诺伯特·维纳Norbert Wiener

诺伯特·维纳的父亲是Leo Wiener，他是一位俄罗斯犹太人。由于Leo Wiener对儿子影响如此之大，我们应当介绍一下他的教育和职业背景。Leo Wiener曾就读于华沙大学医学院，但他对这个职业并不满意，于是前往柏林，开始接受工程师训练。这个职业对他来说似乎只比医学职业稍微有趣一点，他先抵达英国，随后移民到美国。我们应当注意到，在整个受教育过程中，Leo一直对数学感兴趣，尽管他从未在任何工作中运用过自己的数学技能，但这是他一生中深藏的业余爱好。

利奥于1880年抵达新奥尔良，在工厂和农场尝试过各种工作，之后成为堪萨斯城的一名学校教师。他从学校语言教师晋升为密苏里大学现代语言教授。在那里，他遇到并娶了百货商店老板的女儿伯莎·卡恩。伯莎来自一个德国犹太家庭，是[7]：-

……一个身材矮小的女人，健康、精力充沛且活泼。

她与丈夫一同住进了密苏里州哥伦比亚的一家寄宿公寓，次年他们的儿子维纳在那里出生。

维纳出生后不久，就决定将密苏里大学的现代语言系拆分为独立的法语系和德语系。利奥在拆分后加入了德语系，但他在一些政治操作中失利，所以全家离开了哥伦比亚，搬到了波士顿。在那里，利奥通过担任各种教学和其他职位赚钱，最终被任命为哈佛大学的斯拉夫语言讲师。这份工作的薪水不足以养家，所以利奥保留了各种其他职位来增加收入。他在哈佛大学度过了余下的职业生涯，最终被提升为教授。

幼年的维纳有一个保姆。大约四岁时，第二个孩子康斯坦丝出生了；维纳的第二个妹妹生于1901年。他在[7]中这样描述自己的成长经历：——

我是在一个充满学问的家庭中长大的。我的父亲是好几本书的作者，从我记事起，打字机的声音和浆糊罐的气味对我来说就十分熟悉。……我可以自由地在父亲那个极为广博而杂乱的藏书室里漫游。在某个时期，我父亲的科学兴趣几乎涵盖了所有能想到的研究主题。……我是一个无所不读的读者……

维纳在学业上遇到了一些问题，部分原因是他在家里读的书使他在某些领域超前，而在其他领域则远远落后。他七岁时，父母把他送进皮博迪学校，在为他该上哪个年级而费心之后，让他从三年级开始。不久之后，他的父母和老师都觉得他更适合四年级，于是让他跳了一级。然而，他在哪个年级都显然不适应学校，而他的老师对一个如此年幼却已上四年级、却又缺乏这一教育阶段学生应有的某些技能的孩子，几乎没有多少同情。他在[7]中写道：——

我最大的短板是算术。在这方面，我的理解力远远超出我的运算能力，而我的运算能力确实很差。我父亲非常正确地看出，我的主要困难之一在于机械性的练习让我感到厌烦。他决定让我离开学校，不学算术而学代数，目的是给我的想象力提供更大的挑战和刺激。

从这时起，维纳的父亲接管了他的教育，对于如此年幼的孩子来说，他进步迅速。然而，维纳在动作方面有问题，显然非常笨拙。这部分源于协调性差，但也部分因为视力不佳。医生建议他停止阅读六个月以让眼睛恢复，但他仍然定期从父亲那里上课，父亲现在教他心算数学。六个月结束后，维纳重新开始阅读，但在此期间他发展出了一些精细的心智技能，这些技能他终生保留。

1903年秋天，九岁时，他再次被送到学校，这次是艾尔高中。学校同意进行实验，为维纳找到合适的水平，很快他被安排到高三年级，与比他大七岁的学生一起。然而，学校只是他教育的一部分，因为他的父亲继续辅导他。他于1906年从艾尔毕业，年仅十一岁，并与他的十八岁同学7一起庆祝：-

