数学家传记
理查德·卫斯里·汉明是一位美国数学家和计算机科学家，以所谓的理查德·卫斯里·汉明码最为著名。
理查德·卫斯里·汉明的父母是汉明 J 汉明和梅布尔·G·雷德菲尔德。他在芝加哥长大，在那里上学，并很快意识到自己是一个比老师更有能力的数学家。他决定在大学学习工程学，但唯一提供的奖学金来自芝加哥大学，而该校没有工程系。他进入芝加哥大学，主修数学，于1937年获得学士学位。本科学习后，他去了内布拉斯加大学，于1939年获得硕士学位。他于1942年在伊利诺伊大学厄巴纳-香槟分校获得数学博士学位。他的博士论文Some Problems in the Boundary Value Theory of Linear 微分方程由瓦尔德马·特里茨津斯基（1901-1973）指导。特里茨津斯基出生于俄罗斯，于1920年代移民到美国，并于1934年至1969年在伊利诺伊大学担任数学教授。他的兴趣在分析，特别是测度论、积分和微分方程。在1936年发表在Acta Mathematica上的一篇论文中，特里茨津斯基研究了具有特定类型两点边界条件的线性微分系统，汉明在他的论文中扩展了这项工作。特别是，汉明研究了乔治·格林函数以及特征解，并获得了渐近表达式。他进一步发展了雅各布·塔马尔金引入的方法来研究特征数，并证明所使用的级数一致收敛。然而，汉明确实对与他的微分方程研究相去甚远的思想产生了兴趣[15]：-
作为一名研究生，我发现并研究了乔治·布尔的《思维定律》，我觉得它有趣、相关且可信。
获得博士学位后，汉明于1942年9月5日与旺达·利特尔结婚。他曾在伊利诺伊大学短暂任教，随后在路易斯维尔大学任教。1945年，在一位朋友的鼓励下，他加入了曼哈顿计划，这是一个美国政府旨在制造原子弹的研究项目。它被称为曼哈顿计划，是因为最初的研究是在曼哈顿的哥伦比亚大学进行的。然而，当汉明被招募到该项目时，它正在洛斯阿拉莫斯进行。他到达洛斯阿拉莫斯一个月后，他的妻子旺达也加入了，她也受雇于曼哈顿计划。他被赋予运行IBM计算机的任务，这些计算机在项目中发挥了至关重要的作用[9]：-
在洛斯阿拉莫斯，我被带进来运行其他人已经启动的计算机器，这样那些科学家和物理学家就可以回去工作了。……我近距离看到了理查德·费曼。我看到了恩里科·费米和泰勒。我看到了奥本海默。我看到了汉斯·贝特：他是我的老板。
旺达开始使用台式计算机工作，后来为恩里科·费米和爱德华·泰勒工作。汉明在[15]中讲述了一个有趣的插曲：-
……在洛斯阿拉莫斯……我们正在设计原子弹。在第一次实地测试前不久（你意识到无法进行小规模实验——要么有临界质量，要么没有），一个人让我检查他做的一些算术，我同意了，想着把它推给某个下属。当我问那是什么时，他说：“这是测试炸弹点燃整个大气的概率。”我决定自己检查！第二天他来取答案时，我对他说：“算术显然是正确的，但我不知道氧和氮的俘获截面的公式——毕竟，在所需的能级上不可能进行实验。”他像物理学家对数学家说话那样回答说，他要我检查的是算术而不是物理，然后就离开了。我对自己说：“你做了什么，汉明，你卷入了危及宇宙中所有已知生命的风险，而你却对关键部分知之甚少？”我在走廊里来回踱步，一个朋友问我有什么烦恼。我告诉了他。他的回答是：“没关系，汉明，没有人会责怪你。”
曼哈顿计划结束后，汉明在洛斯阿拉莫斯又待了六个月，撰写已完成计算的详细资料。他觉得重要的是要试图确切理解那里取得了什么成就以及为什么如此成功。正是在这个时候，他意识到自己没有学习工程学是正确的——工程师做了很多常规工作，但像他这样的数学家更多地参与了前沿创新。他从这次洛斯阿拉莫斯的经历中形成了一种数学观，即计算至关重要，但这使他对通过抽象数学理论的标准方法的重要性产生了怀疑（例如，见[19]）。他已经获得了新泽西州贝尔电话实验室的一个职位，并于1946年开始在那里数学系工作。然而，他并没有切断与洛斯阿拉莫斯的联系，每年夏天继续作为顾问到洛斯阿拉莫斯科学实验室进行为期两周的访问。在贝尔实验室，他能够与克劳德·香农和约翰·图基一起工作[15]：-
[在]贝尔实验室，我进入了一个非常富有成效的部门。当时博德是部门主管；克劳德·香农在那里……我曾与克劳德·香农共用一间办公室一段时间。同时他在做信息论，我在做编码理论。我们两个人在同一个地方、同一时间做这件事，这很可疑——这是氛围使然。
贝尔实验室数学研究部的负责人是 Hendrik Wade Bode（1905-1982），他于1929年首次加入贝尔实验室该部门，并于1944年成为负责人。除了克劳德·香农和约翰·图基之外，还有一些年轻数学家恰在汉明之前加入了贝尔实验室数学研究部。这些人包括 Donald Percy Ling（1912-1981）和 Brockway McMillan（1915-），他们曾与汉明同时在洛斯阿拉莫斯。克劳德·香农、Ling、McMillan 和汉明自称“青年土耳其党人”，汉明曾谈到这个群体（见[8]）：-
我们在大萧条中长大，所以我们相信我们欠世界一条生路。战争期间，我们都不得不学习我们不想学的东西以赢得战争，所以我们都被交叉授粉了。我们对常规不耐烦，而且常常很早就担任要职。我们是一流的麻烦制造者。我们以非常规的方式做非常规的事情，仍然得到了有价值的结果。因此，管理层不得不容忍我们，很多时候让我们自行其是。
他在贝尔实验室物理研究部工作的 Alan Chynoweth 强调了其非常规方法[9]：-
