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出版社： Touchstone

ISBN：9780671628185

商品编码：10403156603

Men of Mathematics

内容简介
Here is the classic, much-read introduction to the craft and history of mathematics by E.T. Bell, a leading figure in mathematics in America for half a century. Men of Mathematics accessibly explains the major mathematics, from the geometry of the Greeks through Newton's calculus and on to the laws of probability, symbolic logic, and the fourth dimension. In addition, the book goes beyond pure mathematics to present a series of engrossing biographies of the great mathematicians -- an extraordinary number of whom lived bizarre or unusual lives. Finally, Men of Mathematics is also a history of ideas, tracing the majestic development of mathematical thought from ancient times to the twentieth century. This enduring work's clear, often humorous way of dealing with complex ideas makes it an ideal book for the non-mathematician.

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Nature Professor E.T. Bell has written a fascinating book. The amount of biographical details and of mathematics that he has compressed into a volume of 600 pages is extraordinary...he carries the reader along; he whets the appetite. -- Review

作者简介

Eric Temple Bell was born in 1883 in Aberdeen, Scotland. His early education was obtained in England. Coming to the United States in 1902, he entered Stanford University and took his A.B. degree in 1904. In 1908 he was teaching fellow at the University of Washington, where he took his A.M. degree in 1909. In 1911 he entered Columbia University, where he took his Ph.D. degree in 1912. He returned to the University of Washington as instructor in mathematics and became full professor in 1921. During the summers of 1924-28 he taught at the University of Chicago, and in 1926 (first half) at Harvard University, when he was appointed Professor of Mathematics at the California Institute of Technology.

Dr. Bell was a former President of the Mathematical Association of America, a former Vice President of the American Mathematical Society and of the American Association for the Advancement of Science. He was on the editorial staffs of the Transactions of the American Mathematical Society, the American Journal of Mathematics, and the Journal of the Philosophy of Science. He belonged to The American Mathematical Society, the Mathematical Association of America, the Circolo Matematico di Palermo, the Calcutta Mathematical Society, Sigma Xi, and Phi Beta Kappa, and was a member of the National Academy of Sciences of the United States. He won the Bôcher Prize of the American Mathematical Society for his research work. His twelve published books include The Purple Sapphire (1924), Algebraic Arithmetic (1927), Debunking Science, and Queen of the Sciences (1931), Numerology (1933), and The Search for Truth (1934).

Dr. Bell died in December 1960, just before the publication of his latest book, The Last Problem.

目录

Contents

1. INTRODUCTION

For the reader's comfort. The beginning of modern mathematics. Are mathematicians human? Witless parodies. Illimitable scope of mathematical evolution. Pioneers and scouts. A clue through the maze. Continuity and discreteness. Remarkable rarity of common sense. Vivid mathematics or vague mysticism? Four great ages of mathematics. Our own the Golden Age.

2. MODERN MINDS IN ANCIENT BODIES

Zeno (fifth century B.C.), Eudoxus (408-355 B.C.), Archimedes (287?-212 B.C.)

Modern ancients and ancient moderns. Pythagoras, great mystic, greater mathematician. Proof or intuition? The taproot of modern analysis. A bumpkin upsets the philosophers. Zeno's unresolved riddles. Plato's needy young friend. Inexhaustible exhaustion. The useful conics. Archimedes, aristocrat, greatest scientist of antiquity. Legends of his life and personality. His discoveries and claim to modernity. A sturdy Roman. Defeat of Archimedes and triumph of Rome.

3. GENTLEMAN, SOLDIER, AND MATHEMATICIAN

Descartes (1596-1650)

The good old days. A child philosopher but no prig. Inestimable advantages of lying in bed. Invigorating doubts. Peace in war. Converted by a nightmare. Revelation of analytic geometry. More butchering. Circuses, professional jealousy, swashbuckling, accommodating lady friends. Distaste for hell-fire and respect for the Church. Saved by a brace of cardinals. A Pope brains himself. Twenty years a recluse. The Method. Betrayed by fame. Doting Elisabeth. What Descartes really thought of her. Conceited Christine. What she did to Descartes. Creative simplicity of Ms geometry.

4. THE PRINCE OF AMATEURS

Fermat (1601-1665)

Greatest mathematician of the seventeenth century. Fermat's busy, practical life. Mathematics his hobby. His flick to the calculus. His profound physical principle. Analytic geometry again. Arithmetica and logistica. Fermat's supremacy in arithmetic. An unsolved problem on primes. Why are some theorems "important"? An intelligence test. "Infinite descent." Fermat's unanswered challenge to posterity.