我非常感谢我的艾尔朋友们。我有机会在同情和理解的氛围中度过成长过程中一些最尴尬的阶段。

1906年9月，仍然只有十一岁，维纳进入塔夫茨学院。在社会上是个孩子，但在教育方面已是成人，所以他的学生时代并不轻松。尽管选修了各种科学课程，他获得了数学学位。维纳的父亲继续辅导他数学，显示出对本科水平主题的完全掌握。1909年，维纳从塔夫茨毕业，年仅十四岁，进入哈佛开始研究生学习。

相当违背父亲的建议，维纳在哈佛开始动物学研究生学习。然而事情进展得不太顺利，到一年结束时，部分由维纳、部分由他父亲作出决定，他将改换专业到哲学。他获得了康奈尔大学的奖学金，于1910年入学，开始哲学研究生学习。维纳修了数学和哲学课程，这一年并不成功，还没结束，他父亲就已经做好了必要的安排，让他回到哈佛继续学哲学。

回到哈佛后，维纳深受Edward Huntington关于数学哲学的出色教学影响。他18岁时从哈佛获得博士学位，学位论文是关于数学逻辑的，由Karl Schmidt指导。从哈佛，维纳前往英国剑桥，在伯特兰·罗素指导下学习，后者告诉他，为了研究数学哲学，他需要了解更多数学，因此他参加了戈弗雷·哈罗德·哈代的课程。1914年，他前往哥廷根，在大卫·希尔伯特指导下学习微分方程，还参加了埃德蒙·朗道的群论课程。他受到大卫·希尔伯特、埃德蒙·朗道和伯特兰·罗素的影响，但也许更大程度上受到戈弗雷·哈罗德·哈代的影响。在哥廷根，他了解到[7]：-

……数学不仅是一门在书房里做的学科，更是一门需要讨论和共同生活的学科。

维纳在第一次世界大战爆发前几天回到美国，但又回到剑桥跟随伯特兰·罗素进一步学习。回到美国后，他于1915年在哈佛大学教授哲学课程，为通用电气公司工作了一段时间，然后加入《美国百科全书》在奥尔巴尼担任撰稿人。在那里工作期间，他收到奥斯瓦尔德·维布伦的邀请，前往马里兰州的阿伯丁试验场从事弹道学的战争工作。在从事这项战争工作时与同事讨论数学，重新激发了他对数学的兴趣。战争结束时，威廉·佛格·奥斯古德告诉他麻省理工学院有一个空缺职位，他被任命为数学讲师。

他在麻省理工学院的第一项数学工作促使他研究布朗运动。事实上，正如维纳在[7]中所解释的，这第一项工作将为他后来大部分研究提供一条贯穿的线索：-

……这项研究将我引向概率论。此外，它非常直接地引导我研究周期图，以及研究比经典的Fourier series和约瑟夫·傅里叶积分更一般的调和分析形式。所有这些概念与麻省理工学院数学研究所教授的工程关注相结合，引导我在通信理论方面取得了理论和实践上的进展，并最终创立了控制论这门学科，它本质上是一种对通信理论的统计方法。因此，尽管我的科学兴趣看似多样，但从我第一项成熟工作开始，就有一条单一的线索将它们全部连接起来……

他于1920年参加了在斯特拉斯堡举行的国际数学家大会，并在那里与莫里斯·弗雷歇合作。在接下来的几年里，他频繁返回欧洲，拜访英国、法国和德国的数学家。尤其重要的是他与保罗·莱维以及哥廷根的接触，在那里他的工作被认为与量子力学有重要联系。这促成了与马克斯·玻恩的合作。

1926年，维纳与Margaret Engemann结婚，婚后维纳作为古根海姆学者前往欧洲。在剑桥拜访戈弗雷·哈罗德·哈代后，他回到哥廷根，他的妻子在完成宾夕法尼亚州朱尼亚塔学院的现代语言教学职责后也来到那里与他团聚。维纳数学发展的另一个重要年份是1931-32年，他主要在英格兰度过，在剑桥拜访戈弗雷·哈罗德·哈代。在那里，他开设了一门关于他自己对约瑟夫·傅里叶积分贡献的讲座课程，但剑桥也为他提供了一个基地，使他能够拜访欧洲大陆的许多数学同行。其中包括威海姆·布拉希开、卡尔·门格尔和菲利普·弗兰克，他们邀请他访问，同时他还结识了汉斯·哈恩、埃米尔·阿廷和库尔特·弗雷德里希·哥德尔。

