什么是数学:对思想和方法的基本研究(英文版·第2版)(图灵原版数学·统计学系列)
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品牌: Richard Courant
基本信息·出版社：人民邮电出版社
·页码：566 页
·出版日期：2009年
·ISBN：7115206937/9787115206930
·条形码：9787115206930
·包装版本：2版
·装帧：平装
·开本：32
·正文语种：英语
·丛书名：图灵原版数学·统计学系列
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内容简介《什么是数学:对思想和方法的基本研究(英文版·第2版)》是世界著名的数学科普读物。它荟萃了许多数学的奇珍异宝，对数学世界做了生动而易懂的描述。内容涵盖代数、几何、微积分、拓扑等领域，其中还穿插了许多相关的历史和哲学知识。《什么是数学:对思想和方法的基本研究(英文版·第2版)》不仅是数学专业人员的必读之物，也是任何愿意做科学思考者的优秀读物。对于中学数学教师、高中生和大学生来说，这都是一本极好的参考书。
作者简介Richard Courant（1888-1972）20世纪杰出的数学家，哥廷根学派重要成员。曾担任纽约大学数学系主任和数学科学研究院院长，为了纪念他，纽约大学数学科学研究院1964年改名为柯朗数学科学研究院！成为世界上最大的应用数学研究中心。他写的书《数学物理方程》为每一个物理学家所熟知，而他的《微积分学》也被认为是该学科的代表作。
Herbert Robbins（1915－2001） 美国著名数学家和统计学家.他的研究涉及拓扑学. 测度论. 统计学等诸多领域.经验贝叶斯方法中的 Robbins引理,图论中的Robbins定理,还有Robbins 代数和Robbins问题都以他的名字命名。
媒体推荐“对整个数学领域申盼基本概念和方法透彻清晰的阐述……通俗易懂。”
——爱因斯坦
“本书是对整个数学领域中的基本概念及方法的透彻清晰的阐述。”
——爱因斯坦
“毫无疑问，这本书将会有深远的影响，它应当人手一册，无论是专业人员还是喜欢科学思考的任何人。”
——《纽约时报》
“一本极为完美的著作。”
——《数学评论》
“太妙了……这本书是巨大愉快和满足感的源泉。”
—— 应用物理杂志
“这本书是一部艺术著作。”
——Marston Morse（美国著名数学家）
“这是一本非常完美的著作……被数学家们视作科学的鲜血的一切基本思路和方法在这本书中用最简单的例子使之清晰明了，已经达到令人惊讶的程度。”
——Herman Weyl（著名数学家、物理学家）..
20世纪的数学已经发展到了让人望洋兴叹的地步，如何在一本可以带出去郊游时随便翻翻的作品中，把这门异常发达的学科的面貌体现在读者面前呢？柯朗的做法是搜集很多数学上的“珍品”，每个方面的讲述并非深不见底，但也不是蜻蜓点水。适当地深入，然后在该结束的时候结束。这种既非盲人摸象、亦非解剖大象的方法，可以让普通读者也能粗略领悟到数学无比精巧的结构之美……
好作品要让读者常读常新。例如《西游记》，比起那些佛教典籍，太容易读懂了，但好玩的故事和浅显的文字背后，其思想上的玄妙实在不是一语、一人可以道破、穷尽的，故而历来评论绵绵不断；即便是普通读者，碰到一些社会现象，与小说中的情节做些类比，也有新的感悟. 那么科学著作能否也达到同样的功效呢？至少，《什么是数学》这本书是做到了。
——《中华读书报》
编辑推荐《什么是数学:对思想和方法的基本研究(英文版·第2版)》是享有世界声誉的不朽名著，由Richard Courant和Herbert Robbins两位数学大家合著。原版初版于1941年，几十年来一直畅销不衰。书中充满了数学的奇珍异品，生动有趣地描绘出一幅数学世界的画卷。让你如入宝山。目不暇给。第2版由著名数学家lan Stewart增写了新的一章，阐述了数学的最新进展，包括四色定理和费马大定理的证明等。
这是一本人人都能读的数学书，将为你开启一扇认识数学世界的窗口。无论你是初学者还是专家，学生还是教师，哲学家还是工程师，通过这本书，你都将领略到数学之美，最终迷上数学。
目录
PREFACE TO SECOND EDITION
PREFACE TO REVISED EDITIONS
PREFACE TO FIRST EDITION
HOW TO USE THE BOOK
WHAT IS MATHEMATICS?
