乔治·波利亚George Pólya

乔治·波利亚的父母是Anna Deutsch和Jakab 波利亚，两人都是犹太人。Anna来自一个在布达居住了许多代的家庭，1872年布达、老布达和佩斯三镇行政合并成为布达佩斯市时，她十九岁。也许我们应该稍微谈谈波利亚的姓名，因为情况并不完全像表面那样。事实上，尽管Jakab 波利亚在儿子György（或后来所知的波利亚）出生时姓“波利亚”，但在此前的五年里他一直自称波利亚。在那之前他的名字是Jakab Pollák，但是，为了理解Jakab Pollák为什么改名为波利亚，我们需要看看他的职业生涯以及一点匈牙利历史。

Jakab受过律师训练，经营过自己的律师事务所但失败了，然后为的里雅斯特的Assicurazioni Generali国际保险公司工作。然而他真正想要的是一份大学职位，以便能对他真正感兴趣的学科——即经济学和统计学——进行研究。1867年后，匈牙利在奥匈帝国君主制内获得了充分的内部自治，该国的政治理念是朝着一个在精神和制度上都是马扎尔人的匈牙利国家迈进。对于Jakab Pollák来说，还有什么比把一个听起来像犹太人的名字改成一个听起来真正匈牙利化的名字更好的办法来提高获得大学职位的机会呢？他正是在1882年这样做了，这是否有助于他成功获得布达佩斯大学Privatdozent的任命，无法断言，但他在五十出头去世前不久获得了这样一个职位，当时波利亚十岁。

事实上，尽管波利亚的父母是犹太人，但他在出生后不久就受洗加入了罗马天主教会。这是怎么回事呢？原来Jakab、Anna和他们当时的三个孩子在1886年，即波利亚出生的前一年，从犹太教改信了罗马天主教。

波利亚于1897年去世，留下了妻子Anna（当时44岁）和五个孩子。波利亚有一个哥哥Jenö，父亲去世时他21岁，正在学医；两个姐姐Ilona（比波利亚大10岁）和Flóra（比波利亚大8岁），她们去Assicurazioni Generali保险公司工作以帮助养家；还有一个弟弟Lásló（比波利亚小4岁）。值得指出的是，Jenö热爱数学，一直后悔没有从事这门学科，他在医学界也许正如波利亚在数学界一样知名。然而，Lásló被认为是孩子们中最聪明的，但遗憾的是他在第一次世界大战中阵亡，未能成名。也许考虑到他父亲为进入学术界付出了多少努力，波利亚的母亲竟然催促他继承父亲的律师职业，这有点令人惊讶，但这正是她所做的。

波利亚在布达佩斯上小学，并于1894年获得证书，证书上记载（例如见[2]）：-

……勤奋且品行良好。

此后，他进入Dániel Berzsenyi 文理中学（Gymnasium），学习希腊语和拉丁语等古典语言，以及德语等现代语言，当然还有匈牙利语。在学校里，波利亚最喜欢的科目是生物和文学，在后一科目上他获得了“杰出”的成绩，地理和其他科目也是如此。一个后来一生对如此多不同数学分支如此着迷的人，竟然没有在学校爱上这门学科，这相当不寻常，但在波利亚的情况下，这正是所发生的事。他在中学的数学成绩并不特别高，他的几何作业仅被评为“满意”。然而，他的算术成绩确实要好一些。他在数学上缺乏成功的原因很可能归咎于糟糕的教学，他后来将中学的三位数学老师中的两位描述为“可鄙的老师”。

波利亚于1905年进入布达佩斯大学，由他的兄弟Jenö资助，后者当时已是一名外科医生。他开始学习法律，但觉得太无聊，一个学期后就放弃了。然后他学习了两年他最喜欢的学校科目语言和文学，获得了证书，允许他在中学教授拉丁语和匈牙利语。这是一项他引以为豪但从未使用的资格。之后他对哲学产生了浓厚兴趣，但他的教授Bernát Alexander建议他学习物理和数学课程以帮助他理解这门学科，所以最终他被迫学习数学。他说了一句俏皮话，不应认真对待[4]：-

