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History,人物,维纳,控制论,维纳滤波器,布朗运动,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc="" name="description" content="诺伯特·维纳是一位传奇的数学家与科学家，在概率论、信号处理等多个领域有开创性贡献。他最为人知的成就是创立了控制论，深刻影响了自动化与信息科学的发展。 - MHuiG - Magicland"><meta property="og:type" content="website"><meta property="og:title" content="Magicland"><meta property="og:url" content="https://blog.mhuig.top/notes/person/17"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="诺伯特·维纳是一位传奇的数学家与科学家，在概率论、信号处理等多个领域有开创性贡献。他最为人知的成就是创立了控制论，深刻影响了自动化与信息科学的发展。"><meta property="og:locale"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" content="2026-05-13T05:07:00.000Z"><meta property="article:modified_time" content="2026-05-13T05:10:00.000Z"><meta property="article:author" content="MHuiG"><meta property="article:tag" content="History"><meta property="article:tag" content="人物"><meta property="article:tag" content="布朗运动"><meta 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itemscope="" itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="Wiener诺伯特·维纳：从神童到控制论之父"><meta itemprop="description" content="诺伯特·维纳是一位传奇的数学家与科学家，在概率论、信号处理等多个领域有开创性贡献。他最为人知的成就是创立了控制论，深刻影响了自动化与信息科学的发展。"></span><span hidden=""><meta itemprop="image" content="/lib/favicon/android-chrome-192x192.png"></span><div class="article-meta" id="top"><span hidden="" itemprop="name headline"></span></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p> <span class="p logo center large">Wiener</span></p><br><h1 hidden="">Wiener</h1><div class="story post-story"><h2 id="生平"><a href="#生平" class="headerlink" title="生平"></a>生平</h2><p>诺伯特・维纳（Norbert Wiener，1894-1964）是一位杰出的美国计算机科学家、数学家和哲学家。Wiener 的家族融合了深厚的东欧与德国背景。</p><p>Wiener 的父亲利奥・维纳（Leo Wiener，1862-1939）是对他一生影响最深远的人。Leo Wiener 是立陶宛的犹太人，早年前往华沙的医学院求学。由于对医学不感兴趣，Leo Wiener 转赴柏林的理工学院学习工程学。</p><p>毕业后，Leo Wiener 前往美国，做过农场和工厂工人，后来前往 Missouri 州 Kansas 城。凭借卓越的语言天赋（据说能说三十种语言），他成为 Kansas 大学的日耳曼语和罗曼语系讲师，并最终晋升为现代语言学教授。</p><p>在那里，Leo 与伯莎・卡恩（Bertha Kahn）结为连理。1894 年 11 月 26 日，Norbert Wiener 出生于 Missouri 州的 Columbia，是他们的第一个孩子。不久后，全家搬往 Boston。</p><p>为养家，Leo Wiener 兼任了多份教职，最终在 Harvard 大学获得斯拉夫语教职并晋升为教授。Wiener 有两个妹妹。Wiener 从小在充满学术氛围的家庭中长大，能够在父亲包罗万象的书房中自由漫游和阅读。10 岁时，他写下了《无知论》（The theory of Ignorance），反驳了 “人类知识没有限度” 的傲慢假设，指出所有知识 “都基于一种近似（approximation）”，不可能有绝对的确定性。</p><p>然而，他的正规学校教育并不顺利。7 岁时，父母将他送入 Peabody School。尽管年龄很小，但 Wiener 一入学就读三年级，并很快升入四年级，而他对此并无任何不适应。尽管 Wiener 的阅读和理解力远超同龄人，但他的算术运算能力却十分薄弱。</p><p>父亲 Leo Wiener 敏锐地发现，机械枯燥的算数练习让儿子感到厌烦，于是决定让他退学，亲自在家教授他代数，以提供更大的挑战并激发他的想象力。</p><p>后来，因 Wiener 视力不佳且身体协调性差，医生建议他停止阅读六个月以保护眼睛。</p><p>在这期间，父亲通过口授教他进行数学心算。