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<!DOCTYPE html><html lang="zh-Hans"><head hexo-theme="https://github.com/volantis-x/hexo-theme-volantis/#6.0.3"><meta name="generator" content="Hexo 8.1.2"><meta name="Volantis" content="6.0.3"><meta charset="utf-8"><meta name="robots" content="index,follow,max-image-preview:large"><link rel="canonical" href="https://blog.mhuig.top/notes/person/18"><meta http-equiv="x-dns-prefetch-control" content="on"><link rel="dns-prefetch" href="https://static.mhuig.top"><link rel="preconnect" href="https://static.mhuig.top" crossorigin=""><meta name="renderer" content="webkit"><meta name="force-rendering" content="webkit"><meta http-equiv="X-UA-Compatible" content="IE=Edge,chrome=1"><meta name="HandheldFriendly" content="True"><meta name="mobile-web-app-capable" content="yes"><meta name="viewport" content="width=device-width,initial-scale=1,maximum-scale=5"><meta content="black-translucent" name="apple-mobile-web-app-status-bar-style"><meta content="telephone=no" 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History,人物,香农,信息论,数字电路,人工智能先驱,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc="" name="description" 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 - 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/18"><meta property="og:site_name" content="Magicland"><meta property="og:description" 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"><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="2026-05-13T06:07:00.000Z"><meta property="article:modified_time" content="2026-05-13T06:10:00.000Z"><meta property="article:author" content="MHuiG"><meta property="article:tag" content="History"><meta 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itemscope="" itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="Shannon克劳德·香农：信息时代的奠基者与科学巨匠"><meta itemprop="description" 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"></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">Shannon</span></p><br><h1 hidden="">Shannon</h1><div class="story post-story"><h2 id="生平"><a href="#生平" class="headerlink" title="生平"></a>生平</h2><p>克劳德・埃尔伍德・香农（Claude Elwood Shannon, 1916-2001）被誉为 “信息论之父”，是当今信息时代的奠基人之一。</p><p>1916 年 4 月 30 日，Shannon 出生于美国 Michigan 州的佩托斯基（Petoskey），是爱迪生（Thomas Alva Edison, 1847-1931）的远亲。他的母亲是一名语言教师以及高中的校长，他的父亲则是一名商人以及遗嘱检验法官。</p><p>年轻的 Shannon 在公立学校系统接受教育，并于 16 岁那年高中毕业。在他的童年时期，他度过了一段正常而快乐的时光，尽管当时几乎没有迹象表明他日后会展现出惊世骇俗的天才。</p><p>与他后来的生活状态相似，年幼的 Shannon 并不十分外向，但当别人主动接近他时，他总是表现得非常友好。他从小就对各种机械和电子设备表现出浓厚的兴趣，并且总是对各种设备是如何运作的充满极大的好奇心。</p><p>高中毕业后，Shannon 进入 Michigan 大学深造。1936 年，他在 Michigan 大学同时获得了电气工程和数学两个学士学位。他对这两个领域的双重兴趣贯穿了他的整个职业生涯，并成为他日后取得诸多交叉学科突破的关键。</p><p>在本科期间，Shannon 接触到了 Boolean 代数（Boolean algebra）这一数学对象，并对其产生了兴趣，为日后的研究埋下伏笔。</p><p>在大学毕业并思考下一步该做什么时，Shannon 在一个公告栏上看到了一则招聘启事，招募人员来操作布什（Vannevar Bush, 1890-1974）在 MIT 发明的微分分析仪（differential analyzer，一种早期的模拟计算机）。</p><p>Shannon 申请了这份工作，并成功被录取为 MIT 电气工程系的物理研究助理和研究生。</p><p>来到 MIT 后，Shannon 被控制微分分析仪的复杂开关电路（switching circuit）深深吸引，他开始探索使用 Boolean 代数来理解这些开关电路的可能途径。</p><p>当时他已经注意到电话交换电路与 Boolean 代数之间的类似性，即把 Boolean 代数的 “真” 与 “假” 和电路系统的 “开” 与 “关” 对应起来，并用 1 和 0 表示。</p><p>入学一年后，Shannon 于 1937 年的夏天前往 Bell 电话实验室实习。</p><p>同年秋天回到 MIT 后，Shannon 开始完善他关于如何使用 Boolean 代数来进行继电器（relay）电路的分析和综合设计的想法。</p><p>这项工作成为了他在 MIT 的硕士毕业论文以及他发表的第一篇学术论文：《继电器与开关电路的符号分析》（A Symbolic analysis of relay and switching circuits）。</p><p>该论文至今被广泛认为是开关电路领域的奠基之作，也是有史以来最重要的一篇硕士论文之一。</p><p>Harvard 大学的加德纳（Howard Gardner, 1943-）教授说，“这可能是本世纪（二十世纪）最重要、最著名的一篇硕士论文。”</p><p>Shannon 的这篇硕士论文获得了 1939 年的 Alfred Noble 奖 —— 这是由美国土木工程师学会为了纪念前主席诺贝尔（Alfred Noble, 1844-1914）而在 1929 年设立的。</p><p>在 Bush 的建议下，Shannon 开始在遗传学领域寻找他的博士研究课题。他将专业从电气工程转向了数学，并致力于为遗传学建立一个数学基础。</p><p>1940 年，他在希区柯克（Frank Lauren Hitchcock, 1875-1957）的指导下完成博士学位论文《理论遗传学的代数学》（An algebra for theoretical genetics）。在进行博士研究的同时，他也开始对通信的根本问题产生兴趣并继续在开关理论方面开展工作。