约翰·康威John Horton Conway

康威的父母是Agnes Boyce和Cyril 约翰·康威。弗瑞兹·约翰有两个姐姐，Sylvia和Joan。Cyril Conway是一名化学实验室助理。约翰很小就对数学产生了兴趣，他的母亲Agnes回忆说，他四岁时就能背诵2的幂。约翰的幼年很艰难，因为他成长于英国战时物资短缺的时期。在小学时，约翰表现出色，几乎在每一个班级都名列前茅。

尽管康威在小学时对数学是什么还没有清晰的概念，但他一定已经坚定地认为自己将来会成为数学家。十一岁进入中学前接受面试时，他被问到长大后想做什么，他回答说想成为剑桥的数学家。他在中学表现出色，并非所有科目都优秀，但他在数学上的表现无疑是所有学生中最好的。然而，数学并不是唯一让他感兴趣的科目，他还曾对天文学产生过浓厚兴趣，另一些时候则对化石着迷。对天文学的兴趣一直伴随着他，他将其列为自己今天的兴趣之一。

离开中学后，康威进入剑桥大学冈维尔与凯斯学院学习数学。他于1959年获得学士学位，并开始在哈罗德·达文波特的指导下从事数论方面的研究。在解决了哈罗德·达文波特提出的关于将数写成五次幂之和的未解决问题后，康威开始对无穷序数产生兴趣。他对games的兴趣似乎始于在剑桥大学学习的那些年，当时他成为一名狂热的十五子棋玩家，在公共休息室里花数小时玩这个游戏。他于1964年获得博士学位，并被任命为剑桥大学纯数学讲师。他实现了十一岁时的抱负。

同样在1964年，康威被选为剑桥大学西德尼·苏塞克斯学院的会士。在这个阶段，他正在研究数理逻辑，但事情进展得并不顺利。他写道：-

……我变得非常沮丧。我觉得我没有在做真正的数学；我没有发表东西，而且我为此感到非常内疚。

对康威来说，事情突然发生了变化。大约在1965年，John Leech发现了一个24维球体的密集堆积，其格现在被称为Leech格。Leech知道对称群会很有趣，并且他研究了一段时间，给出了其阶的下界（后来证明这实际上是该群的阶）。知道自己不具备证明其猜想所需的群论技能，他试图引起其他人的兴趣，见[1]：-

我把这个问题悬在各种人的鼻子底下，包括哈罗德·斯科特·麦克唐纳·考克斯特、约翰·阿瑟·托德和格雷厄姆·希格曼的鼻子，但康威是第一个吞下诱饵的人……

这一发现的详细描述见[1]。康威的这项工作将改变他的数学生涯。他证明了Leech格的对称群$G$，当模去一个2阶中心子群后，是一个此前未被发现的8,315,553,613,086,720,000阶有限单群。但$G$具有更为非凡的性质。它拥有大量非常有趣的子群，其中包括另外两个此前未知的单群，以及以当时已知的几乎所有有限零星单群作为同态像的群。有限零星单群是指不属于任何标准无限族之一的有限单群。

康威于1968年宣布了他的发现，并于1969年在Bulletin of the London Mathematical Society第一卷中发表了全部细节。正是在这为他出版生涯带来启动之后，康威才“开始做真正的数学”，当时他因这一当时最受瞩目的数学领域中最非凡的发现之一而信心大增。我[EFR]无法为这种信心的转变作证，因为我没有听过康威在发现新单群之前的讲座。然而，在这一发现之后，我确实参加过他的许多讲座，它们精彩、令人难忘，由一位伟大的表演者主讲。

在此之后不久，他因发明Game of Life而闻名于数学界之外。冯·诺伊曼曾在1940年寻找一种通用构造器。他试图找到一种能够自我复制的假想机器，并成功地为这样一种在笛卡尔网格上具有复杂规则的机器找到了数学模型。康威试图简化冯·诺伊曼的想法，并最终成功[3]：-