5. "GREATNESS AND MISERY OF MAN"

Pascal (1625-1662)

An infant prodigy buries his talent. At seventeen a great geometer. Pascal's wonderful theorem. Vile health and religious inebriety. The first calculating Frankenstein. Pascal's brilliance in physics. Holy sister Jacqueline, soul-saver. Wine and women? "Get thee to a nunnery.!" Converted on a spree. Literature prostituted to bigotry. The Helen of Geometry. A celestial toothache. What the post-mortem revealed. A gambler makes mathematical history. Scope of the theory of probability. Pascal creates the theory with Fermat. Folly of betting against God or the Devil.

6. ON THE SEASHORE

Newton (1642-1727)

Newton's estimate of himself. An uncertified youthful genius. Chaos of his times. On the shoulders of giants. His one attachment. Cambridge days. Young Newton masters futility of suffering fools gladly. The Great Plague a greater blessing. Immortal at twenty four (or less). The calculus. Newton unsurpassed in pure mathematics, supreme in natural philosophy. Gnats, hornets, and exasperation. The Principia. Samuel Pepys and other fussers. The flattest anticlimax in history. Controversy, theology, chronology, alchemy, public office, death.

7. MASTER OF ALL TRADES

Leibniz (1646-1716)

Two superb contributions. A politician's offspring. Genius at fifteen. Seduced by the law. The "universal characteristic." Symbolic reasoning. Sold out to ambition. A master diplomat. Diplomacy being what it is, the diplomatic exploits of the master are left to the historians. Fox into historian, statesman into mathematician. Applied ethics. Existence of God. Optimism. Forty years of futility. Discarded like a dirty rag.

8. NATURE OR NURTURE?

The Bernoullis (seventeenth-and eighteenth centuries)

Eight mathematicians in three generations. Clinical evidence for heredity. The calculus of variations.

9. ANALYSIS INCARNATE

Euler (1707-1783)

The most prolific mathematician in history. Snatched from theology. Rulers foot the bills. Practicality of the unpractical. Celestial mechanics and naval warfare. A mathematician by chance and foreordination. Trapped in St. Petersburg. The virtues of silence. Half blind in his morning. Flight to liberal Prussia. Generosity and boorishness of Frederick the Great. Return to hospitable Russia. Generosity and graciousness of Catherine the Great. Total blindness at noon. Master and inspirer of masters for a century.

10. A LOFTY PYRAMID

Lagrange (1736-1813)

Greatest and most modest mathematician of the eighteenth century. Financial ruin his opportunity. Conceives his masterpiece at nineteen. Magnanimity of Euler. Turin, to Paris, to Berlin: a grateful bastard aids a genius. Conquests in celestial mechanics. Frederick the Great condescends. Absent-minded marriage. Work as a vice. A classic in arithmetic. The Mécanique analytique a living masterpiece. A landmark in the theory of equations. Welcomed in Paris by Marie Antoinette. Nervous exhaustion, melancholia, and universal disgust in middle life. Reawakened by the French Revolution and a young girl. What Lagrange thought of the Revolution. The metric system. What the revolutionists thought of Lagrange. How a philosopher dies.

11. FROM PEASANT TO SNOB

Laplace (1749-1827)

Humble as Lincoln, proud as Lucifer. A chilly reception and a warm welcome. Laplace grandiosely attacks the solar system. The Mécanique céleste. His estimate of himself. What others have thought of him. The "potential" fundamental in physics. Laplace in the French Revolution. Intimacy with Napoleon. Laplace's political realism superior to Napoleon's.

12. FRIENDS OF AN EMPEROR

Monge (1746-1818), Fourier (1768-1830)

A knife grinder's son and a tailor's boy help Napoleon to upset the aristocrats' applecart. Comic opera in Egypt. Monge's descriptive geometry and the Machine Age. Fourier's analysis and modern physics. Imbecility of trusting in princes or proletarians. Boring to death and bored to death.

13. THE DAY OF GLORY

Poncelet (1788-1867)

Resurrected from a Napoleonic shambles. The path of glory leads to jail. Wintering in Russia in 1812. What genius does in prison. Two years of geometry in hell. The rewards of genius: stupidities of routine. Poncelet's projective geometry. Principles of continuity and duality.

