# G. H. Hardy

**Godfrey Harold Hardy** FRS[1] (7 February 1877 – 1 December 1947)[2] was an English mathematician, known for his achievements in number theory and mathematical analysis.[3][4] In biology, he is known for the Hardy–Weinberg principle, a basic principle of population genetics. In addition to his research, he is remembered for his 1940 essay on the aesthetics of mathematics, titled *A Mathematician's Apology*. Hardy also was the mentor of the Indian mathematician Srinivasa Ramanujan.[5][6]

## Overview

G. H. Hardy is usually known by those outside the field of mathematics for his essay from 1940 on the aesthetics of mathematics, *A Mathematician's Apology*, which is often considered one of the best insights into the mind of a working mathematician written for the layperson.

Starting in 1914, Hardy was the mentor of the Indian mathematician Srinivasa Ramanujan, a relationship that has become celebrated.[7] Hardy almost immediately recognised Ramanujan's extraordinary albeit untutored brilliance, and Hardy and Ramanujan became close collaborators. In an interview by Paul Erdős, when Hardy was asked what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan.[8] He called their collaboration "the one romantic incident in my life."[7][9]

## Early life and career

G. H. Hardy was born on 7 February 1877, in Cranleigh, Surrey, England, into a teaching family.[10] His father was Bursar and Art Master at Cranleigh School; his mother had been a senior mistress at Lincoln Training College for teachers. Both parents were mathematically inclined.

Hardy's own natural affinity for mathematics was perceptible at an early age. When just two years old, he wrote numbers up to millions, and when taken to church he amused himself by factorising the numbers of the hymns.[11]

After schooling at Cranleigh, Hardy was awarded a scholarship to Winchester College for his mathematical work. In 1896 he entered Trinity College, Cambridge.[12] After only two years of preparation under his coach, Robert Alfred Herman, Hardy was fourth in the Mathematics Tripos examination.[13] Years later, he sought to abolish the Tripos system, as he felt that it was becoming more an end in itself than a means to an end. While at university, Hardy joined the Cambridge Apostles, an elite, intellectual secret society.

Hardy cited as his most important influence his independent study of *Cours d'analyse de l'École Polytechnique* by the French mathematician Camille Jordan, through which he became acquainted with the more precise mathematics tradition in continental Europe. In 1900 he passed part II of the Tripos and was duly elected to a college fellowship, a prestigious honor reserved for the top students and one that could last the rest of their lives.[14] In 1903 he earned his M.A., which was the highest academic degree at English universities at that time. From 1906 onward he held the position of a lecturer where teaching six hours per week left him time for research. In 1919 he left Cambridge to take the Savilian Chair of Geometry (and thus become a Fellow of New College[15]) at Oxford in the aftermath of the Bertrand Russell affair during World War I. Hardy spent the academic year 1928–1929 at Princeton in an academic exchange with Oswald Veblen, who spent the year at Oxford.[3] Hardy gave the Josiah Willards Gibbs lecture for 1928.[16][17] Hardy left Oxford and returned to Cambridge in 1931, where he was Sadleirian Professor until 1942.

He was on the governing body of Abingdon School from 1922-1935.[18]

### Work

Hardy is credited with reforming British mathematics by bringing rigour into it, which was previously a characteristic of French, Swiss and German mathematics. British mathematicians had remained largely in the tradition of applied mathematics, in thrall to the reputation of Isaac Newton (see Cambridge Mathematical Tripos). Hardy was more in tune with the *cours d'analyse* methods dominant in France, and aggressively promoted his conception of pure mathematics, in particular against the hydrodynamics that was an important part of Cambridge mathematics.

From 1911 he collaborated with John Edensor Littlewood, in extensive work in mathematical analysis and analytic number theory. This (along with much else) 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 notable conditional results. This was a major 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 among the most successful and famous collaborations in mathematical history. In a 1947 lecture, the Danish mathematician Harald Bohr reported a colleague as saying, "Nowadays, there are only three really great English mathematicians: Hardy, Littlewood, and Hardy–Littlewood."[19]^{:xxvii}

Hardy is also known for formulating the Hardy–Weinberg principle, a basic principle of population genetics, independently from Wilhelm Weinberg in 1908. He played cricket with the geneticist Reginald Punnett who introduced the problem to him, and Hardy thus became the somewhat unwitting founder of a branch of applied mathematics.