我以为我物理不够好，哲学又太好。数学介于两者之间。

在布达佩斯大学，波利亚的物理由厄特沃什·罗兰教授，数学由费耶尔教授。波利亚说[4]：-

我深受费耶尔的影响，我这一代的所有匈牙利数学家都是如此，事实上，有一两次在一些小事上我与费耶尔合作过。在他的一两篇论文中我有一些评论，他在我的一两篇论文中也做了评论，但这并不是真正深刻的影响。

1910-11学年，波利亚在维也纳大学学习，他通过给当地一位重要显贵的儿子授课来赚钱（他的这个学生显然毫无天赋）。在维也纳，他听了威廉·维廷格和弗朗茨·梅滕斯的数学讲座，但对物理学仍保持着浓厚的兴趣，参加了相对论、光学和其他主题的讲座。次年他回到布达佩斯，在那里获得了数学博士学位，他基本上是在没有导师指导的情况下研究了几何概率论中的一个问题。随后，他在1912年和1913年的大部分时间都在哥廷根度过，在那里他与众多顶尖数学家交往，如菲利克斯·克莱因、康斯坦丁·卡拉西奥多里、大卫·希尔伯特、卡尔·龙格、埃德蒙·朗道、赫尔曼·外尔、埃里希·赫克、理查·科朗特和奥托·特普利茨。

事实上，波利亚是在相当不幸的情况下离开哥廷根的。他在1921年写给路德维希·比贝尔巴赫的一封信中解释了这一事件（例如见[2]）：-

1913年圣诞节，我乘火车从苏黎世前往法兰克福，当时我与一个坐在火车车厢对面的年轻人——关于我掉下来的篮子——发生了口角。我当时情绪过度激动，挑衅了他。当他对我的挑衅没有回应时，我打了他一记耳光。后来才知道，这个年轻人是某位Geheimrat的儿子；他偏偏还是哥廷根的学生。在一些误会之后，我被大学评议会要求离开。

他收到了法兰克福的聘任邀请，但在接受这一聘任之前，他于1914年初前往巴黎短暂访问，会见了埃米尔·皮卡和雅克·阿达马，但这次访问并不十分愉快，主要是因为住宿条件糟糕。在波利亚遇到过的众多数学明星中，对他影响最大的数学家是阿道夫·赫维兹。因此，当波利亚在巴黎逗留期间得知阿道夫·赫维兹已为他安排了苏黎世联邦理工学院的Privatdozent职位，而阿道夫·赫维兹本人就在那里担任数学讲席时，波利亚决定接受[4]：-

我……深受阿道夫·赫维兹的影响。事实上，我去苏黎世就是为了靠近阿道夫·赫维兹，从我1914年到达苏黎世到他……1919年去世，我们密切交往了大约六年。我们有一篇合著论文，但这并不是全部。他给我留下了非常深刻的印象，我编辑了他的著作。他的手稿也给我留下了深刻印象。

在苏黎世，除了阿道夫·赫维兹之外，波利亚的同事还有Geiser、保罗·贝尔奈斯、恩斯特·策梅洛和赫尔曼·外尔。当然，他到达苏黎世是在第一次世界大战开始的那一年，但起初这并没有给波利亚带来真正的问题，因为他学生时代所受的足球伤意味着他在医学上不适合在匈牙利军队服役。这对他来说是幸运的，因为此时他已持有坚定的和平主义观点。然而，随着战争的推进，生活变得更加困难，因为匈牙利军队随着战争的进展越来越急需士兵，要求波利亚返回匈牙利，参军并为他的国家而战；他拒绝了。这确实导致了战争结束多年后，波利亚才能回到匈牙利而不必担心因未服兵役而被捕。他取得了瑞士国籍，尽管这并未使他免受匈牙利当局的追究，1918年他与一位瑞士姑娘Stella Vera Weber结婚，她是纳沙泰尔大学物理学教授的女儿。事实上，尽管很难理解他为何等了这么久，波利亚直到1967年才回到匈牙利，距他上一次访问故土已54年。

