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AOccasionally, in some difficult musical compositions, there are beautiful, but easy parts - parts so simple a beginner could play them. So it is with mathematics as well. There are some discoveries in advanced mathematics that do not depend on specialized knowledge, not even on algebra, geometry, or trigonometry. Instead they may involve, at most, a little arithmetic, such as 'the sum of two odd numbers is even', and common sense. Each of the eight chapters in this book illustrates this phenomenon. Anyone can understand every step in the reasoning.

A 偶尔，在一些难于演绎的复杂乐章中，会有一些美妙但却容易上手的部分——这些部分如此简单，即使一个初学者也可以演奏它们。数学里也有这样的情况。高等数学中有一些发现并不仰仗专业的知识，甚至并不依赖代数、几何或三角函数。正相反，它们可能最多只涉及一点点算术知识，比如“两个奇数之和为偶数”，再加上常识即可。这本书八个章节中的每一章都能证明这一现象。任何人都能理解这种推理过程中的每一个步骤。

The thinking in each chapter uses at most only elementary arithmetic, and sometimes not even that. Thus all readers will have the chance to participate in a mathematical experience, to appreciate the beauty of mathematics, and to become familiar with its logical, yet intuitive, style of thinking.

每一章里的思维过程都最多只用到基本算术，有时候甚至连那个也用不上。这样一来所有的读者都将有机会参与一场数学的体验，体会数学的美妙，并逐渐熟悉它那富有逻辑性的然而也是发乎直觉的思考风格。

BOne of my purposes in writing this book is to give readers who haven't had the opportunity to see and enjoy real mathematics the chance to appreciate the mathematical way of thinking. I want to reveal not only some of the fascinating discoveries, but, more importantly, the reasoning behind them.

B 我写这本书的目的之一，就是为那些到目前为止还从未有机会看到和欣赏什么才是真正数学的读者提供一个机会，借此玩味数学的思考方式。我希望展示给读者的，不仅仅是一些引人入胜的发现，而且更重要的还是这些发现背后的思考推理行为。

In that respect, this book differs from most books on mathematics written for the general public. Some present the lives of colorful mathematicians. Others describe important applications of mathematics. Yet others go into mathematical procedures, but assume that the reader is adept in using algebra.

从以上角度来说，这本书不同于大多数为大众写就的关于数学的书籍。一些书描绘了某些数学家丰富多彩的人生。另一些叙述了数学的重大用途。还有一些虽则深入讲解了数学推演过程，但却假定读者必定在代数运用方面相当娴熟。(就是说，运用了太多术语和行话，写得太专业，不够深入浅出)。

CI hope this book will help bridge that notorious gap that separates the two cultures: the humanities and the sciences, or should I say the right brain (intuitive) and the left brain (analytical, numerical). As the chapters will illustrate, mathematics is not restricted to the analytical and numerical; intuition plays a significant role. The alleged gap can be narrowed or completely overcome by anyone, in part because each of us is far from using the full capacity of either side of the brain. To illustrate our human potential, I cite a structural engineer who is an artist, an electrical engineer who is an opera singer, an opera singer who published mathematical research, and a mathematician who publishes short stories.

C 我希望这本书将能有助于架起一座桥梁，跨越那道臭名昭著的裂隙，从而沟通两种文化：人文与科学，或者我也许应该将之称为右脑(直觉性的)与左脑(分析性的，数字性的)。正如以下书中章节将会展示的那样，数学并不仅仅局限于分析性和数字性；直觉扮演了一个重要角色。那道所谓的鸿沟可以被任何人缩短或完全弥合，部分原因在于我们中的每个人都还远没有充分运用大脑任何一侧的全部能力。为了说明我们人类的潜能，我列举了若干例证：一个结构工程师同时也是一位艺术家，一名电气工程师身兼歌剧演唱家，一位歌剧演唱家发表过数学研究专著，而一个数学家则出版了若干短篇小说。

DOther scientists have written books to explain their fields to non-scientists, but have necessarily had to omit the mathematics, although it provides the foundation of their theories. The reader must remain a tantalized spectator rather than an involved participant, since the appropriate language for describing the details in much of science is mathematics, whether the subject is expanding universe, subatomic particles, or chromosomes. Though the broad outline of a scientific theory can be sketched intuitively, when a part of the physical universe is finally understood, its description often looks like a page in a mathematics text.

