戈弗雷·哈罗德·哈代G.H. Hardy

戈弗雷·哈罗德·哈代的父亲Isaac Hardy是克兰利学校的财务主管和美术教师。他的母亲索菲亚曾是林肯教师培训学校的教师。父母双方都极为聪明，具有一定的数学技能，但出身贫寒，未能接受大学教育。哈代（除了一个或两个称他为哈罗德的密友外，他一直被称为哈代）在克兰利学校就读到十二岁，成绩非常成功[6]：-

他的父母知道他聪明绝顶，他自己也知道。他在所有科目上都名列班级第一。但是，由于名列班级第一，他不得不到学校前面去领奖：而他无法忍受这一点。

哈代似乎没有许多数学家在年轻时对数学所怀有的那种热情。哈代本人在[5]中写道：-

我不记得自己小时候对数学有过任何热情，而我可能曾有过的关于数学家生涯的想法也远非高尚。我是从考试和奖学金的角度来考虑数学的：我想打败其他男孩，而这似乎是我能最果断地做到这一点的方式。

确实，他在1889年赢得了温切斯特公学的奖学金，并于次年入学。温切斯特是英格兰数学训练最好的学校，然而，尽管晚年承认自己在那里受到了良好的教育，哈代除了所接受的学术训练外，对学校的一切都不喜欢。像所有公学一样，对于像哈代这样体弱、害羞的男孩来说，那是个艰难的地方。值得注意的是，尽管他确实对球类运动，尤其是板球，怀有热情，但他在温切斯特从未接受过体育训练。不知怎的，他未能充分参与非学术活动。

在温彻斯特期间，哈代赢得了剑桥大学三一学院的公开奖学金，并于1896年进入该学院。在剑桥，哈代被分配给最著名的教练R R Webb。他很快意识到，训练的目的仅仅是通过学习所有行业技巧来在考试中取得尽可能好的分数。他震惊地发现Webb对数学学科不感兴趣，只对考试技巧感兴趣。

哈代认为他或许可以改换专业，改学历史。然而，他设法将他的辅导教师换成了乐甫。哈代在[6]中表达了对乐甫的感激：-

我的眼界最初是由乐甫打开的，他首先教了我几个学期，并给了我关于分析的第一个严肃概念。但我欠他的巨大恩情是他建议我阅读卡米耶·若尔当的《分析教程》；我永远不会忘记我阅读那部非凡著作时的惊讶，它是我这一代许多数学家的第一个灵感，我在阅读时第一次了解到数学的真正含义。

哈代在1898年的数学荣誉学位考试中位列第四名数学荣誉学位考试一等及格者（Wrangler），这一结果一直令他烦恼，因为尽管觉得这个制度非常愚蠢，他仍然觉得自己本应名列榜首。哈代于1900年被选为三一学院会士，然后在1901年，他与詹姆斯·琼斯共同被授予史密斯奖，“相对功绩未作说明”。

哈代职业生涯的下一个阶段一直持续到1911年，正如约翰·查理·伯基尔在[1]中所写，他：-

……撰写了多篇关于级数与积分的收敛性及相关主题的论文。尽管这项工作确立了他作为分析学家的声誉，但他在这一早期阶段对数学最大的贡献是《纯数学教程》（1908年）。这部著作是第一部面向本科生、以严谨的英文阐述数、函数、极限等概念的著作，从而改变了大学教学。

这是一个哈代自己曾写道[4]的时期：-

我写了很多……但几乎没有多少重要的；能让我至今仍带着些许满意回忆起来的论文不超过四五篇。

此时值得注意的是，哈代是一个极其诚实的人，尤其是他对自己的能力、长处和短处非常诚实。

哈代工作的一个重大变化发生在1911年，当时他开始与利特尔伍德合作，这种合作持续了35年。然后在1913年初，他收到了拉马努金从印度寄来的第一封信，这开启了他的第二次重大合作。到1914年第一次世界大战爆发时，拉马努金已在剑桥，这为哈代缓解了本将非常困难的时期。