维纳的论文很难读。有时困难的结果几乎不加证明就出现，仿佛对维纳来说是显而易见的，而另一些时候他又会对琐碎之事给出冗长的证明。汉斯·弗洛伊登萨写道[1]：-

维纳常常忍不住要把他那渊博头脑中涌现的一切都讲出来，他常常难以将相关的数学与其科学和社会含义乃至个人经历清晰地分开。他似乎针对的读者似乎在门外汉、数学本科生、普通数学家和维纳本人之间以随机顺序交替。

尽管他的论文风格如此，维纳还是贡献了一些极为重要的思想。我们在上文已经提到他1921年在布朗运动方面的工作。他在一维路径空间中引入了一种测度，自然地引进了probability的概念。从1923年起，他研究了约翰·彼得·古斯塔夫·勒热纳·狄利克雷的问题，其工作对potential theory产生了重大影响。

维纳的数学思想很大程度上是由麻省理工学院工程同事向他提出的问题所驱动的。这些问题促使他将关于布朗运动的工作推广到更一般的随机过程。这反过来又引导他在1930年研究调和分析。他在广义调和分析方面的工作引导他在1932年研究陶伯定理，他在这方面的贡献为他赢得了1933年的马克希莫·博谢奖。他从美国数学会获得了该奖项，因为他的回忆录Tauberian theorems于前一年在Annals of Mathematics上发表。关于陶伯定理的工作自然地引导他研究约瑟夫·傅里叶变换，他于1933年发表了The Fourier Integral, and Certain of Its Applications，并于1934年发表了Fourier Transforms。

维纳的兴趣范围极其广泛，除上文提到的领域外，他还对许多领域有所贡献，包括通信理论、控制论（这个术语是他创造的）、量子理论，并且在第二次世界大战期间从事火炮控制工作。很可能正是后一项工作促使他开创了控制论这一新领域，他在Cybernetics: or, Control and Communication in the Animal and the Machine（1948）中对此作了描述。汉斯·弗洛伊登萨在[1]中写道：-

在研究防空火炮控制时，维纳可能已经产生了这样的想法：把人视为操纵机制的一部分，并对他应用反馈和稳定性等概念，这些概念原本是为机械系统和电路设计的。……随着时间的推移，这类洞见被更有意识地用于某种生物学研究……[控制论]有助于普及一种以通信理论术语进行的思维方式，如反馈、信息、控制、输入、输出、稳定性、内稳态、预测和滤波。另一方面，它也有助于传播关于数学真正含义的错误观念。

维纳本人也意识到这些危险，他与其它科学家的广泛交往使他说道：-

数学家作为科学家的顾问时，其主要职责之一就是劝阻他们不要对数学期望过高。

维纳还有一些我们未提及的出版物，包括Nonlinear Problems in Random Theory（1958）和God and Golem, Inc.: A Comment on Certain Points Where Cybernetics Impinges on Religion（1964）。

我们在上文提到了汉斯·弗洛伊登萨对维纳糟糕写作风格的评论。他最著名的著作Cybernetics尤其受到汉斯·弗洛伊登萨的批评：-

即使按照维纳的标准来衡量，《控制论》也是一部组织糟糕的著作——一堆印刷错误、错误的数学陈述、错误的公式、精彩但互不相关的想法，以及逻辑上的荒谬。令人遗憾的是，这部著作使维纳获得了其公众声誉的绝大部分，但这只是事后之见。当时，数学读者更多是被其思想的丰富性所吸引，而非其缺点。

汉斯·弗洛伊登萨在[1]中描述了维纳的外貌和性格：-

在外貌和行为上，维纳是一个巴洛克式的人物，矮小、圆胖、近视，将这些以及许多特质以极端程度结合在一起。他的谈话是自负与放纵的奇特混合。他不善于倾听。他的自我夸耀带有戏谑意味，令人信服，从不令人反感。他会说多种语言，但没有一种容易听懂。他是出了名的糟糕讲师。