D 其他科学家们也曾出书向非科学专业人员解说他们的研究领域，但却都不得不省略其中的数学专业知识，即使这些知识构成了他们理论的基石。读者只好全程做一个跃跃欲试而不得的旁观者，而不是加入其中的参与者，因为描述大部分科学领域中细节内容的恰当语言是数学语言，无论话题是膨胀宇宙、亚原子粒子，还是染色体。虽然某个科学理论的大致轮廓可以通过直觉性思维来进行粗略描述，可一旦实体宇宙的某个组成部分最终为人们所理解，对这部分的描述往往还是看起来很像数学课本中的某一页。

EStill, the non-mathematical leader can go far in understanding mathematical reasoning. This book presents the details that illustrate the mathematical style of thinking, which involves sustained, step-by-step analysis, experiments, and insights. You will turn these pages much more slowly than when reading a novel or a newspaper. It may help to have a pencil and paper ready to check claims and carry out experiments.

E 没有数学专业背景的读者仍然可以在理解数学分析方面走得很远。这本书中给出的细节展示了数学风格的思维方式，这涉及耐心的、一步接一步的分析、实验和深入思考。你在翻动本书页码的时候，会比阅读一部小说或一份报纸时缓慢得多。准备好一支笔和一张纸会有助于你来测试书中理论和展开各种实验。

FAs I wrote, I kept in mind two types of readers: those who enjoyed mathematics until they were turned off by an unpleasant episode, usually around fifth grade, and mathematics aficionados, who will find much that is new throughout the book.

F 我在写作的时候，脑海中构想了两种类型的读者：有些人本来一直挺喜欢数学的，直到他们被某个不愉快的小插曲转变了看法，通常是在五年级左右；另外一些则是数学狂热爱好者，他们将在整本书内找到许多全新的东西。

This book also serves readers who simply want to sharpen their analytical skills. Many careers, such as law and medicine, require extended, precise analysis. Each chapter offers practice in following a sustained and closely argued line of thought. That mathematics can develop this skill is shown by these two testimonials:

这本书同时也能服务于那些仅仅只是想要锻炼自身分析能力的读者。许多职业，例如法律和医药，都需要从业者具备全面、精确的分析能力。每一章都提供了一些可供读者沿一条持之以恒、逻辑严密的思路线索一路探究的练习。数学可以帮你开发这方面的技能，不信请看以下两份大力推荐：

GA physician wrote, 'The discipline of analytical thought processes [in mathematics] prepared me extremely well for medical school. In medicine one is faced with a problem which must be thoroughly analyzed before a solution can be found. The process is similar to doing mathematics.'

G 一位医生写道：“(数学中)分析性思维加工的训练令我为医学学习做足了准备。在医学领域，一个人在遇到问题时，必须先仔仔细细地分析清楚才能找到解决办法。这个过程与学习数学是类似的。”

A lawyer made the same point, ‘Although I had no background in law - not even one political science course - I did well at one of the best law schools. I attribute much of my success there to having learned, through the study of mathematics, and, in particular, theorems, how to analyze complicated principles. Lawyers who have studied mathematics can master the legal principles in a way that most others cannot.'

一位律师也提出了同样的观点：“尽管我没有任何法律知识背景”——甚至连一门政治科学课也不曾上过，但却在一所顶级的法律学校里成绩优异。我将自己在那里取得成功的很大一部分归功于通过学习数学，特別是各种定理，掌握了如何分析复杂的原理。学过数学的律师们有能力以一种大多数其他律师所无法上手的方式掌握法律原则。”

I hope you will share my delight in watching as simple, even naïve, questions lead to remarkable solutions and purely theoretical discoveries find unanticipated applications.

我希望你能分享我的这一份喜悦，去看简单的、有时甚至是幼稚的各种问题引向非同凡响的解决之道，同时纯理论的发现则能找到意料之外的应用之途。