……我们习惯作为一个物理学小组一起午餐，不知为何这个来自数学的奇怪家伙总是乐于加入我们。我们总是很高兴有他在，因为他带来了那么多非正统的想法和观点。那些午餐很刺激，我可以向你保证。
然而，并非所有汉明的同事都乐于容忍他的‘非常规方式’[8]：-
贝尔实验室的一些前同事回忆汉明自负，并评论说他偶尔会“半生不熟地，追逐某个半生不熟的想法”，而且他迟迟才意识到自己的错误方向。“他很难共事，”一位前贝尔科学家说，“因为他大量广播，很少倾听。”
下面我们将讨论汉明在纠错码方面一些极为重要的工作，但这里我们先指出他在贝尔实验室[11]研究过的众多不同问题中的一些：-
……由于我在贝尔系统工作，我做过许多电话计算以及其他数学工作，涉及诸如行波管、电视线路均衡、复杂通信系统的稳定性、通过电话中心局的呼叫阻塞等各种各样的课题，这里仅举几例。
他继续为贝尔电话公司工作直到1976年，尽管他对教学越来越感兴趣，并在1960年至1976年间在斯坦福大学、纽约市立学院、加州大学欧文分校和普林斯顿大学担任访问或兼职教授。1976年从贝尔实验室退休后，他接受了加利福尼亚州蒙特雷海军研究生院计算机科学教授的职位。此时他放弃了研究生涯，决定专注于教学和写书。他形成了这样一种信念：数学的教学方式是错误的，但改变这一点的唯一途径是编写可用于新方法的教科书。这里我们引用[17]中的两个例子来说明他对数学教学的看法：-
他试图转向一种新的微积分教学方式，这体现在他的书Methods of mathematics applied to calculus, probability, and statistics（1985）中。他在[17]中谈到这本书：-
这本书……与标准教材非常不同，它的成功或失败将告诉我们变革与创新的前景。如果我们要转向任何一种新课程……这本书是一个开端。
他写的其他教材，都试图改变各自领域中的传统方法，包括Numerical Methods for Scientists and Engineers（1962）、Introduction to applied numerical analysis（1971）、Computers and Society（1972）、Digital filters（1977）、 Coding and information theory（1980）、The Art of Probability for Scientists and Engineers（1991）和The Art of Doing Science and Engineering : Learning to Learn（1997）。
这些书的大部分书评摘录见THIS LINK。
汉明最著名的是他在检错与纠错码方面的工作。他关于这一主题的基础性论文Error detecting and error correcting codes于1950年4月发表在Bell System Technical Journal上。凭借这篇论文，他在信息论中开创了一个新领域。汉明码、汉明距离和汉明度量是当今编码理论中使用的标准术语，但它们也用于数学的许多其他领域。这些在编码理论中具有根本重要性的思想，都起源于这篇经典论文，并在计算机设计中具有实际用途。汉明是如何开始这项工作的？他在论文中告诉我们，他：-
……是从考虑大规模计算机器而被引导到这项研究的，在这些机器中必须执行大量运算，而最终结果中不能出现一个错误。
事实上，1947年，汉明让贝尔实验室的计算机在周末处理一个特定问题。他的同事需要这个结果，但到了星期一，他发现计算早期就出现了一个错误，他没有什么可报告的。他断定，如果计算机能够检测到何时出现了错误，那么它必定能够检测到错误出现在哪里。由于这些机器以二进制工作，每个条目要么是0要么是1，所以知道某个特定条目是错的就意味着计算机可以纠正它——如果0是错的，正确的条目必定是1，反之亦然。检错是通过对每个符号块进行奇偶校验来完成的——添加一个额外的数字（0或1），使得该块中数字之和为偶数。由于奇偶校验会失败，因此可以检测出一个不正确的条目。汉明想出一种方法，不仅能确定是否发生了单个错误，而且通过添加额外的奇偶校验，还能检测出错误发生在哪里。然后他设计出一种方法，既能纠正一个块中的单个错误，又能检测出第二个错误。这些码在论文的第一部分中作了描述，而第二部分则使用一个几何模型来表明他所描述的这些码是最优的。
编码方面的工作与装填问题有关，而汉明发现的纠错码促成了有限域上矩阵装填问题的解决。1956年，汉明在早期计算机IBM 650上工作。他在这里的工作促成了一门编程语言的发展，这门语言已经演变为今天用于计算机编程的高级计算机语言。汉明还从事数值分析、微分方程积分以及汉明谱窗方面的工作，后者在计算中被广泛用于在Fourier analysing数据之前对数据进行平滑处理。
汉明因其开创性工作获得了许多奖项。1968年，他成为电气与电子工程师协会的会士。IEEE于1979年授予汉明伊曼纽尔·R·皮奥雷奖：-
因引入纠错码、在操作系统和编程语言方面的开创性工作，以及推进数值计算。
IEEE以他的名义命名了一枚奖章“汉明 W. Hamming奖章”，他是1988年首位获得这枚价值10,000美元奖章的人：-
因对信息科学与系统的杰出贡献。
同样在1968年，他获得了艾伦·图灵协会颁发的奖项：-
因其在数值方法、自动编码系统以及检错与纠错码方面的工作。
他于1994年成为⟦L1⟧协会的会士。