Hardy's collected papers have been published in seven volumes by Oxford University Press.[20]

### Pure mathematics

Hardy preferred his work to be considered *pure mathematics*, perhaps because of his detestation of war and the military uses to which mathematics had been applied. He made several statements similar to that in his *Apology*:

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.[21]

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

Moreover, Hardy deliberately pointed out in his *Apology* that mathematicians generally do not "glory in the uselessness of their work," but rather – because science can be used for evil ends as well as good – "mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean."[22]^{:33} Hardy also rejected as a "delusion" the belief that the difference between pure and applied mathematics had anything to do with their utility. Hardy regards as "pure" the kinds 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 "has permanent aesthetic value" and "is eternal because the best of it may, like the best literature, continue to cause intense emotional satisfaction to thousands of people after thousands of years." Although he admitted that what he called "real" mathematics may someday become useful, he asserted that, at the time in which the *Apology* was written, only the "dull and elementary parts" of either pure or applied mathematics could "work for good or ill."[22]^{:39}

### Attitudes and personality

Socially, Hardy was 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 exchange for half an hour every day with as much interest and attention as he did the day's cricket scores, he would have become a rich man.[23]

He was at times politically involved, if not an activist. He took part in the Union of Democratic Control during World War I, and For Intellectual Liberty in the late 1930s.

Hardy was an atheist. Apart from close friendships, he had a few platonic relationships with young men who shared his sensibilities, and often his love of cricket.[23] A mutual interest in cricket led him to befriend the young C. P. Snow.[24][25] Hardy was a lifelong bachelor and in his final years he 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 top of his class in most subjects, and won many prizes and awards but hated having to receive them in front of the entire school. He was uncomfortable being introduced to new people, and could not bear to look at his own reflection in a mirror. It is said that, when staying in hotels, he would cover all the mirrors with towels.[24]

## Hardy's aphorisms

- 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.[26]
- 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*.[22]^{:84} - We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.[22]
^{:43} - Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three,[lower-alpha 1] Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty.[22]
^{:6–7} - Hardy once told Bertrand Russell "If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof".[27]

## Cultural references

Hardy is a key character, played by Jeremy Irons, in the 2015 film *The Man Who Knew Infinity*, based on the biography of Ramanujan with the same title.[28] Hardy is a major character in David Leavitt's fictive biography, *The Indian Clerk* (2007), which depicts his Cambridge years and his relationship with John Edensor Littlewood and Ramanujan.[29] Hardy is a secondary character in *Uncle Petros and Goldbach's Conjecture* (1992), a mathematics novel by Apostolos Doxiadis.[30]

## Bibliography

- Hardy, G. H. (2004) [1940].
*A Mathematician's Apology*. Cambridge: University Press. ISBN 978-0-521-42706-7. PDF The reprinted*Mathematician's Apology*with an introduction by C.P. Snow was recommended by Marcus du Sautoy in the BBC Radio program*A Good Read*in 2007. [31] - Hardy, G. H. (1940)
*Ramanujan*, Cambridge University Press: London (1940). Ams Chelsea Pub. (1999) ISBN 0-8218-2023-0. - Hardy, G. H.; Wright, E. M. (2008) [1938]. Heath-Brown, D. R.; Silverman, J. H.; Andrew Wiles (eds.).
*An Introduction to the Theory of Numbers*(6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921985-8. - Hardy, G. H. (1952) [1908].
*A Course of Pure Mathematics*(10th ed.). Cambridge University Press. ISBN 978-0-521-72055-7. - Hardy, G. H. (1949).
*Divergent Series*. Clarendon Press. xvi+396. ISBN 978-0-8218-2649-2. LCCN 49005496. MR 0030620. OCLC 808787. 2nd Ed. published by Chelsea Pub. (1991) ISBN 0-828-40334-1.[32] - Hardy, G. H.; London Mathematical Society (1966).
*Collected papers of G.H. Hardy; including joint papers with J.E. Littlewood and others*. Oxford: Clarendon Press. ISBN 0-19-853340-3. OCLC 823424. - Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952) [1934].
*Inequalities*(2nd ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-35880-4.

## See also

- Critical line theorem
- Hardy hierarchy
- Hardy notation
- Hardy space
- Hardy–Littlewood inequality
- Hardy–Littlewood maximal function
- Hardy–Littlewood tauberian theorem
- Hardy–Littlewood zeta-function conjectures
*Hardy–Ramanujan Journal*- Hardy–Ramanujan theorem
- Hardy's inequality
- Hardy's theorem
- Pisot–Vijayaraghavan number
- Hardy field

## Notes

- Sic. Ramanujan died aged 32.

