# Godfrey Harold Hardy

Godfrey Harold Hardy

Godfrey Harold Hardy (born February 7, 1877 in Cranleigh , Surrey , † December 1, 1947 in Cambridge , England ) was a British mathematician . His fields of work were analysis and number theory . Close friends addressed him as "Harold", but otherwise he is commonly called "GH Hardy".

Hardy was with John Edensor Littlewood , with whom he entered into a close scientific collaboration, the dominant figure in mathematics in Great Britain in the first half of the 20th century. In the English-speaking world, he is known to non-mathematicians for his essay A Mathematician's Apology , an essay on the beauty of mathematics. It is considered to be one of the best depictions of the work of professional mathematicians aimed at laypeople.

His discovery of and later collaboration with the Indian mathematician S. Ramanujan has become famous. Hardy recognized Ramanujan's extraordinary, albeit untrained, talent almost immediately in 1913. When Hardy was once asked by Paul Erdős what his greatest contribution to mathematics was, he answered without hesitation that it was Ramanujan's discovery; he called it "the only romantic incident in my life".

## biography

GH Hardy was on 7 February 1877 in Cranleigh in the county of Surrey south-west of London born into a family of teachers. His father taught geography and drawing, his mother had been assistant director of a teacher training institute; both parents showed mathematical inclinations. Hardy's talent was palpable from a young age. At the age of two he was already writing down numbers in the millions, and while going to church he occupied himself with factoring the hymn book numbers .

After graduating from Cranleigh in 1890, at the age of twelve, Hardy received a scholarship to Winchester College , then the school with the best mathematics classes in England; apparently he was taught individually in this subject at both schools. He studied from 1896 at Trinity College , Cambridge . After only two years of preparation, he passed the notorious Tripos test there in fourth place; Years later he endeavored to abolish the tripos because, in his opinion, it had become an end in itself.

Hardy mentions the study of the book Cours d'analysis de l'Ecole Polytechnique by the French mathematician Camille Jordan , from which he got to know the much more precise mathematics tradition of continental Europe, as a formative influence during this time . In 1900 he became a Fellow of Trinity College and in 1901 he won the Smith Prize with James Jeans . In 1903 he received his MA , the highest academic degree at English universities at the time; from 1906 he was a lecturer ( lecturer ) with six hours of lectures per week, which gave him plenty of time for research. As a result of the Bertrand Russell affair (Russell was a pacifist and just as Hardy staunch opponents of the war, so lost in Cambridge his Fellowship of Trinity College and was later imprisoned) he left Cambridge and moved in 1919 as a geometry professor ( Savilian Professor of Geometry ) to the Oxford University , where he stayed until 1931. He then returned to Cambridge, where he held a professorship until 1942. One reason for this was the higher reputation that Cambridge enjoyed in mathematics, but also (after CP Snow ) that, unlike Oxford, he could continue to live in college even after retirement.

Hardy was a cricket fan.

## Honors

In 1910 Hardy was accepted as a member (" Fellow ") in the Royal Society , which gave him the Royal Medal in 1920 , the Sylvester Medal in 1940 and the Copley Medal in 1947 . In 1921 he was elected to the Göttingen Academy of Sciences and the American Academy of Arts and Sciences . In 1925 he was elected a member of the Leopoldina , in 1927 a member of the National Academy of Sciences and in 1929 a corresponding member of the Prussian Academy of Sciences . Since 1924 he was a corresponding and since 1934 honorary member of the then Soviet Academy of Sciences . In 1939 he was admitted to the American Philosophical Society , 1945 to the Académie des Sciences and in 1946 as an honorary member ( Honoray Fellow ) to the Royal Society of Edinburgh .

Hardy was twice President of the London Mathematical Society .

The asteroid (24935) Godfreyhardy was named after him.

## plant

Hardy's early works, now almost forgotten, deal with the properties of functions that are given as definite integrals ; later he published articles on the theory of integral equations . From around 1906 he began to work with infinite series , especially Fourier series , the boundary behavior of power series and the summation of divergent series. From around 1912, results in the field of number theory were also added, for example on the Diophantine approximation , on the zeros of the zeta function (1914) and the prime number theorem (1915).

During this time he worked with the Indian natural talent S. Ramanujan , whom he had discovered and for whose mathematical theorems he found through inspiration he worked out exact proofs. From this, for example, an asymptotic formula for the partition function (with the beginning of the circle method) and the theorem of Hardy and Ramanujan emerged . However, his cooperation with his colleague John Edensor Littlewood from 1912 onwards was much more productive and lasting ; both together became the namesake of several theorems and a famous unproven conjecture about the distribution of prime numbers and one about prime twins and other prime number constellations. His work on analytical and additive number theory (e.g. Goldbach's conjecture , Waring's problem ) has remained important because Hardy and Littlewood used a new method (the so-called circle method ) here from around 1917 , which is still a standard method today.

In addition to these major topics, there are also many smaller papers, mainly on analysis. Around the mid-1930s, a phase of book publications began that lasted until the end of his life. His textbooks on number theory (with Edward Maitland Wright ), inequalities (with George Pólya and John Edensor Littlewood), and divergent series are still used today. His collected works are available in print.