利特尔伍德离开剑桥，加入皇家炮兵团服役参加战争。哈代自愿参军服役，但因健康原因被拒绝。然而，哈代对战争的看法使他与剑桥的大多数同事意见不合。他对德国怀有极大的敬意[6]：-

……他对德国怀有深厚的感情。毕竟，德国曾是19世纪伟大的教育力量。对东欧、对俄罗斯、对美国而言，正是德国的大学教会了它们研究的真谛。……在大多数方面，德国文化，包括其社会福利，在他看来都比自己的祖国更为优越。……哈代，如同伯特兰·罗素……并不认为这场战争本应发生。此外，由于他对英国政客根深蒂固的不信任，他认为错误的天平偏向英国一方。

在剑桥深感不幸的哈代，于1919年获得机会离开，当时他被任命为牛津大学的萨维尔几何学讲席教授。从许多方面来说，这些年是他最快乐的时期，也是他与利特尔伍德合作产出最优秀数学成果的时期。这一合作是在利特尔伍德在剑桥、哈代在牛津的时期实现的，这使得共同研究成为一项相当困难的后勤工作。正如哈代在[5]中所写：-

我在刚过四十岁时处于最佳状态，当时我是牛津大学的教授。

尽管有着这样的背景和所担任的职位，哈代更喜欢穷人和弱势群体，而不是他称之为“大屁股”的那些人，其中包括[6]：-

……那些自信满满、声如洪钟、帝国主义式的英国资产阶级。这一称号涵盖了大多数主教、校长、法官，以及所有政客，唯一的例外是劳合·乔治。

在剑桥时，他选择不住最好的房间，而大卫·希尔伯特如此担心哈代没有得到应有的待遇，以至于他写信给学院院长，指出英格兰最优秀的数学家应该拥有最好的房间。然而，哈代并不这么认为。他曾担任工会职务两年（1924-26），担任科学工作者协会主席。在一个似乎很难做到这一点的时代，哈代对美国和对俄罗斯同样喜爱。他在1928-29学年在普林斯顿度过，与奥斯瓦尔德·维布伦交换，后者在牛津度过了这一年。

尽管在剑桥曾不快乐，哈代于1931年欧内斯特·威廉·霍布森退休时回到那里的萨德莱尔讲席。斯诺在[6]中说，哈代回到剑桥有两个原因，首先是他仍然认为剑桥是英国数学的中心，萨德莱尔讲席是英国最重要的数学讲席，其次是他可以保留在剑桥学院的房间，而在牛津则不可能。对未婚的哈代来说，这具有吸引力，因为他开始展望老年。

哈代的兴趣涵盖纯数学的许多主题——Diophantine analysis、发散级数的求和、Fourier series、Riemann zeta function以及primes的分布。他与利特尔伍德的长期合作产生了最高质量的数学。在这种合作中，哈代承认利特尔伍德更精湛的技术数学技能，但同时哈代带来了巨大的数学洞察力天赋和将他们的工作以非常清晰的方式写成论文的伟大能力。

更引人注目的是哈代与拉马努金的合作。哈代从1913年拉马努金从印度寄给他的手稿中立即发现了拉马努金的天才。另外两位顶级数学家此前未能发现这一天才。哈代将拉马努金带到剑桥，他们一起写了五篇杰出的论文。

不仅与利特尔伍德和拉马努金，哈代还与其他许多人合作。他天生善于合作，还与爱德华·查尔斯·蒂奇马什、艾伯特·英厄姆、埃德蒙·朗道、乔治·波利亚、爱德华·梅特兰·赖特、W W Rogosinski和马塞尔·里斯合写过论文。

哈代是一位纯粹数学家，他希望自己的数学永远不被应用。然而在1908年，在他职业生涯初期，他给出了一个定律，描述显性和隐性遗传性状的比例将如何在大量人群中传播。哈代认为它不重要，但它在血型分布中已被证明具有重大意义。