这种独特的训练不仅保护了他的视力，还为他培养了终生受用的卓越心智推演能力。</p><p>1903 年秋天（9 岁），他进入 Ayer High School 读高三（同学大他七岁），并于 1906 年（11 岁）顺利毕业。</p><p>1906 年 10 月 7 日，世界首次知道了 Wiener 的名字。当时他的照片出现在《纽约世界报》的头版，标题是《全世界最杰出的男孩》（The most remarkable boy in the world）。</p><p>11 岁时，Wiener 进入塔夫特学院（Tufts College，即如今的 Tufts 大学）。在那里，他学习了希腊语和德语、物理学和数学，以及生物学。</p><p>他自己回忆说：“尽管对生物学很感兴趣，但我毕业时拿到的是数学学士学位。在大学的每一年中，我都学习数学…… 我发现微积分和微分方程很简单，过去我常和父亲探讨大学数学的内容。”</p><p>1909 年（14 岁），Wiener 获得数学学士学位。</p><p>之后他进入 Harvard 大学攻读动物学研究生，这遭到了父亲 Leo 的反对，但是，动物学强调实验室工作，而 Wiener 的眼疾使他难以做实验，动物学对他而言有些吃力。</p><p>不久，Wiener 决定听从父亲的建议改修哲学。和以往一样，这还是他父亲做出的决定 —— 他父亲认为 Wiener 在 Tufts 读本科时的哲学成绩暗示了他应该成为一名哲学家。</p><p>Wiener 获得了 Cornell 大学哲学院的奖学金，并于 1910 年转到 Cornell 学习哲学。</p><p>但是在 1911 年，他回到 Harvard 攻读数学与哲学，深受亨廷顿（Edward Vermilye Huntington，1874-1952）的数理哲学启发。</p><p>1913 年，18 岁的 Wiener 在施密特（Karl Schmidt）的指导下获得了 Harvard 大学博士学位。</p><p>他的博士论文探讨了数理逻辑，比较了施罗德（Friedrich Wilhelm Karl Ernst Schröder，1841-1902）与怀特海（Alfred North Whitehead，1861-1947）和罗素（Bertrand Arthur William Russell，1872-1970）的工作。</p><p>1914 年，这篇论文的结果被写入论文《A simplification in the logic of relations》，并发表在《剑桥哲学学会会刊》上。</p><p>1914 年，Wiener（19 岁）远赴欧洲游学并进行博士后研究，希望能够最终拿到美国知名大学的终身教职职位。</p><p>在 Cambridge 大学，他师从 Bertrand Russell。Russell 对 Wiener 的印象似乎是相互的：</p><ul><li>Wiener: Russell 的态度似乎是冷漠掺杂着轻蔑，我觉得自己应该满足于在课堂上看到的他。</li><li>Russell: 在阅读 Wiener 的博士论文后，他给出了这样的评价 “这是一份不错的技术研究”。</li></ul><p>他在 Cambridge 还旁听了哈代（Godfrey Harold Hardy，1877-1947）的课程。当时的 Wiener 认为 Hardy 对自己的影响最大：Hardy 的课程及其严谨性启发了我。在我学习数学课程的那么多年里，我从未见过像 Hardy 这样思维清晰、兴趣盎然、智力超群的人。如果让我选一个人作为数理思维的导师，我一定会选 Hardy。</p><p>Hardy 引导 Wiener 学习了 Lebesgue 积分。</p><p>在 Göttingen 大学，他跟随希尔伯特（David Hilbert，1862-1943）学习微分方程，参加了兰道（Edmund Georg Hermann Landau，1877-1938）的群论课程，并听了胡塞尔（Edmund Gustav Albrecht Husserl，1859-1938）的现象学研讨班。在 Göttingen 他领悟到：</p><p>“数学不仅是一门在书房里研究的学问，更是一门需要讨论和与之共存的生活方式。”</p><p>一战爆发前夕他返回美国，并在 New York 做短暂停留，当时他在 Columbia 大学跟随哲学家杜威（John Dewey，1859-1952）继续研究哲学。之后，他在 Harvard 讲授过哲学，在通用电气（General Electric）当过工程师，也做过撰稿人和记者。</p><p>尽管 Wiener 后来成为坚定的和平主义者，但一战期间他极度渴望参军。但因视力差被拒收。1918 年夏天，应维布伦（Oswald Veblen，1880-1960）的邀请，他前往 Maryland 州的阿伯丁试验场（Aberdeen Proving Ground）从事弹道学（ballistics）研究，这重新唤醒了他对数学的兴趣。</p><p>战后，各种原因的驱使下，Wiener 未能获得 Harvard 的终身职位。</p><p>在被澳大利亚 Melbourne 大学拒绝后，经奥斯古德（William Fogg Osgood，1864-1943）举荐，MIT 聘请他担任数学讲师。此后他一生都在 MIT 任教，成为该校传奇的人物。</p><p>1926 年，他与玛格丽特（Margaret Engemann）结婚（育有两女）。整个 20、30 年代，他频繁往返欧洲，结识莱维（Paul Lévy，1886-1971）、玻恩（Max Born，1882-1970）、布拉施克（Wilhelm Blaschke，1885-1962）、阿廷（Emil Artin，1898-1962）和哥德尔（Kurt Gödel，1906-1978）等顶尖学者。