Beauty is the first test: There is no permanent place in the world for ugly mathematics. — GH Hardy in A Mathematician’s Apology

Rarely does a quote provide such an apt description of the author’s personality. Still rarer for it to come from a mathematician who despised delving into commentary on his subject, believing that first-class minds should focus on developing the subject further rather than acting as critics. But then men like GH Hardy are rare after all, perhaps best described by his friend CP Snow as “eccentric, radical, ready to talk about anything” yet at the same time “shy, self-conscious and having a dread of introductions”.

Godfrey Harold Hardy (1877–1947) was an English mathematician best remembered for his work in analytic number theory and mathematical analysis. He almost single-handedly injected rigour (then a feature limited to continental mathematics) into mainstream British mathematics. But when asked (in an interview with Paul Erdős) to single out his greatest contribution to mathematics, he unreservedly opted for his discovery of Srinivasa Ramanujan, whom he elevated from a clerical position in Madras (modern day Chennai, India) to the elite mathematical stature the genius of Ramanujan richly deserved.

In this article, we explore the life and mathematics of Hardy. Majority of the information is drawn from his famous autobiographical essay, A Mathematician’s Apology, and CP Snow’s brilliant foreword which it has been bestowed with. So, first up is a brief synopsis of the sources themselves.

As the title implies, more than an autobiography, the essay is a defence of Hardy’s profession, i.e., a defence of the pursuit of mathematical truths. In it, Hardy blatantly accepts that real mathematics’ (the meaning of real is discussed in detail subsequently) practical usefulness is almost non-existent, proclaiming that -

I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world… Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow.

Subsequently, Hardy addresses the question — “Why then should real mathematics be tolerated in the societal structure?” To this end, he argues that mathematics, especially that of the highest class, has an aesthetic beauty of its own, which grants it the freedom and authority to survive just for its own sake. Hardy likens the work of a real mathematician to that of an artist, saying that –

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

And that

Real mathematics is eternal because it may continue to cause intense emotional satisfaction to thousands of people after thousands of years.

So, in a way, just like da Vinci’s Mona Lisa, the world wouldn’t have turned upside down if (a major chunk of) real mathematics didn’t exist. But it caters to a niche group of people who continue to derive emotional satisfaction by delving into and contributing towards the same. And that is justification enough for its existence.

CP Snow enjoyed a close, sixteen-year friendship with Hardy. His foreword, acting like a guide, lays out the perspective Hardy’s essay should be read with. Hardy was an eccentric man, who measured himself and others ruthlessly on the basis of cold facts. For him, rigour wasn’t confined to mathematics - it was his lifestyle.

Hardy wrote the essay in 1940 at the age of 63, when both his mental as well as physical prowess had deserted him. Keeping this in mind, it is no surprise that the work is dominated by gloom and pessimism (e.g., Hardy saying that ‘most men are good at nothing’), of which the reader must beware. Snow pinpoints Hardy’s mentality at the time of writing the essay when he says –

It is very rare for a writer to realize, with the finality of truth, that he is absolutely finished. — Snow

So, without further ado, here’s the story of G. H. Hardy. We follow a chronological development, intertwining Hardy’s essay with Snow’s foreword, to bring forth a holistic view.

Childhood

Hardy was born in 1877 to parents working in the academic sector. As is typical of top-notch mathematicians, he displayed extraordinary intelligence right from the off, topping each subject in each standard. Thanks to the academic know-how of his parents, Hardy was always assured of having the correct path (in the form of scholarships and the like) laid out for him towards a career befitting his intelligence. He just had to oblige, so to say.

Hardy completed his schooling from Winchester (as a boarding student), the best mathematical school in England at the time. This is where he met JE Littlewood, with whom he would go on to form the most remarkable of research partnerships. But beyond mathematics, Winchester was a typically cold and gloomy Victorian school. And perhaps this is when Hardy decided to turn atheist, a choice he adhered to for the rest of his life.