大卫·乔治·肯德尔写道[3]：-

作为一个人，维纳首先是令人振奋的。我认识一些觉得这种刺激不受欢迎的人。他会在公开场合冒犯人，比如在讲座中打鼾，然后在讨论中提出一个尴尬的问题；在私下场合也会冒犯人，比如向一位显赫的晚宴同伴提供远离其自身领域的资讯和建议。我喜欢回忆维纳，正如我曾在牛津大学莫德林学院深夜见到他，被一群着迷的本科生包围着，谈论着，无休止地谈论着。

Norbert Wiener's father was Leo Wiener who was a Russian Jew. Because Leo Wiener was such a major influence on his son, we should give some background to his education and career. Leo Wiener attended medical school at the University of Warsaw but was unhappy with the profession, so he went to Berlin where he began training as an engineer. This profession seemed only a little more interesting to him than the medical profession, and he emigrated to the United States having first landed in England. We should note that throughout his education Leo was interested in mathematics and, although he never used his mathematical skills in any jobs he held, it was a deep amateur interest to him all through his life.

Arriving in New Orleans in 1880, Leo tried his hand at various jobs in factories and farms before becoming a school teacher in Kansas City. He progressed from being a language teacher in schools to becoming Professor of Modern Languages at the University of Missouri. While there he met and married Bertha Kahn, who was the daughter of a department store owner. Bertha, from a German Jewish family, was [7]:-

... a small woman, healthy, vigorous and vivacious.

She joined her husband in the boarding house in Columbia, Missouri where their son Norbert was born in the following year.

Not long after Norbert's birth a decision was taken to split the Modern Languages Department at the University of Missouri into separate departments of French and German. Leo was to join the German Department after the split but he lost out in some political manoeuvring so the family left Columbia and they moved to Boston. There Leo brought in money by taking a variety of teaching and other positions and eventually was appointed as an Instructor in Slavic Languages at Harvard. This did not pay well enough to provide for his family, so Leo kept various other positions to augment his salary. He remained at Harvard University for the rest of his career, being eventually promoted to professor.

As a young child Norbert had a nursemaid. When he was about four years old, a second child Constance was born; Wiener's second sister was born on 1901. He writes in [7] about his upbringing:-

I was brought up in a house of learning. My father was the author of several books, and ever since I can remember, the sound of the typewriter and the smell of the paste pot have been familiar to me. ... I had full liberty to roam in what was the very catholic and miscellaneous library of my father. At one period or other the scientific interests of my father had covered most of the imaginable subjects of study. ... I was an omnivorous reader ...

Wiener had problems regarding his schooling, partly because the reading which he had done at home had meant that he was advanced in certain areas but much less so in others. His parents sent him to the Peabody School when he was seven years old and, after worrying about which class he should enter, had him begin in the third grade. After a short time his parents and teachers felt he would be better suited to the fourth grade and he was moved up a year. However, he certainly did not fit into the school in either grade and his teacher had little sympathy with so young a boy in the fourth grade yet lacking certain skills which would be expected the pupils at this stage in their education. He writes [7]:-

My chief deficiency was arithmetic. Here my understanding was far beyond my manipulation, which was definitely poor. My father saw quite correctly that one of my chief difficulties was that manipulative drill bored me. He decided to take me out of school and put me on algebra instead of arithmetic, with the purpose of offering a greater challenge and stimulus to my imagination.

From this time on Wiener's father took over his education and he made rapid progress for so young a child. However, Wiener had problems relating to his movements and was obviously very clumsy. This stemmed partly from poor coordination but also partly for poor eyesight. Advised by a doctor to stop reading for six months to allow his eyes to recover, he still had regular lessons from his father who now taught him to do mathematics in his head. After the six months were up Wiener went back to reading but he had developed some fine mental skills during this period which he retained all his life.

