爱德华·卡斯勒Edward Kasner

爱德华·卡斯勒 的父母 Fanny Ritterman 和 Bernard Kasner 分别于 1843 年和 1841 年出生在奥地利。他们在 1860 年代末移居美国，定居在纽约。他们有八个孩子，卡斯勒 是第六个孩子，年长的兄弟姐妹有 Annie、Marcus、Alexander、Lottie 和 Adolf。卡斯勒 的 [4]：-

多年来，[对家人的]深情与忠诚构成了他性格的基石之一。

他在纽约长大，就读于曼哈顿下城（离家不远）的第2公立学校[4]：-

这个男孩卡斯勒觉得小学算术不足以挑战他的能力，于是找来一本代数课本，常常在算术课上用石板解联立二次方程……

他于1891年十三岁时完成了这一阶段的教育。他因综合表现最佳而获得小学颁发的金质奖章，还因成绩优异获得许多其他最高奖项。随后，在1891年，他进入纽约市立学院。这所学校于1847年作为纽约市自由学院创办，同时提供高中和学位层次的教育。它仅以学业成绩为依据接收移民和穷人的孩子，在纽约市列克星敦大道与第23街的校舍中提供免费教育。卡斯勒于1896年获得理学学士学位，学习了一系列科目，包括数学、天文学、逻辑学、物理学和政治学。他在所有这些科目中都获得了奖项，并被授予数学金质奖章。

完成第一个学位后，卡斯勒前往哥伦比亚大学攻读研究生。他于1897年获得硕士学位，并继续攻读博士学位，导师是法兰克·尼尔森·寇尔，后者于1895年被任命为哥伦比亚大学教授。当时，美国人在美国攻读博士学位并不常见——德国是美国研究生首选的留学地。法兰克·尼尔森·寇尔曾在莱比锡跟随菲利克斯·克莱因学习多年，能够将这一经验用于指导卡斯勒，后者于1899年凭借学位论文The Invariant Theory of the Inversion Group: Geometry upon a Quadric Surface获得博士学位。他成为哥伦比亚大学最早授予数学博士学位的人之一。该学位论文于1900年发表在美国数学会的Transactions上。

尽管他没有按惯例前往德国就获得了博士学位，卡斯勒并未完全打破传统，因为他在1899至1900年间在哥廷根旁听了菲利克斯·克莱因和大卫·希尔伯特的课程。在国外待了一年后，他回到纽约，被任命为巴纳德学院的数学导师，该学院自1889年建校起便隶属于哥伦比亚大学。1904年，他受邀在圣路易斯举行的国际艺术与科学大会上发表演讲。该大会是路易斯安那购地博览会（圣路易斯世界博览会）的一部分，旨在庆祝路易斯安那领地成为美国一州100周年。卡斯勒作了题为Present Problems of Geometry的演讲，并很高兴看到儒勒·昂利·庞加莱——大会的另一位演讲者——也在听众席中。他于1905年成为巴纳德学院的数学讲师，次年晋升为副教授。他一直担任此职，直到1910年被任命为哥伦比亚大学正教授。1937年，他被授予哥伦比亚大学Robert Adrain数学讲席教授。他一直担任该讲席，直到1949年退休，并被授予Adrain荣休教授称号。

卡斯勒最著名的学生杰西·道格拉斯于1916年至1920年间在哥伦比亚大学从事研究。他将卡斯勒的主要研究贡献总结如下[4]：-

卡斯勒的主要数学贡献在几何学领域，主要是微分几何。他富有想象力、直觉、具体、视觉化的基本思维方式和素养，使他最适合从事数学的这一分支。他观察到，数学物理、分析、代数中的许多命题都有一种纯粹几何的本质，可以完全用空间关系来表述，而不必涉及该特殊学科所特有的其他概念，如力、质量、时间或数。他的计划便是提炼出这种几何本质，并从各种角度分析它，特别是针对可能由此提出的任何有趣问题。按照卡斯勒当时的主要兴趣，他的科学生涯大致可细分为四个时期：动力学的微分几何方面（1905—1920）、阿尔伯特·爱因斯坦相对论的几何方面（1920—1927）、多基因函数（1927—1940）、角角（1940—1955）。