波利亚大约在1913年在布达佩斯第一次见到加博尔·塞格，当时他在国外各种学习之间回到那里。加博尔·塞格此时是布达佩斯的学生，波利亚与加博尔·塞格讨论了他关于约瑟夫·傅里叶系数的一个猜想。事实上，加博尔·塞格后来证明了波利亚的猜想，这成为他的第一篇出版物。几年后，当波利亚决定写一本分析问题书时，他知道没有帮助就无法完成这项任务，于是他求助于加博尔·塞格，经过多年，两人汇编了一本精彩的问题集。在[4]中，波利亚解释了他为什么以不同于以往的方式传达数学思想：-

我很晚才接触数学。……当我接触数学并对其有所了解时，我想：嗯，事情就是这样，我明白了，证明看起来是确凿的，但人们是如何发现这样的结果的呢？我在理解数学上的困难：它是如何被发现的？

是什么伟大的创新使得波利亚和加博尔·塞格的分析问题集如此与众不同？那是波利亚的想法，即不按主题而是按解题方法来分类问题。波利亚和加博尔·塞格在1923年向出版商Springer提出了他们关于两卷本著作的想法，1925年Aufgaben und Lehrsätze aus der AnalysisⓉ（《微积分中的问题与定理》）问世。这部著作是[5]：-

……一部数学杰作，确立了他们的声誉。

波利亚于1920年在苏黎世的ETH被提升为编外教授。1924年，他获得洛克菲勒奖学金，得以在英国与戈弗雷·哈罗德·哈代一起学习。1924年，他部分时间在牛津，部分时间在剑桥，与戈弗雷·哈罗德·哈代和利特尔伍德合作，他们开始合作撰写Inequalities，该书于1934年出版。在撰写这本书期间，波利亚继续发表了一系列引人注目的论文，在1926-28这三年间共发表了31篇论文。鉴于这些出版物的范围、深度和数量，他在1928年被提升为ETH的正教授也就不足为奇了。

在评论3时，波利亚对波利亚的数学成就作了如下总结：-

波利亚可以说是20世纪最有影响力的数学家。他的基础研究贡献横跨复分析、数学物理、概率论、几何学和组合学。他是一位杰出的教师，在其漫长的职业生涯中始终对教学问题保持着浓厚的兴趣。

在苏黎世期间，他的数学产出非常丰富且范围广泛。例如，1918年他发表了关于级数、数论、组合学和投票系统的论文。第二年，除了关于这些主题的论文外，他还发表了关于天文学和概率的论文。在进行这些广泛工作的同时，他还在整函数研究中证明了他的一些最深刻的结果。

1933年，波利亚获得了第二笔洛克菲勒奖学金，这次是让他访问普林斯顿。他在美国期间，Blichfeldt邀请他访问斯坦福，他去了，并且非常喜欢那里。他回到了苏黎世，但1940年欧洲的政治局势迫使波利亚移居美国，在那里，在布朗大学工作两年、史密斯学院短暂工作一段时间后，他接受了斯坦福的职位。在去美国之前，波利亚有一本用德语写的书How to solve it的草稿。他不得不尝试了四家出版社，才找到一家在美国出版英文版，但多年来它售出了超过一百万册，并被翻译成17种语言。Schoenfeld在[24]中描述了它的重要性：-

对于数学教育和问题求解领域来说，它标志着两个时代之间的分界线，即波利亚之前和之后的问题求解。

波利亚在How to solve it中解释说，解决问题需要研究启发法：-

启发法的目的是研究发现与发明的方法和规则……启发法作为形容词，意思是“用于发现的”。……其目的是发现当前问题的解法。……什么是好的教育？系统地给学生机会，让他自己发现事物。

他还给出了明智的建议：——

如果你不能解决一个问题，那么就有一个更容易的问题你能解决：找到它。

波利亚出版了更多关于解决数学问题艺术的书籍。例如Mathematics and plausible reasoning（1954），以及分两卷出版的Mathematical discovery（1962，1965）。