14. THE PRINCE OF MATHEMATICIANS

Gauss (1777-1855)

Gauss the mathematical peer of Archimedes and Newton. Humble origin. Paternal brutality. Unequalled intellectual precocity. His chance, at ten. By twelve he dreams revolutionary discoveries, by eighteen achieves them. The Disquisitiones Arithmeticae. Other epochal works summarized. The Ceres disaster. Napoleon, indirectly robbing Gauss, takes second best. Fundamental advances in all branches of mathematics due to Gauss too numerous for citation: see the account given. A sage of sages. Unwelcome death.

15. MATHEMATICS AND WINDMILLS

Cauchy (1789-1857)

Change in nature of mathematics with nineteenth century. Childhood in the French Revolution. Cauchy's early miseducation. Lagrange's prophecy. The young Christian engineer. Prophetic acuteness of Malus. The theory of groups. In the front rank at twenty seven. One of Fermat's enigmas solved. The pious hippopotamus. Butted by Charles the Goat. Memoirs on astronomy and mathematical physics. Sweetness and obstinacy invincible. The French Government makes a fool of itself. Cauchy's place in mathematics. Drawbacks of an irreproachable character.

16. THE COPERNICUS OF GEOMETRY

Lobatchewsky (1793-1856)

The widow's mite. Kazan. Appointed professor and spy. Universal ability. Lobatchewsky as an administrator. Reason and incense combat the cholera. Russian gratitude. Humiliated in his prime. Blind as Milton, Lobatchewsky dictates his masterpiece. His advance beyond Euclid. Non-Euclidean geometry. A Copernicus of the intellect.

基本信息
出版社: Touchstone; Reissue (1986年10月15日)
丛书名: Touchstone Book
平装: 608页
语种： 英语
ISBN: 0671628186
条形码: 9780671628185
商品尺寸: 13.5 x 4.1 x 21 cm
商品重量: 499 g
ASIN: 0671628186