……只有在否定了许多图案——包括三角形、六边形以及正方形格——以及许多其他生死法则，包括引入两种甚至三种性别之后。成片的方格纸被写满，他和他的研究生随从们摆弄着扑克筹码、外国硬币、贝壳、围棋棋子或任何手边的东西，直到在生与死之间找到了一种可行的平衡。

康威把这个游戏展示给他的朋友马丁·加德纳，后者在1970年10月他为Scientific American撰写的专栏中描述了它。这个游戏一经推出便大获成功，康威成了家喻户晓的名字。人们常常声称，自1970年以来，全世界用于生命游戏的计算机时间比任何其他单一活动都多。马丁·加德纳写道：-

这个游戏使康威一举成名，但它也开辟了一个全新的数学研究领域，即元胞自动机领域。

同样在1970年，康威当选为剑桥大学冈维尔与凯斯学院的会士，三年后，他从讲师晋升为剑桥大学纯粹数学与数理统计的准教授。

康威另一个著名的主题是超现实数的发现，这也可以追溯到1970年左右。人们总是惊讶于数系的发展仍在继续。人们普遍认为复数是数的发展的终点，而康威的发现完美地说明了即使是数系也是不断演变的主题的一部分。

也许令人惊讶的事实是，康威并不是在试图发展数系，而是在分析围棋游戏。康威研究了剑桥两位具有国际水平的围棋选手的对局。他注意到，在对局接近尾声时，它看起来像是许多较小对局的总和。分析这种情况，康威发现某些对局表现得像数字，超现实数就此诞生。然而，超现实数这个名字不是康威发明的，而是高德纳发明的，他对康威的发现印象深刻，以至于他以中篇小说形式写了Surreal Numbers（1974年），旨在向学生介绍数学研究的思想。

在这样一篇短文中，不可能审视康威研究过的全部主题范围，甚至不可能让人了解他在如此多不同领域中非凡的独创性贡献。让我们提一下，除了他对群论的创新性贡献和上面提到的超现实数的创造之外，他还在纽结理论、数论、博弈论、二次型、编码理论和镶嵌方面进行了领先研究。

康威于1971年被伦敦数学会授予Berwick奖。1981年3月，康威当选为伦敦皇家学会会士。然后，在1983年，他被任命为剑桥大学数学教授。大约在这个时候，Guy绘制了下面这幅康威的画像[3]：-

康威极其不整洁。他在剑桥纯粹数学与数学统计系的房间里的桌子上堆满了论文、书籍、未回复的信件、笔记、模型、图表、表格、图解、喝剩的咖啡杯以及各种各样令人惊叹的小摆设，这些已经溢满了大部分地板和所有的椅子，以至于很难走进房间超过一两步，也不可能坐下。如果你能走到黑板前，那里有各种颜色的粉笔，但没有空间写字。他在学院的房间也是类似的状态。尽管他记忆力极好，但他常常找不到几天前发现的重要结果的那张纸，而这个结果没有记录在别处。

1986年，康威在接受美国普林斯顿大学冯·诺伊曼数学讲席职位后离开了剑桥，在那里他的大部分工作集中在几何学，特别是研究晶体格的对称性。1987年，康威被授予伦敦数学会的乔治·波利亚奖。最近，他被授予西北大学1997-98年度弗雷德里克·埃塞尔·内默斯数学奖。该奖项授予在其学科中通过对新知识做出重大贡献而表现出杰出成就的数学家。他还于2000年被美国数学会授予勒罗伊·P凯瑟琳·斯蒂尔数学阐述奖，2001年被迪金森学院（宾夕法尼亚州卡莱尔）授予约瑟夫·普里斯特利奖，并于2001年被利物浦大学授予荣誉理学博士学位。