在我们审视波利亚对教学的贡献时——许多人认为这是他对数学最大的贡献——让我们再引用一些波利亚关于这一主题的论述。首先引用一段关于小学数学教学的演讲：——

小学数学有一个良好而狭窄的目标，这在小学里相当清楚。……然而，我们有一个更高的目标。我们希望发展成长中儿童的一切资源。而数学所起的作用主要关乎思维。数学是思维的好学校。但什么是思维？你在数学中能学到的思维，例如，是处理抽象。数学关乎数。数是抽象。当我们解决一个实际问题时，那么从这个实际问题出发，我们必须首先把它变成一个抽象问题。……但我认为有一点甚至更重要。你看，数学不是一项观赏性运动。理解数学意味着能够做数学。而做数学意味着什么？首先，它意味着能够解决数学问题。

下面我们引用波利亚关于一般教学的一段话：-

教学不是一门科学，而是一门艺术。如果教学是一门科学，那就会存在一种最佳的教学方式，每个人都必须那样教。既然教学不是一门科学，就存在很大的自由度和许多个人差异的可能性。……让我告诉你们我对教学的看法。也许第一点是广泛接受的，即教学必须是主动的，或者更确切地说是主动学习。……数学教学的主要点是发展解决问题的策略。

让我们简要讨论波利亚在数学的许多不同领域中进行的一些研究。其范围如此广泛，数量如此丰富，我们只能提及几个方面。在概率论中，波利亚研究了概率测度的约瑟夫·傅里叶变换，于1923年证明它是一个特征函数。他撰写了关于正态分布的论文，并于1920年创造了“中心极限定理”这一术语，这已成为现在的标准用法。1921年，他证明了关于整数格上随机游走的著名定理。他考虑了一个$d$维的格点阵列，其中一个点以等概率移动到其任何邻居。他问，给定格中任意一点$A$，从原点开始执行随机游走的点是否会以概率1到达$A$。波利亚令人惊讶的答案是，对于$d = 1$和$d = 2$会，但对于$d ≥ 3$则不会。在后来的工作中，他研究了两个执行独立随机游走的点，以及满足移动点从不两次经过同一格点条件的随机游走。

几何对称性以及对象的对称类的枚举多年来是波利亚感兴趣的主要领域。1924年，他通过用平面的镶嵌来图示17个平面晶体群，增进了对它们的理解。这篇论文启发了莫里茨·科内利斯·埃舍尔，使他创作了关于周期图画的著名工作。波利亚使用生成函数和置换群来枚举有机化学中的异构体，这项工作具有根本的重要性。

他对组合数学的主要贡献是他的枚举定理，发表于1937年。Read在[18]中将其描述为：-

……一篇非凡论文中的一个非凡定理，是组合分析史上的一个里程碑。

该定理解决了具有某些性质的构型有多少个的问题。它有诸如化合物枚举和图论中有根树的枚举等应用。事实上，基于波利亚的思想，图论中一个全新的领域——枚举图论——发展起来了。

波利亚对复分析的兴趣促使他研究幂级数的奇点、间隙定理、整系数幂级数以及在正整数上取整数值的幂级数、指数型整函数的波利亚表示以及零点的位置。他还研究共形映射和potential theory，并因此被引导去研究偏微分方程的边值问题以及与它们相关的各种泛函理论。他的方法特别适用于具有高度对称性的区域中的等周问题。1951年，他与加博尔·塞格合写了如今已成为经典的著作Isoperimetric inequalities in mathematical physics。Schiffer在[23]中写道：

整部著作展现了作者们对具体而明确的结果、对优雅与巧妙方法的品味。

1953年，波利亚从斯坦福退休，但仍继续着极其活跃的数学活动，尤其关注数学教育。他继续以荣休教授的身份与斯坦福保持联系，1977年12月13日，那里为他90岁生日举行了晚宴，许多朋友和同事都致以热烈的颂词。然而，他的教学生涯仍未结束，1978年他在斯坦福计算机科学系讲授了一门组合数学课程。

他因其杰出贡献而获得许多荣誉，我们这里只提及少数几个。他被选为Hungarian Academy、London Mathematical Society、大不列颠的数学协会以及瑞士数学会的名誉成员。他被选入国家科学院 of the United States、American Academy of Arts and Sciences、布鲁塞尔国际科学哲学科学院以及加利福尼亚数学委员会。他是巴黎Académie des Sciences的通讯成员。