## References

- Titchmarsh, E. C. (1949). "Godfrey Harold Hardy. 1877–1947".
*Obituary Notices of Fellows of the Royal Society*.**6**(18): 446–461. doi:10.1098/rsbm.1949.0007. - GRO Register of Deaths: DEC 1947 4a 204 Cambridge – Godfrey H. Hardy, aged 70
- O'Connor, John J.; Robertson, Edmund F., "G. H. Hardy",
*MacTutor History of Mathematics archive*, University of St Andrews. - G. H. Hardy at the Mathematics Genealogy Project
- Kanigel, Robert (1991).
*The Man Who Knew Infinity: a Life of the Genius Ramanujan*. New York: Charles Scribner's Sons. p. 80. ISBN 0-684-19259-4. - Hardy, G. H. (1937). "The Indian mathematician Ramanujan" (PDF).
*The American Mathematical Monthly*.**44**(3): 137–155. doi:10.2307/2301659. JSTOR 2301659. - THE MAN WHO KNEW INFINITY: A Life of the Genius Ramanujan. Retrieved 2 December 2010.
- Alladi, Krishnaswami (19 December 1987), "Ramanujan—An Estimation",
*The Hindu*, Madras, India, ISSN 0971-751X. Cited in Hoffman, Paul (1998),*The Man Who Loved Only Numbers*, Fourth Estate, pp. 82–83, ISBN 1-85702-829-5 - Freudenberger, Nell (16 September 2007). "Lust for Numbers".
*The New York Times*. Retrieved 2 December 2010. - GRO Register of Births: MAR 1877 2a 147 Hambledon – Godfrey Harold Hardy
- Robert Kanigel,
*The Man Who Knew Infinity*, p. 116, Charles Scribner's Sons, New York, 1991. ISBN 0-684-19259-4. - "Hardy, Godfrey Harold (HRDY896GH)".
*A Cambridge Alumni Database*. University of Cambridge. - In the 1898 Tripos competition, R. W. H. T. Hudson was 1st, J. F. Cameron was 2nd, and James Jeans was 3rd. "What became of the Senior Wranglers?" by D. O. Forfar
- Wolfram, Stephen (2016).
*Idea Makers: Personal Perspectives on the Lives & Ideas of Some Notable People*. Wolfram Media, Inc. p. 174. ISBN 978-1-5795-5-003-5. - "G H Hardy's Oxford Years" (PDF). Oxford University Mathematical Institute. Retrieved 16 April 2016.
- Josiah Willard Gibbs Lectures. American Mathematical Society
- Hardy, G. H. (1929). "An introduction to the theory of numbers".
*Bull. Amer. Math. Soc.***35**(6): 778–818. doi:10.1090/s0002-9904-1929-04793-1. MR 1561815. - "School Notes" (PDF). The Abingdonian.
- Bohr, Harald (1952). "Looking Backward".
*Collected Mathematical Works*.**1**. Copenhagen: Dansk Matematisk Forening. xiii–xxxiv. OCLC 3172542. - Hardy, Godfrey Harold (1979).
*Collected Papers of G. H. Hardy – Volume 7*. Oxford: Oxford University Press. ISBN 0-19-853347-0. - Titchmarsh, E.C. (1950). "Godfrey Harold Hardy".
*J. London Math. Soc*.**25**(2): 81–138. doi:10.1112/jlms/s1-25.2.81. - Hardy, G. H.
*A Mathematician's Apology*, 1992 [1940] - Khan, Haider Riaz (18 September 2014). "GH Hardy, the mathematician who loved cricket".
*Cricket Blogs*. ESPNcricinfo. Retrieved 19 September 2014. - C. P. Snow, Foreword, in:
G. H. Hardy,
*A Mathematician's Apology*, Cambridge University Press, 1967, pp 26–27. - C. P. Snow, Variety of Men, Penguin books, 1969, pp 25–56.
- Gaither, Carl C.; Cavazos-Gaither, Alma E. (2012).
*Gaither's Dictionary of Scientific Quotations*. Springer. p. 1645. ISBN 9781461411130. - Quoted in Bertrand Russell,
*Logical and Philosophical Papers, 1909–13*, Routledge, 1992, p. xxix. - George Andrews (February 2016). "Film Review: 'The Man Who Knew Infinity'" (PDF).
*Notices of the American Mathematical Society*. - Taylor, D. J. (26 January 2008). "Adding up to a life. Review of The Indian Clerk by David Leavitt".
*The Guardian*. Retrieved 21 April 2016. - Devlin, Keith (1 April 2000). "Review: Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis". Mathematical Association of America. Retrieved 21 April 2016.
- "A Good Read - Marcus du Sautoy and David Dabydeen - BBC Sounds".
*www.bbc.co.uk*. - Szász, Otto (1950). "Book Review: G. H. Hardy,
*Divergent series*".*Bull. Amer. Math. Soc*.**56**(5): 472–473. doi:10.1090/s0002-9904-1950-09415-4.

## Further reading

- Kanigel, Robert (1991).
*The Man Who Knew Infinity: A Life of the Genius Ramanujan*. New York: Washington Square Press. ISBN 0-671-75061-5. - Snow, C. P. (1967). "Variety of Men". London: Macmillan. Cite journal requires
`|journal=`

(help) - 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.

## External links

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- Works by G. H. Hardy at Project Gutenberg
- Works by or about G. H. Hardy at Internet Archive
- Works by G. H. Hardy at LibriVox (public domain audiobooks)
- O'Connor, John J.; Robertson, Edmund F., "G. H. Hardy",
*MacTutor History of Mathematics archive*, University of St Andrews. - Quotations of G. H. Hardy
- Hardy's work on Number Theory
- Weisstein, Eric Wolfgang (ed.). "Hardy, Godfrey Harold (1877–1947)".
*ScienceWorld*. - I. Grattan-Guinness, "The interest of G.H. Hardy, F.R.S. in the philosophy and the history of mathematics"