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齣版社： Touchstone

ISBN：9780671628185

商品編碼：10403156603

Men of Mathematics

內容簡介
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.

媒體推薦

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

作者簡介

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.

目錄

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
齣版社: Touchstone; Reissue (1986年10月15日)
叢書名: Touchstone Book
平裝: 608頁
語種： 英語
ISBN: 0671628186
條形碼: 9780671628185
商品尺寸: 13.5 x 4.1 x 21 cm
商品重量: 499 g
ASIN: 0671628186

探尋思維的奧秘：現代邏輯與計算的基石 一書深潛，洞察理性之源流，解析思維之演進。 本書並非聚焦於宏大敘事或曆史的側麵迴響，而是深入現代數學邏輯與計算理論的核心構造與內在機製。它以嚴謹的學術態度，剖析瞭人類理性如何從模糊的直覺邁嚮精確的符號係統，並最終催生瞭我們今日習以為常的計算技術。全書結構精密，層層遞進，旨在為讀者構建一個關於“可計算性”、“可證明性”以及“形式係統”的完整知識框架。 第一部分：形式係統的奠基與危機 本部分著重迴顧瞭十九世紀末至二十世紀初，數學基礎領域經曆的深刻“危機”與隨之而來的結構性變革。我們不談論那些已經成為傢喻戶曉的數學傢傳記，而是將聚光燈投嚮那些定義瞭“什麼是數學”的公理化嘗試與形式化構建。 1.1 集閤論的勃興與悖論的陰影： 討論瞭弗雷格（Frege）的《概念文字》如何嘗試為算術建立純邏輯基礎，以及羅素（Russell）的悖論如何無情地揭示瞭樸素集閤論的內在脆弱性。重點分析瞭策梅洛-弗蘭剋爾（ZFC）公理化係統的建立過程，並非簡單敘述公理，而是深入探討選擇公理（Axiom of Choice）在構造數學對象時的必要性與爭議性，以及它如何影響瞭不同數學分支的基石。 1.2 形式語言的精確化： 本章細緻闡述瞭如何將自然語言的模糊性剝離，構建齣一階邏輯（First-Order Logic）的精確形式語言。這包括對項、公式、閤式公式（WFF）的嚴格定義，以及如何通過形式推導規則（如自然演繹或相繼演算）來定義“證明”。重點在於理解一個證明的句法（Syntax）結構與它所錶達的語義（Semantics）之間的張力與聯係。 1.3 希爾伯特綱領的輝煌與破滅： 這一節是全書的理論高潮之一。它詳細介紹瞭大衛·希爾伯特（David Hilbert）試圖將整個數學置於一個有限的、可靠的公理係統之下的宏偉計劃。我們探討瞭該綱領的兩個核心目標：完備性（Completeness）——所有真命題皆可證明；以及可靠性/一緻性（Soundness/Consistency）——係統中不會導齣矛盾。理解這些概念，是通往圖靈和哥德爾思想的必經之路。 第二部分：可計算性的邊界與圖靈的機器 在形式係統試圖證明一切的同時，另一個關鍵問題浮現：哪些問題是原則上可以被解決的？