Hardy would go on to attend Trinity College (Cambridge), against the Winchester tradition of joining New College (Oxford). Comically, this decision was based on his reading of Alan St Aubyn’s A Fellow of Trinity, a book glorifying Cambridge lifestyle. As per his own admission, at this stage, he hadn’t yet discovered his zeal for mathematics –

I thought of mathematics in terms of examinations and scholarships. I wanted to beat other boys, and this seemed to be the way I could do so most decisively.

The perfect Anti-Narcissist

Socially awkward, he hated having to collect the first prize medal in front of the entire school. So much so that he perfunctorily tried coming second in order to escape the horror. He never kept a mirror, and despised having his photograph taken. As a result, only five photographs of him are in existence. No wonder this article is short on pictures!

Early Cambridge years

During those times, mathematics undergraduates at Cambridge were measured by the revered Tripos in its original form. It was an orthodox and heavily out-of-date examination system, with stress placed on rote learning and cumbersome procedures rather than originality. This profoundly troubled Hardy, to say the least. He gave a serious thought to changing his subject to history. However, he was rescued by one of his teachers, Professor Love, who introduced him to the famous Jordan’s Cours d’analyse. This is where Hardy got his first taste of realmathematics. In his own words –

I shall never forget the astonishment with which I read that remarkable work …, and learned for the first time as I read it what mathematics really meant.

Hardy topped Part II of the Tripos in 1900, and his life was essentially set thereon. His passion had led him to his purpose. There was to be no looking back now (though he forever remained vocal in his opposition to Tripos, seeking its abolishment). Over the next 47 years of his existence, he would go on to contribute significantly to number theory and analysis, all with his trademark rigour.

However, Hardy wasn’t a quick starter. Among his publications from 1900–1911, very few rank in the elite class he would become synonymous with later on.

Littlewood, Ramanujan and the First World War

1911 witnessed the start of the Hardy-Littlewood collaboration, a partnership remarkable for its productivity and longevity. In a span of 35 years, the duo published almost 100 papers, most of them, as Hardy would put it, in the Bradman class (he was a cricket fanatic, more on this later). Such was there collaboration that intellectuals of the age would often joke — “The top three mathematicians in England are Hardy, Littlewood and Hardy-Littlewood”.

In 1913, Hardy received a parcel containing sample works from an Indian clerk, Srinivasa Ramanujan. He was totally blown apart upon his first reading of the material, but was cautiously pragmatic over its authenticity (another trademark characteristic of Hardy). Had he just unearthed a genius or was it just another trickster? That night, he looped in Littlewood for a closer inspection. And the result was crystal clear. In Snow’s words –

Before midnight they knew, and knew for certain. The writer of these manuscripts was a man of genius. — Snow

The obvious followed. Hardy arranged for Ramanujan to come to Trinity. The ensuing five years saw Hardy-Ramanujan produce remarkable work. The pair produced five papers of the highest class, before of course, Ramanujan had to return homeland owing to ill health. Ramanujan would die prematurely in 1920, aged just 32. However, there is no doubt regarding the impression he left on Hardy.

Hardy describes his discovery of Ramanujan as the “one romantic incident” of his life — a “singular experience” which gave him a sense of -

What did modern mathematics look like to someone who had the deepest insight, but who had literally never heard of most of it?

Hardy rated both Littlewood and Ramanujan a lot higher than himself, stating that –

I still say to myself when I am depressed and find myself forced to listen to pompous and tiresome people, “Well, I have done one thing you could never have done, and that is to have collaborated with Littlewood and Ramanujan on something like equal terms.”

It must be noted that during the period of collaboration with Ramanujan, Hardy was devoid of majority of his friends, including Littlewood, who had gone on to assume duties in the first world war (Hardy had applied to, but was rejected due to medical reasons). So, in a way, Ramanujan was his only solace during the times.