</p><p>Wiener 与在 Göttingen 访学的冯・诺伊曼（John von Neumann，1903-1957）和奥本海默（Julius Robert Oppenheimer，1904-1967）相遇，并与 von Neumann 建立了私人联系。</p><p>1925 年夏，Wiener 向 Göttingen 大学的在职和访学数学家介绍其工作，后在家书中写道：“Hilbert 认为他的工作非常优美”。在快离开这里时，Göttingen 大学数学系主任库朗（Richard Courant，1888-1972）告诉 Wiener，如果下一年他返回 Göttingen，他将得到客座教授的职位。</p><p>1935 年，Wiener 应邀到清华大学进行为期一学年的访问讲学。Wiener 到清华后，系统讲授了 Fourier 级数和 Fourier 积分的理论。除此外，还作了一些其他专题讲座。</p><p>在清华期间，Wiener 展现了科学伯乐的眼光，他非常赏识当时还只是年轻助理的青年学者华罗庚。Wiener 亲自写信，向自己曾在 Cambridge 大学游学时的导师 Hardy 极力推荐华罗庚。正是得益于这封关键推荐信，促成了华罗庚于 1936 年远赴 Cambridge 大学跟随 Hardy 深造。</p><p>1941 年，Wiener 加入国防研究委员会致力于高射炮自动瞄准系统的研发。这一需要将机器与人类操作员结合起来的研究，促使他与罗森布鲁斯（Arturo Rosenblueth Stearns，1900-1970）和助手比格洛（Julian Himely Bigelow，1913-2003）在 1943 年发表了经典文章《行为、目的与目的论》（Behavior, Purpose and Teleology）。</p><p>在 1948 年，Wiener 出版巨著《控制论：或动物与机器中的控制与通信》（Cybernetics: Or Control and Communication in the Animal and the Machine）。</p><p>Wiener 在自传《我是一个数学家》（I am a Mathematician）中称：“他宁愿选择在清华大学任访问教授的 1935 年作为创立控制论的起点”。</p><p>作为一名学者，Wiener 拥有极具个性的特质。荷兰数学家弗赖登塔尔（Hans Freudenthal，1905-1990）曾这样描述他：<br>“在外表和举止上，Norbert Wiener 是一个巴洛克式的人物，身材矮胖、高度近视，极端地结合了诸多特质。他的谈吐奇特地混合着浮夸与随性。他不是个好的倾听者。他的自我夸耀带有游戏色彩，令人信服且从不惹人反感。他会说多种语言，但无论哪种都很难让人听懂。他还是一个出了名的糟糕讲演者。”</p><p>统计学家肯德尔（David George Kendall，1918-2007）也回忆道：“作为一个人，Wiener 极具启发性…… 他可能会在讲座上打呼噜，醒来后却在讨论环节提出令人尴尬的刁钻问题；私下里，他也可能对同席晚宴的显要人物就其毫不了解的领域提出建议。但我更喜欢回忆曾经深夜在 Oxford 大学的 Magdalen College 看到的场景：Wiener 被一群听得入了迷的本科生围在中间，滔滔不绝地讲着。”</p><p>1964 年 3 月 18 日，Wiener 在瑞典斯德哥尔摩（Stockholm）因心脏病发作逝世，享年 69 岁。</p><p>作为跨学科的导师，Wiener 培养了诸如 Bose 音响创始人博斯（Amar Gopal Bose，1929-2013）、莱文森（Norman Levinson，1912-1975）等一大批杰出博士生。</p><p>此外，Wiener 的学术思想还深深影响了克劳德・埃尔伍德・香农（Claude Elwood Shannon，1916-2001）。在 Shannon 1948 年宣告信息论诞生的著作《通信的数学理论》（A mathematical theory of communication）的第三部分（Part III）的开头，Shannon 极其谦逊地写下了对 Wiener 的致敬：“通信理论在很大程度上受惠于 Wiener 的基础哲学与理论（Communication theory is heavily indebted to Wiener for much of its basic philosophy and theory.）。”</p><p>Shannon 在文中明确指出，是 Wiener 在二战期间的国防机密报告中，首次将通信理论明确表述为一个统计学问题，这直接启发了他的信息论。</p><p>他一生获誉无数，1963 年获颁美国国家科学奖章；其探讨控制论与宗教边界的著作《God & Golem, Inc.》在 1965 年被追授美国国家图书奖。</p><p>为纪念他，应用数学界在 1967 年设立了 “Norbert Wiener 应用数学奖”（Peter Lax 在 1975 年获得过）；月球背面的 “Wiener 环形山（Crater Wiener）” 以及 Maryland 大学的调和分析及应用中心（Norbert Wiener Center for Harmonic Analysis and Applications）均以他命名。