在4中，杰西·道格拉斯讲述了一件有趣的事：他为自己的研究学生组织了周六下午的远足卡斯勒：——

我们……会在李堡渡口与卡斯勒会合，乘船到泽西海岸，然后坐有轨电车到帕利塞德崖顶。去树林的路上，我们可能会停下来买些饼干；这些饼干要配着茶吃，茶是卡斯勒用壶煮的，壶是从一根圆木下藏匿的地方挖出来的，那是上次远足结束时留在那里的。有趣的是，尽管我们看不出任何标记，卡斯勒却能准确无误地把我们带到这个藏匿处。水由卡斯勒从附近的小溪取来，用我们收集的枯枝和落叶生火加热。然后我们会围着火坐一下午，谈论我们在哥伦比亚大学的教学、数学研究以及各种感兴趣的话题。当太阳西沉、黑暗开始降临时，卡斯勒会在谈话停顿所标志的心理时刻起身，用备用的水小心地把火熄灭——这时我们就知道该动身回家了。

卡斯勒今天最为人铭记的是“googol”一词以及他与James R 马克斯·纽曼合著的杰出著作Mathematics and the Imagination（1940年）。我们引用卡斯勒在Mathematics and the Imagination中对googol一词来源的描述：——

智慧之言出自孩子之口的频率，至少不亚于出自科学家之口。“googol”这个名字是由一个孩子（卡斯勒博士九岁的侄子）发明的，他被要求想出一个非常大数的名字，即1后面跟一百个零。他非常确定这个数不是无穷大，因此同样确定它必须有个名字。在他提出“googol”的同时，他还为一个更大的数取了个名字：“Googolplex”。一个googolplex比一个googol大得多，但仍然是有限的，正如这个名字的发明者立刻指出的那样。有人建议，一个googolplex应该是1，后面写零，直到你写累了为止。这是对一个人如果真的试图写出一个googolplex时会发生什么的描述，但不同的人在不同的时间会累，而仅仅因为Carnera更有耐力，就让他成为比阿尔伯特·爱因斯坦博士更好的数学家，那是绝对不行的。因此，googolplex是一个特定的有限数，1后面有如此多的零，以至于零的个数是一个googol。一个googolplex比一个googol大得多。从这个非常大但有限的数有多大，你可以得到一些概念：如果你到最远的恒星去，游遍所有星云，沿途每一英寸都写下一个零，也没有足够的空间把它写下来。

网络搜索引擎，也许是最著名的网页，叫做Google，只不过是googol的拼写错误。

在Mathematics and the Imagination中，作者们的目标是：——

……以其丰富多样来展示数学的某些特性，展示其大胆、不羁的精神，展示它作为一门艺术和一门科学，如何不断引领创造力超越想象和直觉。

Herbert Turnbull在其评论[9]中给出了该书内容的概要：-

它信息丰富，涉及数学的所有基本要素——数、形、线、级数、模式、无穷、悖论、机遇、变化，并配有大量有趣的视觉辅助材料：论述令人耳目一新。这本书既是一部数学谜题与惊奇之作的合集，又是一部文笔流畅的数学思想史，也是一次真正的尝试，要把数学家的典型发现呈现给有兴趣但非专业的读者。

I B Cohen，如同下文所引的所有评论者一样，对这本书赞誉有加，他写道：-

……这是我们拥有的关于现代数学的最佳叙述。它以优雅的风格写成，将阐述的清晰与良好的幽默感结合在一起。

G W Dunnington在这本书出版几个月后意识到它大获成功：-

本书已经跻身当前的畅销书之列。显然，它成功地向外行传达了富有创造力的数学家在解决难题时所体验到的一部分乐趣。

埃里克·坦普尔·贝尔在该书出版一年后写道：-

这是对现代数学某些思想的一次极为成功的通俗介绍。它摆脱了通常那些陈腐的数学通俗化读物……我们相信它将拥有长久而兴旺的生命。在数学通俗化方面，它是别具一格的新事物。