哈代一生中除了数学之外只有一种爱好，那就是板球。事实上，在他一生的大部分时间里，至少是在板球赛季，他的一天会从早餐开始，期间他阅读The Times，饶有兴趣地研究板球比分。早餐后，他会从9点工作到1点，进行自己的数学研究。然后，在清淡的午餐后，他会步行到大学板球场观看比赛。傍晚时分，他会慢慢走回学院的房间。在那里吃晚餐，随后喝一杯葡萄酒。当板球赛季不在时，他会在The Times上阅读澳大利亚板球比分，并在下午打网球。

哈代以其怪癖闻名。他无法忍受被人拍照，已知仅存五张快照。他也讨厌镜子，进入任何酒店房间的第一件事就是用毛巾盖住任何镜子。他总是玩一个有趣的游戏，试图愚弄上帝（这也很奇怪，因为他一生都声称不信上帝）。例如，在去丹麦旅行时，他寄回一张明信片，声称他证明了黎曼假设。他的推理是，上帝不会允许船在回程中沉没，并让他获得与皮埃尔·德·费马通过其“last theorem”所取得的同样名声。

他试图愚弄上帝的另一个例子是，当他去看板球比赛时，他会带上他所谓的“反上帝电池”。这包括厚毛衣、雨伞、要审阅的数学论文、学生考试卷等。他的理论是，上帝会认为他预计会下雨，这样他就可以继续工作。由于哈代认为上帝会因此让太阳整天照耀来刁难他，他就能在完美的阳光下享受板球。

正如第一次世界大战对哈代来说是痛苦的，第二次世界大战同样如此。直到1939年，他在62岁时心脏病发作之前，他的身心都保持着非凡的年轻状态。他非凡的智力开始离他而去，他此前一直喜欢参与的体育运动也变得不可能。他对欧洲再次陷入战争的疯狂充满愤怒。然而，哈代还有一份礼物留给世界，那就是A mathematicians apology，它激励了许多人走向数学。

哈代的著作A mathematicians apology写于1940年。它是对数学家如何思考以及数学之乐趣的最生动描述之一。但正如⟦N2⟧在[6]中所写，这本书的意义不止于此：-

《一个数学家的辩白》如果以它应得的文本细读方式来阅读，是一本萦绕着悲伤的书。是的，它机智而犀利，充满智识上的高昂精神：是的，那种水晶般的清晰与坦率依然存在：是的，它是一位创造性艺术家的证言。但它同时也以一种含蓄的坚忍方式，是对曾经拥有、再也不会回来的创造力的激情哀悼。我在语言中不知道有类似的作品：部分原因是大多数具备表达这种哀悼的文学天赋的人并不会感受到它：一位作家以真理般的终结性意识到自己绝对已经完结，这是非常罕见的。

以下引自A mathematicians apology（[5]）的引文清楚地表明了哈代关于数学的思想：-

数学家的模式，就像画家或诗人的模式一样，必须是美的，思想，就像色彩或词语一样，必须以和谐的方式组合在一起。丑陋的数学在世界上没有永久的位置。

到1945年战争结束时，哈代的健康正在迅速衰退。他渴望再次具有创造力，因为那对他的人生来说才是真正重要的，但他知道自己的创造力已经消失，他变得非常抑郁。到1946年，他只能靠乘坐出租车出行，走几步就会让他气短。1947年初夏，他试图通过服用大剂量巴比妥类药物自杀。然而，他服用了太多，结果呕吐并活了下来。⟦N2⟧写道[6]：-

在伊夫林疗养院，哈代躺在床上。作为一点闹剧式的插曲，他有一只黑眼圈。由于药物导致呕吐，他的头撞到了盥洗盆上。他在自嘲。他把这件事搞砸了。……

他谈了一点，几乎每次我见到他，都会谈到死亡。他想要它。他不怕它：虚无之中有什么可怕的呢？他那顽固的理智禁欲主义又回来了。他不会再试图自杀了。他不擅长这个。他准备等待。带着一种可能会让他痛苦的矛盾——因为他……对理性的信仰达到了一种我认为不理性的程度——他对自己的症状表现出一种强烈的疑病症式的好奇。