我们应该提到康威写过的几本书，并用评论者的简短评论来说明它们的重要性和原创性。

1. On numbers and games（1976）：-

……这本书是数学文献中的一个重大补充：一个全新的、令人兴奋的、高度原创的理论由其创造者阐述，风格既简洁、文雅又令人愉悦地异想天开。

2. （与 E R Berlekamp 和 R K Guy 合著）Winning ways for your mathematical plays（2卷）（1982）：-

这两卷书塞满了信息、彩色插图和例子……要真正理解并证明书中的一切，更不用说尝试解决书中每一页所启发的众多问题，将需要许多人花费许多年。……它很可能在未来许多年里仍将是这一领域的杰出引领者。

此后又出现了Winning ways for your mathematical plays的两卷，其中第3卷第二版于2003年9月出版，第4卷第二版于2004年5月出版。

3. （与 N J A Sloane 合著），Sphere packings, lattices and groups（1988）：-

这本书是球堆积文献中的一个里程碑。

4. （与 R K Guy 合著）The book of numbers（1996）：-

尽管读者不需要具备特定的数学背景，但每位读者都能在这趟愉快的旅程中轻松找到感兴趣的内容，探索“数”这个词的各种含义与延伸。……这本书非常“对读者友好”。为了对哪怕对周围世界只有一丁点兴趣的潜在读者公平起见，出版商本应被要求在封面上贴一个警告标签，指明这本书含有极易令人上瘾的内容。

5. （与D A Smith合著）On Quaternions and Octonions（2003）：-

这是一本关于四元数代数与八元数代数的几何与算术的优美而引人入胜的书。……这本薄薄的书读起来极为有趣：它是对极具吸引力主题的出色阐述，并包含若干新的重要结果。

6. （与H Burgiel、Goodman-Strauss合著）Symmetries of things（2008）

总体而言，这本书是一个宝库，充满了新旧乐趣。其中大部分内容对于具有本科数学背景的任何人来说都应当可以理解，并且很可能激发进一步的兴趣。

最后我们注意到，康威自2001年起与妻子Diana结婚，有一个儿子Gareth生于2001年。他们的家在美国新泽西州普林斯顿。与前几任妻子，他有儿子Oliver生于1988年、Alex生于1983年；女儿Susan生于1962年、Rose生于1963年、Elena生于1965年和Ann-Louise生于1968年。他有三个孙辈：约翰、Ellen和Joseph Wayman。他还有两个曾孙辈。

John Conway's parents were Agnes Boyce and Cyril Horton Conway. John had two older sisters, Sylvia and Joan. Cyril Conway was a chemistry laboratory assistant. John became interested in mathematics at a very early age and his mother Agnes recalled that he could recite the powers of two when aged four years. John's young years were difficult for he grew up in Britain at a time of wartime shortages. At primary school John was outstanding and he topped almost every class.

Despite not having any clear idea of what mathematics was when he was at primary school, John Conway must have had it firmly fixed in his mind that he would become a mathematician. When he went to be interviewed at age eleven before entering secondary school he was asked what he wanted to be when he grew up and he replied that he wanted to be a mathematician at Cambridge. He excelled at secondary school, not in all his subjects but certainly his performance in mathematics was by far the best of any of the pupils. Mathematics was not the only subject which interested him, however, for he also had spells of deep interest in astronomy and at other times in fossils. The interest in astronomy has remained with him and he lists it as one of his interests today.

After leaving seconday school, Conway entered Gonville and Caius College Cambridge to study mathematics. He was awarded his BA in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals. It appears that his interest in games began during his years studying at Cambridge, where he became an avid backgammon player spending hours playing the game in the common room. He was awarded his doctorate in 1964 and was appointed as Lecturer in Pure Mathematics at the University of Cambridge. He had achieved the ambition which he had as an eleven year old.

Also in 1964 Conway was elected to a fellowship at Sidney Sussex College, Cambridge. At this stage he was working on mathematical logic but things were not going well. He wrote:-

... I became very depressed. I felt that I wasn't doing real mathematics; I hadn't published, and I was feeling very guilty because of that.