的确，埃里克·坦普尔·贝尔说得对，这本书在首次出版70多年后仍然广受欢迎。

我们应当注意到卡斯勒在各个层次上作为教师的才能。他常被邀请到小学和中学谈论数学。他能够通过精心挑选的问题把深奥的数学思想讲清楚，在孩子们受教育的早期阶段就向他们展现出数学研究的韵味。作为给本科生授课的大学讲师，他出类拔萃，同样能把直观的理解讲清楚而不过分讲究严格性。他常声称“严格是给迟钝的人准备的”，但正如杰西·道格拉斯所指出的[4]：-

……他在科学工作或其表述上绝无任何粗疏之处；他只是认为，精确程度足以清楚理解手头的理论或问题即可。超出这一点的任何东西，他都视为做作，而且其倾向是阻碍对本质的清晰洞察以及对相关问题的把握，因而绝对有害，而他认为这种洞察和把握才是最重要的。

对于他的研究生，其中包括约瑟夫·里特、杰西·道格拉斯和弗瑞兹·约翰 De Cicco，卡斯勒举办了他著名的微分几何讨论班[4]：-

我想可以毫不夸张地说，本世纪上半叶在纽约地区成长起来的几乎每一位数学家——无论他的主要兴趣是否为微分几何——都在其学习过程中的某个时候参加过这个讨论班，并从中获得启迪和灵感。

卡斯勒获得了许多荣誉。他于1906年担任美国科学促进会副主席，并于1906年担任美国数学会副主席。他获得的最大荣誉是于1917年当选为国家科学院会士。哥伦比亚大学于1954年庆祝其建校二百周年，作为庆祝活动的一部分，他们于10月31日授予卡斯勒荣誉博士学位，期间[4]：-

……一场在校园附近圣约翰大教堂举行的极为隆重的仪式。

此时他已身患重病，于1951年7月中风。获得荣誉学位后，他的健康状况持续恶化，六个月后去世。

杰西·道格拉斯在结束他为卡斯勒所写的传记时指出，他：-

……在本世纪初开始了他的数学研究生涯，当时美国在世界数学中只占次要地位。如果今天这一地位显然已是领先的，那么其中相当一部分功劳必须归于卡斯勒的工作和影响。

Edward Kasner's parents, Fanny Ritterman and Bernard Kasner, were born in Austria in 1843 and 1841 respectively. They emigrated to the United States in the late 1860s, settling in New York. They had eight children, Edward being the sixth child with older siblings Annie, Marcus, Alexander, Lottie and Adolf. Edward's [4]:-

... affection and loyalty [for his family] over the years formed one of the cornerstones of his character.

He was brought up in New York where he attended Public School No 2 in lower Manhattan (close to his home) [4]:-

Finding elementary school arithmetic an insufficient challenge to his ability, the boy Kasner obtained an algebra text and would often occupy himself during the arithmetic lesson by solving simultaneous quadratics on his slate ...

He completed this stage of his education in 1891 at the age of thirteen. He received the gold medal from the elementary school for the best overall performance as well as numerous other top awards for excellence. Then, in 1891, he entered the College of the City of New York. This school, founded as the Free Academy of the City of New York in 1847, provided both high school and degree level education. Accepting children of immigrants and of the poor solely on academic merit, it provided free education in its buildings in Lexington Avenue and 23rd Street in New York City. Kasner was awarded a B.S. in 1896 having studied a range of subjects including mathematics, astronomy, logic, physics, and political science. He received awards in all these subjects and was awarded the gold medal in mathematics.