</p><p>1940 年，毕业后的 Shannon 到 IAS 工作了一年，并开始认真致力于发展他那初具雏形的通信数学理论。</p><p>到了 1941 年的夏天，Shannon 回到了 Bell 实验室。受第二次世界大战影响，Shannon 的工作包括了协助开发防空火控系统、防空指挥仪以及保密系统。</p><p>同一时期，Shannon 开始涉及密码学。Shannon 敏锐地意识到，密码学中的根本问题与他正在发展的通信理论中的思想有着极其密切的联系。</p><p>在 Bell 实验室，Shannon 遇到了担任数值分析员的贝蒂（Betty）。两人于 1949 年结婚。</p><p>1943 年，英国数学家图灵（Alan Mathison Turing, 1912-1954）被派到华盛顿和美国海军交流破译德国的北大西洋潜艇舰队密码的成果，并在 Bell 实验室待了一段时间。Turing 向 Shannon 介绍了现在被称为 “通用 Turing 机” 的概念。Shannon 对此很感兴趣，因为 Turing 机的概念和 Shannon 自己的很多想法相吻合。</p><p>1945 年，Shannon 将他的密码学研究成果撰写成了机密论文《密码学的数学理论》（A mathematical theory of cryptography）；这篇论文直到 1949 年才解密并在公开文献中以《保密系统的通信理论》（Communication theory of secrecy systems）为题发表。</p><p>这篇论文为保密系统奠定了坚实的数学基础，并对现代密码学产生了巨大影响。</p><p>1948 年，经过长达八年断断续续地思考，Shannon 在《Bell 系统技术期刊》（Bell System Technical Journal）上发表了《通信的数学理论》（A mathematical theory of communication）。</p><p>在这一篇论文中，Shannon 提出熵（entropy）的概念，给出了可量化的信息的定义，推导出一系列定理。这篇论文奠定了信息论的基础，提出了一个通信系统的线性示意图模型，在当时的科技界引起了巨大的轰动。</p><p>Shannon 借用了一个极具启发性的术语 “信息（information）” 来描述被通信的事物。更重要的是，他成功地对信源和信道的 “信息” 进行了精确的量化。</p><p>这种关于信息的新概念导致了我们今天习以为常的通过发送 1 和 0 组成的信号流来传输图像、文字、声音等的想法的产生，彻底改变了人们对电信系统的理解方式和设计范式。</p><p>1949 年，Shannon 发表了《编程实现计算机下棋》（Programming a computer for playing chess），这是人工智能的一个先驱工作。</p><p>1950 年，Shannon 发明了会自我学习走迷宫的机械老鼠 “Theseus”，成为第一台人工智能装置的雏形。</p><p>1951 年，他发表了论文《一个走迷宫机器的介绍》（Presentation of a maze solving machine），这是一篇机器学习的先驱著作。</p><p>1953 年，他曾设计了 “心灵阅读（Mind Reading）” 机，可通过观察、记忆和分析对方过去所做选择的样本，试图猜测对方下一次可能选择。</p><p>Shannon 一直留在 Bell 实验室的数学研究小组工作直到 1956 年。随后，Shannon 在 MIT 做了一年的访问学者，并于 1958 年正式接受 MIT 的永久任命成为 Donner 科学教授，同时在电气工程系和数学系任职。</p><p>Shannon 在 MIT 一直工作到 1978 年。进入八十年代，Shannon 被确诊患有阿尔茨海默症（Alzheimer's disease）。他在一家私立医院度过了生命的最后几年，于 2001 年 2 月 24 日与世长辞。</p><p>除了学术研究，Shannon 爱好杂耍、骑独轮脚踏车和下棋。</p><p>Shannon 因其杰出贡献荣获诸多荣誉，其中包括 1939 年美国工程师协会颁的 Alfred Noble Prize（与 Nobel Prize 无关，由美国土木工程师学会设立）、1966 年美国国家科学奖章、1985 年音频工程协会金奖以及 1985 年京都奖。</p><p>2000 年，他被 Guglielmo Marconi 国际奖学金基金会授予 Marconi 终身成就奖，这是该基金会首次颁发此项殊荣。Shannon 被授予包括 Princeton 大学、Oxford 大学等多所大学的荣誉博士学位。</p><p>Shannon 还是包括美国国家科学院在内多个科学院、学会的成员、院士。</p><p>为了纪念 Shannon 对信息论的奠基性贡献，IEEE 信息理论协会设立了以 Shannon 命名的 Claude Elwood Shannon 奖用以表彰信息论领域里持久且深远的贡献，是通信理论领域最高奖，被誉为 “信息科学诺贝尔奖”。</p><p>1973 年的首届 Claude Elwood Shannon 奖被授予 Shannon 本人。</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、Boolean-代数与开关电路"><a href="#一、Boolean-代数与开关电路" class="headerlink" title="一、Boolean 代数与开关电路"></a>一、Boolean 代数与开关电路</h3><p>在二十世纪三十年代，电话交换机和早期的模拟计算机正变得越来越复杂，工程师们在设计继电器电路时主要依靠经验、直觉和繁杂的试错。Shannon 意识到，这些复杂的继电器和开关系统在本质上是一种逻辑载体。在他 1938 年发表的论文《继电器与开关电路的符号分析》（A symbolic analysis of relay and switching circuits）中，Shannon 创造性地将 Boolean 代数引入了电路设计中。</p><p>在文中，Shannon 详细论述了如何利用 Boolean 代数的加法、乘法和否定运算来描述并简化复杂的电路连接。通过这种方法，工程师不再需要面对一团乱麻的物理连线图，而是可以将其转化为简洁的数学方程进行化简，然后再根据化简后的结果重新构建最高效的物理结构。</p><p>Shannon 还展示了电路如何执行基本的算术运算。这一突破直接定义现代数字计算机的运行准则，将电路设计转变为一门严谨的科学，为随后数字时代的开启奠定首块基石。</p><h3 id="二、创立信息论框架"><a href="#二、创立信息论框架" class="headerlink" title="二、创立信息论框架"></a>二、创立信息论框架</h3><p>1948 年，Shannon 在《Bell 系统技术杂志》（Bell System Technical Journal）上分两部分发表了长篇论文《通信的数学理论》（A mathematical theory of communication）。</p><p>这部著作标志着 “信息论（information theory）” 这一全新学科的诞生，它为现代通信系统中的各个组成部分以及整体架构建立了一个清晰的概念基础。</p><p>在 Shannon 之前，通信工程界对于 “消息” 究竟是什么只有一个非常模糊的概念。Shannon 剥离了消息的语义（即 “意义”），指出消息应当被视为在多个选项之间的 “选择”。</p><p>他强调，重要的是进行选择，而不是选项的具体表现形式（如字母、象形文字或二进制 “0, 1” 代码）。这一想法构成了我们今天进行信息传输、存储的基础。</p><p>Shannon 进一步给出了通信过程的模型。他将整个通信过程抽象为一个包含信源（Source）、编码器（Encoder）、信道（Channel）、解码器（Decoder）和接收者的模型，并在历史上首次明确提出了 “编码（encoding）” 的概念。</p><p>他引入了人为数学模型（如 Markov 信源）来模拟具有统计规律的真实数据源，开启了利用随机过程来建立通信模型的先河。</p><h3 id="三，离散信源时的信息论"><a href="#三，离散信源时的信息论" class="headerlink" title="三，离散信源时的信息论"></a>三，离散信源时的信息论</h3><p>在他 1948 年论文的第一部分，Shannon 研究了信源取值为离散变量（比如 0, 1）时的信息论。