Things were to change suddenly for Conway. Around 1965 John Leech found a dense packing of spheres in 24 dimensions with a lattice now known as the Leech lattice. Leech knew that the symmetry group would be interesting, and he worked on it for some time giving a lower bound for its order (which later proved to be the actual order of the group). Knowing that he did not have the group theory skills necessary to prove his conjectures he tried to interest others, see [1]:-

I dangled the problem under various noses, including those of Coxeter, Todd, and Graham Higman, but Conway was the first to swallow the bait ...

A detailed description of this discovery is given in [1]. This work by Conway was to transform his mathematical career. He showed that the symmetry group $G$ of the Leech lattice, when factored by a central subgroup of order 2, was a previously undiscovered finite simple group of order 8,315,553,613,086,720,000. But $G$ had even more remarkable properties. It had a great number of very interesting subgroups, including two more previously unknown simple groups, as well as groups having as homomorphic images almost all the finite sporadic simple groups known at that time. A finite sporadic simple group is a finite simple group which is not a member of one of the standard infinite families.

Conway announced his discovery in 1968 and published full details in the first volume of the Bulletin of the London Mathematical Society in 1969. It was only after this kick-start to his publishing career that Conway "started doing real mathematics" having received a great boost to his confidence with one of the most remarkable discoveries in the area of mathematics which had the highest profile at that time. I [EFR] cannot vouch for this change in confidence since I did not hear Conway lecture before his discovery of new simple groups. However, I did attend many of his lectures after this discovery and they were brilliant, memorable, and given by a great showman.

He became well known outside the world of mathematics shortly after this with his invention of the Game of Life. John von Neumann had looked for a universal constructor in the 1940. He tried to find a hypothetical machine that could build copies of itself and succeeded when he found a mathematical model for such a machine with complicated rules on a cartesian grid. Conway tried to simplify von Neumann's ideas and eventually succeeded [3]:-

... only after the rejection of many patterns, triangular and hexagonal lattices as well as square ones, and of many other laws of birth and death, including the introduction of two and even three sexes. Acres of squared paper were covered, and he and his admiring entourage of graduate students shuffled poker chips, foreign coins, cowrie shells, Go stones or whatever came to hand, until there was a viable balance between life and death.

Conway showed the game to his friend Martin Gardner who described it in the October 1970 column which he wrote in Scientific American. The game became an instant success and Conway became a household name. It has often been claimed that since 1970 more computer time world-wide has been devoted to the Game of Life than any other single activity. Gardner wrote:-

The game made Conway instantly famous, but it also opened up a whole new field of mathematical research, the field of cellular automata.

Also in 1970 Conway was elected to a fellowship at Gonville and Caius College Cambridge and, three years later, he was promoted from lecturer to reader in Pure Mathematics and Mathematical Statistics at Cambridge.

Another topic for which Conway is famous is discovery of surreal numbers, which again dates from around 1970. It always comes as a surprise to people that development of the number systems is still going on. There is a common belief that complex numbers are the end of the road in the development of numbers and Conway's discovery is the perfect illustration of how even the number systems are part of the continually evolving subject.

Perhaps the surprising fact is that Conway was not trying to develop number systems, but rather was analysing the game of Go. Conway studied the games of two players of Go at Cambridge who were of international standard. He noticed that near the end of a game it appeared like the sum of a lot of smaller games. Analysing the situation Conway discovered that certain games behaved like numbers and surreal numbers were born. The name surreal numbers was not invented by Conway, however, but by Donald Knuth who was so impressed with Conway's discovery that he wrote Surreal Numbers (1974) in the form of a novelette aimed at introducing the ideas of mathematical research to students.

It is impossible in a short article such as this to look at the full range of topics that Conway has studied or even to give an idea of the remarkable originality of his contributions in so many different areas. Let us mention that, in addition to his innovative contributions to group theory and his creation of surreal numbers mentioned above, he has done leading research in knot theory, number theory, game theory, quadratic forms, coding theory, and tilings.

