什么是數(shù)學：對思想和方法的基本研究（英文版·第2版）

PREFACE TO SECOND EDITION
PREFACE TO REVISED EDITIONS
PREFACE TO FIRST EDITION
How TO USE THE BOOK
WHAT IS MATHEMATICS?
CHAPTER Ⅰ. THE NATURAL NUMBERS
Introduction
1. Calculation with Integers
1. Laws of Arithraetic. 2. The Representation of Integers. 3. Computation in Systems Other than the Decimal.
2. The Infinitude of the Number System, Mathematical Induction
1. The Principle of Mathematical .Induction. 2. The Arithmetical Progression. 3. The Geometrical Progression. 4. The Sum of the First n Squares. 5. An Important Inequality. 6. The Binomial Theorem. 7. Further Remarks on Mathematical Induction.
SUPPLEMENT TO CHAPTER I. THE THEORY OF NUMBERS
Introduction
1. The Prime Numbers
1. Fundamental Facts. 2. The Distribution of the Primes. 3. Formulas Producing Primes. b. Primes in Aritlunetical Progressions. c. The Prime Number Theorem. d. Two Unsolved Problems Concerning Prime Numbers.
2. Congruences
1. General Concepts. 2. Fermat's Theorem. 3. Quadratic Residues.
3. Pythagorean Numbers and Fermat's Last Theorem
4. The Euclidean Algorithm
1. General Theory. 2. Application to the Fundamental Theorem of Arithmetic. 3. Euler's Function. Fermat's Theorem Again. 4. Continued Fractions. Diophantine Equations.
CHAPTER Ⅱ. THE NUMBER SYSTEM OF MATHEMATICS
Introduction
1. The Rational Numbers
1. Rational Numbers as a Device for Measuring. 2. Intrinsic Need for the Rational Numbers. Principal of Generation. 3. Geometrical Interpretation of Rational Numbers.
2. Incommensurable Segments, Irrational Numbers, and the Concept of Limit
1. Introduction. 2. Decimal Fractions. Infinite Decimals. 3. Limits. Infinite Geometrical Series. 4. Rational Numbers and Periodic Deci- maiN. 5. General Definition of Irrational Numbers by Nested
Intervals 6. Alternative Methods of Defining Irrational Numbers. Dedekind Cuts.
3. Remarks on Analytic Geometry
1. The Basic Principle. 2. Equations of Lines and Curves.
4. The Mathematical Analysis of Infinity
1. Fundamental Concepts. 2. The Denumerability of the Rational Numbers and the Non-Denumerability of the Continuum. 3. Cantor's "Cardinal Numbers." 4. The Indirect Method of Proof. 5. The Paradoxes of the Infinite. 6. The Foundations of Mathematics.
5. Complex Numbers
1. The Origin of Complex Numbers. 2. The Geometrical Interpretation of Complex Numbers. 3. De Moivre's Formula and the Roots of Unity. 4. The Fundamental Theorem of Algebra.
6. Algebraic and Transcendental Numbers
1. Definition and Existence. 2. Liouville's Theorem and the Construction of Transcendental Numbers.
SUPPLEMENT TO CHAPTER II. THE ALGEBRA OF SETS
1. General Theory. 2. Application to Mathematical Logic. 3. An Application to the Theory of Probability.
CHAPTER Ⅰ. GEOMETRICAL CONSTRUCTIONS. THE ALGEBRA OF NUMBER FIELDS
Introduction
Part Ⅰ. Impossibility Proofs and Algebra
1. Fundamental Geometrical Constructions
1. Construction of Fields and Square Root Extraction. 2. Regular Polygons. 3. Apollonius' Problem.
2. Constructible Numbers and Number Fields
1. General Theory. 2. All Constructible Numbers are Algebraic.
3. The Unsolvability of the Three Greek Problems
1. Doubling the Cube. 2. A Theorem on Cubic Equations. 3. Trisecting the Angle. 4. The Regular Heptagon. 5. Remarks on the Problem of Squaring the Circle.
Part Ⅱ. Various Methods for Performing Constructions
4. Geometrical Transformations. Inversion
1. General Remarks. 2. Properties of Inversion. 3. Geometrical Constrnction of Inverse Points. 4. How to Bisect a Segment and Find the Center of a Circle with the Compass Alone.
5. Constructions with Other Tools. Mascheroni Constructions with Compass Alone
1. A Classical Construction for Doubling the Cube. 2. Restriction to the Use of the Compass Alone. 3. Drawing with Mechanical Instruments. Mechanical Curves. Cycloids. 4. Linkages. PeauceUier's and Hart's Inversors.
6. More About Inversions and its Applications
1. Invariance of Angles. Families of Circles. 2. Application to the Problem of Apollonius. 3. Repeated Reflections.
CHAPTER Ⅳ. PROJECTIVE GEOMETRY. AXIOMATICS. NON-EucLIDEAN GEOMETRIES .
1. Introduction
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CHAPTER Ⅴ TOPOLOGY
CHAPTER Ⅵ FUNCTIONS AND LIMITS
CHAPTER Ⅶ MAXIMA AND MINIMA
CHAPTER Ⅷ THE CALCULUS
CHAPTER Ⅸ RECENT DEVELOPMENTS
APPENDIX: SUPPLEMENTARY REMARKS, PROBLEMS, AND EXERCISES
SUGGESTIONS FOR FURTHER READING
SUGGESTIONS FOR ADDITIONAL READING
INDEX