</p></div><div class="story post-story"><h2 id="贡献"><a href="#贡献" class="headerlink" title="贡献"></a>贡献</h2><h3 id="一、数理逻辑与集合论（Mathematical-logic-and-set-theory）"><a href="#一、数理逻辑与集合论（Mathematical-logic-and-set-theory）" class="headerlink" title="一、数理逻辑与集合论（Mathematical logic and set theory）"></a>一、数理逻辑与集合论（Mathematical logic and set theory）</h3><p>在 1913 年的博士论文中，Wiener 首次公开提出可以用初等集合论来定义有序对（ordered pairs）<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.561ex" xmlns="http://www.w3.org/2000/svg" width="5.169ex" height="2.253ex" role="img" focusable="false" viewBox="0 -748 2284.7 996"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="(x,y)"><g data-mml-node="mo" data-latex="("><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="mi" data-latex="x" transform="translate(389,0)"><path data-c="1D465" d="M527 373C527 419 482 442 432 442 389 442 355 419 329 373 308 419 273 442 222 442 173 442 133 419 101 374 74 335 60 306 60 287 60 278 65 273 75 273 84 273 90 278 92 287 111 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78-245Z"></path></g></g></g></svg></mjx-container> 。这意味着关系理论不再需要独立于集合论的公理。</p><p>1921 年，波兰数学家库拉托夫斯基（Kazimierz Kuratowski，1896-1980）将其定义简化为：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:30.714ex"><svg style="vertical-align:-.566ex;min-width:30.714ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math" data-latex="(x,y):= \{\{x\}, \{x,y\}\}"><g data-mml-node="mtable" data-latex="(x,y):= \{\{x\}, \{x,y\}\}" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 750) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="4709.8 -750 1 1000"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mo" data-latex="("><path data-c="28" 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data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>该定义被全球数学界沿用至今。</p><h3 id="二、概率论与布朗运动（Probability-theory-and-Brownian-motion）"><a href="#二、概率论与布朗运动（Probability-theory-and-Brownian-motion）" class="headerlink" title="二、概率论与布朗运动（Probability theory and Brownian motion）"></a>二、概率论与布朗运动（Probability theory and Brownian motion）</h3><p>从 1921 年起，Wiener 将原本属于物理概念的 Brownian 运动（Brownian motion）进行了严密的数学化。他在一维连续路径组成的空间中引入了一种测度 ——Wiener 测度（Wiener measure），首次自然地将概率概念引入连续函数空间，并证明了 Brownian 运动路径的不可微性。</p><p>一维 Brownian 运动也因此被命名为 Wiener 过程（Wiener process），它是最具代表性的 Lévy 过程（Lévy process，一类具有右连续左极限 càdlàg 特性的独立增量随机过程）之一。</p><p>在此基础上，衍生出了描述流体粒子速度随机波动的 Wiener 方程（Wiener equation），以及概率拓扑中代表 Brownian 运动轨迹管状邻域的 Wiener 香肠（Wiener sausage）。</p><h3 id="三、分析学与抽象空间（Analysis-and-abstract-spaces）"><a href="#三、分析学与抽象空间（Analysis-and-abstract-spaces）" class="headerlink" title="三、分析学与抽象空间（Analysis and abstract spaces）"></a>三、分析学与抽象空间（Analysis and abstract spaces）</h3><p>在空间理论上，Wiener 与巴拿赫（Stefan Banach，1892-1945）几乎在同一时期独立发现了 Banach 空间（Banach space）的概念；他构建的经典 Wiener 空间后来被推演为抽象 Wiener 空间（Abstract Wiener space）。</p><p>此外，在 1923 年至 1925 年间，Wiener 对偏微分方程中的 Dirichlet 问题（Dirichlet’s problem）进行了深入探讨。