Having completed his first degree, Kasner went to Columbia University for his graduate studies. He was awarded a Master's degree in 1897 and continued to work towards his doctorate advised by Frank Nelson Cole, who had been appointed to a professorship at Columbia in 1895. At this time it was unusual for an American to study for a doctorate in the United States - Germany was the preferred place for Americans to go for graduate studies. Cole, who had studied for several years with Felix Klein in Leipzig, was able to use this experience in advising Kasner who was awarded his Ph.D. in 1899 for his thesis The Invariant Theory of the Inversion Group: Geometry upon a Quadric Surface. He became one of the first people to be awarded a Ph.D. in mathematics by Columbia University. The thesis was published in the Transactions of the American Mathematical Society in 1900.

Although he had been awarded a doctorate without making the usual visit to Germany, Kasner did not totally break with tradition for he spent the year 1899-1900 in Göttingen attending courses by Felix Klein and David Hilbert. He returned to New York after the year abroad and was appointed as a mathematics tutor at Barnard College, which had been affiliated to Columbia University from its founding in 1889. In 1904 he was invited to address the International Congress of Arts and Sciences meeting at St Louis. This Congress was organised as part of the Louisiana Purchase Exposition (St Louis World's Fair) put on to celebrate the 100th anniversary of Louisiana Territory becoming a US State. Kasner gave the lecture Present Problems of Geometry and was delighted to have Henri Poincaré, another of the speakers at the Congress, in the audience. He became an Instructor in Mathematics at Barnard College in 1905, and was promoted to adjunct professor in the following year. He held this post until 1910 when he was appointed as a full professor at Columbia University. In 1937 he was named Robert Adrain Professor of Mathematics at Columbia. He held this Chair until he retired in 1949 when he was named Adrain Professor Emeritus.

Jesse Douglas, Kasner's most famous student, undertook research at Columbia between 1916 and 1920. He summarises Kasner's main research contributions as follows [4]:-

Kasner's principal mathematical contributions were in the field of geometry, chiefly differential geometry. His basic mental outlook and equipment, imaginative, intuitive, concrete, visual, fitted him most naturally for work in this branch of mathematics. He observed that many propositions in mathematical physics, in analysis, in algebra, had an essence that was purely geometrical, and could be stated entirely in terms of spatial relations without reference to other concepts peculiar to the special subject, such as force, mass, time, or number. His program was then to distill this geometric essence, and to analyze it from various points of view, particularly with regard to any interesting problems that might be suggested in this way. A rough subdivision of Kasner's scientific career might be made into four periods according to his dominant interest at the time: Differential-Geometric Aspects of Dynamics (1905-1920), Geometric Aspects of the Einstein Theory of Relativity (1920-1927), Polygenic Functions (1927-1940), Horn Angles (1940-1955).

In [4] Douglas relates an interesting account of Saturday afternoon excursions Kasner organised for his research students:-

We ... would meet Kasner at the Fort Lee ferry and take the ride to the Jersey shore, then the trolley car to the top of the Palisades. On the way to the woods we might stop to buy some cookies; these were to consume with the tea which Kasner brewed in a pot, dug up from its hiding place under a log, where it had been left at the end of the previous excursion. It was amusing that Kasner could direct us unerringly to this cache in spite of the absence of any markers that we could notice. Water was obtained by Kasner from a nearby stream, and heated over a fire built from dead branches and leaves which we had gathered. Around this we would then sit through the afternoon, conversing about our teaching at Columbia, about mathematical research, and about random topics of interest. As the sun faded and darkness began to fall, Kasner would arise at the psychological moment signalled by a halt in the conversation, and carefully extinguish the fire with some water kept in reserve - we then knew it was time to start for home.