探寻思维的奥秘：现代逻辑与计算的基石 一书深潜，洞察理性之源流，解析思维之演进。 本书并非聚焦于宏大叙事或历史的侧面回响，而是深入现代数学逻辑与计算理论的核心构造与内在机制。它以严谨的学术态度，剖析了人类理性如何从模糊的直觉迈向精确的符号系统，并最终催生了我们今日习以为常的计算技术。全书结构精密，层层递进，旨在为读者构建一个关于“可计算性”、“可证明性”以及“形式系统”的完整知识框架。 第一部分：形式系统的奠基与危机 本部分着重回顾了十九世纪末至二十世纪初，数学基础领域经历的深刻“危机”与随之而来的结构性变革。我们不谈论那些已经成为家喻户晓的数学家传记，而是将聚光灯投向那些定义了“什么是数学”的公理化尝试与形式化构建。 1.1 集合论的勃兴与悖论的阴影： 讨论了弗雷格（Frege）的《概念文字》如何尝试为算术建立纯逻辑基础，以及罗素（Russell）的悖论如何无情地揭示了朴素集合论的内在脆弱性。重点分析了策梅洛-弗兰克尔（ZFC）公理化系统的建立过程，并非简单叙述公理，而是深入探讨选择公理（Axiom of Choice）在构造数学对象时的必要性与争议性，以及它如何影响了不同数学分支的基石。 1.2 形式语言的精确化： 本章细致阐述了如何将自然语言的模糊性剥离，构建出一阶逻辑（First-Order Logic）的精确形式语言。这包括对项、公式、合式公式（WFF）的严格定义，以及如何通过形式推导规则（如自然演绎或相继演算）来定义“证明”。重点在于理解一个证明的句法（Syntax）结构与它所表达的语义（Semantics）之间的张力与联系。 1.3 希尔伯特纲领的辉煌与破灭： 这一节是全书的理论高潮之一。它详细介绍了大卫·希尔伯特（David Hilbert）试图将整个数学置于一个有限的、可靠的公理系统之下的宏伟计划。我们探讨了该纲领的两个核心目标：完备性（Completeness）——所有真命题皆可证明；以及可靠性/一致性（Soundness/Consistency）——系统中不会导出矛盾。理解这些概念，是通往图灵和哥德尔思想的必经之路。 第二部分：可计算性的边界与图灵的机器 在形式系统试图证明一切的同时，另一个关键问题浮现：哪些问题是原则上可以被解决的？本部分是关于计算理论的开端，探讨了人类对“算法”这一概念的精确捕捉过程。 2.1 算法概念的直觉与形式化： 在计算机诞生之前，“算法”更多是一个操作性的直觉概念。本章追溯了数学家们（如图灵、丘奇、波斯特）如何独立地、但却惊人地一致地，将“可计算性”这一模糊概念转化为一个严格的数学模型。我们不会停留在对图灵机历史的简单描述，而是聚焦于图灵机（Turing Machine）的模型本身——磁带、读写头、状态寄存器，以及它如何精确模拟了人类的机械性计算步骤。 2.2 通用性与不可判定性： 深入分析了通用图灵机（Universal Turing Machine, UTM）的概念，理解为何一台机器能够模拟任何其他图灵机。随后，本部分的核心——停机问题（Halting Problem）的不可判定性——将被完整且深入地证明。我们将使用对角线论证法，严谨地展示为什么存在一个明确定义的数学问题，是任何算法都无法解决的。这不仅是计算理论的基石，更是对人类理性能力边界的首次精确测定。 2.3 丘奇-图灵论题（Church-Turing Thesis）： 讨论了这一“论题”（而非定理）的重要性，即所有直觉上可计算的过程，都可以被图灵机所捕获。剖析了lambda演算（λ-Calculus）作为另一种等价的计算模型，以及它与图灵机之间的可相互模拟性。 第三部分：哥德尔的深刻洞察与算术的局限 如果说图灵的工作划定了计算的外部边界，那么哥德尔的工作则揭示了任何足够强大的形式系统内部的局限性。本部分是本书在理论深度上最密集的部分。 3.1 算术的编码与“哥德尔编码”： 详细解释了哥德尔如何通过精妙的数论编码（Gödel Numbering），将关于证明、公式和逻辑结构的元语言概念，转化为关于自然数的纯代数陈述。理解这一编码过程是理解哥德尔定理的钥匙，它使得逻辑问题可以被转化为算术问题。 3.2 不完备性定理的剖析（第一定理）： 在建立好编码系统后，我们将严格证明哥德尔第一不完备性定理：任何包含一阶算术且一致的形式系统，都必然存在一个在该系统内既不能被证明亦不能被证否的命题。重点解析了如何构造这个“自我指涉”的命题 $G$（“此命题不可被证明”）。 3.3 第二不完备性定理与一致性： 随后，分析了哥德尔第二不完备性定理，即一个足够强大的系统不能证明自身的一致性。这直接宣告了希尔伯特纲领的最终失败，并强调了外部验证在数学基础中的不可替代性。 第四部分：递归论、可计算性与复杂性初探 最后一部分将理论成果引向更广阔的计算领域，探讨了递归函数、程序语言的根源，并触及了现代计算机科学中最核心的难题。 4.1 递归函数的等级与程序语言： 探讨了比图灵机更基础的模型——递归函数的定义（如原始递归、μ-递归）。展示了这些函数如何等价于图灵机的可计算性。同时，对比了现代编程语言的结构与这些早期形式模型的对应关系，理解诸如“循环”和“递归调用”的数学根源。 4.2 判定问题与复杂性分类的萌芽： 在理解了“可解性”之后，我们转向“效率”。简要介绍判定问题（Decision Problem）的概念，以及如何区分那些可解但效率极低的问题。虽然本书不深入现代复杂性理论的细节（如P与NP），但会奠定理解计算资源限制的理论基础，指出有些问题不仅难，而且可能比不可解性问题更难处理——因为它们需要指数级或更长的资源来求解。 本书力求以一种结构清晰、逻辑严密的方式，带领读者穿越逻辑的迷宫，直面人类理性工具的局限与伟大，是理解现代数学、理论计算机科学和哲学逻辑交汇点的必读之作。

评分☆☆☆☆☆

这本《数学精英：改变世界的数学巨匠们》的英文原版，着实是一部引人入胜的传记合集。我拿到这本书时，首先被其厚重的质感和精心设计的封面所吸引，它散发着一种经典学术著作的韵味。阅读的体验非常流畅，作者的叙述功力深厚，他不仅仅是在罗列数学家的生平事迹和他们的主要成就，更深入地挖掘了这些天才的内心世界、他们的挣扎、他们的激情，以及他们是如何在那个特定的时代背景下，一步步构建起现代数学的宏伟殿堂。特别是对早期数学家，比如那些古希腊的哲人，他们的思想是如何萌芽、如何在那个缺乏现代工具的环境下进行严谨的逻辑推演，描述得栩栩如生。我印象最深的是其中对笛卡尔和牛顿之间那段充满火药味又相互成就的描述，那种天才之间的碰撞与竞争，读来令人热血沸腾。它让我意识到，数学的发展从来不是一条平坦的直线，而是充满了人性的复杂性与历史的偶然性。这本书成功地将枯燥的数学概念，转化成了有血有肉的故事，即便是对数学不太“感冒”的读者，也会被这些伟大灵魂所散发出的光芒所感染。它不仅仅是写给数学专业人士的，更是一部关于人类智慧如何突破自身局限的史诗。