</p><p>Shannon 引入了熵（entropy）的概念，将其定义为离散随机变量<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.925ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 851 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="X"><g data-mml-node="mi" data-latex="X"><path data-c="1D44B" d="M834 683C817 683 758 680 741 680 721 680 651 683 631 683 616 683 608 675 608 659 608 650 613 645 624 644 648 641 660 633 660 618 660 609 654 598 642 585L486 417C457 485 429 552 400 620 401 622 402 624 405 626 414 636 431 642 455 644 471 646 479 654 479 667 479 678 473 683 460 683 436 683 359 680 335 680 314 680 245 683 224 683 209 683 202 675 202 659 202 649 211 644 229 644 274 644 288 644 300 615L416 342 208 119C205 116 201 112 196 109 155 66 108 42 53 39 36 38 27 29 27 14 30 5 31 0 44 0 61 0 120 3 137 3 158 3 227 0 248 0 263 0 270 8 270 24 270 33 265 38 254 39 230 41 218 50 218 65 218 75 227 89 244 107 306 173 369 239 430 306 464 225 499 145 532 63 531 61 530 59 527 56 518 47 502 42 478 39 463 37 455 29 455 16 455 5 461 0 474 0L599 3C620 3 690 0 708 0 723 0 731 8 731 23 731 33 724 38 709 39 678 39 658 40 651 44 644 48 636 60 628 80L501 383C569 455 620 509 653 546 677 571 693 588 702 595 739 626 780 642 825 644 846 645 851 650 851 669 848 678 847 683 834 683Z"></path></g></g></g></svg></mjx-container> 的平均不确定度。对于一个具有<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" 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240 329 244 347 244 360 244 409 210 445 161 445 118 445 84 420 59 369 39 328 29 301 29 288 29 279 34 275 45 275 58 275 60 281 64 295 87 375 118 415 158 415 171 415 178 406 178 387 178 373 175 357 168 338 145 275 121 208 101 155 86 115 78 88 78 74 78 25 114-11 162-11 205-11 239 14 264 64 283 103 293 130 293 145 293 154 288 159 277 159 274 159 259 150 259 138Z"></path></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1361 1 2222.1"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:1" transform="translate(0,1159)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(1)}"><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><path 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></g></svg></g></g></g></g></svg></mjx-container></p><p>这一定义不仅赋予了信息以精确的数学度量，还指出了信息量与概率分布之间的负对数关系：概率越小的事件发生时带来的信息量越大。</p><p>在此基础上，Shannon 提出了著名的信源编码定理（source coding theorem）。</p><p>该定理的粗略陈述为：对于熵为<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.561ex" xmlns="http://www.w3.org/2000/svg" width="5.679ex" height="2.253ex" role="img" focusable="false" viewBox="0 -748 2510 996"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="H(X)"><g data-mml-node="mi" data-latex="H"><path data-c="1D43B" d="M881 668C881 678 875 683 863 683L736 680 609 683C593 683 586 674 586 659 586 652 589 647 594 646 604 645 612 644 617 644 648 643 665 641 670 640 675 639 678 636 678 631 677 628 676 622 674 613L615 375 321 375 379 602C384 623 393 636 406 641 413 643 430 644 458 644 485 644 496 644 496 668 496 678 490 683 478 683L351 680 223 683C207 683 200 674 200 659 200 652 203 647 209 646 219 645 227 644 232 644 263 643 281 641 286 640 291 639 293 636 293 631 293 629 292 623 289 613L156 82C151 60 141 47 126 42 119 40 101 39 70 39 48 39 39 37 39 15 39 5 45 0 57 0L183 3 246 2C257 2 299 0 310 0 326 0 334 8 334 24 334 34 323 39 302 39 262 39 242 43 242 52 242 52 243 55 245 68L312 336 605 336 538 68C535 53 523 43 503 40 498 39 481 39 452 39 433 39 424 31 424 15 424 5 430 0 442 0L568 3 631 2C642 2 683 0 696 0 712 0 720 8 720 24 720 34 709 39 688 39 647 39 627 43 627 52 627 52 628 55 630 68L764 602C769 623 778 636 791 641 797 643 814 644 843 644 869 644 881 644 881 668Z"></path></g><g data-mml-node="mo" data-latex="(" transform="translate(881,0)"><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(1270,0)"><path data-c="1D44B" d="M834 683C817 683 758 680 741 680 721 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239 430 306 464 225 499 145 532 63 531 61 530 59 527 56 518 47 502 42 478 39 463 37 455 29 455 16 455 5 461 0 474 0L599 3C620 3 690 0 708 0 723 0 731 8 731 23 731 33 724 38 709 39 678 39 658 40 651 44 644 48 636 60 628 80L501 383C569 455 620 509 653 546 677 571 693 588 702 595 739 626 780 642 825 644 846 645 851 650 851 669 848 678 847 683 834 683Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(2121,0)"><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"></path></g></g></g></svg></mjx-container> 个二进制位才能实现无损表征；若分配的比特率低于此值，则无法实现无误差的复原。