His doctoral students include Irving Good , Sydney Chapman , Mary Cartwright , S. Ramanujan (as a bachelor's degree for his research in 1916, Ph. D. were only introduced in England from around 1917), Edward Charles Titchmarsh , Donald Spencer , EM Wright , K. Ananda Rau , Lancelot Bosanquet , Richard Rado , Frank Smithies .

## Pure versus applied math

Hardy is credited with reforming British mathematics, into which he introduced the strict disambiguation and reasoning customary in continental Europe . British mathematics had long relied on Isaac Newton's reputation and was mainly concerned with applied problems. Hardy opposed the tradition of the French cours d'analysis and aggressively advocated “pure mathematics”, with which he particularly set himself apart from the mathematical physics and applied mathematics (for example hydrodynamics ) operated in Cambridge .

Hardy insisted that his own work was entirely "pure mathematics", possibly resulting from his rejection of all military applications of mathematics. In his A Mathematician's Apology , written in 1940, he emphasizes: “I have never done anything 'useful'. None of my discoveries has ever or is likely to, directly or indirectly, make a difference, for better or for worse, to the welfare of the world. ” However, he had an idiosyncratic conception of“ true mathematics ”, which he also works on because of its mathematical elegance von Einstein and Maxwell counted.

Strangely enough, however, a relatively simple consideration in a letter to the editor of Science was enough to make him well-known among evolutionary biologists. This Hardy-Weinberg rule , according to which the relative frequency of the alleles in a gene pool remains constant over the generations, was formulated independently of Wilhelm Weinberg . So, of all people, the despiser of all applied mathematics became the founder of a branch of applied mathematics, population genetics . Apart from the Hardy-Weinberg rule, some of his results from number theory are now also to be used in cryptography .

## Views

Hardy was already an avowed atheist in his youth , later he went so far that he refused to enter the university chapel on formal occasions. According to Littlewood, Hardy was also a "non-practicing homosexual". However, this statement is relativized by Robert Kanigel (see literature), who points out that university life in Cambridge and Oxford was generally largely free of women. In any case, Hardy never married and was looked after by his sister during the last years of his life.

During his student days Hardy joined the elite Cambridge Apostles secret society ; later he also worked in the Bloomsbury group . He was friends with GE Moore , Bertrand Russell, CP Snow and John Maynard Keynes . At times he was also politically active. He participated in the Union of Democratic Control during the First World War and in the For Intellectual Liberty campaign in the late 1930s . His only great passion besides mathematics, however, was cricket , which he often watched from the stands for long afternoons. He said of cricket that it was the only game in which you played against eleven players from the opposing team and ten players from your own team at the same time.

## Anecdotes

One of Hardy's quotations from CP Snow in the preface to A Mathematician's Apology reads: “Young men should be convinced of themselves, but have no obligation to act imbecile ” (“ Young men ought to be conceited, but they oughtn't to be imbecile "). Hardy was referring to a young man who gave him Finnegans Wake by James Joyce touted as the greatest literary works of all time. Hardy reports similar experiences in an anecdote: One day he was sitting across from a schoolboy on the train who was reading an elementary algebra textbook. Out of politeness, he asked him after his reading, whereupon he received the condescending answer: “ It's advanced math, you wouldn't understand” (“ It's advanced algebra, you won't understand ”).

## Aphorisms

“The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. "

“The mathematician's patterns, like those of the painter or poet, must be beautiful; ideas, like colors or words, must fit together in a harmonious way. Beauty is the first criterion: there is no place in this world for ugly math. "

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

“A mathematician creates - like a painter or a poet - patterns. If his patterns are more permanent, it is because they are made with ideas. "

"Sometimes one has to say difficult things, but one ought to say them as simply as one knows how."

"Sometimes you have to say difficult things, but you should say them as simply as you can."

“It is never worth a first class man's time to express a majority opinion. By definition, there are plenty of others to do that. "

“For an intelligent person, it is a waste of time to express majority opinions. By definition, there are already enough other people for it. "

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

"For any serious matter, intelligence is a very minor gift."

## Fonts

• with EM Wright's introduction to number theory. Oldenbourg 1958, engl. An introduction into the theory of numbers. 5th ed., Oxford 1993.
• A Mathematician's Apology. Cambridge University Press, Cambridge 2006, ISBN 0-521-42706-1 (contains a detailed foreword by CP Snow on the life of GH Hardy, first 1941).
• A course in pure mathematics. 10th edition, Cambridge 1960 (first 1908).
• Collected papers. 7 vols., Clarendon Press, Oxford 1966-1979.
• Divergent series. Oxford 1973.
• Fourier Series. 3rd ed., Cambridge 1956.
• with George Pólya , John Edensor Littlewood Inequalities. 1934.
• Orders of infinity. Cambridge 1924.
• Ramanujan - 12 lectures suggested by the subject of his life and work. Cambridge University Press (1940) and AMS (1999).

