3. Index of Terminology 282