</p><p>为了证明这一定理，Shannon 引入了典型序列（typical sequence）的概念，即每种符号出现频率与其先验概率极其接近的序列。这一定理及其证明中引入的概念构成了今天各类有损、无损压缩技术的理论基础。</p><p>接下来，Shannon 考虑了噪声环境下的传输可靠性问题，他提出了信道容量（channel capacity）的概念及其核心定理：噪声信道编码定理（noisy channel coding theorem）。</p><p>这一定理推翻了传统认为提高传输可靠性必须降低速率的直觉。</p><p>该定理陈述为：只要信息传输速率<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.595ex" role="img" focusable="false" viewBox="0 -683 759 705"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="R"><g data-mml-node="mi" data-latex="R"><path data-c="1D445" d="M739 531C739 582 713 621 662 649 621 672 572 683 517 683L235 683C212 683 202 682 202 659 202 652 205 647 211 646 221 645 229 644 234 644 264 643 281 641 286 640 291 639 294 636 294 631 294 629 293 623 290 613L158 82C153 60 143 46 128 41 121 39 103 38 72 38 50 38 41 37 41 15 41 4 47-1 59 0L183 3 309 0C325-1 333 8 333 23 333 33 322 38 301 38 261 38 241 43 241 52 241 52 242 54 244 68L308 327 423 327C492 327 527 298 527 241 527 232 522 210 513 174 502 132 497 104 497 89 497 13 556-22 632-22 660-22 687-9 714 16 741 41 755 68 755 96 755 106 750 111 739 111 732 111 726 106 723 95 712 64 698 41 682 28 666 15 651 8 636 8 613 8 601 27 601 64 601 88 604 125 611 176 614 197 615 212 615 223 615 276 587 315 531 339 625 362 739 429 739 531M609 616C629 603 639 581 639 550 639 530 635 507 626 480 599 398 531 357 422 357L316 357 379 610C384 631 392 642 403 643 408 644 428 644 463 644 528 644 566 642 609 616Z"></path></g></g></g></svg></mjx-container> 小于信道容量<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.719ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 760 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="C"><g data-mml-node="mi" data-latex="C"><path data-c="1D436" d="M148 218C148 346 204 476 267 548 318 607 406 666 505 666 609 666 660 587 660 479 660 470 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84 438 115 478 224 498 296 498 333 498 403 451 442 381 442 322 442 271 416 230 363 222 407 187 442 136 442 80 442 58 390 44 345 34 313 29 294 29 287 29 278 34 273 45 273 50 273 53 274 56 276 61 285 64 292 65 299 83 375 106 413 133 413 151 413 160 399 160 371 160 358 155 331 144 290L87 63C84 50 78 24 78 19 78-1 89-11 111-11 130-11 144-1 151 19 153 24 159 49 170 92L191 181 221 295C232 318 249 341 270 365 299 397 335 413 378 413 411 413 427 391 427 348 427 310 406 234 363 120 356 102 353 87 353 74 353 25 390-11 438-11 482-11 517 14 542 64 561 104 571 131 571 144 571 153 566 158 555 158 552 158 537 149 537 137Z"></path></g></g></g></svg></mjx-container> 趋于无穷大，接收端的译码错误概率<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.439ex" xmlns="http://www.w3.org/2000/svg" width="2.071ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 915.5 636"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g 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433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 433L377 433C383 474 411 625 420 644 430 665 444 675 462 675 477 675 490 671 501 664 470 657 454 639 454 608 454 582 467 569 494 569 528 569 552 599 552 633Z"></path></g><g data-mml-node="mo" data-latex="(" transform="translate(552,0)"><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(941,0)"><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 92 287 111 345 153 413 220 413 253 413 269 392 269 351 269 330 251 252 216 118 199 51 169 18 126 18 112 18 99 21 88 26 114 36 127 54 127 80 127 106 114 119 87 119 54 119 29 91 29 58 29 12 76-11 125-11 167-11 201 12 228 58 247 12 283-11 335-11 383-11 423 12 455 57 482 96 496 125 496 144 496 153 491 158 481 158 472 158 467 153 464 144 447 87 402 18 337 18 304 18 287 38 287 79 287 92 292 120 303 165L337 300C356 375 387 413 431 413 445 413 458 410 469 405 442 396 429 378 429 351 429 325 443 312 470 312 502 312 527 341 527 373Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(1513,0)"><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"></path></g></g></g></svg></mjx-container> 是信号的概率密度函数。