CHARTER I. THE NATURAL NUMBERS
Introduction
1. Calculation with Integers
1. Laws of Arithmetic. 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
I. Fundamental Facts. 2. The Distribution of the Primes. a. Formulas Producing Primes. b. Primes in Arithmetical 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 Generalization. 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 Decimals. 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 III. 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 Construction 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. Peaucellier'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 Apoilonius. 3. Repeated Reflections
CHAPTER IV. PROJECTIVE GEOMETRY. AXIOMATICS. NoN-EUCLIDEAN GEOMETRIES
I. Introduction
1. Classification of Geometrical Properties. lnvariance under Transformations. 2. Projective Transformations
2. Fundamental Concepts
1. The Group of Projective Transformations. 2. Desargues's Theorem
3.Cross-Ratio
1. Definition and Proof of Invariance. 2. Application to the Complete Quadrilateral
4. Parallelism and Infinity
1. Points at Infinity as "Ideal Points." 2. Ideal Elements and Projection. 3. Cross-Ratio with Elements at Infinity
5. Applications
1. Preliminary Remarks. 2. Proof of Desargues's Theorem in the Plane. 3. Pascal's Theorem. 4. Brianchon's Theorem. 5. Remark on Duality
6. Analytic Representation
1. Introductory Remarks. 2. Homogeneous Co6rdinates. The Algebraic Basis of Duality
7. Problems on Constructions with the Straightedge Alone
8. Conics and Quadric Surfaces
1. Elementary Metric Geometry of Conics. 2. Projective Properties of Conics. 3. Conics as Line Curves. 4. Pascal's and Brianchon's General Theorems for Conics. 5. The Hyperboloid
9. Axiomatics and Non-Euclidean Geometry
1. The Axiomatic Method. 2. Hyperbolic Non-Euchdean Geometry. 3. Geometry and Reality. 4. Poincare's Model. 5. Elliptic or Riemannian Geometry
APPENDIX. GEOMETRY IN MORE THAN THREE DIMENSIONS
1. Introduction. 2. Analytic Approach. 3. Geometrical or Combinatorial Approach
CHAPTER V. TOPOLOGY
Introduction
1. Euler's Formula for Polyhedra
2. Topological Properties of Figures
1. Topological Properties. 2. Connectivity
3. Other Examples of Topological Theorems
1. The Jordan Curve Theorem. 2. The Four Color Problem. 3. The Concept of Dimension. 4. A Fixed Point Theorem. 5. Knots
4. The Topological Classification of Surfaces
1. The Genus of a Surface. 2. The Euler Characteristic of a Surface. 3. One-Sided Surfaces
APPENDIX
1. The Five Color Theorem. 2. The Jordan Curve Theorem for Polygons. 3. The Fundamental Theorem of Algebra
CHAPTER VI. FUNCTIONS AND LIMITS
Introduction
1. Variable and Function
1. Definitions and Examples. 2. Radian Measure of Angles. 3. The Graph of a Function. Inverse Functions. 4. Compound Func
tions. 5. Continuity. 6. Functions of Several Variables. 7. Functions and Transformations