本部分是關於計算理論的開端，探討瞭人類對“算法”這一概念的精確捕捉過程。 2.1 算法概念的直覺與形式化： 在計算機誕生之前，“算法”更多是一個操作性的直覺概念。本章追溯瞭數學傢們（如圖靈、丘奇、波斯特）如何獨立地、但卻驚人地一緻地，將“可計算性”這一模糊概念轉化為一個嚴格的數學模型。我們不會停留在對圖靈機曆史的簡單描述，而是聚焦於圖靈機（Turing Machine）的模型本身——磁帶、讀寫頭、狀態寄存器，以及它如何精確模擬瞭人類的機械性計算步驟。 2.2 通用性與不可判定性： 深入分析瞭通用圖靈機（Universal Turing Machine, UTM）的概念，理解為何一颱機器能夠模擬任何其他圖靈機。隨後，本部分的核心——停機問題（Halting Problem）的不可判定性——將被完整且深入地證明。我們將使用對角綫論證法，嚴謹地展示為什麼存在一個明確定義的數學問題，是任何算法都無法解決的。這不僅是計算理論的基石，更是對人類理性能力邊界的首次精確測定。 2.3 丘奇-圖靈論題（Church-Turing Thesis）： 討論瞭這一“論題”（而非定理）的重要性，即所有直覺上可計算的過程，都可以被圖靈機所捕獲。剖析瞭lambda演算（λ-Calculus）作為另一種等價的計算模型，以及它與圖靈機之間的可相互模擬性。 第三部分：哥德爾的深刻洞察與算術的局限 如果說圖靈的工作劃定瞭計算的外部邊界，那麼哥德爾的工作則揭示瞭任何足夠強大的形式係統內部的局限性。本部分是本書在理論深度上最密集的部分。 3.1 算術的編碼與“哥德爾編碼”： 詳細解釋瞭哥德爾如何通過精妙的數論編碼（Gödel Numbering），將關於證明、公式和邏輯結構的元語言概念，轉化為關於自然數的純代數陳述。理解這一編碼過程是理解哥德爾定理的鑰匙，它使得邏輯問題可以被轉化為算術問題。 3.2 不完備性定理的剖析（第一定理）： 在建立好編碼係統後，我們將嚴格證明哥德爾第一不完備性定理：任何包含一階算術且一緻的形式係統，都必然存在一個在該係統內既不能被證明亦不能被證否的命題。重點解析瞭如何構造這個“自我指涉”的命題 $G$（“此命題不可被證明”）。 3.3 第二不完備性定理與一緻性： 隨後，分析瞭哥德爾第二不完備性定理，即一個足夠強大的係統不能證明自身的一緻性。這直接宣告瞭希爾伯特綱領的最終失敗，並強調瞭外部驗證在數學基礎中的不可替代性。 第四部分：遞歸論、可計算性與復雜性初探 最後一部分將理論成果引嚮更廣闊的計算領域，探討瞭遞歸函數、程序語言的根源，並觸及瞭現代計算機科學中最核心的難題。 4.1 遞歸函數的等級與程序語言： 探討瞭比圖靈機更基礎的模型——遞歸函數的定義（如原始遞歸、μ-遞歸）。展示瞭這些函數如何等價於圖靈機的可計算性。同時，對比瞭現代編程語言的結構與這些早期形式模型的對應關係，理解諸如“循環”和“遞歸調用”的數學根源。 4.2 判定問題與復雜性分類的萌芽： 在理解瞭“可解性”之後，我們轉嚮“效率”。簡要介紹判定問題（Decision Problem）的概念，以及如何區分那些可解但效率極低的問題。雖然本書不深入現代復雜性理論的細節（如P與NP），但會奠定理解計算資源限製的理論基礎，指齣有些問題不僅難，而且可能比不可解性問題更難處理——因為它們需要指數級或更長的資源來求解。 本書力求以一種結構清晰、邏輯嚴密的方式，帶領讀者穿越邏輯的迷宮，直麵人類理性工具的局限與偉大，是理解現代數學、理論計算機科學和哲學邏輯交匯點的必讀之作。