</p><p>Shannon 证明了在给定的平均功率时，Gaussian 分布具有最大的熵，这一特性使得 Gaussian 噪声成为通信系统中最具破坏性的干扰形式。</p><p>为了将连续波形与离散比特流联系起来，香农应用并推广了采样定理（sampling theorem），指出一个带宽限制在<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="2.371ex" height="1.595ex" role="img" focusable="false" viewBox="0 -683 1048 705"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="W"><g data-mml-node="mi" data-latex="W"><path data-c="1D44A" d="M956 680C937 680 872 683 853 683 838 683 830 675 830 660 830 650 836 645 848 644 885 643 903 632 903 610 903 604 899 594 891 579L629 125 594 602C590 638 614 644 659 644 680 644 690 652 690 668 690 678 684 683 671 683 649 683 572 680 550 680 531 680 465 683 446 683 431 683 423 675 423 659 423 649 432 644 451 644 482 644 498 637 499 623L505 548 261 125 225 613C222 638 264 644 296 644 313 644 321 652 321 668 321 678 315 683 303 683 281 683 203 680 181 680 162 680 96 683 77 683 62 683 55 675 55 660 55 649 65 644 84 644 126 644 130 642 132 612L177 7C178-12 185-22 197-22 210-20 215-17 223-2 224-1 224 0 224 1L509 494 545 7C546-12 553-22 565-22 575-22 584-15 592-1L918 562C945 611 964 639 1025 644 1043 646 1048 651 1048 668 1048 678 1043 683 1032 683 1018 683 970 680 956 680Z"></path></g></g></g></svg></mjx-container> 赫兹、持续时间为<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.593ex" height="1.532ex" role="img" focusable="false" viewBox="0 -677 704 677"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="T"><g 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642 132 612L177 7C178-12 185-22 197-22 210-20 215-17 223-2 224-1 224 0 224 1L509 494 545 7C546-12 553-22 565-22 575-22 584-15 592-1L918 562C945 611 964 639 1025 644 1043 646 1048 651 1048 668 1048 678 1043 683 1032 683 1018 683 970 680 956 680Z"></path></g></g></g></svg></mjx-container> 个等间隔的离散样本点完全确定。这一定理建立了连续信号与离散信号之间的一个基本桥梁。</p><p>Shannon 给出了连续信道的加性白高斯噪声（AWGN）容量公式。对于带宽为<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="2.371ex" height="1.595ex" role="img" focusable="false" viewBox="0 -683 1048 705"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="W"><g data-mml-node="mi" data-latex="W"><path data-c="1D44A" d="M956 680C937 680 872 683 853 683 838 683 830 675 830 660 830 650 836 645 848 644 885 643 903 632 903 610 903 604 899 594 891 579L629 125 594 602C590 638 614 644 659 644 680 644 690 652 690 668 690 678 684 683 671 683 649 683 572 680 550 680 531 680 465 683 446 683 431 683 423 675 423 659 423 649 432 644 451 644 482 644 498 637 499 623L505 548 261 125 225 613C222 638 264 644 296 644 313 644 321 652 321 668 321 678 315 683 303 683 281 683 203 680 181 680 162 680 96 683 77 683 62 683 55 675 55 660 55 649 65 644 84 644 126 644 130 642 132 612L177 7C178-12 185-22 197-22 210-20 215-17 223-2 224-1 224 0 224 1L509 494 545 7C546-12 553-22 565-22 575-22 584-15 592-1L918 562C945 611 964 639 1025 644 1043 646 1048 651 1048 668 1048 678 1043 683 1032 683 1018 683 970 680 956 680Z"></path></g></g></g></svg></mjx-container>、信号功率为<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.706ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 754 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="P"><g data-mml-node="mi" data-latex="P"><path data-c="1D443" d="M555 683 235 683C212 683 201 682 201 660 201 649 212 644 234 644 254 644 294 647 294 631 294 