Hardy, Godfrey Harold

(1877-1947), mathematician

by Béla Bollobás

© Oxford University Press 2004 All rights reserved

Hardy, Godfrey Harold (1877-1947), mathematician, was born on 7 February 1877, at Cranleigh, Surrey, the elder of two children of Isaac Hardy (1842-1901), a master at Cranleigh School, and his wife, Sophia Hall (b. 1846). His mathematical talent was evident early: by the time he was two he could write down numbers up to millions, and later in church he amused himself by factorizing the numbers of the hymns. Throughout his life he frequently played about with numbers of taxicabs, railway carriages, and the like. In 1890, after Cranleigh School, he went to Winchester College, where he was never taught mathematics in a class: Dr George Richardson, head of College, always coached him privately. Hardy strongly disliked public-school life; Winchester at that time was a rough place and not to his taste. Although he was grateful to the school for the excellent education it gave him, once having left it he never again visited it.

Fellowship at Trinity
As a Wykehamist Hardy was expected to go to New College, Oxford. However, when he was about fifteen, he read a highly coloured novel of Cambridge life and set his heart on becoming, like its hero, a fellow of Trinity. He went up to Trinity College, Cambridge, as an entrance scholar in 1896, his tutor being Dr A. W. Verral. Even in Trinity, he thought of mathematics as an essentially 'competitive' subject. His eyes were opened by A. E. H. Love, who not only gave him his first serious conception of analysis, but also introduced him to Camille Jordan's Cours d'analyse de l'école polytechnique. Hardy was fourth wrangler in 1898 and in 1900 he achieved his childhood ambition when, on the basis of a dissertation, he was elected into a prize fellowship at Trinity. Hardy and J. H. Jeans, in that order, were awarded Smith's prizes in 1901.

Having secured a fellowship in Trinity, Hardy could spend all his time on mathematical research and in 1900 he published the first of his more than 350 research papers. In 1906 when his fellowship was to expire, he was made a college lecturer in mathematics, a position he held until 1919. His success in research was soon recognized: he became a fellow of the Royal Society in 1910, and in 1914 the University of Cambridge gave him the honorary title of Cayley lecturer. His book A Course in Pure Mathematics (1908; 9th edn, 1948) determined the outlook of English analysts for the next fifty years.

Although Hardy's earlier work had been good enough to earn his election as FRS his research really took off the following year, in 1911, when he began a collaboration with J. E. Littlewood (1885-1977), a colleague in Trinity. The Hardy-Littlewood partnership is the most famous in the history of mathematics: together they wrote a hundred papers, the first published in 1912 and the last in 1948, after Hardy's death.

Hardy and Littlewood made fundamental contributions to many branches of number theory and analysis: the summation of divergent series; the theory of Fourier series; the theory of the Riemann zeta-function and the distribution of prime numbers; the solution of Waring's problem; and the theory of Diophantine approximation. Their series of eight papers, Partitio Numerorum, on Waring's problem and the Goldbach conjecture, was especially influential. Later they worked on inequalities: with George Pólya they wrote Inequalities (1934), which became an instant best-seller, and a set of twenty-four papers on the theory of series.

In 1914 Hardy embarked on another successful collaboration, this time with the Indian genius Srinivasa Ramanujan, whom Hardy considered, rightly, to be his own 'discovery'. Ramanujan had no formal university education, and worked unaided in India until he was twenty-five. In 1913 he sent some of his results in a letter to Hardy in Trinity College, and Hardy and Littlewood soon decided that he was a mathematician of the highest calibre. Hardy arranged a scholarship for Ramanujan, who arrived in Cambridge in April 1914. Hardy and Ramanujan wrote five papers together, but the collaboration was cut short by Ramanujan's return to Madras in 1919, and his early death in April 1920.

In his political views and in his interest in mathematical philosophy, Hardy was a disciple of Bertrand Russell. Before 1914 Russell, Hardy, and Littlewood spent several summer holidays together, discussing mathematics, physics, philosophy, and politics. Like Russell, Hardy was strongly opposed to the First World War, although he did not go to the lengths which resulted in Russell's gaol sentence. He tried to prevent the rift between Trinity College and Russell, and was instrumental in the later reconciliation. He described the affair in a little book entitled Bertrand Russell and Trinity, which he had printed for private circulation in 1942 (reissued by Cambridge University in 1970).