Kasner is best remembered today for the term 'googol' and for his remarkable book Mathematics and the Imagination (1940) co-authored with James R Newman. We quote Kasner's description of where the term googol came from as given in Mathematics and the Imagination:-

Words of wisdom are spoken by children as least as often by scientists. The name "googol" was invented by a child (Dr Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex." A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was suggested that a googolplex should be 1, followed by writing zeros until you get tired. This is a description of what would happen if one actually tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr Einstein, simply because he had more endurance. The googolplex then, is a specific finite number, with so many zeros after the 1 that the number is a googol. A googolplex is much bigger than a googol. You will get some idea of the size of this very large but finite number from the fact that there would not be enough room to write it, if you went to the farthest star, touring all the nebulae and putting down zeros every inch of the way.

The web search engine, perhaps the most well-known web page, called Google is simply a misspelling of googol.

In Mathematics and the Imagination the authors aim:-

... to show by its very diversity something of the character of mathematics, of its bold, untrammeled spirit, of how, as both an art and a science, it has continued to lead the creative faculties beyond even imagination and intuition.

Herbert Turnbull gives an indication of the contents of the book in his review [9]:-

It is packed with information and deals with all the elementary ingredients of mathematics, number, shape, line, series, pattern, infinity, paradox, chance, change, with copious and entertaining visual aids: and the treatment is refreshing. The book is at once a collection of mathematical puzzles and surprises, a pleasantly written history of mathematical thought, and a real attempt to bring before the interested but unprofessional reader the characteristic findings of mathematicians.

I B Cohen, like all the reviewers referenced below, is full of praise for the book and writes:-

... it is the best account of modern mathematics that we have. It is written in a graceful style, combining clarity of exposition with good humour

G W Dunnington realised a few months after publication of the book that it was a big hit:-

This volume has already taken its place among the current best sellers. Apparently it has succeeded in communicating to the layman something of the pleasure experienced by the creative mathematician in difficult problem solving.

E T Bell writes a year after the book was published:-

This is a highly successful popular presentation of certain ideas of modern mathematics. It departs from the usual trite popularizations of mathematics ... we trust that it will have a long and prosperous career. It is something new, of its own kind, in the popularization of mathematics.

Indeed Bell was right and the book continues to be popular more than 70 years after its first publication.

We should note Kasner's talents as a teacher at all levels. He was often invited to talk about mathematics in elementary schools and high schools. He was able to get across deep mathematical ideas with a careful selection of problems which brought out the flavour of mathematical research to children at an early stage in their education. As a university lecturer to undergraduates he excelled, again getting an intuitive understanding across without undue rigour. He often claimed "Rigour is for dull people" but as Douglas points out [4]:-

... he was not in any sense careless in his scientific work or its presentation; he simply believed in such a degree of accuracy as was sufficient for the clear comprehension of the theory or problem at hand. Anything more than this he regarded as affectation, positively harmful in its tendency to block the road to that clear insight into essentials, and view of related problems, which he considered paramount.

For his graduate students, who included Joseph Ritt, Jesse Douglas and John De Cicco, Kasner put on his famous Seminar in Differential Geometry [4]:-

I think it may truly be said that nearly every mathematician who arose in the New York area during the first half of this century - whether his main interest was differential geometry or not - attended this seminar at some time during his course of study, and derived from it illumination and inspiration.

Kasner received many honours. He was vice-president of the American Association for the Advancement of Science in 1906, and vice-president of the American Mathematical Society in 1906. The greatest honour he received was election to the National Academy of Sciences in 1917. The University of Columbia celebrated its bicentenary in 1954 and, as part of these celebrations, they gave Kasner an honorary doctorate on 31 October during [4]:-

... a most impressive ceremony in the Cathedral of St John the Divine, near the campus.

By this time he was already seriously ill, having suffered a stroke in July 1951. Following receiving his honorary degree, his health steadily declined and he died six months later.

Douglas ends his biography of Kasner by pointing out that he:-

... began his mathematical career at the turn of the century, America occupied a minor position in world mathematics. If today this position is patently a leading one, then some significant portion of the credit must be assigned to the work and influence of Edward Kasner.