They proved prime number results and conditional results which were of primary importance in number theory. Hardy—Littlewood circle method came forward from their joint work and they are considered the most illustrious duo to collaborate in mathematical history. Aside from the number theory, Hardy is highly recognized for formulating the Hardy—Weinberg principle. It is a basic principle of population genetics which Wilhelm Weinberg and he worked on.
It was G. It has been speculated that the reason behind it is his disdain for the application of mathematics in the warfare instead of utilizing it for humanity. His another one of famous collaboration was with Ramanujan and with whom he created the Hardy—Ramanujan asymptotic formula.
It is applied as to derive quantum partition functions of atomic nuclei. The Oxford University Press published his collective works in seven volumes. When met in person, I have little doubt that he will render himself as rude and arrogant and pathetic to me and the many who will misunderstand him and many significant ideas.
He is honest, neither haughty nor humble, and he speaks of his word without any discretion or regret, all the time knowing that he will be disliked but nevertheless not caring since those who take offense are not too worthy of talking with anyway. For instance, he dismisses exposition, criticism, appreciation as work of "second-rate mind" which, ironically but deliberately, includes himself who is writing about his view on mathematics and suggests thta "most people can do nothing at all well Is his not his mathematics that impresses me, and I apologize Hardy for not understanding his work and appreciating him for that.
He is frank with himself and his thoughts. He regards his study highly, and justly so, but never despise for other subjects lest the people who represents them are idiots. It is the expression of the limit of human capacity, be it in mathematics or cricket, that is worth our time. And it is with selfishness that a "good work" is done, not with a humble mind.
I find it interesting, that I find many ideas echoing his own, particularly of ideas I found in the Fountainhead by Ayn Rand. I used to be rather lost by when I was rather certain that I did not have ingenious mind to be a great mathematician, physicist, computer scientist, poet, or a writer, professions which I most admire.. Here is what Hardy says on that note:.
I am a lawyer, or a stockbroker, or a professional cricketer, because I have some real talent for that particular job. I am a lawyer because I have a fluent tongue, and am interested in legal subtleties; I am a stockbroker because my judgement of the markets is quick and sound; I am professional cricketer because I can bat unusually well.
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 , 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 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 , 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 then, in , 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 when, as Burkill writes in [ 1 ] , he Although this work established his reputation as an analyst, his greatest service to mathematics in this early period was "A course of pure mathematics" 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 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 when he began his collaboration with J E Littlewood which was to last 35 years. Then in early he received Ramanujan 's first letter from India which was to start his second major collaboration. By the time World War I started in , 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 ] 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.
Hardy, like Russell 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 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 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.
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