評分☆☆☆☆☆

說實話，我原本以為這會是一本讀起來會有些吃力的書，畢竟涉及的人物跨度很大，而且很多數學成果的背景知識需要一定的儲備。但這本書的處理方式非常巧妙，作者似乎有一種魔力，能將那些晦澀難懂的數學概念，用一種近乎文學化的語言進行解讀，讓門外漢也能窺見其精妙之處。我尤其欣賞作者在處理“人”與“學問”之間的平衡。比如，他沒有迴避那些著名數學傢性格上的缺陷——他們的傲慢、偏執，甚至是道德上的瑕疵。正是這些不完美，使得這些站在人類智力巔峰的人物顯得更加真實可觸。閱讀過程中，我多次停下來，去查閱一些相關的曆史背景資料，這反而成瞭一種探索的樂趣，而不是負擔。這本書提供瞭一個絕佳的切入點，讓我得以一窺數學史的壯闊圖景。它不是一本教科書，更像是一部引人入勝的“群像劇”，每一個數學傢都是一個立體而鮮活的角色，他們的貢獻如同一個個裏程碑，共同鋪就瞭我們今天的科學之路。我甚至覺得，它比許多專門的曆史著作還要精彩，因為它聚焦於那些驅動曆史前進的“人”的力量。

評分☆☆☆☆☆

這本書的敘事節奏把握得極其精準，高潮迭起，張弛有度。從早期的幾何學奠基者到微積分的誕生，再到近現代抽象代數和集閤論的興起，作者總能在關鍵的轉摺點設置引人深思的論斷。我特彆喜歡作者對那些“跨界”思考者的描述，比如那些同時在哲學、物理和數學領域都有深厚造詣的天纔。他們的思維方式顯然與常人不同，那種不受既有學科壁壘限製的自由探索精神，是這本書給我最大的啓發。我感覺自己不是在讀一本關於過去的記錄，而是在進行一場與曆史上最偉大頭腦的深度對話。作者在描述那些突破性發現時，常常會引用當時的信件、日記片段，這極大地增強瞭現場感和曆史的真實感。這種細緻入微的考證，讓那些被時間塵封的思考過程重新鮮活瞭起來。對於那些渴望瞭解“靈光乍現”是如何發生的讀者來說，這本書提供瞭寶貴的洞察。它證明瞭偉大的數學發現往往不是孤立的靈感，而是長期積纍、激烈思辨和對世界深刻洞察的必然結果。

評分☆☆☆☆☆

作為一名對知識有著永不滿足的讀者，我一直在尋找那種能夠拓展思維邊界的作品，而這本《數學精英》無疑達到瞭我的預期，甚至有所超越。它讓我明白瞭，數學並非一門僵化的、隻有公式和定理的學科，而是一門充滿創造力、甚至帶有浪漫主義色彩的藝術。作者的筆觸細膩入微，尤其在描寫那些“失之交臂”的發現和未竟的理論時，那種曆史的遺憾感撲麵而來。書中對那些被時代誤解、或是在生前未能得到充分認可的數學傢的著墨尤為動人，這讓我對“曆史的評價”多瞭一層反思。它不僅僅是頌揚成功者，更是對那些默默耕耘、或是在錯誤的時間齣現瞭正確思想的人的緻敬。這本書的價值在於，它提供瞭一種宏觀的曆史視角，讓讀者能夠理解數學知識體係是如何在競爭、閤作與傳承中不斷自我完善的。讀完之後，我感覺自己對“嚴謹性”和“想象力”在科學中的地位有瞭全新的認識，這兩者並非對立，而是相輔相成的驅動力。

評分☆☆☆☆☆

這本《數學精英：改變世界的數學巨匠們》的英文原版，著實是一部引人入勝的傳記閤集。我拿到這本書時，首先被其厚重的質感和精心設計的封麵所吸引，它散發著一種經典學術著作的韻味。閱讀的體驗非常流暢，作者的敘述功力深厚，他不僅僅是在羅列數學傢的生平事跡和他們的主要成就，更深入地挖掘瞭這些天纔的內心世界、他們的掙紮、他們的激情，以及他們是如何在那個特定的時代背景下，一步步構建起現代數學的宏偉殿堂。特彆是對早期數學傢，比如那些古希臘的哲人，他們的思想是如何萌芽、如何在那個缺乏現代工具的環境下進行嚴謹的邏輯推演，描述得栩栩如生。我印象最深的是其中對笛卡爾和牛頓之間那段充滿火藥味又相互成就的描述，那種天纔之間的碰撞與競爭，讀來令人熱血沸騰。它讓我意識到，數學的發展從來不是一條平坦的直綫，而是充滿瞭人性的復雜性與曆史的偶然性。這本書成功地將枯燥的數學概念，轉化成瞭有血有肉的故事，即便是對數學不太“感冒”的讀者，也會被這些偉大靈魂所散發齣的光芒所感染。它不僅僅是寫給數學專業人士的，更是一部關於人類智慧如何突破自身局限的史詩。

評分☆☆☆☆☆

這本書的英文原版，在語言的駕馭上達到瞭極高的水準，即便是一些相對專業的術語，作者也能通過精準的上下文描述，讓讀者無需過度依賴專業背景知識就能領會其核心思想。我被作者強大的整閤能力所摺服，他成功地將不同世紀、不同文化背景下的數學傢們串聯成一條清晰的、邏輯嚴密的敘事綫索。這種跨越時空的對話，讓人感覺數學史是一部連續的、未完待續的史詩。我尤其欣賞作者在介紹某些數學概念的起源時，那種追本溯源的嚴謹態度，他總是會迴到最初提齣問題的那一刻，去感受先驅者們麵對未知時的那種敬畏與興奮。這種敘述方式極大地提升瞭閱讀的沉浸感。它不是一本快餐式的讀物，更像是一壇需要慢慢品味的陳釀，每一次重讀都會有新的體會。它不僅僅是關於數學傢的故事，更是關於人類理性探索精神的贊歌，激勵著每一個追求真理的靈魂去敢於質疑、勇於創造。