让我们以Frank Harary对波利亚的颂词来结束本文[11]：

毫不犹豫地说，波利亚是我个人心目中的数学家英雄。……[他]不仅是一位杰出的绅士，而且是一位最善良温和的人：他那洋溢的热情、眼中的光芒、巨大的好奇心、慷慨地付出时间、轻快有力的步伐、温暖真诚的友善、欢迎访客到他家中并向他们展示他所认识的伟大数学家的照片——这些都是他快乐性格的组成部分。作为一名数学家，他的深度、速度、才华、多才多艺、力量和普遍性都令人鼓舞。但愿有办法教授和学习这些品质。

George Pólya's parents were Anna Deutsch and Jakab Pólya who were both Jewish. Anna was from a family who had lived for many generations in Buda, and she had been nineteen years old in 1872 when the towns of Buda, Obuda, and Pest had administratively merged to become the city of Budapest. Perhaps we should say a little about George Pólya's names, for the situation is not quite as it appears. In fact, although Jakab Pólya had the name "Pólya" when his son György (or George as he was later known) was born, he had only called himself Pólya for the five preceding years. Before that his name had been Jakab Pollák but, in order to understand why Jakab Pollák changed his name to Pólya, we need to look at both his career and at a little Hungarian history.

Jakab was trained as a lawyer, ran his own law firm which failed, and then worked for the international insurance company Assicurazioni Generali of Trieste. However what he really wanted was a university post in which he could conduct research into the subjects which really interested him, namely economics and statistics. After 1867 Hungary had gained full internal independence within the Austro-Hungarian Monarchy and the political philosophy of the country was to move towards a Hungarian state that was both Magyar in spirit, and in its institutions. What better way for Jakab Pollák to improve his chances of a university post than to change his name from a Jewish sounding one to one which sounded really Hungarian. He did just that in 1882 and whether it contributed to his success in getting an appointment as a Privatdozent at the University of Budapest, one cannot say but he received such a post shortly before he died in his early fifties when George was ten years old.

In fact although George's parents were Jewish, he was baptized into the Roman Catholic Church shortly after his birth. How did this come about? Well Jakab, Anna, and their three children at the time, converted from the Jewish faith to the Roman Catholic faith in 1886, the year before George's birth.

When Jakab Pólya died in 1897 he left a wife, Anna aged 44 at the time, and five children. George had an older brother Jenö, who was 21 years old and studying medicine when his father died, two older sisters Ilona (10 years older than George) and Flóra (8 years older than George) who went to work for the insurance company Assicurazioni Generali to help support the family, and a younger brother Lásló (4 years younger than George). It is worth pointing out that Jenö, who loved mathematics and always regretted not having pursued that subject, is perhaps as well known to medical people as George is to mathematicians. However, it was Lásló who was considered the brightest of the children, but sadly he was killed in World War I before making a name for himself. Perhaps given how much effort his father had put in trying to enter the academic profession, it is slightly surprising that George's mother should press him to follow his father's profession of law but this is exactly what she did.

George attended elementary school in Budapest and received his certificate in 1894 which recorded (see for example [2]):-

... diligence and good behaviour.

Following this he entered the Dániel Berzsenyi Gymnasium studying the classical languages of Greek and Latin as well as the modern language of German and of course Hungarian. At school Pólya's favourite subjects were biology and literature and in this latter subject he received "outstanding" grades as he did in geography and other subjects. It is rather unusual that someone who went on to spend their life being so fascinated by so many different branches of mathematics should not have fallen in love with the subject at school but in Pólya's case this is exactly what happened. He did not score particularly high marks in mathematics at the Gymnasium, his work in geometry being graded as merely "satisfactory". He did score rather better in arithmetic, however. The reason for his lack of success in mathematics may well have been due to poor teaching, and he would later describe two of his three mathematics teachers at the gymnasium as "despicable teachers".