他还获得了更多荣誉，包括1980年当选美国国家工程院院士，以及1981年获得宾夕法尼亚大学颁发的Harold Pender奖。1996年，在慕尼黑，汉明因他在纠错码方面的工作而获得了享有盛誉的、奖金为130,000美元的Eduard Rheim技术成就奖。
1997年，汉明从海军研究生院退休，并被授予杰出荣休教授称号。退休前不久，他说（见[8]）：-
一位朋友最近告诉我：“汉明，你停止教书的那天，你会垮掉。”他大概说得对。当我离开贝尔实验室时，我知道那是我科学生涯的终结。当我从这里退休时，从另一种意义上说，那真的是终结。
确实，他教到1997年12月是对的，他在下个月因心脏病发作去世。在汉明去世时，蒙特雷海军研究生院的汉明 Franke写道：-
人们将长久铭记他对科学和计算的许多方面的敏锐洞察力。我也会长久长久地记住他那件红色格子运动外套和他那些糟糕的笑话。
James F Kaiser在汉明的简短讣告中写道：-
我们都会怀念他那引人入胜的头脑以及他对科学、工程和日常生活事务的深刻洞察。
Richard Hamming's parents were Richard J Hamming and Mabel G Redfield. He was brought up in Chicago where he attended school and soon realised that he was a more able mathematician than his teacher. He decided to study engineering at university but the only offer of a scholarship came from the University of Chicago which had no engineering department. He entered the University of Chicago receiving his B.S. in 1937 after majoring in mathematics. Following his undergraduate studies, he went to the University of Nebraska where he was awarded his M.A. in 1939. He received his Ph.D. in mathematics in 1942 from the University of Illinois at Urbana-Champaign. His doctoral dissertation Some Problems in the Boundary Value Theory of Linear Differential Equation was supervised by Waldemar Trjitzinsky (1901-1973). Trjitzinsky, a Russian by birth, emigrated to the United States in the 1920s and was a professor of Mathematics at the University of Illinois from 1934 to 1969. His interests were in analysis, particularly measure theory, integration and differential equations. In a paper published in Acta Mathematica in 1936, Trjitzinsky studied linear differential systems with two point boundary conditions of a particular type and Hamming extended this work in his thesis. In particular, Hamming investigated the Green's function and also the characteristic solutions for which he obtained asymptotic expressions. He further developed methods introduced by Jacob D Tamarkin to investigate the characteristic numbers and to show that the series used converged uniformly. Hamming did, however, develop interests in ideas that were quite far removed from his study of differential equations [15]:-
As a graduate student I found, and studied, Boole's 'Laws of Thought', and I found it interesting, relevant, and believable.