Life at Oxford
The tragedy of the war and the fight in Trinity over the Russell affair depressed Hardy, and he was relieved when, in 1919, he was elected to the Savilian chair of geometry at Oxford, and could migrate to New College. This was the happiest period of his life: his collaboration with Littlewood rose to new heights. The Oxford-Cambridge distance caused no difficulty since most of the collaboration was done by mail. According to their unwritten 'axioms', it was completely immaterial whether what they wrote was right or wrong; the recipient of a letter was under no obligation to open it, let alone read it and work on it; it was preferable that they should not both think about the same detail; and it was quite irrelevant if one of them made no contribution to the contents of a joint paper.

The informality and friendliness of New College suited Hardy--he was more at home in Oxford than he had ever been in Cambridge. He lectured on numerous subjects, including geometry (to fulfil the conditions of his chair) and mathematics for philosophers. He was an admirable research supervisor, always willing to help and always full of ideas, and many an Oxford DPhil dissertation owes much to him. He made many visits to American universities, and loved the country, but the only time he spent an extended period abroad was in 1928-9, when he visited Princeton and the California Institute of Technology.

Hardy liked all forms of ball games, and was obsessed with cricket: he would not miss a test match at Lord's and spent some of his happiest hours in Cambridge at Fenners, watching university cricket. He was an excellent batsman and, had he had any coaching at Winchester, could have come close to being first-class. His highest term of praise of a human achievement was 'in the Bradman class': thus Archimedes, Newton, and Gauss were in the Bradman class. Even in his sixties Hardy was a good real tennis player, and excelled at bowls.

Sadleirian professorship
In 1931 Hardy succeeded E. W. Hobson in the Sadleirian chair at Cambridge, and became again a fellow of Trinity. As the most influential British mathematician Hardy secured jobs for many young people, and from 1933 he was deeply concerned with the fate of his fellow mathematicians on the continent, and directed attempts to find places for those whom persecution had driven out. He also took up his pen to ridicule the view that there is a strong link between mathematical creative style and race. The highlight of Hardy's later years was the Hardy-Littlewood 'conversation class', in which mathematicians of all ages lectured on a great variety of topics.

Throughout his life Hardy took a great interest in education. He was an implacable enemy of the mathematical tripos at Cambridge which he blamed for the dearth of great English mathematicians and so wished to abolish. He worked much for the success of the London Mathematical Society; he was president twice and left the royalties from his books to the society. He held honorary degrees from many universities, and was an associé étranger of the Paris Académie des Sciences. The Royal Society awarded him a royal medal in 1920 and its Sylvester medal in 1940; the Copley medal, its highest award, was to be presented to him on the day he died.

As an editor, Hardy was 'in the Bradman class'. It was largely due to him that the Quarterly Journal was started in Oxford, and as an editor of the Cambridge Tracts in Mathematics and Mathematical Physics from 1914 to 1946, he was instrumental in establishing it as a major series. He contributed four volumes to the Tracts: Integration of Functions of a Single Variable (no. 2, 1905); Orders of Infinity (no. 12, 1910); The General Theory of Dirichlet's Series (with M. Riesz, no. 18, 1915); and Fourier Series (with W. W. Rogosinski, no. 38, 1944). His last textbook, Divergent Series, was published posthumously, in 1949.

From the second half of the thirties Hardy's mathematical output greatly diminished, due partly to the onset of Littlewood's depression, and this reinforced his belief that mathematics was a young person's game. Nevertheless, he wrote two masterpieces: An Introduction to the Theory of Numbers (1938), with E. M. Wright, and A Mathematician's Apology (1940), the most poetic writing about being a mathematician.

Character and philosophy
Writing at the outbreak of the Second World War, in his Apology Hardy was preoccupied with war and the role of the sciences: 'There is one comforting conclusion which is easy for a real mathematician. Real mathematics has no effect on war' (Hardy, Apology, 140). On a personal note, he claimed, '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' (ibid., 150). Later there came at least two major exceptions to this claim: in 1908, in a letter to Science entitled 'Mendelian proportions in a mixed population', he made an important contribution to population genetics, which is now in every textbook as the Hardy-Weinberg law. Another example is modern 'control theory', in which a number of problems are best formulated in terms of the Hardy space, H∞, of the right half-plane and require, both theoretically and computationally, a fair amount of deep theory, whose origin is in Hardy's work.