629 293 623 290 613L158 82C152 60 142 47 128 42 121 40 103 39 72 39 50 39 40 37 40 16 40 5 46 0 59 0L184 3 248 2C259 2 299 0 312 0 328 0 336 8 336 24 336 34 325 39 304 39 264 39 244 43 244 52 244 52 245 55 247 68L307 312 472 312C537 312 599 332 657 371 722 414 754 467 754 530 754 629 660 683 555 683M524 644C611 644 654 614 654 554 654 498 626 423 597 397 559 363 509 346 447 346L313 346 379 610C387 644 387 644 429 644Z"></path></g></g></g></svg></mjx-container>、噪声功率为<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.993ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 881 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="N"><g data-mml-node="mi" data-latex="N"><path data-c="1D441" d="M864 683C845 683 783 680 764 680 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data-c="1D436" d="M148 218C148 346 204 476 267 548 318 607 406 666 505 666 609 666 660 587 660 479 660 470 657 438 657 429 657 420 663 415 676 415 681 415 685 416 688 417 693 424 696 431 698 438L760 691C760 700 755 705 745 705 741 705 735 701 727 692L662 619C623 676 568 705 497 705 442 705 388 692 333 667 222 615 141 533 89 421 63 366 50 310 50 253 50 173 75 108 126 56 177 4 242-22 322-22 401-22 473 7 538 64 565 87 588 115 607 146 634 191 648 222 648 241 648 250 643 255 632 255 623 255 618 251 616 242 595 177 562 125 517 88 460 41 400 17 338 17 218 17 148 98 148 218Z"></path></g></g></g></svg></mjx-container> 的精确陈述为</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:31.029ex"><svg style="vertical-align:-1.872ex;min-width:31.029ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="4.876ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1327.5)"><g data-mml-node="math" data-latex="
C = W \log_2\left(1 + \frac{P}{N}\right)
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590 0 599 0 608 0 614 8 618 24L757 574C770 625 798 643 860 644 874 645 881 653 881 669 878 678 877 683 864 683Z"></path></g><rect width="1081" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" data-latex-item="\right)" data-latex="\right)" transform="translate(3706.4,0)"><path data-c="29" d="M86-792C198-707 290-571 363-386 429-217 462-54 462 101L462 399C462 554 429 717 363 886 290 1071 198 1207 86 1292 83 1295 80 1296 75 1296 62 1296 55 1289 55 1276 55 1269 57 1264 62 1260 163 1183 243 1052 302 865 353 707 378 552 378 399L378 101C378-51 353-206 302-365 243-552 163-684 62-760 57-764 55-769 55-776 55-789 62-796 75-796 80-796 83-795 86-792Z"></path></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1327.5 1 2155"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:3" transform="translate(0,716.5)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(3)}"><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><path data-c="33" d="M303 353C369 378 431 441 431 526 431 569 410 604 369 631 333 654 292 666 246 666 201 666 162 654 127 631 88 605 68 571 68 528 68 495 90 472 122 472 154 472 176 495 176 527 176 560 157 578 119 580 145 615 186 633 242 633 302 633 332 598 332 527 332 485 324 450 309 421 282 373 245 364 183 364 171 362 165 357 165 348 165 333 172 333 192 333L235 333C310 333 348 280 348 173 348 88 317 14 241 14 176 14 128 36 99 80 134 79 160 105 160 139 160 173 135 198 101 198 62 198 42 178 42 137 42 88 64 49 108 18 147-9 193-22 244-22 301-22 350-3 393 34 436 71 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transform="scale(1,-1)"><g data-mml-node="math" data-latex="D"><g data-mml-node="mi" data-latex="D"><path data-c="1D437" d="M567 683 235 683C212 683 201 682 201 660 201 649 212 644 234 644 254 644 294 647 294 631 294 629 293 623 290 613L158 82C153 60 143 47 128 42 121 40 103 39 72 39 50 39 40 37 40 16 40 5 51 0 72 0L399 0C467 0 532 20 595 61 702 130 804 265 804 429 804 577 713 683 567 683M710 466C710 353 659 216 607 149 550 76 475 39 381 39L270 39C259 39 252 39 248 40 242 40 