After the award of his doctorate, Hamming married Wanda Little on 5 September 1942. He taught, for a short time, at the University of Illinois and then at the University of Louisville. In 1945, encouraged by a friend, he joined the Manhattan Project, a U.S. government research project to produce an atomic bomb. It was called the Manhattan Project because the first research had been done at Columbia University in Manhattan. However by the time that Hamming was recruited to the project it was being undertaken at Los Alamos. A month after he arrived at Los Alamos, he was joined by his wife Wanda who was also employed on the Manhattan Project. He was given the task of running the IBM computers which played a vital role in the project [9]:-
At Los Alamos I was brought in to run the computing machines which other people had got going, so those scientists and physicists could get back to business. ... I saw Feynman up close. I saw Fermi and Teller. I saw Oppenheimer. I saw Hans Bethe: he was my boss.
Wanda began working with desk calculators and, later, worked for Enrico Fermi and Edward Teller. Hamming relates an interesting episode in [15]:-
... at Los Alamos ... we were designing atomic bombs. Shortly before the first field test (you realize that no small scale experiment can be done - either you have a critical mass or you do not), a man asked me to check some arithmetic he had done, and I agreed, thinking to fob it off on some subordinate. When I asked what it was, he said, "It is the probability that the test bomb will ignite the whole atmosphere." I decided I would check it myself! The next day when he came for the answers I remarked to him, "The arithmetic was apparently correct but I do not know about the formulas for the capture cross sections for oxygen and nitrogen - after all, there could be no experiments at the needed energy levels." He replied, like a physicist talking to a mathematician, that he wanted me to check the arithmetic not the physics, and left. I said to myself, "What have you done, Hamming, you are involved in risking all of life that is known in the Universe, and you do not know much of an essential part?" I was pacing up and down the corridor when a friend asked me what was bothering me. I told him. His reply was, "Never mind, Hamming, no one will ever blame you."