Conway was awarded the Berwick Prize by the London Mathematical Society in 1971. In March 1981 Conway was elected a fellow of the Royal Society of London. Then, in 1983, he was appointed professor of mathematics at Cambridge. It was around this time that the following picture of Conway was painted by Guy [3]:-

Conway is incredibly untidy. The tables in his room at the Department of Pure Mathematics and Mathematical Statistics in Cambridge are heaped high with papers, books, unanswered letters, notes, models, charts, tables, diagrams, dead cups of coffee and an amazing assortment of bric-à-brac, which has overflowed most of the floor and all of the chairs, so that it is hard to take more than a pace or two into the room and impossible to sit down. If you can reach the blackboard there is a wide range of coloured chalk, but no space to write. His room in College is in a similar state. In spite of his excellent memory he often fails to find the piece of paper with the important result that he discovered some days before, and which is recorded nowhere else.

In 1986 Conway left Cambridge after accepting appointment to the John von Neumann Chair of Mathematics at Princeton in the United States where much of his work has focused on geometry, in particular studying the symmetries of crystal lattices. In 1987 Conway was awarded the Polya Prize of the London Mathematical Society. More recently he was awarded the 1997-98 Frederic Esser Nemmers Prize in Mathematics from Northwestern University. This Prize is awarded to mathematicians who display outstanding achievement in their discipline by making major contributions to new knowledge. He was also awarded the Leroy P Steele Prize for Mathematical Exposition by the American Mathematical Society, 2000, the Joseph Priestley Award from Dickinson College (Carlisle, PA) in 2001, and awarded an Honorary DSc by the University of Liverpool in 2001.

We should mention a few of the books Conway has written, and illustrate their importance and originality with a brief comment from a reviewer.

1. On numbers and games (1976):-

... this book is a momentous addition to the mathematical literature: a new, exciting, and highly original theory is expounded by its creator in a style that is at once concise, literate, and delightfully whimsical.

2. (with E R Berlekamp and R K Guy) Winning ways for your mathematical plays (2 volumes) (1982):-

The two volumes are crammed to the brim with information, coloured illustrations and examples ... To really understand and prove everything in the book, not to mention to attempt solutions of the many questions inspired on every page of the book, will engage many people for many years. ... It is likely to remain an eminent leader in this field for many years to come.

Two further volumes of Winning ways for your mathematical plays have since appeared with the second edition of Volume 3 appearing in September 2003 and the second edition of Volume 4 appearing in May 2004.

3. (with N J A Sloane), Sphere packings, lattices and groups (1988):-

The book is a landmark in the literature on sphere packings.

4. (with R K Guy) The book of numbers (1996):-

Although no particular mathematical background is required of the reader, nevertheless every reader will easily find something of interest in this delightful tour of the various meanings and extensions of the word "number". ... The book is very "reader friendly". In fairness to any potential readers having even the least interest in the world around them, the publishers should have been required to post a warning label on the front cover indicating that this book contains extremely addictive material.

5. (with D A Smith) On Quaternions and Octonions (2003):-

This is a beautiful and fascinating book on the geometry and arithmetic of the quaternion algebra and the octonion algebra. ... This slim book is most intriguing to read: it is an excellent exposition of very attractive topics, and it contains several new and significant results.

6. (with H Burgiel, Goodman-Strauss) Symmetries of things (2008)

Overall, the book is a treasure trove, full of delights both old and new. Much of it should be accessible for anyone with an undergraduate-level background in mathematics, and is likely to stimulate further interest.

Finally we note that Conway has been married to wife Diana since 2001 and has a son Gareth born 2001. Their home is in Princeton, New Jersey, USA. With previous wives he has sons Oliver born 1988 and Alex born 1983; daughters Susan born 1962, Rose born 1963, Elena born 1965 and Ann-Louise born 1968. He has three grandchildren: John, Ellen and Joseph Wayman. He also has two great-grandchildren.