2. Limits
1. The Limit of a Sequence an 2. Monotone Sequences. 3. Euler's Number e. 4. The Number π 5. Continued Fractions
3. Limits by Continuous Approach
1. Introduction. General Definition. 2. Remarks on the IAmit Concept. 3. The Limit of sin x/x. 4. Limits as x
4. Precise Definition of Continuity .
5. Two Fundamental Theorems on Continuous Functions
1. Bolzano's Theorem. 2. Proof of Bolzano's Theorem. 3. Weierstrass' Theorem on Extreme Values. 4. A Theorem on Sequences. Compact Sets
6. Some Applications of Bolzano's Thoerem
1. Geometrical Applications. 2. Application to a Problem in Mechanics
SUPPLEMENT TO CHAPTER VI. MORE EXAMPLES ON LIMITS AND CONTINUITY
1. Examples of Limits
1. General Remarks. 2. The Limit of q. 3. The IAmit of p. 4. Discon tinuous Functions as Limits of Continuous Functions. 5. Limits by Iteration
2. Example on Continuity
CHAPTER VII. MAXIMA AND MINIMA
Introduction
1. Problems in Elementary Geometry
1. Maximum Area of a Triangle with Two Sides Given. 2. Heron's Thoerem. Extremum Property of Light Rays. 3. Applications to Problems on Triangles. 4. Tangent Properties of Ellipse and Hyperboll Corresponding Extremum Properties. 5. Extreme Distances to a Given Curve
2. A General Principal Underlying Extreme Value Problems
1. The Principle. 2. Examples
3. Stationary Points and the Differential Calculus
1. Extrema and Stationary Points. 2. Maxima and Minima of Functions of Several Variables. Saddle Points. 3. Minimax Points and Topology. 4. The Distance from a Point to a Surface
4. Schwax-z's Triangle Problem
1. Schwarz's Proof. 2. Another Proof. 3. Obtuse Triangles. 4. Triangles Formed by Light Rays. 5. Remarks Concerning Problems of Reflection and Ergodic Motion
5. Steiner's Problem
1. Problem and Solution. 2. Analysis of the Alternatives. 3. A Complementary Problem. 4. Remarks and Exercises. 5. Generalization to the Street Network Problem
6. Extrema and Inequalities
1. The Arithmetical and Geometrical Mean of Two Positive Quantifies. 2. Generalization to n Variables. 3. The Method of Least Squares
7. The Existence of an Extremum. Dirichlet's Principle
1. General Remarks. 2. Examples. 3. Elementary Extremum Problems 4. Difficulties m Higher Cases
8. The Isoperimetric Problem
9. Extremum Problems with Boundary Conditions. Connection Between Steiner's Problem and the Isoperimetric Problem
10. The Calculus of Variations
1. Introduction. 2. The Calculus of Variations. Fermat's Principle in Optics. 3. Bemoulli's Treatment of the Brachistochrone Problem, 4. Geodesics on a Sphere. Geodesics and Maxi-Minima
11. Experimental Solutions of Minimum Problems. Soap Film Experiments.
1. Introduction. 2. Soap Film Experiments. 3. New Experiments on Plateau's Problem. 4. Experimental Solutions of Other Mathematical Problems
CHAPTER VIII. THE CALCULUS
Introduction
1. The Integral
1. Area as a Limit. 2. The Integral. 3. General Remarks on the Integral Concept. General Definition. 4. Examples of Integration. Integration of x. 5. Rules for the "Integral Calculus"
2. The Derivative
1. The Derivative as a Slope. 2. The Derivative as a Limit. 3. Examples. 4. Derivatives of Trigonometrical Functions. 5, Differentiation and Continuity. 6. Derivative and Velocity. Second Derivative and Acceleration. 7. Geometrical Meaning of the Second Derivative. 8. Maxima and Minima
3. The Technique of Differentiation
4. Leibniz' Notation and the "Infinitely Small"
5. The Fundamental Theorem of the Calculus
1. The Fundamental Theorem. 2. First Applications. Integration of x, cos x, sin x. Arc tan x. 3. Leibniz' Formula for π
6. The Exponential Function and the Logarithm
1. Definition and Properties of the Logarithm. Euler's Number e. 2. The Exponential Function. 3. Formulas for Differentiation of ex, ax . 4. Explicit Expressions for e, ex, and log x as Limits. 5. Infinite Series for the Logarithm. Numerical Calculation
7. Differential Equations
1. Definition. 2. The Differential Equation of the Exponential Function. Radioactive Disintegration. Law of Growth. Compound Interest. 3. Other Examples. Simplest Vibrations. 4. Newton's Law of Dynamics
SUPPLEMENT TO CHAPTER VIII
1. Matters of Principle
1. Differentiability. 2. The Integral. 3. Other Applications of the Concept of Integral. Work. Length
2. Orders of Magnitude
1. The Exponential Function and Powers of x. 2. Order of Magnitude of log (n!)
3. Infinite Series and Infinite Products
1. Infinite Series of Functions. 2. Euler's Formula, cos x + i sin x = eix. 3. The Harmonic Series and the Zeta Function. Euler's Product for the Sine
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
Arithmetic and Algebra
Analytic Geometry
Geometrical Constructions
Projective and Non-Euclidean Geometry
Topology
Functions, Limits, and Continuity
Maxima and Minima
The Calculus
Technique of Integration
SUGGESTIONS FOR FURTHER READING
SUGGESTIONS FOR ADDITIONAL READING
INDEX
……[看更多目录]
序言What Is Mathematics? is one of the great classics, a sparkling collection of mathematical gems, one of whose aims was to counter the idea that "mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician." In short, it wanted to put the meaning back into mathematics. But it was meaning of a very different kind from physical reality, for the meaning of mathematical objects states "only the relationships between mathematically 'undefined objects' and the rules governing operations with them." It doesn't matter what mathematical things are: it's what they do that counts. Thus mathematics hovers uneasily between the real and the not-real; its meaning does not reside in formal abstractions, but neither is it tangible. This may cause problems for philosophers who like tidy categories, but it is the great strength of mathematics——what I have elsewhere called its "unreal reality." Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely In either. .