After the Manhattan Project ended, Hamming remained at Los Alamos for six months, writing up details of the calculations that had been made. He felt that it was important to try to understand exactly what had been achieved there and why it had been so successful. It was at this time that he realised that he had done the right thing by not studying engineering - the engineers did much of the routine work but mathematicians like himself were more involved in the cutting edge innovations. He formed a view of mathematics, arising from this Los Alamos experience, that computation was of major importance but it made him doubt the significance of the standard approach through abstract mathematical theories (see, for example, [19]). He had already the offer of a position at the Bell Telephone Laboratories in New Jersey and, in 1946, he began working in the mathematics department there. However, he did not break his link with Los Alamos, continuing to make two week visits each summer to Los Alamos Scientific Laboratories as a consultant. At Bell Labs he was able to work with both Claude Shannon and John Tukey [15]:-
[At] Bell Labs I came into a very productive department. Bode was the department head at the time; Shannon was there ... I shared an office for a while with Shannon. At the same time he was doing information theory, I was doing coding theory. It is suspicious that the two of us did it at the same place and at the same time - it was in the atmosphere.
The head of the Mathematical Research Department at Bell Labs was Hendrik Wade Bode (1905-1982) who first joined the Department at Bell Labs in 1929 and became head in 1944. In addition to Shannon and Tukey, some other young mathematicians had joined the Mathematical Research Department at Bell Labs just before Hamming. These included Donald Percy Ling (1912-1981) and Brockway McMillan (1915-) who had been at Los Alamos at the same time as Hamming. Shannon, Ling, McMillan and Hamming called themselves the Young Turks and Hamming spoke about the group (see [8]):-
We grew up in the great depression, so we believed we owed the world a living. During the war, we all had to learn things we didn't want to learn to get the war won, so we were all cross-fertilized. We were impatient with conventions and had often had responsible jobs very early. We were first-class troublemakers. We did unconventional things in unconventional ways and still got valuable results. Thus, management had to tolerate us and let us alone a lot of the time.
His unconventional approach was emphasised by Alan Chynoweth who worked for the Physics Research Department at Bell Labs [9]:-
... we were in the habit of lunching together as a physics group, and for some reason this strange fellow from mathematics was always pleased to join us. We were always happy to have him with us because he brought so many unorthodox ideas and views. Those lunches were stimulating, I can assure you.
Not all of Hamming's colleagues were as happy to tolerate his 'unconventional ways', however [8]:-
Some former colleagues from Bell Labs recall Hamming as egotistical and comment that he occasionally went off "half-cocked, after some half-baked idea," and he was slow to pick up on his misdirection. "He is very hard to work with," one former Bell scientist said, "because he does a lot of broadcasting and not a lot of listening."
We will discuss below some of Hamming's highly significant work on error-correcting codes, but here we note some of the many varied problems he worked on in Bell Labs [11]:-
... working as I did for the Bell System, I did many telephone computations and other mathematical work on such varied things as traveling wave tubes, the equalization of television lines, the stability of complex communication systems, the blocking of calls through a telephone central office, to name but a few.
He was to continue to work for Bell Telephones until 1976 although he became increasingly interested in teaching and held visiting or adjunct professorships at Stanford University, the City College of New York, the University of California at Irvine, and Princeton University between 1960 and 1976. After retiring from Bell Labs in 1976, he accepted a professorship of computer science at the Naval Postgraduate School at Monterey, California. At this point he gave up his research career, deciding to concentrate on teaching and writing books. He formed the belief that the way mathematics is being taught is wrong, but the only way to change this was to write textbooks that can be used for a new approach. Here we quote two examples from [17] illustrating his views on mathematics teaching:-
His attempt to move to a new way of teaching calculus is exhibited in his book Methods of mathematics applied to calculus, probability, and statistics (1985). He wrote about this book in [17]:-
This book ... is very different from the standard texts and its success or failure will tell us something about the prospects for change and innovation. If we are to move to any sort of new curriculum ... this book is a first start.
