数学无处不在，无所不能。它渗透在所有领域，能解释宇宙万物，完全超越了人类的能力。本书按照数学关键概念的演化过程来组织结构，引经据典地从哲学角度全方位阐释数学的本质，以及数学和我们的物质世界、人类思维之间的关系。其间，传奇的历史人物和神秘的古老传说让深奥的哲学论证趣味横生。 本书适合所有对数学感兴趣的读者阅读。 博大精深的数学究竟是人类的发明还是发现？为什么它竟能描述宇宙万物的规律？是否真的有一位无所不能的上帝，他本身就是数学家呢？ 蜚声世界的科普名家Mario Livio，回顾了千百年来人类所进行的深层次思考，试图从哲学角度阐释数学的本质，揭示数学与物质世界和人类思维之间的紧密联系。从信奉万物皆数的毕达 哥拉斯、刀斧之下依然从容演算的阿基米德，到自制天文望远镜探索宇宙的近代科学始祖伽利略、坚持“我思故我在”的解析几何之父笛卡儿，再到极力反对柏拉图 主义的现代数学家阿蒂亚，数学思想几经发展又已一脉相承。历史上这些伟大的科学家在各自领域的重要贡献、他们在数学发展史上的远见卓识，以及他们的智慧人 生和传奇故事，绘成一幅幅五彩斑斓的历史画卷，自Livio的丹青妙笔下缓缓流露舒展。

引言 一、故事与统计之间 二、主观视角与非主观概率之间 三、无形式聊天与形式逻辑之间 四、含义与信息之间 五、为缺口架桥 文献精选 索引

The leading reference on probabilistic methods in combinatorics-now expanded and updated When it was first published in 1991, The Probabilistic Method became instantly the standard reference on one of the most powerful and widely used tools in combinatorics. Still without competition nearly a decade later, this new edition brings you up to speed on recent developments, while adding useful exercises and over 30ew material. It continues to emphasize the basic elements of the methodology, discussing in a remarkably clear and informal style both algorithmic and classical methods as well as modern applications. The Probabilistic Method, Second Edition begins with basic techniques that use expectation and variance, as well as the more recent martingales and correlation inequalities, then explores areas where probabilistic techniques proved successful, including discrepancy and random graphs as well as cutting-edge topics in theoretical computer science. A series of proofs, or "probabilistic lenses," are interspersed throughout the book, offering added insight into the application of the probabilistic approach. New and revised coverage includes: * Several improved as well as new results * A continuous approach to discrete probabilistic problems * Talagrand's Inequality and other novel concentration results * A discussion of the connection between discrepancy and VC-dimension * Several combinatorial applications of the entropy function and its properties * A new section on the life and work of Paul Erdös-the developer of the probabilistic method

This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.

《20世纪数学的五大指导理论》向读者展示了20世纪数学家的艺术的五个最精致的成就：极小极大定理(对策论)、布劳威尔不动点定理(拓扑学)、莫尔斯定理(奇点理论)、停机定理(计算的理论)、单纯形法(最优化理论)。