Godfrey Harold Hardy

Godfrey Harold Hardy (also known as G. H. Hardy) (1877-1947) was a British mathematician who formulated the inequality that bears his name.

Starting in 1914, Hardy mentored the self-taught Indian mathematician Srinivasa Ramanujan (1887-1920), a relationship that has become celebrated. He was Ramanujan's foremost advocate in Britain and thesis advisor, known for some of his astonishing formulas and his innate mathematical intuition. Hardy recognized almost immediately Ramanujan's extraordinary if inexperienced brilliance, and Hardy and Ramanujan became close collaborators.. In an interview conducted by Paul Erdős, when Hardy was asked what his greatest contribution to mathematics was, he answered without hesitation that it was Ramanujan's discovery. In a lecture on Ramanujan, Hardy said that "my relationship with the it's the only romantic incident of my life".{rp|2}}

Biography

Godfrey Hardy was born on February 7, 1877 in Cranleigh, Surrey, England. His parents, both schoolteachers, had a penchant for pure mathematics.

Hardy's natural predisposition towards mathematics kicked in very early. When he was two years old he would write numbers in excess of two million, and would test himself by factoring the numbers of hymns in Church.

After going to school in Cranleigh, Hardy entered Winchester College. In 1896 he went on to Trinity College (Cambridge). He held the Sadleirian professorship from 1931 to 1942; he had left Cambridge in 1919 to take up the Savilian chair of geometry at Oxford.

In the early 20th century, the British mathematicians Hardy and John E. Littlewood proved the conjecture to be true for numbers odds larger than some unspecified constant.

Work

Hardy is credited with reforming British mathematics by introducing into it rigor, which until then had been a characteristic of French, Swiss and German mathematics. British mathematicians had largely remained in the tradition of mathematics. applied mathematics, enslaved by the reputation of Isaac Newton. Hardy was more in tune with the cours d'analyse methods dominant in France, and aggressively promoted his conception of pure mathematics, particularly against hydrodynamics which was an important part of the mathematics of Cambridge.

Beginning in 1911, he collaborated with John Edensor Littlewood, on extensive work in mathematical analysis and analytic number theory. This (along with many other things) led to quantitative progress on Waring's problem, as part of the Hardy-Littlewood circle method, as it became known. In prime number theory, they proved results and some remarkable conditional results. This was an important factor in the development of number theory as a system of conjectures; examples are the first and second Hardy-Littlewood conjectures. Hardy's collaboration with Littlewood is one of the most successful and famous in the history of mathematics. In a 1947 lecture, the Danish mathematician Harald Bohr told a colleague: "Today, there are only three great English mathematicians: Hardy, Littlewood, and Hardy-Littlewood."

Hardy is also known for formulating the Hardy-Weinberg principle, a basic principle of population genetics, independently of Wilhelm Weinberg in 1908. He played cricket with geneticist Reginald Punnett, who presented the problem to him in purely mathematical terms. Hardy, who had no interest in genetics and described the mathematical argument as "very simple", may never have realized how important the result came to be.

Hardy's collected papers have been published in seven volumes by Oxford University Press.

Pure Mathematics

Hardy preferred his work to be considered pure mathematics, perhaps because he detested war and the military uses to which mathematics had been put. He made several similar statements in his Apology:

I've never done anything helpful. No discovery of mine has made, or is likely to make, directly or indirectly, for good or for evil, the least difference in the amenity of the world.

However, aside from formulating the Hardy-Weinberg principle in population genetics, his famous work on integer partitioning with his collaborator Ramanujan, known as the Hardy-Ramanujan Asymptotic Formula, has been widely applied in physics for to find quantum partition functions of atomic nuclei (first used by Niels Bohr) and to derive thermodynamic functions of non-interacting Bose-Einstein systems. Although Hardy wanted his math to be "pure" and devoid of any application, much of his work has found applications in other branches of science.

Furthermore, Hardy pointedly pointed out in his Apology that mathematicians, in general, do not "glory about the uselessness of their work," but - since science can be used for both evil and good ends- "mathematicians may be justified in rejoicing that there is a science anyway, and that their own, whose very remoteness from ordinary human activities should keep it smooth and clean"." Hardy also dismissed as a "hoax" the belief that the difference between pure and applied mathematics had something to do with its usefulness. Hardy considers "pure" classes of mathematics that are independent of the physical world, but also considers some "applied" mathematicians, such as the physicists Maxwell and Einstein, to be among the "real" mathematicians, whose work & #34;has permanent aesthetic value" and "it is eternal because the best of it can, like the best literature, continue to give thousands of people intense emotional satisfaction after thousands of years." Although he admitted that what he called "real" mathematics could one day be useful, he claimed that, at the time the Apology was written, only the "boring and elementary parts"; of pure or applied mathematics could "work for better or worse"."