哈代因其工作获得了许多荣誉。他于1910年当选为皇家学会会士，1920年获得该学会的皇家奖章，1940年获得该学会的詹姆斯·约瑟夫·西尔维斯特奖章：-

……因其对纯数学许多分支的重要贡献。

他还在1947年获得了皇家学会的科普利奖章：-

……因其在过去三十年中在英格兰数学分析发展中所起的杰出作用。

哈代在去世前几周才得知这一奖项。

在3中，他被描述如下：-

他体现了人们心目中那种心不在焉的教授形象。但那些认为他仅仅是个心不在焉的教授的人，在与他交谈时会大吃一惊，因为他在谈论天下任何话题时都展现出惊人的活力。……他对国际象棋感兴趣，但坦率地说，他对棋的某种本质感到困惑，这种本质似乎与他的数学原则相冲突。

他于1926年至1928年以及1939年至1941年担任伦敦数学会主席。1929年，他获得了该学会的奥古斯塔斯·德摩根奖章。

G H Hardy's father, Isaac Hardy, was bursar and an art master at Cranleigh school. His mother Sophia had been a teacher at Lincoln Teacher's Training School. Both parents were highly intelligent with some mathematical skills but, coming from poor families, had not been able to have a university education. Hardy (he was always known as Hardy except to one or two close friends who called him Harold) attended Cranleigh school up to the age of twelve with great success [6]:-

His parents knew he was prodigiously clever, and so did he. He came top of his class in all subjects. But, as a result of coming top of his class, he had to go in front of the school to receive prizes: and that he could not bear.

Hardy did not appear to have the passion for mathematics that many mathematicians experience when young. Hardy himself writes in [5]:-

I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.

Indeed he did win a scholarship to Winchester College in 1889, entering the College the following year. Winchester was the best school in England for mathematical training yet, despite admitting later in life that he had been well-educated there, Hardy disliked everything about the school other than the academic training he received. Like all public schools it was a rough place for a frail, shy boy like Hardy. It is significant that although he did have a passion for ball games in general and cricket in particular, he was never coached in sport at Winchester. Somehow he failed to take part fully in the non-academic activities.

While at Winchester Hardy won an open scholarship to Trinity College, Cambridge, which he entered in 1896. At Cambridge Hardy was assigned to the most famous coach R R Webb. He quickly realised that the point of the training was simply to achieve the best possible marks in the examinations by learning all the tricks of the trade. He was shocked to discover that Webb was not interested in the subject of mathematics, only in the tricks of examinations.

Briefly Hardy thought he might change topics and study history instead. However, he managed to change his coach to A E H Love. Hardy expresses his gratitude to Love in [6]:-

My eyes were first opened by Professor Love, who first taught me a few terms and gave me my first serious conception of analysis. But the great debt which I owe to him was his advice to read Jordan's "Cours d'analyse"; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.

Hardy was placed as fourth wrangler in the Mathematical Tripos of 1898, a result which continued to annoy him for, despite feeling that the system was very silly, he still felt that he should have come out on top. Hardy was elected a fellow of Trinity in 1900 then, in 1901, he was awarded a Smith's prize jointly with J H Jeans 'with unspecified relative merit'.

The next period of Hardy's career was up to 1911 when, as Burkill writes in [1], he:-

... wrote many papers on the convergence of series and integrals and allied topics. Although this work established his reputation as an analyst, his greatest service to mathematics in this early period was "A course of pure mathematics" (1908). This work was the first rigorous English exposition of number, function, limit, and so on, adapted to the undergraduate, and thus it transformed university teaching.

This was a period of which Hardy wrote himself [4]:-

I wrote a great deal... but very little of any importance; there are not more than four or five papers which I can still remember with some satisfaction.

It is worth noting at this point that Hardy was a remarkably honest man, and in particular he was very honest about his own abilities, strengths and weaknesses.

A major change in Hardy's work came about in 1911 when he began his collaboration with J E Littlewood which was to last 35 years. Then in early 1913 he received Ramanujan's first letter from India which was to start his second major collaboration. By the time World War I started in 1914, Ramanujan was in Cambridge and this eased for Hardy what was to be a very difficult period.