他提出了判断边界点是否满足特定连续性条件的 Wiener 准则（Wiener criterion），并定义了广义的容度（capacity）。这些理论成果极大地拓展了经典的位势论（potential theory）。</p><p>为了解答工程中复杂的信号处理问题，Wiener 试图将随机过程的研究推广到更一般的形式。这促使他在 1930 年左右系统地发展了广义调和分析（generalized harmonic analysis）。在研究过程中，他利用 Fourier 变换的工具，深入探讨了 Tauberian 定理（Tauberian theorem）—— 这是一类研究无穷级数或积分收敛条件的定理。</p><p>1932 年，他证明了许多复杂的收敛问题可以被纳入一个更本质的调和分析框架中。凭借这篇发表在《Annals of Mathematics》上的经典论文《Tauberian 定理》（Tauberian Theorems），他在 1933 年荣获了美国数学会的 Bôcher 奖。</p><p>此外，他与佩利（Raymond Edward Alan Christopher Paley，1907-1933）合作证明的 Paley-Wiener 定理深刻揭示了复平面上整函数（entire functions）的增长性与其 Fourier 变换的支集之间的内在联系。</p><p>他证明的 Wiener - Khinchin 定理则指出，宽平稳随机过程的功率谱密度是其自相关函数的 Fourier 变换（The power spectral density of a wide-sense stationary (WSS) random process is the Fourier transform of its autocorrelation function）。</p><h3 id="四、信号处理与-Wiener-滤波器（Signal-processing-and-Wiener-filter）"><a href="#四、信号处理与-Wiener-滤波器（Signal-processing-and-Wiener-filter）" class="headerlink" title="四、信号处理与 Wiener 滤波器（Signal processing and Wiener filter）"></a>四、信号处理与 Wiener 滤波器（Signal processing and Wiener filter）</h3><p>第二次世界大战期间，Wiener 参与了防空火炮控制系统的研发。由于敌机的飞行轨迹带有不可预知的随机性，且雷达接收到的信号夹杂着环境噪声（noise），系统必须基于不完整的含噪数据来预测飞机未来的位置。</p><p>基于对时间序列的统计分析，Wiener 开发出了一种数学滤波器（filter）。该模型能够从含有随机干扰的观测数据中提取出最优的真实信号估计，使得提取结果与真实信号之间的均方误差（mean square error）最小化。</p><p>这一技术被称为 Wiener 滤波器（Wiener filter），它成为了现代通信理论（communication theory）和数字信号处理的基础工具。</p><h3 id="五、创立控制论与非线性控制理论（Cybernetics-and-nonlinear-control）"><a href="#五、创立控制论与非线性控制理论（Cybernetics-and-nonlinear-control）" class="headerlink" title="五、创立控制论与非线性控制理论（Cybernetics and nonlinear control）"></a>五、创立控制论与非线性控制理论（Cybernetics and nonlinear control）</h3><p>1948 年，他出版名著《控制论：或动物与机器中的控制与通信》（Cybernetics: or, control and communication in the animal and the machine），正式创立了控制论（cybernetics，该词由他本人创造）。在这门新学科中，他探讨了输入（input）、输出（output）、信息（information）、预测（prediction）和稳态（homeostasis）等概念。</p><p>他指出，无论是自动寻的导弹还是维持机体平衡的生物体，其核心都在于通过环境的反馈来纠正自身的行为。这一理论打破了生命体与机器的绝对界限，直接启发了后来的自动化控制和早期人工智能的研究。</p><p>他留下了那句改变现代科学世界观的名言：“信息就是信息，不是物质，也不是能量。（Information is information, not matter or energy）”</p><p>在非线性控制领域，Wiener 晚年还深入研究了多项式混沌（polynomial chaos），并将 Hermite-Laguerre 展开（Hermite-Laguerre expansion）成功应用于非线性控制理论中的系统辨识（system identification）与控制。</p></div></div><div class="footer"></div><div class="article-meta" id="bottom"><div class="new-meta-box"></div></div><div class="prev-next"><a class="prev" href="/notes/person/16"><p class="title"><i class="fa-solid fa-chevron-left" aria-hidden="true"></i>Birkhoff乔治·大卫·伯克霍夫：美国数学巨擘与美学探路者</p><p class="content">乔治·大卫·伯克霍夫是美国杰出的数学家，他在动力系统与遍历定理方面做出了奠基性贡献，并因证明庞加莱最后几何定理而闻名。