239 42 239 45 239 47 241 54 244 67L379 610C388 643 387 644 429 644L535 644C647 644 710 578 710 466Z"></path></g></g></g></svg></mjx-container> ）的前提下，信源所需的最小信息率<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.561ex" xmlns="http://www.w3.org/2000/svg" width="5.351ex" height="2.253ex" role="img" focusable="false" viewBox="0 -748 2365 996"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="R(D)"><g 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data-mml-node="mo" data-latex=")" transform="translate(1976,0)"><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"></path></g></g></g></svg></mjx-container> 。</p><p>该理论指出，对于给定的失真测度，存在一个最小的比特率，使得解码后的信号与原始信号的平均失真不超过<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.873ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 828 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="D"><g data-mml-node="mi" data-latex="D"><path data-c="1D437" d="M567 683 235 683C212 683 201 682 201 660 201 649 212 644 234 644 254 644 294 647 294 631 294 629 293 623 290 613L158 82C153 60 143 47 128 42 121 40 103 39 72 39 50 39 40 37 40 16 40 5 51 0 72 0L399 0C467 0 532 20 595 61 702 130 804 265 804 429 804 577 713 683 567 683M710 466C710 353 659 216 607 149 550 76 475 39 381 39L270 39C259 39 252 39 248 40 242 40 239 42 239 45 239 47 241 54 244 67L379 610C388 643 387 644 429 644L535 644C647 644 710 578 710 466Z"></path></g></g></g></svg></mjx-container> 。这一发现为现代语音编码、图像压缩（如 JPEG）和视频流媒体技术（如 H.264 / H.265）提供了核心理论依据。</p><h3 id="五，密码学"><a href="#五，密码学" class="headerlink" title="五，密码学"></a>五，密码学</h3><p>在 1945 年撰写并于 1949 年公开发表的《保密系统的通信理论》（Communication theory of secrecy systems）中，Shannon 观察到保密系统在本质上就是一种特殊的通信系统：加密者试图通过引入干扰来防止非法监听者获取信息。</p><p>利用自己创立的信息论工具，Shannon 对密码系统的安全性进行了定量的分析。</p><p>他提出了 “完美保密性” 的概念，并用严格的数学语言证明了，要实现这种绝对无法破译的保密，唯一的办法是让密钥的熵不小于被加密消息的熵。</p><p>这一结论确立 OTP（One-time Pad）在理论上的安全性。</p><p>除此之外，Shannon 还考虑了保密性与消息冗余度之间的权衡。他引入了 “唯一解距离” 等概念，分析了攻击者在拥有多少密文的情况下可以通过统计方法还原原始信息。</p><p>他提出的 “扩散” 与 “混乱” 两大原则，即通过复杂的置换和代换将明文的统计特征抹去，至今仍是现代对称加密算法（如 DES 和 AES）设计的核心准则。</p><h3 id="六、对人工智能、计算机博弈与自动机的早期探索"><a href="#六、对人工智能、计算机博弈与自动机的早期探索" class="headerlink" title="六、对人工智能、计算机博弈与自动机的早期探索"></a>六、对人工智能、计算机博弈与自动机的早期探索</h3><p>在完成了信息论的构建后，Shannon 的兴趣转向了更具挑战性的领域：如何让机器像人类一样思考和学习。他在 1950 年发表的论文《编程实现计算机下棋》（Programming a computer for playing chess）是这一方向的早期探索。</p><p>在那个计算机计算能力极其有限的年代，Shannon 就预见到了启发式搜索和博弈树的概念。</p><p>他详细描述了如何通过评估函数来判断棋局的优劣，并提出了两种不同的搜索策略：暴力搜索和具有选择性的深度搜索。</p><p>尽管他本人没有亲手写出能够击败世界冠军的代码，但他的思想框架为后续工作提供了重要参考。</p><p>不仅在理论上，Shannon 还是一位动手能力极强的发明家。他设计的 “Theseus” 机器老鼠是最早的机器学习和自动机实验之一。通过继电器电路实现的逻辑存储，这一装置能够在迷宫中自主探索并 “记住” 路径，甚至在迷宫结构发生改变时重新学习。</p><p>此外，他还对自动机理论做出了重要贡献，证明了仅需两个内部状态就能构造出一台通用的 Turing 机，极大地简化了计算理论中的物理模型。</p><p>他在机器人、概率逻辑以及数据结构等方面的零星尝试，虽然大多未正式发表为宏篇巨著，但其展现出的跨学科融合能力和对机器智能本质的探索，深刻地启发了后来的人工智能学者。</p><p>Shannon 始终认为，机器的价值在于其处理信息和做出决策的能力，这一愿景至今仍指引着现代智能技术的发展方向。</p><p>除了自身研究，Shannon 与麦卡锡（John McCarthy，1927-2011）、明斯基（Marvin Minsky，1927-2016）和罗切斯特（Nathaniel Rochester，1919-2001）共同组织并参与了 1956 年的 Dartmouth 研讨会（1956 Dartmouth workshop），该研讨会被认为是人工智能领域的奠基性会议。</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/17"><p class="title"><i class="fa-solid fa-chevron-left" aria-hidden="true"></i>Wiener诺伯特·维纳：从神童到控制论之父</p><p class="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。</p></a><a class="next" href="/notes/person/19"><p class="title">Hedlund符号动力学与拓扑动力学的奠基人古斯塔夫·赫德伦德<i class="fa-solid fa-chevron-right" aria-hidden="true"></i></p><p class="content">古斯塔夫·赫德伦德是符号动力学和拓扑动力学的创始人之一。他揭示了圆周变换的旋转数与莫尔斯类型数的深刻联系，证明了常负曲率曲面上测地流的遍历性，并与莫尔斯共同创立了符号动力学。其研究为动力系统理论奠定了坚实基础。