Littlewood left Cambridge for war service in the Royal Artillery. Hardy volunteered for war service but was rejected on medical grounds. However Hardy's views on the war left him at odds with most of his colleagues at Cambridge. He had great respect for Germany [6]:-

... he had a strong feeling for Germany. Germany had, after all, been the great educating force of the nineteenth century. To Eastern Europe, to Russia, to the United States, it was the German universities which had taught the meaning of research. ... in most respects the German culture, including its social welfare, appeared to him higher than his own. ... Hardy, like Russell ... did not believe that the war should have been fought. Further, with his ingrained distrust of English politicians, he thought the balance of wrong was on the English side.

Deeply unhappy at Cambridge, Hardy took the opportunity to leave in 1919 when he was appointed as Savilian professor of geometry at Oxford. These were in many ways the years when he was happiest and also the years when he produced his best mathematics in the collaboration with Littlewood. This collaboration was achieved during a period when Littlewood was in Cambridge and Hardy was in Oxford, making joint research a quite difficult logistical exercise. As Hardy wrote in [5]:-

I was at my best at a little past forty, when I was a professor at Oxford.

Despite his background and the positions he held, Hardy preferred the poor and disadvantaged to those he called the 'large bottomed' who included [6]:-

... the confident, booming, imperialist bourgeois English. The designation included most bishops, headmasters, judges, and all politicians, with the single exception of Lloyd George.

He had chosen not to live in the best rooms while at Cambridge, and Hilbert was so concerned that Hardy was not being properly treated that he wrote to the Master of the College pointing out that the best mathematician in England should have the best rooms. However, Hardy did not think that way. He held a trade union office for two years (1924-26) as President of the Association of Scientific Workers. At a time when it seemed difficult to do so, Hardy liked equally both the United States and Russia. He spent the academic year 1928-29 at Princeton in an exchange with Veblen, who spent the year in Oxford.

Despite having been unhappy at Cambridge, Hardy returned to the Sadleirian chair there in 1931 when Hobson retired. Snow in [6] says that Hardy returned to Cambridge for two reasons, firstly that he still considered Cambridge the centre of English mathematics and the Sadleirian chair the foremost mathematics chair in England, and secondly, that he could keep his rooms in College at Cambridge while this was not possible at Oxford. To the unmarried Hardy, this held an attraction as he began to look toward old age.

Hardy's interests covered many topics of pure mathematics - Diophantine analysis, summation of divergent series, Fourier series, the Riemann zeta function, and the distribution of primes. His long collaboration with Littlewood produced mathematics of the highest quality. It was a collaboration in which Hardy acknowledged Littlewood's greater technical mathematical skills, but at the same time Hardy brought great talents of mathematical insight and a great ability to write their work up in papers with great clarity.

Even more remarkable was Hardy's collaboration with Ramanujan. Hardy instantly spotted Ramanujan's genius from a manuscript sent to him by Ramanujan from India in 1913. Two other top class mathematicians had previously failed to spot the genius. Hardy brought Ramanujan to Cambridge and they wrote five remarkable papers together.

It was not only with Littlewood and Ramanujan that Hardy collaborated. He was a natural collaborator who also wrote joint papers with Titchmarsh, Ingham, Edmund Landau, Pólya, E M Wright, W W Rogosinski and Marcel Riesz.

Hardy was a pure mathematician who hoped his mathematics could never be applied. However in 1908, near the beginning of his career, he gave a law describing how the proportions of dominant and recessive genetic traits would be propagated in a large population. Hardy considered it unimportant but it has proved of major importance in blood group distribution.

There was only one passion in Hardy's life other than mathematics and that was cricket. In fact for most of his life his day, at least during the cricket season, would consist of breakfast during which he read The Times studying the cricket scores with great interest. After breakfast he would work on his own mathematical researches from 9 o'clock till 1 o'clock. Then, after a light lunch, he would walk down to the university cricket ground to watch a game. In the late afternoon he would walk slowly back to his rooms in College. There he took dinner, which he followed with a glass of wine. When cricket was not in season, it was the Australian cricket scores he would read in The Times and he would play real tennis in the afternoons.

