The Princeton Companion to Mathematics
出版时间
2008-09-28
ISBN
9780691118802
评分
This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more.
Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics, providing the context and broad perspective that are vital at a time of increasing specialization in the field. Packed with information and presented in an accessible style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.
* Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors
* Presents major ideas and branches of pure mathematics in a clear, accessible style
* Defines and explains important mathematical concepts, methods, theorems, and open problems
* Introduces the language of mathematics and the goals of mathematical research
* Covers number theory, algebra, analysis, geometry, logic, probability, and more
* Traces the history and development of modern mathematics
* Profiles more than ninety-five mathematicians who influenced those working today
* Explores the influence of mathematics on other disciplines
* Includes bibliographies, cross-references, and a comprehensive index
Timothy Gowers is the Rouse Ball Professor of Mathematics at the University of Cambridge. He received the Fields Medal in 1998, and is the author of Mathematics: A Very Short Introduction. June Barrow-Green is lecturer in the history of mathematics at the Open University. Imre Leader is professor of pure mathematics at the University of Cambridge.
- 本书由菲尔兹奖得主高尔斯主编，汇集全球顶尖数学家撰写近两百篇条目，全面覆盖现代数学的核心概念、历史脉络及前沿分支，是了解数学全貌的权威参考，深度与广度均无出其右。
- 内容不仅介绍抽象定义，更强调数学思想与方法论，如通过具体例子理解结构、利用代数工具解决几何问题、探讨证明的本质与数学基础的危机，帮助读者建立严谨的数学思维框架。
- 书中包含对著名数学家成就的介绍及数学在生物、金融、音乐等跨学科领域的应用，同时提供对青年数学家的建议，旨在纠正对数学的支离破碎认知，展现数学的内在美与逻辑力量。
- 适合对数学有严肃兴趣、希望系统了解现代数学分支与核心概念的高中生、大学生及科研人员，特别是那些缺乏竞赛背景但渴望接触真实数学研究态度的读者，可作为入门指引。
- 适合理工科背景人士，旨在纠正对数学片面或错误的理解，帮助厘清各分支关系，提升数学素养，适合希望从宏观视角把握数学发展史及基础理论演变的专业学习者。
- 不适合寻求轻松阅读或具体解题技巧的读者，因其内容高度抽象且专业性强，仅适合具备一定数学基础、愿意深入思考数学哲学与方法论、并能接受高密度信息挑战的硬核爱好者。
- 切勿试图从头到尾线性阅读，本书为百科全书式结构，建议根据兴趣挑选特定章节或概念进行查阅，避免陷入晦涩细节导致挫败感，保持对数学整体架构的好奇心即可。
- 阅读时需具备批判性思维，书中强调定义背后的直觉与例子的重要性，读者应主动思考概念间的联系，而非死记硬背公式，遇到不懂之处可跳过，重点领会数学家的思维方式。
- 注意区分不同章节的难度差异，部分基础介绍相对易懂，但涉及高阶抽象代数、拓扑等内容的章节极具挑战性，建议结合其他通俗读物辅助理解，切勿因个别难点而放弃全书。
- 读者普遍认可本书为数学领域的“神书”与权威指南，认为其由名家执笔，内容严谨且富有启发性，能让人窥见现代数学概貌，领略数学之美，是了解数学研究现状的极佳窗口。
- 多数读者反映全书难度极大，阅读过程极其吃力，后半部分趣味性下降且晦涩难懂，建议仅作为工具书查阅或泛读，不建议非专业人士精读，否则极易产生畏难情绪而放弃。
- 尽管翻译存在个别术语别扭之处，但整体质量获肯定，读者赞赏其重思想轻全面性的编写理念，认为其有助于打破对数学的误解，但警告初学者若无扎实基础，切勿盲目挑战，以免被劝退。
本导读基于书籍简介、目录、原文摘录、短评和书评生成，不等同于全文精读。
- "要理解一个数学结构的定义，重要的第一步是找到足够多的例子。有了例子，就开始找到了对于这个结构的感觉，而仅仅定义是提供不了这种感觉的。"
- "时常是从一个大得令人绝望而又极为复杂的对象出发，但是“把绝大部分的乱七八糟都分出来除掉了”，结果得到的商结构却足够简单而能够处理，而仍旧传递重要的信息."
- "紧流形 X 上的相当大一部分几何学可以纯粹地以 C*-代数 C(X) 的语言来重新陈述. 这里的“纯粹地”这个词意味着并无必要讲到流形 X，而 C(X) 本来是参照着流形 X 来定义的——我们需要的仅是 C(X) 是一个代数. 这就意味着有可能有这样的不是几何地生成的代数，但是对于它们，经过重新陈述的几何概念仍然可用."
- "把函数在一个点的值理解为同态导致了环论的一个一般观点，据此，交换环经常可以被理解为一个集合上的函数环，该集合的“点”对应于原来那个环到域的同态."
- "……真正有趣的定义是不那么显然，而是一旦有了它，就会用新的方式来思考的那种定义。"
- "It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange—if he has completed a str"
- "The idea is both to make general arguments that are then applicable to all categories possessing partic- ular structural features, and also to be able to make arguments in specific environments without having to go into the details of the structures in question. The use of the former to achieve the "
- "只有平庸之辈才对自己的能力极为自信。"
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