I first encountered What Is Mathematics? in 1963. I was about to take up a place at Cambridge University, and the book was recommended reading for prospective mathematics students. Even today, anyone who wants an advance look at university mathematics could profitably skim through its pages. However, you do not have to be a budding mathematician to get a great deal of pleasure and insight out of Courant and Robbins's masterpiece. You do need a modest attention span, an interest in mathematics for its own sake, and enough background not to feel out of your depth. High-school algebra, basic calculus, and trigonometric functions are enough, although a bit of Euclidean geometry helps.
One might expect a book whose most recent edition was prepared nearly fifty years ago to seem old-fashioned, its terminology dated, its viewpoint out of line with current fashions. In fact, What Is Mathematics? has worn amazingly well. Its emphasis on problem-solving is up to date, and its choice of material has lasted so well that not a single word or symbol had to be deleted from this new edition.
In case you imagine this is because nothing ever changes in mathematics, I direct your attention to the new chapter, "Recent Developments," which will show you just how rapid the changes have been. No, the book has worn well because although mathematics is still growing, it is the sort of subject in which old discoveries seldom become obsolete. You cannot "unprove" a theorem. True, you might occasionally find that a long-accepted proof is wrong——it has happened. But then it was never proved in the first place. However, new viewpoints can often render old proofs obsolete, or old facts no longer interesting. What Is Mathematics? has worn well because Richard Courant and Herbert Robbins displayed impeccable taste in their choice of material. ..
Formal mathematics is like spelling and grammar——a matter of the correct application of local rules. Meaningful mathematics is like journalism-it tells an interesting story. Unlike some journalism, the story has to be true. The best mathematics is like literature——it brings a story to life before your eyes and involves you in it, intellectually and emotionally. Mathematically speaking, What Is Mathematics? is a very literate work. The main purpose of the new chapter is to bring Courant and Robbins's stories up to date——for example, to describe proofs of the Four Color Theorem and Fermat's Last Theorem. These were major open problems when Courant and Robbins wrote their masterpiece, but they have since been solved. I do have one genuine mathematical quibble (see 9 of "Recent Developments"). I think that the particular issue involved is very much a case where the viewpoint has changed. Courant and Robbins's argument is correct, within their stated assumptions, but those assumptions no longer seem as reasonable as they did.
I have made no attempt to introduce new topics that have recently come to prominence, such as chaos, broken symmetry, or the many other intriguing mathematical inventions and discoveries of the late twentieth century. You can find those in many sources, in particular my book From Here to Infinity, which can be seen as a kind of companionpiece to this new edition of What Is Mathematics?. My rule has been to add only material that brings the original up to date——although I have bent it on a few occasions and have been tempted to break it on others.
What Is Mathematics?
Unique.
Ian Stewart
Coventry
June 1995
文摘插图：
In quite a different way mathematical induction is used to establish the truth of a mathematical theorem for an inf.mite sequence of cases. the first，the second，the third，and so on without exception.Let US denote by A a statement that involves an arbitrary integer For example,A may be the statement，“The sum of the angles in a convex polygon of n+2 sides is n times 180 degrees.”or A may be the as- sertion.“By drawing n lines in a plane we cannot divide the plane into more than 2”parts."To prove such a theorem for every integer n it does not suffice to prove it separately for the first 10 or 100 or even t000 values of n.This indeed would correspond to the attitude of empirical induction.Instead，we must use a method of strictly mathematical and non-empirical reasoning whose character will be indicated by the following proofs for the special examples A and A.In the case A，we know that for n=1 the polygon is a triangle.and from elementary geometry the sum of the angles iS known to be 1·180。.For a quadri- lateral，n=2，we draw a diagonal which divides the quadrilateral into two triangles.TMs shows immediately that the sum of the angles of the quadrilateral is equal to the sum of the angles in the two triangles， which yields 180。+180。=2·180。.