Hardy had many peculiarities. He strongly disliked mechanical gadgets, in particular the telephone, and never used a watch or a fountain pen. He was unashamedly heliotropic and loathed the English climate. He loved conversation, odd little word games, walking, and gentle climbing; he liked cats and was meticulously orderly in everything but dress. He delighted in good literature but did not appreciate music. He hated war, cruelty of all kinds, dogs, hot roast mutton (memories of Winchester!), politicians, and any kind of sham. He preferred the downtrodden and those handicapped by race to the people he called 'large bottomed'. He was indifferent to noise, even when doing creative work, and his best time of work was the long vacation, with tennis or cricket in the early afternoon. Hardy was violently opposed to religion, although he had many clerical friends: he considered God his personal enemy, and went to great lengths to foil him.

Until he was about thirty, Hardy looked incredibly young; later everyone thought him striking. According to Russell, he had the bright eyes that only very clever men have. Nevertheless, he hated mirrors: the first thing he did on entering a hotel room was to cover up the mirror. He disliked having his photograph taken--there are hardly a dozen snapshots of him. Hardy had no women friends: according to Littlewood, he was a 'non-practicing homosexual' (private information) and P. A. M. Dirac found him 'uncomfortably unusual'. He was shy throughout his life, but in his remarks he could be merciless. He was wont to pass people in the street without any sign of recognition. The Second World War and the deterioration of his creative powers and health drove Hardy into depression. After a suicide attempt and a long illness, he died at the Evelyn Nursing Home, Cambridge, on 1 December 1947, with his devoted sister, Gertrude, at his bedside.

Mathematics books age notoriously quickly; nevertheless, three of Hardy's books are not only classics, but are constantly reprinted and remain best-sellers more than fifty years after their first appearances. Hardy described himself as a problem solver rather than a theory builder, but he had a profound influence on modern mathematics and ranks as one of the greatest English mathematicians of the twentieth century. Together with Littlewood, he brought pure mathematics in England to the highest level, and was instrumental in improving the teaching of mathematics throughout the world. He was fiercely proud of pure mathematics; as he wrote in A Mathematician's Apology:

If intellectual curiosity, professional pride, and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of gratifying them than a mathematician ... as history proves abundantly, mathematical achievement, whatever its intrinsic value, is the most enduring of all. (Hardy, Apology, 80)


E. C. Titchmarsh, Obits. FRS, 6 (1948-9), 47-61
E. C. Titchmarsh, Journal of the London Mathematical Society, 25 (1950), 81-101
G. H. Hardy, A mathematician's apology (1940)
B. Bollobás, ed., Littlewood's miscellany (1986)
Trinity Cam., Hardy MSS
private information (2004)
Collected papers of G. H. Hardy: including joint papers with J. E. Littlewood and others, ed. London Mathematical Society, 7 vols. (1966-79)
J. Todd, 'G. H. Hardy as an editor', Mathematical Intelligencer, 16 (1994), 32-7
I. Grattan-Guinness, 'Russell and G. H. Hardy: a study of their relationship', Russell, n. s., 11 (1991-2), 165-79
G. H. Hardy, 'The case against the mathematical tripos', Mathematical Gazette, 13 (1926-7), 61-71
G. H. Hardy, 'The J-Type and the S-type among mathematicians', Nature, 134 (1934), 250
d. cert.

Bodl. Oxf., corresp. with British Association mathematical tables committee
CUL, asymptotic formulae and papers on Waring's problem
Trinity Cam., notebooks and papers |  BL, corresp. with Albert Mansbridge, Add. MS 65258
CUL, letters to G. E. Moore
L. Cong., O. Veblen MSS
McMaster University, Hamilton, Ontario, corresp. with Bertrand Russell
New College, Oxford, letters to E. C. Titchmarsh
Trinity Cam., corresp. with Harold Davenport
Trinity Cam., corresp. with A. E. Ingham

G. Bollobás, bronze bust, 1989, priv. coll.
photograph, repro. in D. C. Russell, Bulletin of the London Mathematical Society, 18 (1986), 403-20
photograph, repro. in D. J. Albers, G. L. Alexanderson, and C. Reid, eds., More mathematical people (1990)
photograph, repro. in Obits. FRS, 6 (1948-9) [see illus.]
six photographs, repro. in London Mathematical Society, ed., Collected papers
two photographs, repro. in Todd, 'G. H. Hardy'

Wealth at death  
£18,187 11s. 10d.: probate, 12 Feb 1948, CGPLA Eng. & Wales

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