In the autumn of 1903, at age nine, he was sent to school again, this time to Ayer High School. The school agreed to experiment and to find the right level for Wiener who was soon put into senior third year class with pupils who were seven years older than he was. The school only formed part of his education, however, for his father continued to coach him. He graduated in 1906 from Ayer at the age of eleven and celebrated with his eighteen year old fellow students [7]:-

I owe a great deal to my Ayer friends. I was given a chance to go through some of the gawkiest stages of growing up in an atmosphere of sympathy and understanding.

In September 1906, still only eleven years old, Wiener entered Tufts College. Socially a child, he was an adult in educational terms so his student days were not easy ones. Although taking various science courses, he took a degree in mathematics. Wiener's father continued to coach him in mathematics showing complete mastery of undergraduate level topics. In 1909 Wiener graduated from Tufts at age fourteen and entered Harvard to begin graduate studies.

Rather against his father's advice, Wiener began graduate studies in zoology at Harvard. However things did not go too well and by the end of a year a decision was taken, partly by Wiener partly by his father, that he would change topic to philosophy. Having won a scholarship to Cornell he entered in 1910 to begin graduate studies in philosophy. Taking mathematics and philosophy courses, Wiener did not have a successful year and before it was finished his father had made the necessary arrangements to return to Harvard to continue philosophy.

Back at Harvard Wiener was strongly influenced by the fine teaching of Edward Huntington on mathematical philosophy. He received his Ph.D. from Harvard at the age of 18 with a dissertation on mathematical logic supervised by Karl Schmidt. From Harvard Wiener went to Cambridge, England, to study under Russell who told him that in order to study the philosophy of mathematics he needed to know more mathematics so he attended courses by G H Hardy. In 1914 he went to Göttingen to study differential equations under Hilbert, and also attended a group theory course by Edmund Landau. He was influence by Hilbert, Landau and Russell but also, perhaps to an even greater degree, by Hardy. At Göttingen he learned that [7]:-

... mathematics was not only a subject to be done in the study but one to be discussed and lived with.

Wiener returned to the United States a couple of days before the outbreak of World War I, but returned to Cambridge to study further with Russell. Back in the United States he taught philosophy courses at Harvard in 1915, worked for a while for the General Electric Company, then joined Encyclopedia Americana as a staff writer in Albany. While working there he received an invitation from Veblen to undertake war work on ballistics at the Aberdeen Proving Ground in Maryland. Taking about mathematics with his fellow workers while undertaking this war work revived his interest in mathematics. At the end of the war Osgood told him of a vacancy at MIT and he was appointed as an instructor in mathematics.

His first mathematical work at MIT led him to examine Brownian motion. In fact, as Wiener explained in [7], this first work would provide a connecting thread through much of his later studies:-

... this study introduced me to the theory of probability. Moreover, it led me very directly to the periodogram, and to the study of forms of harmonic analysis more general than the classical Fourier series and Fourier integral. All these concepts have combined with the engineering preoccupations of a professor of the Mathematical Institute of Technology to lead me to make both theoretical and practical advances in the theory of communication, and ultimately to found the discipline of cybernetics, which is in essence a statistical approach to the theory of communication. Thus, varied as my scientific interests seem to be, there has been a single thread connecting all of them from my first mature work ...

He attended the International Congress of Mathematicians at Strasbourg in 1920 and while there worked with Fréchet. He returned to Europe frequently in the next few years, visiting mathematicians in England, France and Germany. Especially important was his contacts with Paul Lévy and with Göttingen where his work was seen to have important connections with quantum mechanics. This led to a collaboration with Born.

In 1926 Wiener married Margaret Engemann, and after their marriage Wiener set off for Europe as a Guggenheim scholar. After visiting Hardy in Cambridge he returned to Göttingen where his wife joined him after completing her teaching duties in modern languages at Juniata College in Pennsylvania. Another important year in Wiener's mathematical development was 1931-32 which he spent mainly in England visiting Hardy at Cambridge. There he gave a lecture course on his own contributions to the Fourier integral but Cambridge also provided a base from where he was able to visit many mathematical colleagues on the Continent. Among these were Blaschke, Menger and Frank who invited him to make a visit, while he also met Hahn, Artin and Gödel.