Hardy, like many of his English colleagues, held his German counterparts in high regard. And was thus naturally against the war. Also, he was gradually growing discomforted with the Cambridge intelligentsia. The tensions heightened with the acrimonious expulsion of his close friend, Bertrand Russell. And he grabbed the first opportunity to move away from Cambridge, joining New College (Oxford) in 1920.

Oxford Years

It’s no secret that Hardy enjoyed Oxford life a lot more than Cambridge’s. He had always been held in high regards intellectually, but perhaps the Oxford intelligentsia accommodated his idiosyncrasies really well. Snow describes this as –

There was company eager to listen to him after hall. They could take his eccentricities. He was not only a great and good man, they realized, but an entertaining one. — Snow

Hardy was in his early forties while making the transition from Cambridge to Oxford. An age which, in light of Hardy’s belief that “mathematics is a young man’s game”, seems a bit too late to carry out significant mathematical work. Ironically, the period would prove to be the spring of Hardy’s career, with the Hardy-Littlewood partnership reaching its apex. This was especially remarkable given that Littlewood remained at Cambridge, and the pair’s only mode of communication was snail mail.

Cambridge Return

Hardy was so happy at Oxford that people doubted whether he would ever be seen anywhere near Cambridge again. But he returned. In 1931. The decision was a pragmatic one. Oxford, though unbelievably suitable for him in the present, would force him out of his housing once his term was over. Cambridge, on the other hand, would allow him to reside in his quarters till death.

Hardy’s second stint at Cambridge was the time Snow befriended him, with the friendship stemming from the duo’s shared interest in cricket. Although Hardy might not have been exactly pleased at his return, the newer generation of Cambridge mathematicians was certainly elated to have him back. Again, in Snow’s words –

They were delighted to have him back: he was a real mathematician, they said, not like those Diracs and Bohrs the physicists were always talking about: he was the purest of the pure. — Snow

Needless to say, Hardy continued his partnership with Littlewood. But surely, now he was in the twilight of his career.

Later years

His life remained the life of a brilliant young man until he was old: so did his spirit: his games, his interests, kept the lightness of a young don’s. And, like many men who keep a young man’s interests into their sixties, his last years were the darker for it. — Snow

In 1939, Hardy suffered coronary thrombosis. He recovered, but most of his favourite outdoor past times (especially tennis) were no more feasible. This was a sucker punch to Hardy’s spirits. Keeping in mind that he remained a bachelor for his entire life, such recreations meant a lot more to him than they would have otherwise. His sister accompanied him for the final years of his life (she was unmarried as well), a period he spent in continuous depression (the second world war only worsened things up).

In the early summer of 1947, Hardy attempted suicide. And failed, perhaps for the worse. Snow had continued to pay him regular visits over the last two years. And as he puts it, after the attempt, it was only a matter of time. Hardy had decided that there was no point trying again, as he was not good at it. He was ready to wait patiently for the end. By now he was a hypochondriac as well.

Very near to the end, Snow visited him at least once each week. Around ninety percent of their talk revolved around cricket, with Snow responsible for reading him scores from the latest matches. A few weeks before his death, Hardy was awarded Royal Society’s highest honour, the Copley Medal. A few days later, he went to sleep and never woke up again.

Next, we take a closer look at few aspects of Hardy’s life which deserve a section of their own.

Mathematics — Real, Beautiful, Pure, Applied

Much of Hardy’s essay focusses around separating out real mathematics from the mundane. Like all the other aspects of his life, Hardy’s mathematical taste was very selective. Majority of his work falls under pure mathematics without a shadow of a doubt. Also, it’s no secret that Hardy disliked a good portion of applied mathematics (perhaps the dislike accentuated with the way applied mathematics was being put to use in the world war). All this can lead one to falsely assume that Hardy uses the term real (or beautiful) mathematics as a synonym to pure mathematics. But that is not the case. Hardy effectively quashes this notion by saying –

… [While talking about real mathematics] I was not thinking only of pure mathematics. I count Maxwell and Einstein, Eddignton and Dirac, among ‘real’ mathematicians.