Attitudes and personality

Socially, Hardy associated with the Bloomsbury group and the Cambridge Apostles; G. E. Moore, Bertrand Russell, and J. M. Keynes were friends. He was an avid cricket fan. Maynard Keynes observed that if Hardy had read the stock market for half an hour each day with as much interest and attention as the day's cricket scores, he would have become a rich man.

Sometimes he got politically involved, if he wasn't an activist. He participated in the Democratic Control Union during World War I and in For Intellectual Freedom in the late 1930s.

Aside from close friendships, he had some platonic relationships with young men who shared his sensibilities and often his fondness for cricket. A mutual interest in cricket led to a friendship with the young C. P. Snow. Hardy he was a lifelong bachelor and in his later years was cared for by his sister.

Hardy was extremely shy as a child, and was socially awkward, cold, and eccentric throughout his life. During his school years he was at the top of his class in most subjects and won many awards and recognitions, but he hated having to receive them in front of the whole school. He felt uncomfortable when being introduced to new people and couldn't bear to look at his own reflection in a mirror. It is said that when he stayed in hotels, he covered all the mirrors with towels.

Hardy's Aphorisms

• It is never worth a first-class man to express a majority opinion. By definition, there are many others who do.
• A mathematician, like a painter or a poet, is a pattern creator. If their patterns are more permanent than theirs, it is because they are made with ideas.
• We have come to the conclusion that trivial mathematics is, in general, useful, and that real mathematics, in general, is not.
• Galois died at twenty-one years old, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great works quite later; Gauss' great memory of differential geometry was published when he was fifty years old (although he had the fundamental ideas ten years earlier). I do not know any case of a great mathematical advance initiated by a man who has passed from the age of fifty.
• Hardy once told Bertrand Russell "If I could prove by the logic that you would die in five minutes, I would regret that I would die, but my grief would be greatly mitigated by the pleasure of the test."
• A chess problem is a genuine mathematics, but in a way it is a "trivial" mathematics. As ingenious and intricate as the plays are, as original and astonishing as they are, something essential is missing. Chess problems are not important. The best mathematics are serious besides beautiful, 'important'.

Works

• A Mathematician's Apology (Apology of a mathematician) Cambridge University Press. London, 1940.
• Ramanujan.Twelve Lectures on Subjects Suggested by His Life and Work Cambridge University Press: London, 1940.
• An Introduction to the Theory of Numbers. With E. M. Wright. 1938.
• Hardy, G. H. (2008, 1a. ed. 1908). A Course of Pure Mathematics. With preface by Thomas William Körner (10th edition). Cambridge University Press. ISBN 978-0-521-72055-7.
• Hardy, G. H. (2013, 1a ed. Clarendon Press 1949). Divergent Series (2nd edition). Providence, RI: American Mathematical Society. ISBN 978-0-8218-2649-2. LCCN 49005496. MR 0030620. OCLC 808787. Full text
• Hardy, G. H. (1966–1979). Collected papers of G. H. Hardy; including joint papers with J. E. Littlewood and others. Edited by a committee appointed by the London Mathematical Society. Oxford: Clarendon Press. ISBN 0-19-853340-3. OCLC 823424. (requires registration). Vol.1 Vol.3 Vol.6 Vol.7
• Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952, orig.year 1a ed. 1934). Inequalities (2nd edition). Cambridge: Cambridge University Press. ISBN 978-0-521-35880-4.
• Hardy, G. H. (1970, orig. year 1a ed. 1942). Bertrand Russell and Trinity. With a foreword by C. D. Broad. Cambridge University Press. ISBN 978-0-521-11392-2.
• Autojustification of a mathematician [... ] (1981), Editorial Ariel, Barcelona-Spain, warning of Manuel Sadosky.

Acknowledgments

• Member of the Royal Society.
• Smith Prize (1901), Royal Medal (1920), Medalla De Morgan (1929), Chauvenet Prize (1932), Medalla Sylvester (1940)
• Copley Medal (1947)

Additional bibliography

• Kanigel, Robert (1991). The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Washington Square Press. ISBN 0-671-75061-5. (requires registration).
• Snow, C. P. (1967). "G. H. Hardy." Variety of Men. London: Macmillan. pp. 15-46. Reprinted as Hardy, G. H. (2012). «Foreword». A Mathematician's Apology. Cambridge University Press. ISBN 978-1-107-29559-9.
• Albers, D.J.; Alexanderson, G.L.; Dunham, W., eds. (2015). The G.H. Hardy Reader. Cambridge: Cambridge University Press. ISBN 978-1-10713-555-0.