Wiener's papers were hard to read. Sometimes difficult results appeared with hardly a proof as if they were obvious to Wiener, while at other times he would give a lengthy proof of a triviality. Freudenthal writes [1]:-

All too often Wiener could not resist the temptation to tell everything that cropped up in his comprehensive mind, and he often had difficulty in separating the relevant mathematics neatly from its scientific and social implications and even from his personal experiences. The reader to whom he appears to be addressing himself seems to alternate in a random order between the layman, the undergraduate student of mathematics, the average mathematician, and Wiener himself.

Despite the style of his papers, Wiener contributed some ideas of great importance. We have already mentioned above his work in 1921 in Brownian motion. He introduced a measure in the space of one dimensional paths which brings in probability concepts in a natural way. From 1923 he investigated Dirichlet's problem, producing work which had a major influence on potential theory.

Wiener's mathematical ideas were very much driven by questions that were put to him by his engineering colleagues at MIT. These questions pushed him to generalise his work on Browian motion to more general stochastic processes. This in turn led him to study harmonic analysis in 1930. His work on generalised harmonic analysis led him to study Tauberian theorems in 1932 and his contributions on this topic won him the Bôcher Prize in 1933. He received the prize from the American Mathematical Society for his memoir Tauberian theorems published in Annals of Mathematics in the previous year. The work on Tauberian theorems naturally led him to study the Fourier transform and he published The Fourier Integral, and Certain of Its Applications (1933) and Fourier Transforms in 1934.

Wiener had an extraordinarily wide range of interests and contributed to many areas in addition to those we have mentioned above including communication theory, cybernetics (a term he coined), quantum theory and during World War II he worked on gunfire control. It is probably this latter work which motivated his invention of the new area of cybernetics which he described in Cybernetics: or, Control and Communication in the Animal and the Machine (1948). Freudenthal writes in [1]:-

While studying anti-aircraft fire control, Wiener may have conceived the idea of considering the operator as part of the steering mechanism and of applying to him such notions as feedback and stability, which had been devised for mechanical systems and electrical circuits. ... As time passed, such flashes of insight were more consciously put to use in a sort of biological research ... [Cybernetics] has contributed to popularising a way of thinking in communication theory terms, such as feedback, information, control, input, output, stability, homeostasis, prediction, and filtering. On the other hand, it also has contributed to spreading mistaken ideas of what mathematics really means.

Wiener himself was aware of these dangers and his wide dealings with other scientists led him to say:-

One of the chief duties of the mathematician in acting as an adviser to scientists is to discourage them from expecting too much from mathematics.

Some of Wiener's publications which we have not mentioned include Nonlinear Problems in Random Theory (1958), and God and Golem, Inc.: A Comment on Certain Points Where Cybernetics Impinges on Religion (1964).

We have mentioned above Freudenthal's comments on Wiener's poor writing style. His most famous work Cybernetics comes in for special criticism by Freudenthal:-

Even measured by Wiener's standards "Cybernetics" is a badly organised work -- a collection of misprints, wrong mathematical statements, mistaken formulas, splendid but unrelated ideas, and logical absurdities. It is sad that this work earned Wiener the greater part of his public renown, but this is an afterthought. At that time mathematical readers were more fascinated by the richness of its ideas than by its shortcomings.

Freudenthal, in [1], describes both Wiener's appearance and his character:-

In appearance and behaviour, Norbert Wiener was a baroque figure, short, rotund, and myopic, combining these and many qualities in extreme degree. His conversation was a curious mixture of pomposity and wantonness. He was a poor listener. His self-praise was playful, convincing and never offensive. He spoke many languages but was not easy to understand in any of them. He was a famously bad lecturer.

D G Kendall writes [3]:-

As a human being Wiener was above all stimulating. I have known some who found the stimulus unwelcome. He could offend publicly by snoring through a lecture and then asking an awkward question in the discussion, and also privately by proffering information and advice on some field remote from his own to an august dinner companion. I like to remember Wiener as I once saw him late at night in Magdalen College, Oxford, surrounded by a spellbound group of undergraduates, talking, endlessly talking.