</p></a></div><div class="recommended-article"><div class="recommended-article-header"><i class="fa-solid fa-bookmark fa-fw" aria-hidden="true"></i> <span>推荐阅读</span></div><div class="recommended-article-group"> <a class="recommended-article-item" href="/notes/person/17.html" 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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" active-action="action-notesperson8"><div class="name"> Whitehead</div></a></li><li><a class="flat-box" title="/notes/person/9" href="/notes/person/9" active-action="action-notesperson9"><div class="name"> Iwaniec</div></a></li><li><a class="flat-box" title="/notes/person/10" href="/notes/person/10" active-action="action-notesperson10"><div class="name"> Sarnak</div></a></li><li><a class="flat-box" title="/notes/person/11" href="/notes/person/11" active-action="action-notesperson11"><div class="name"> Ramanujan</div></a></li><li><a class="flat-box" title="/notes/person/12" href="/notes/person/12" active-action="action-notesperson12"><div class="name"> Hardy</div></a></li><li><a class="flat-box" title="/notes/person/13" href="/notes/person/13" active-action="action-notesperson13"><div class="name"> Littlewood</div></a></li><li><a class="flat-box" title="/notes/person/14" href="/notes/person/14" active-action="action-notesperson14"><div class="name"> von Neumann</div></a></li><li><a class="flat-box" title="/notes/person/15" href="/notes/person/15" active-action="action-notesperson15"><div class="name"> Irving Segal</div></a></li><li><a class="flat-box" title="/notes/person/16" href="/notes/person/16" active-action="action-notesperson16"><div class="name"> Birkhoff</div></a></li><li><a class="flat-box" title="/notes/person/17" href="/notes/person/17" active-action="action-notesperson17"><div class="name"> Wiener</div></a></li><li><a class="flat-box" title="/notes/person/18" href="/notes/person/18" active-action="action-notesperson18"><div class="name"> Shannon</div></a></li><li><a class="flat-box" title="/notes/person/19" href="/notes/person/19" active-action="action-notesperson19"><div class="name"> Hedlund</div></a></li><li><a class="flat-box" title="/notes/person/20" href="/notes/person/20" active-action="action-notesperson20"><div class="name"> Gibbs</div></a></li><li><a class="flat-box" title="/notes/person/21" href="/notes/person/21" active-action="action-notesperson21"><div class="name"> Veblen</div></a></li><li><a class="flat-box" title="/notes/person/22" href="/notes/person/22" active-action="action-notesperson22"><div class="name"> Spanier</div></a></li><li><a class="flat-box" title="/notes/person/23" href="/notes/person/23" active-action="action-notesperson23"><div class="name"> Hochschild</div></a></li><li><a class="flat-box" title="/notes/person/24" href="/notes/person/24" active-action="action-notesperson24"><div class="name"> Erdős</div></a></li><li><a class="flat-box" title="/notes/person/25" href="/notes/person/25" active-action="action-notesperson25"><div class="name"> Pólya</div></a></li><li><a class="flat-box" title="/notes/person/26" href="/notes/person/26" active-action="action-notesperson26"><div class="name"> Atiyah</div></a></li><li><a class="flat-box" title="/notes/person/27" href="/notes/person/27" active-action="action-notesperson27"><div class="name"> Singer</div></a></li><li><a class="flat-box" title="/notes/person/28" href="/notes/person/28" active-action="action-notesperson28"><div class="name"> Grothendieck</div></a></li><li><a class="flat-box" title="/notes/person/29" href="/notes/person/29" active-action="action-notesperson29"><div class="name"> Loo-Keng Hua</div></a></li><li><a class="flat-box" title="/notes/person/30" href="/notes/person/30" active-action="action-notesperson30"><div class="name"> Poisson</div></a></li><li><a class="flat-box" title="/notes/person/31" href="/notes/person/31" active-action="action-notesperson31"><div class="name"> Dyson</div></a></li></ul></div></section><div class="widget-sticky pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div 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