Pólya enrolled at the University of Budapest in 1905 supported financially by his brother Jenö who was by now a surgeon. He began to study law but found it so boring that he gave up that topic after one semester. He then studied his favourite school subjects of languages and literature for two years, gaining his certificate which allowed him to teach Latin and Hungarian in a gymnasium. It was a qualification of which he was proud but never put it to use. He then became very interested in philosophy but his professor, Bernát Alexander, advised him to take physics and mathematics courses to help him understand this subject, so eventually he was made to study mathematics. He made the witty remark, which should not be taken seriously [4]:-

I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.

At the University of Budapest Pólya was taught physics by Eötvös and mathematics by Fejér. Pólya said [4]:-

I was greatly influenced by Fejér, as were all Hungarian mathematicians of my generation, and, in fact, once or twice in small matters I collaborated with Fejér. In one or two papers of his I have remarks and he made remarks in one or two papers of mine, but it was not really a deep influence.

The academic year 1910-11 Pólya spent studying at the University of Vienna where he earned money by teaching the son of an important local dignitary (his pupil, apparently, lacking any talent whatsoever). In Vienna he attended mathematics lectures by Wirtinger and Mertens but continued to have a strong interest in physics attending lectures in relativity, optics and other topics. In the following year he returned to Budapest where he was awarded a doctorate in mathematics having studied, essentially without supervision, a problem in the theory of geometric probability. He then spent much of 1912 and 1913 at Göttingen where he mixed with a whole host of leading mathematicians such as Klein, Carathéodory, Hilbert, Runge, Edmund Landau, Weyl, Hecke, Courant and Toeplitz.

In fact Pólya left Göttingen in rather unfortunate circumstances. He explained the incident in a letter to Bieberbach in 1921 (see for example [2]):-

On Christmas 1913 I travelled by train from Zürich to Frankfurt and at that time I had a verbal exchange - about my basket that had fallen down - with a young man who sat across from me in the train compartment. I was in an overexcited state of mind and I provoked him. When he did not respond to my provocation, I boxed his ear. Later on it turned out that the young man was the son of a certain Geheimrat; he was a student, of all things, in Göttingen. After some misunderstandings I was told to leave by the Senate of the University.

He received an offer of an appointment at Frankfurt but, before taking up this appointment, he went to Paris for a short visit early in 1914, meeting Émile Picard and Hadamard but not enjoying his visit a great deal mainly due to dreadful accommodation. From the wide range of mathematical stars that Pólya had met the mathematician who was the greatest influence on him was Hurwitz. Therefore when Pólya learnt during his stay in Paris that Hurwitz had arranged an appointment as Privatdozent for him at Eidgenössische Technische Hochschule Zürich, where Hurwitz himself held the chair of mathematics, Pólya decided to accept [4]:-

I was... deeply influenced by Hurwitz. In fact I went to Zürich in order to be near Hurwitz and we were in close touch for about six years, from my arrival in Zürich in 1914 to his passing in ... 1919. And we have one joint paper, but that is not the whole extent. I was very much impressed by him and edited his works. I was also impressed by his manuscripts.

In Zürich, in addition to Hurwitz, Pólya had Geiser, Bernays, Zermelo and Weyl as colleagues. Of course his arrival in Zürich was in the year that World War I started, but at first this caused Pólya no real problems since a soccer injury he had received as a student meant that he was not deemed medically fit for service in the Hungarian army. This was fortunate for him since, by this time, he held firm pacifist views. Life became more difficult as the war progressed, however, since the Hungarian army, becoming more desperate for soldiers as the war progressed, required Pólya to return to Hungary, to join the army, and to fight for his country; he refused. This did have the consequence that it would be many years after the war ended before Pólya was able to return to Hungary without fear of arrest for failing to undertake war service. He took Swiss citizenship, although this did not protect him from the Hungarian authorities, and in 1918 he married a Swiss girl, Stella Vera Weber, who was a daughter of the professor of physics at the University of Neuchâtel. In fact, although it is difficult to see why he waited so long, Pólya did not return to Hungary until 1967, 54 years after his last visit to his native land.

