What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

FOREWORD

PREFACE

WHAT IS MATHEMATICS?

CHAPTER I. THE NATURAL NUMBERS

§1. CALCULATION WITH INTEGERS

§2. THE INFINITUDE OF THE NUMBER SYSTEM. MATHEMATICAL INDUCTION

SUPPLEMENT TO CHAPTER I. THE THEORY OF NUMBERS

§1. THE PRIME NUMBERS

§2. CONGRUENCES

§3. PYTHAGOREAN NUMBERS AND FERMAT’S LAST THEOREM

§4. THE EUCLIDEAN ALGORITHM

CHAPTER II. THE NUMBER SYSTEM OF MATHEMATICS

§1. THE RATIONAL NUMBERS

§2. INCOMMENSURABLE SEGMENTS, IRRATIONAL NUMBERS, AND THE CONCEPT OF LIMIT

§3. REMARKS ON ANALYTIC GEOMETRY

§4. THE MATHEMATICAL ANALYSIS OF INFINITY

§5. COMPLEX NUMBERS

§6. ALGEBRAIC AND TRANSCENDENTAL NUMBERS

SUPPLEMENT TO CHAPTER II. THE ALGEBRA OF SETS

CHAPTER III. GEOMETRICAL CONSTRUCTIONS. THE ALGEBRA OF NUMBER FIELDS

PART I. IMPOSSIBILITY PROOFS AND ALGEBRA

§1. FUNDAMENTAL GEOMETRICAL CONSTRUCTIONS

§2. CONSTRUCTIBLE NUMBERS AND NUMBER FIELDS

§3. THE UNSOLVABILITY OF THE THREE GREEK PROBLEMS

PART II. VARIOUS METHODS FOR PERFORMING CONSTRUCTIONS

§4. GEOMETRICAL TRANSFORMATIONS. INVERSION

§5. CONSTRUCTIONS WITH OTHER TOOLS. MASCHERONI CONSTRUCTIONS WITH COMPASS ALONE

§6. MORE ABOUT INVERSION AND ITS APPLICATIONS

CHAPTER IV. PROJECTIVE GEOMETRY. AXIOMATICS. NON-EUCLIDEAN GEOMETRIES

§1. INTRODUCTION

§2. FUNDAMENTAL CONCEPTS

§3. CROSS-RATIO

§4. PARALLELISM AND INFINITY

§5. APPLICATIONS

§6. ANALYTIC REPRESENTATION

§7. PROBLEMS ON CONSTRUCTIONS WITH THE STRAIGHT-EDGE ALONE

§8. CONICS AND QUADRIC SURFACES

§9. AXIOMATICS AND NON-EUCLIDEAN GEOMETRY

APPENDIX. GEOMETRY IN MORE THAN THREE DIMENSIONS

CHAPTER V. TOPOLOGY

§1. EULER’S FORMULA FOR POLYHEDRA

§2. TOPOLOGICAL PROPERTIES OF FIGURES

§3. OTHER EXAMPLES OF TOPOLOGICAL THEOREMS

§4. THE TOPOLOGICAL CLASSIFICATION OF SURFACES

APPENDIX

CHAPTER VI. FUNCTIONS AND LIMITS

§1. VARIABLE AND FUNCTION

§2. LIMITS

§3. LIMITS BY CONTINUOUS APPROACH

§4. PRECISE DEFINITION OF CONTINUITY

§5. TWO FUNDAMENTAL THEOREMS ON CONTINUOUS FUNCTIONS

§6. SOME APPLICATIONS OF BOLZANO’S THEOREM

SUPPLEMENT TO CHAPTER VI. MORE EXAMPLES ON LIMITS AND CONTINUITY

§1. EXAMPLES OF LIMITS

§2. EXAMPLE ON CONTINUITY

CHAPTER VII. MAXIMA AND MINIMA

§1. PROBLEMS IN ELEMENTARY GEOMETRY

§2. A GENERAL PRINCIPLE UNDERLYING EXTREME VALUE PROBLEMS

§3. STATIONARY POINTS AND THE DIFFERENTIAL CALCULUS

§4. SCHWARZ’S TRIANGLE PROBLEM

§5. STEINER’S PROBLEM

§6. EXTREMA AND INEQUALITIES

§7. THE EXISTENCE OF AN EXTREMUM. DIRICHLET’S PRINCIPLE

§8. THE ISOPERIMETRIC PROBLEM

§9. EXTREMUM PROBLEMS WITH BOUNDARY CONDITIONS. CONNECTION BETWEEN STEINER’S PROBLEM AND THE ISOPERIMETRIC PROBLEM

§10. THE CALCULUS OF VARIATIONS

§11. EXPERIMENTAL SOLUTIONS OF MINIMUM PROBLEMS. SOAP FILM EXPERIMENTS

CHAPTER VIII. THE CALCULUS

§1. THE INTEGRAL

§2. THE DERIVATIVE

§3. THE TECHNIQUE OF DIFFERENTIATION

§4. LEIBNIZ’ NOTATION AND THE “INFINITELY SMALL”

§5. THE FUNDAMENTAL THEOREM OF THE CALCULUS

§6. THE EXPONENTIAL FUNCTION AND THE LOGARITHM

§7. DIFFERENTIAL EQUATIONS

SUPPLEMENT TO CHAPTER VIII

§1. MATTERS OF PRINCIPLE

§2. ORDERS OF MAGNITUDE

§3. INFINITE SERIES AND PRODUCTS

§4. THE PRIME NUMBER THEOREM OBTAINED BY STATISTICAL METHODS

CHAPTER IX. RECENT DEVELOPMENTS

§1. A FORMULA FOR PRIMES

§2. THE GOLDBACH CONJECTURE AND TWIN PRIMES

§3. FERMAT’S LAST THEOREM

§4. THE CONTINUUM HYPOTHESIS

§5. SET-THEORETIC NOTATION

§6. THE FOUR COLOR THEOREM

§7. HAUSDORFF DIMENSION AND FRACTALS

§8. KNOTS

§9. A PROBLEM IN MECHANICS

§10. STEINER’S PROBLEM

§11. SOAP FILMS AND MINIMAL SURFACES

§12. NONSTANDARD ANALYSIS

APPENDIX. SUPPLEMENTARY REMARKS, PROBLEMS, AND EXERCISES

SUGGESTIONS FOR FURTHER READING

SUGGESTIONS FOR ADDITIONAL READING