评分☆☆☆☆☆

这本书的英文原版，在语言的驾驭上达到了极高的水准，即便是一些相对专业的术语，作者也能通过精准的上下文描述，让读者无需过度依赖专业背景知识就能领会其核心思想。我被作者强大的整合能力所折服，他成功地将不同世纪、不同文化背景下的数学家们串联成一条清晰的、逻辑严密的叙事线索。这种跨越时空的对话，让人感觉数学史是一部连续的、未完待续的史诗。我尤其欣赏作者在介绍某些数学概念的起源时，那种追本溯源的严谨态度，他总是会回到最初提出问题的那一刻，去感受先驱者们面对未知时的那种敬畏与兴奋。这种叙述方式极大地提升了阅读的沉浸感。它不是一本快餐式的读物，更像是一坛需要慢慢品味的陈酿，每一次重读都会有新的体会。它不仅仅是关于数学家的故事，更是关于人类理性探索精神的赞歌，激励着每一个追求真理的灵魂去敢于质疑、勇于创造。

评分☆☆☆☆☆

这本书的叙事节奏把握得极其精准，高潮迭起，张弛有度。从早期的几何学奠基者到微积分的诞生，再到近现代抽象代数和集合论的兴起，作者总能在关键的转折点设置引人深思的论断。我特别喜欢作者对那些“跨界”思考者的描述，比如那些同时在哲学、物理和数学领域都有深厚造诣的天才。他们的思维方式显然与常人不同，那种不受既有学科壁垒限制的自由探索精神，是这本书给我最大的启发。我感觉自己不是在读一本关于过去的记录，而是在进行一场与历史上最伟大头脑的深度对话。作者在描述那些突破性发现时，常常会引用当时的信件、日记片段，这极大地增强了现场感和历史的真实感。这种细致入微的考证，让那些被时间尘封的思考过程重新鲜活了起来。对于那些渴望了解“灵光乍现”是如何发生的读者来说，这本书提供了宝贵的洞察。它证明了伟大的数学发现往往不是孤立的灵感，而是长期积累、激烈思辨和对世界深刻洞察的必然结果。

评分☆☆☆☆☆

作为一名对知识有着永不满足的读者，我一直在寻找那种能够拓展思维边界的作品，而这本《数学精英》无疑达到了我的预期，甚至有所超越。它让我明白了，数学并非一门僵化的、只有公式和定理的学科，而是一门充满创造力、甚至带有浪漫主义色彩的艺术。作者的笔触细腻入微，尤其在描写那些“失之交臂”的发现和未竟的理论时，那种历史的遗憾感扑面而来。书中对那些被时代误解、或是在生前未能得到充分认可的数学家的着墨尤为动人，这让我对“历史的评价”多了一层反思。它不仅仅是颂扬成功者，更是对那些默默耕耘、或是在错误的时间出现了正确思想的人的致敬。这本书的价值在于，它提供了一种宏观的历史视角，让读者能够理解数学知识体系是如何在竞争、合作与传承中不断自我完善的。读完之后，我感觉自己对“严谨性”和“想象力”在科学中的地位有了全新的认识，这两者并非对立，而是相辅相成的驱动力。

评分☆☆☆☆☆

说实话，我原本以为这会是一本读起来会有些吃力的书，毕竟涉及的人物跨度很大，而且很多数学成果的背景知识需要一定的储备。但这本书的处理方式非常巧妙，作者似乎有一种魔力，能将那些晦涩难懂的数学概念，用一种近乎文学化的语言进行解读，让门外汉也能窥见其精妙之处。我尤其欣赏作者在处理“人”与“学问”之间的平衡。比如，他没有回避那些著名数学家性格上的缺陷——他们的傲慢、偏执，甚至是道德上的瑕疵。正是这些不完美，使得这些站在人类智力巅峰的人物显得更加真实可触。阅读过程中，我多次停下来，去查阅一些相关的历史背景资料，这反而成了一种探索的乐趣，而不是负担。这本书提供了一个绝佳的切入点，让我得以一窥数学史的壮阔图景。它不是一本教科书，更像是一部引人入胜的“群像剧”，每一个数学家都是一个立体而鲜活的角色，他们的贡献如同一个个里程碑，共同铺就了我们今天的科学之路。我甚至觉得，它比许多专门的历史著作还要精彩，因为它聚焦于那些驱动历史前进的“人”的力量。