Pólya first met Szegő on Budapest in around 1913 when he returned there between his various studies abroad. Szegő at this time was a student at Budapest and Pólya discussed a conjecture he had made on Fourier coefficients with Szegő. In fact Szegő went on to prove Pólya's conjecture and this became his first publication. When several years later Pólya decided to write a problem book on analysis he knew that it was not a task he could accomplish without help, so he turned to Szegő and over a number of years the two assembled a wonderful collection of problems. In [4] Pólya explained why he approached putting across mathematical ideas in a different way to that previously used:-

I came very late to mathematics. ... as I came to mathematics and learned something of it, I thought: Well it is so, I see, the proof seems to be conclusive, but how can people find such results? My difficulty in understanding mathematics: How was it discovered?

What was the great novelty which made Pólya and Szegő's book of analysis problems so different? It was Pólya's idea to classify the problems not by their subject, but rather by their method of solution. Pólya and Szegő approached the publisher Springer in 1923 with their idea for a two volume work and in 1925 Aufgaben und Lehrsätze aus der Analysis Ⓣ appeared. This work was [5]:-

... a mathematical masterpiece that assured their reputations.

Pólya had been promoted to extraordinary professor at ETH in Zürich in 1920. He received a Rockefeller Fellowship in 1924 to enable him to study with Hardy in England. He spent 1924 partly in Oxford, partly in Cambridge, working with Hardy and Littlewood and they began a collaboration on the book Inequalities was published in 1934. While the book was being worked on, Pólya continued a remarkable series of publications, with a total of 31 papers appearing during the three years 1926-28. Given the range, depth and number of these publications it is not surprising that he was promoted to Ordinary Professor at ETH in 1928.

In reviewing [3], Duren gave this summary of Pólya's mathematical achievements:-

Pólya was arguably the most influential mathematician of the 20th century. His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics. He was a teacher par excellence who maintained a strong interest in pedagogical matters throughout his long career.

While in Zürich his output of mathematics was very large and wide ranging. For example, in 1918 he published papers on series, number theory, combinatorics and voting systems. The following year, in addition to papers on these topics, he published on astronomy and probability. While he was doing this wide range of work he was also proving some of his deepest results in the study of integral functions.

In 1933 Pólya was awarded a second Rockefeller Fellowship, this time to allow him to visit Princeton. While he was in the United States Blichfeldt invited him to visit Stanford which he did, and greatly enjoyed being there. He returned to Zürich but in 1940 the political situation in Europe forced Pólya to move to the United States where, after working at Brown University for two years and Smith College for a short while, he took up an appointment at Stanford. Before going to the United States Pólya had a draft of a book How to solve it written in German. He had to try four publishers before finding one to publish the English version in the United States but it sold over one million copies over the years and has been translated in 17 languages. Schoenfeld described its importance in [24]:-

For mathematics education and the world of problem solving it marked a line of demarcation between two eras, problem solving before and after Pólya.

Pólya explained in How to solve it that to solve problems required the study of heuristic:-

The aim of heuristic is to study the methods and rules of discovery and invention .... Heuristic, as an adjective, means 'serving to discover'. ... its purpose is to discover the solution of the present problem. ... What is good education? Systematically giving opportunity to the student to discover things by himself.

He also gave the wise advice:-

If you can't solve a problem, then there is an easier problem you can solve: find it.

Pólya published further books on the art of solving mathematical problems. For example Mathematics and plausible reasoning (1954), and Mathematical discovery which was published in two volumes (1962, 1965).

While we are looking at Pólya's contributions to teaching, and many people consider this to be his greatest contribution to mathematics, let us give some further quotes from Pólya on this topic. First a quote from a lecture on teaching mathematics in primary schools:-

Mathematics in the primary schools has a good and narrow aim and that is pretty clear in the primary schools. ... However, we have a higher aim. We wish to develop all the resources of the growing child. And the part that mathematics plays is mostly about thinking. Mathematics is a good school of thinking. But what is thinking? The thinking that you can learn in mathematics is, for instance, to handle abstractions. Mathematics is about numbers. Numbers are an abstraction. When we solve a practical problem, then from this practical problem we must first make an abstract problem. ... But I think there is one point which is even more important. Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems.