## literature

• Donald J. Albers, Gerald Alexanderson, William Dunham : The GH Hardy Reader. MAA Press, Cambridge University Press, 2015.
• John Charles Burkill : Hardy, Godfrey Harold . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 6 : Jean Hachette - Joseph Hyrtl . Charles Scribner's Sons, New York 1972, p. 113-114 .
• Robert Kanigel: Who knew the infinite. The life of the brilliant mathematician Srinivasa Ramanujan. Vieweg, Braunschweig and Wiesbaden 1993, ISBN 3-528-06509-5 (Chapter 4 on pp. 89-127 and large parts of the rest of the book are about GH Hardy).
• Robert Alexander Rankin : Hardy as I knew him. Australian Mathematical Society Gazette, Volume 25, 1998, pp. 73-81.
• Edward Charles Titchmarsh : Godfrey Harold Hardy . In: Notices of Fellows of the Royal Society of London , Volume 6, 1949, pp. 447-470 (obituary).
• Edward Charles Titchmarsh: Godfrey Harold Hardy . In: Journal of the London Mathematical Society , Volume 25, 1950, pp. 82-101 (obituary).
• Robin Wilson : Hardy and Littlewood. In: Cambridge scientific minds . Cambridge University Press, Cambridge 2002, ISBN 0-521-78100-0 , pp. 202-219.

In fiction :

• David Leavitt: The Indian Clerk. Bloomsbury, London 2008, ISBN 978-0-7475-8168-0 (the novel is about Ramanujan and Hardy).

## Individual evidence

1. Hardy: Ramanujan. Cambridge University Press, 1940, p. 2: I owe more to him as to anyone else in the world with one exception, and my association with him is the one romantic incident in my life .
2. ^ CP Snow: Foreword. In: GH Hardy: A Mathematician's Apology. Cambridge University Press, Cambridge 2006, ISBN 0-521-42706-1 , p. 14.
3. Hardy was not a pacifist like Russell, however, in 1914 he applied three times in vain to volunteer for the army. Sanford Segal: Mathematicians under the Nazis. Princeton University Press, p. 267.
4. Hardy later wrote a book about this incident, which was not published, but only circulated among his friends. CP Snow, foreword to Hardy A mathematician's apology.
5. preface to Hardy's A mathematicians apology. Cambridge University Press, 1994.
6. entry on Hardy; Godfrey Harold (1877–1947) in the Archives of the Royal Society , London
7. Holger Krahnke: The members of the Academy of Sciences in Göttingen 1751-2001 (= Treatises of the Academy of Sciences in Göttingen, Philological-Historical Class. Volume 3, Vol. 246 = Treatises of the Academy of Sciences in Göttingen, Mathematical-Physical Class. Episode 3, vol. 50). Vandenhoeck & Ruprecht, Göttingen 2001, ISBN 3-525-82516-1 , p. 103.
8. Godfrey Harold Hardy. In: amacad.org, American Academy of Arts and Sciences, accessed January 28, 2018 (select "Include Deceased Members" to search).
9. ^ Members of the previous academies. Godfrey Harold Hardy. In: bbaw.de. Berlin-Brandenburg Academy of Sciences and Humanities , accessed on April 1, 2015 .
10. ^ Foreign members of the Russian Academy of Sciences since 1724. Godfrey Harold Hardy. In: ras.ru. Russian Academy of Sciences, accessed November 2, 2015 .
11. ^ Member History: Godfrey H. Hardy. American Philosophical Society, accessed September 21, 2018 .
12. ^ Fellows Directory. Biographical Index: Former RSE Fellows 1783–2002. (PDF) Royal Society of Edinburgh, accessed December 15, 2019 .
13. ^ Hardy-Littlewood Conjectures at Wolfram Mathworld . The first assumption concerns twins of prime numbers, the second the number function of prime numbers${\ displaystyle \ pi (x + y) - \ pi (x) \ leq \ pi (y)}$
14. Hardy at the Mathematics Genealogy Project , but sometimes he is also a co-referee there.
15. ^ 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 . Hardy: A mathematicians apology. Cambridge University Press, 1994, p. 49.
16. ^ GH Hardy: Mendelian proportions in a mixed population. In: Science . 28, 1908, pp. 49-50.
17. Reproduced for example in Steven G. Krantz: Mathematical Apocrypha - stories and anecdotes of Mathematicians and the Mathematical. Mathematical Association of America 2002, p. 29.
18. ^ Cricket is the only game in which one plays against eleven opponents - and the ten members of one's own team. In: Hardy: A Mathematician's Apology. Cambridge 1994, p. 46.
19. a b c Hardy: A mathematicians apology. Cambridge University Press, 1994, p. 47.
20. ^ Steven Krantz: Mathematical Apocrypha. MAA Press. 2002, p. 29.
21. ^ Hardy: A mathematicians apology. Cambridge University Press, 1994, p. 14.
22. ^ Hardy: A mathematicians apology. Cambridge University Press, 1994, p. 13.
23. ^ Hardy: A mathematicians apology. Cambridge University Press, 1994, p. 46.