Then he goes on to add –

It is the dull and elementary parts of applied mathematics, as it is the dull and elementary parts of pure mathematics, that work for good or ill.

What then does Hardy exactly mean when he refers to real mathematics? Of course, he himself tries to answer this question to the best of his abilities. Understandably, the task to explain such subtleties to the layman is a daunting one. As Hardy puts it, the situation is very much like trying to figure out what makes a poem beautiful -

We may not know quite what we mean by a beautiful poem, but that does not prevent us from recognising one when we read it.

However, Hardy does give a glimpse of what qualifies as real mathematics in the form of two proofs

· Euclid’s proof of the infinitude of primes

· Proof of the irrationality of root 2

Both the proofs deploy reductio ad absurdum technique, which assumes the statement (to be proved) as false, and then proceeds via logical arguments to arrive at a contradiction. Hardy contrasts this approach against a game of chess, where a player might willingly sacrifice some pieces in the interest of winning the game –

A chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

Hardy attributes the above theorems’ elegance to their ‘generality’ and ‘depth’. He also mentions couple more theorems as worthy candidates, Fermat’s two square theorem, and Cantor’s non-denumerability of R (though grasping these two requires substantial background knowledge).

Friends, Sexual Orientation

Unsurprisingly, Hardy was selective in whom he befriended. His friends needed to pass his personal tests, and possess a certain degree of spin (a cricketing term). It seems these tests acted as his shields. But once someone made it inside, Hardy was an altogether different person. As Snow puts it –

He was tolerant, loyal, extremely high-spirited, and in an undemonstrative way fond of his friends. — Snow

Hardy remained a bachelor throughout his life. It may well have been a personal choice, but naturally, one is bound to wonder whether he was a homosexual which would have necessitated concealment in those times. Certainly, there is a hint of that in Snow’s remark -

He had, scattered through his life, two or three other relationships, different in kind. These were intense affections, absorbing, non-physical but exalted. — Snow

Cricket

It would be criminal to write about Hardy without mentioning cricket. He followed the sport crazily. Not just followed, he studied every aspect of it. The stats, the tactics, the captaincy. It was a sport that fitted seamlessly with his personality — a sport of grace and order.

He held one particular bitterness with Winchester school. In spite of being a “natural ball-games player with a splendid eye”, he was never seriously considered for school cricket. This deprivation continued to trouble him for the entirety of his life. He believed he could have become a decent batsman if proper training was imparted to him early on.

Nonetheless, nothing would stop him from sustaining a child-like devotion to the game. He would watch all the university matches, reserving a spot where every inch of his body could be graced by the sun (he was heliotropic). Sometimes, he would get bored and start conjuring up hypothetical Playing XIs, such as the all-time team of Trinity or Christ’s.

Cricket is what would go on to solace him during the final few days. Every evening, he would have a chapter from the history of Cambridge University cricket read out to him on the deathbed. And, as Snow puts it –

One such chapter contained the last words he heard, for he died suddenly, in the early morning. — Snow

Idiosyncrasies

Perhaps stemming from his love for pure mathematics, Hardy despised most of the mechanical gadgets of his age. He never wore a watch. Neither did he ever use a telephone for communication. He loved keeping it simple (thank god he wasn’t born in today’s times!) In some ways, he lived his life on the same terms on which he did his mathematics -

He believed in the rational to the extent which I thought irrational. — Snow

Hardy did not believe in genius, and opined that even if the word meant anything, he didn’t count himself as one. And it seems he wasn’t a big fan of natural intelligence either -

For any serious purpose, intelligence is a very minor gift.

However, he had a gift common to all shy and introverted people, that of knowing himself exceptionally well. He lived a life true to himself. Yes, his final years were troublesome. But for the majority of his life, he was in a state of happiness and satisfaction most of us can only dream of. I will conclude the article with the following remark of Snow which brilliantly nutshells Hardy in a single sentence –

Underneath his shyness, he just didn’t give a damn.