他成功地将数学思想拓展至美学理论，其研究领域广泛，包括地图着色和广义相对论，深刻影响了20世纪的数学发展</p></a><a class="next" href="/notes/person/18"><p class="title">Shannon克劳德·香农：信息时代的奠基者与科学巨匠<i class="fa-solid fa-chevron-right" aria-hidden="true"></i></p><p class="content">克劳德·香农是信息论之父，其1948年发表的《通信的数学理论》为现代通信奠定了数学基础。他首次将布尔代数应用于开关电路设计，革新了数字计算机的构建原理。此外，他在密码学、人工智能及自动机领域的早期探索也产生了深远影响</p></a></div><div class="recommended-article"><div class="recommended-article-header"><i class="fa-solid fa-bookmark 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id="comments"><span hidden=""><meta itemprop="discussionUrl" content="/notes/person/17#comments"></span><p ct=""><i class="fa-duotone fa-comments"></i> 留言区</p><div id="layoutHelper-comments"></div></article></div><aside id="l_side" itemscope="" itemtype="http://schema.org/WPSideBar"><section class="widget text desktop mobile pjax"><header><a href="/notes/"><i class="fa-duotone fa-book fa-fw" aria-hidden="true"></i> <span class="name">Notes</span></a></header><div class="content"></div></section><section class="widget list group desktop mobile pjax"><header><i class="fa-duotone fa-p fa-fw" aria-hidden="true"></i> <span class="name">History of mathematics</span></header><div class="content"><ul class="list entry navigation"><li><a class="flat-box" title="/notes/person/" href="/notes/person/" active-action="action-notesperson"><div class="name"> Welcome</div></a></li><li><a class="flat-box" title="/notes/person/1" href="/notes/person/1" active-action="action-notesperson1"><div class="name"> Hilbert</div></a></li><li><a class="flat-box" title="/notes/person/2" href="/notes/person/2" active-action="action-notesperson2"><div class="name"> Thompson</div></a></li><li><a class="flat-box" title="/notes/person/3" href="/notes/person/3" active-action="action-notesperson3"><div class="name"> Conway</div></a></li><li><a class="flat-box" title="/notes/person/4" href="/notes/person/4" active-action="action-notesperson4"><div class="name"> Borcherds</div></a></li><li><a class="flat-box" title="/notes/person/5" href="/notes/person/5" active-action="action-notesperson5"><div class="name"> Wall</div></a></li><li><a class="flat-box" title="/notes/person/6" href="/notes/person/6" active-action="action-notesperson6"><div class="name"> Waldhausen</div></a></li><li><a class="flat-box" title="/notes/person/7" href="/notes/person/7" active-action="action-notesperson7"><div class="name"> Dehn</div></a></li><li><a class="flat-box" title="/notes/person/8" href="/notes/person/8" 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