Hardy was known for his eccentricities. He could not endure having his photograph taken and only five snapshots are known to exist. He also hated mirrors and his first action on entering any hotel room was to cover any mirror with a towel. He always played an amusing game of trying to fool God (which is also rather strange since he claimed all his life not be believe in God). For example, during a trip to Denmark he sent back a postcard claiming that he had proved the Riemann hypothesis. He reasoned that God would not allow the boat to sink on the return journey and give him the same fame that Fermat had achieved with his "last theorem".

Another example of his trying to fool God was when he went to cricket matches he would take what he called his "anti-God battery". This consisted of thick sweaters, an umbrella, mathematical papers to referee, student examination scripts etc. His theory was that God would think that he expected rain to come so that he could then get on with his work. Since Hardy thought that God would then have the sun shine all day to spite him, he would be able to enjoy the cricket in perfect sunshine.

As World War I had been painful for Hardy, World War II was equally so. He had remained remarkably youthful in both mind and body until 1939 when, at the age of 62, he had a heart attack. His remarkable mental powers began to leave him and sports which he had loved to participate in up till then became impossible. He was filled with anger that Europe had again entered the lunacy of war. However, Hardy had one further gift to leave to the world, namely A mathematicians apology which has inspired many towards mathematics.

Hardy's book A mathematicians apology was written in 1940. It is one of the most vivid descriptions of how a mathematician thinks and the pleasure of mathematics. But the book is more, as Snow writes in [6]:-

A mathematicians apology is, if read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits: yes, the crystalline clarity and candour are still there: yes, it is the testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again. I know nothing like it in the language: partly because most people with the literary gift to express such a lament don't come to feel it: it is very rare for a writer to realise, with the finality of truth, that he is absolutely finished.

The following quotation from A mathematicians apology ([5]) gives a clear idea of Hardy's thoughts on mathematics:-

The mathematician's patterns, like those of the painter's or the poet's, must be beautiful, the ideas, like the colours or the words, must fit together in a harmonious way. There is no permanent place in the world for ugly mathematics.

By the time the war ended in 1945 Hardy's health was failing fast. He longed to be creative again, for that was all that really mattered to him in life, but he knew that his creativity was gone and that he became very depressed. By 1946 he could only get around by taking taxi rides, a few steps would make him short of breath. In early summer of 1947 he tried to take his own life by taking a large dose of barbiturates. He took so many, however, that he was sick and survived. Snow writes [6]:-

In the Evelyn nursing home, Hardy was lying in bed. As a touch of farce, he had a black eye. Vomiting from the drugs, he had hit his head on the lavatory basin. He was self-mocking. He had made a mess of it. ...

He talked a little, nearly every time I saw him, about death. He wanted it. He didn't fear it: what was there to fear in nothingness? His hard intellectual stoicism had come back. He would not try to kill himself again. He was not good at it. He was prepared to wait. With an inconsistency which might have pained him - for he ... believed in the rational to an extent that I thought irrational - he showed an intense hypochondriac curiosity about his own symptoms.

Hardy received many honours for his work. He was elected a Fellow of the Royal Society in 1910, he received the Royal Medal of the Society in 1920 and Sylvester Medal of the Society in 1940:-

... for his important contributions to many branches of pure mathematics.

He also received the Copley Medal of the Royal Society in 1947:-

... for his distinguished part in the development of mathematical analysis in England during the last thirty years.

Hardy learnt of the award only a few weeks before his death.

He is described in [3] as follows:-

He personified the popular idea of the absent-minded professor. But those who formed the idea that he was merely an absent-minded professor would receive a shock in conversation, where he displayed amazing vitality on every subject under the sun. ... He was interested in the game of chess, but was frankly puzzled by something in its nature which seemed to come into conflict with his mathematical principles.

He was president of the London Mathematical Society from 1926 to 1928 and again from 1939 to 1941. He received the De Morgan Medal of the Society in 1929.