Next we give a quote from Pólya regarding teaching in general:-

Teaching is not a science; it is an art. If teaching were a science there would be a best way of teaching and everyone would have to teach like that. Since teaching is not a science, there is great latitude and much possibility for personal differences. ... let me tell you what my idea of teaching is. Perhaps the first point, which is widely accepted, is that teaching must be active, or rather active learning. ... the main point in mathematics teaching is to develop the tactics of problem solving.

Let us briefly discuss some of the research which Pólya carried out in many different areas of mathematics. It was so wide ranging and so plentiful that there is no way that we can do more than mention a few aspects. In probability Pólya looked at the Fourier transform of a probability measure, showing in 1923 that it was a characteristic function. He wrote on the normal distribution and coined the term "central limit theorem" in 1920 which is now standard usage. In 1921 he proved his famous theorem on random walks on an integer lattice. He considered a $d$-dimensional array of lattice points where a point moves to any of its neighbours with equal probability. He asked whether given an arbitrary point $A$ in the lattice, a point executing a random walk starting from the origin would reach $A$ with probability 1. Pólya's surprising answer was that it would for $d = 1$ and for $d = 2$, but it would not for $d ≥ 3$. In later work he looked at two points executing independent random walks and also at random walks satisfying the condition that the moving point never passed through the same lattice point twice.

Geometric symmetry and the enumeration of symmetry classes of objects was a major area of interest for Pólya over many years. He added to the understanding of the 17 plane crystallographic groups in 1924 by illustrating each with tilings of the plane. This paper inspired Escher to produce his famous work on periodic drawings. Pólya's work using generating functions and permutation groups to enumerate isomers in organic chemistry was of fundamental importance.

His main contribution to combinatorics is his enumeration theorem, published in 1937. Read, in [18], describes this as:-

... a remarkable theorem in a remarkable paper, and a landmark in the history of combinatorial analysis.

The theorem solves the problem of how many configurations with certain properties exist. It has applications such as the enumeration of chemical compounds and the enumeration of rooted trees in graph theory. In fact a whole new area of graph theory called enumerative graph theory grew up based on Pólya's ideas.

Pólya's interest in complex analysis led him to investigate singularities of power series, gap theorems, power series with integral coefficients and those taking integral values at the positive integers, the Pólya representation for entire functions of exponential type, and the location of zeros. He also worked on conformal mappings and potential theory, and he was led to study boundary value problems for partial differential equations and the theory of various functionals connected with them. His methods applied particularly to isoperimetric problems in domains with a high degree of symmetry. Together with Szegő, he wrote the now classic text Isoperimetric inequalities in mathematical physics in 1951. Schiffer writes in [23]:-

The whole work displays the taste of the authors for the concrete and explicit result, for elegance and ingenious methods.

In 1953 Pólya retired from Stanford, but continued with an exceedingly active mathematical life particularly concerning himself with mathematical education. He continued his association with Stanford as Professor Emeritus and, on 13 December 1977, a dinner was given there to mark his 90th birthday at which many friends and colleagues gave glowing tributes. His teaching career, however, was still not over and in 1978 he taught a course on combinatorics in the Computer Science Department at Stanford.

He received many honours for his outstanding contributions and we only mention a few here. He was elected an honorary member of the Hungarian Academy, the London Mathematical Society, the Mathematical Association of Great Britain, and the Swiss Mathematical Society. He was elected to the National Academy of Sciences of the United States, the American Academy of Arts and Sciences, the Académie Internationale de Philosophie des Sciences de Bruxelles, and the California Mathematics Council. he was a corresponding member of the Académie des Sciences in Paris.

Let us end this article with Frank Harary's tribute to Pólya [11]:-

With no hesitation, George Pólya is my personal hero as a mathematician. ... [he] is not only a distinguished gentleman but a most kind and gentle man: his ebullient enthusiasm, the twinkle in his eye, his tremendous curiosity, his generosity with his time, his spry energetic walk, his warm genuine friendliness, his welcoming visitors into his home and showing them his pictures of great mathematicians he has known - these are all components of his happy personality. As a mathematician, his depth, speed, brilliance, versatility, power and universality are all inspiring. Would that there were a way of teaching and learning these traits.