Other texts he wrote, all attempting to change conventional approaches to the areas they covered, include Numerical Methods for Scientists and Engineers (1962), Introduction to applied numerical analysis (1971), Computers and Society (1972), Digital filters (1977), Coding and information theory (1980), The Art of Probability for Scientists and Engineers (1991), and The Art of Doing Science and Engineering : Learning to Learn (1997).
Extracts from reviews of most of these books are given at THIS LINK.
Hamming is best known for his work on error-detecting and error-correcting codes. His fundamental paper on this topic Error detecting and error correcting codes appeared in April 1950 in the Bell System Technical Journal. With this paper, he started a new subject within information theory. Hamming codes, Hamming distance and Hamming metric are standard terms used today in coding theory but they are also used in many other areas of mathematics. These ideas, of fundamental importance in coding theory, all originated in this classic paper and are of practical use in computer design. How did Hamming come to do this work? He tells us in the paper that he:-
... was led to the study ... from a consideration of large scale computing machines in which a large number of operations must be performed without a single error in the end result.
In fact it was in 1947 when Hamming set computers at Bell Labs to work on a particular problem over the weekend. The result was needed by his colleagues but, come Monday, he discovered that an error had occurred early in the calculations and he had nothing to report. He decided that if the computer could detect when an error had occurred then it must be able to detect where it had occurred. Since the machines worked in binary, every entry was either 0 or 1 so knowing that a particular entry was wrong meant that the computer could correct it - if 0 was wrong the correct entry must be 1 and vice-versa. Error detecting was done by a parity check on each block of symbols - an extra digit was added (either 0 or 1) so that the sum of the digits in the block was even. One incorrect entry could be detected since the parity check would fail. Hamming thought out a way not only to determine if a single error had occurred but also, by adding extra parity checks, to detect where it had occurred. He then devised a way to both correct a single error in a block and detect a second error. These codes were described in the first part of the paper while the second part used a geometrical model to show that the codes he had described were best possible.
Work in codes is related to packing problems and the error-correcting codes discovered by Hamming led to the solution of a packing problem for matrices over finite fields. In 1956 Hamming worked on the early computer, the IBM 650. His work here led to the development of a programming language which has evolved into the high-level computer languages used to program computers today. Hamming also worked on numerical analysis, integrating differential equations, and the Hamming spectral window which is much used in computation for smoothing data before Fourier analysing it.
Hamming has received many awards for his pioneering work. In 1968 he was made a fellow of the Institute of Electrical and Electronics Engineers. The IEEE awarded Hamming the Emanuel R Piore Award in 1979:-
For introduction of error correcting codes, pioneering work in operating systems and programming languages, and the advancement of numerical computation.
The IEEE named a medal "The Richard W. Hamming Medal" in his honour and he was the first recipient of this $10,000 prize medal in 1988:-
For exceptional contributions to information sciences and systems.
Also in 1968 he was presented with the Turing Award from the Association for Computing Machinery:-
For his work on numerical methods, automatic coding systems, and error-detecting and error-correcting codes.
He was made a fellow of the Association for Computing Machinery in 1994.
He has received further honours included being elected a member of the National Academy of Engineering in 1980 and receiving the Harold Pender Award from the University of Pennsylvania in 1981. In 1996, in Munich, Hamming received the prestigious $130,000 Eduard Rheim Award for Achievement in Technology for his work on error correcting codes.
In 1997 Hamming retired from teaching at the Naval Postgraduate School and was made Distinguished Professor Emeritus. Shortly before he retired he said (see [8]):-
A friend told me recently, "Hamming, the day you quit teaching, you are going to fall apart." He's probably right. When I left Bell Labs, I knew that that was the end of my scientific career. When I retire from here, in another sense, it's really the end.
Indeed he was right for having taught up to December 1997, he died of a heart attack in the following month. On Hamming's death Richard Franke of the Naval Postgraduate School at Monterey wrote:-
He will be long remembered for his keen insights into many facets of science and computation. I'll also long long remember him for his red plaid sport coat and his bad jokes.
James F Kaiser, in a brief obituary of Hamming, writes:-
We will all miss his engaging mind and his penetrating insight into matters scientific, engineering, and of everyday living.
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