# G. H. Hardy

**Godfrey Harold** ("**G. H.**") **Hardy** FRS^{} (7 February 1877 – 1 December 1947)^{} was an English mathematician, known for his achievements in number theory and mathematical analysis.^{}^{}

He 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 layman.

Starting in 1914, he was the mentor of the Indian mathematician Srinivasa Ramanujan, a relationship that has become celebrated.^{}^{} 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^{[]}. He called their collaboration "the one romantic incident in my life."^{}^{}

## Early life and career

G. H. Hardy was born on 7 February 1877, in Cranleigh, Surrey, England, into a teaching family.^{} 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.^{}

After schooling at Cranleigh, Hardy was awarded a scholarship to Winchester College for his mathematical work. In 1896 he entered Trinity College, Cambridge.^{} After only two years of preparation under his coach, Robert Alfred Herman, Hardy was fourth in the Mathematics Tripos examination. 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.

As the most important influence Hardy cites the self-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 awarded a fellowship. 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 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.^{} Hardy gave the Josiah Willards Gibbs lecture for 1928.^{}^{} Hardy left Oxford and returned to Cambridge in 1931, where he was Sadleirian Professor until 1942.

*The Indian Clerk* (2007) is a novel by David Leavitt based on Hardy's life at Cambridge, including his discovery of and relationship with Srinivasa Ramanujan.

### 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 which 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 the 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."^{}^{: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.

His 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 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."*^{}

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 as well as good ends – "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." 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."

### Attitudes and personality

Socially he 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 and befriended the young C. P. Snow who was one also. 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.^{}

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.^{} He 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.^{}

## Hardy's aphorisms

*No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game**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.*^{}*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.**Nothing I have ever done is of the slightest practical use.*- 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". Russell agreed with Hardy wholeheartedly about the delights of proofs, as he himself comments in his
*Autobiography*. *There's no place in this world for ugly mathematics.*Quoted in Robert Wilson's*The Company of Strangers.*

See also: en.Wikiquotes.org

## In popular culture

- Main character in David Leavitt's
*The Indian Clerk*(2007), which depicts his Cambridge years and the relationship with John Edensor Littlewood and Ramanujan. - Secondary character in
*Uncle Petros and Goldbach's Conjecture*(1992), a mathematics novel by Apostolos Doxiadis.

## Bibliography

- Hardy, G. H. (2004) [1940].
*A Mathematician's Apology*. Cambridge: University Press. ISBN 978-0-521-42706-7. - Hardy, G. H. (1940)
*Ramanujan*, Cambridge University Press: London (1940). Ams Chelsea Pub. (25 November 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. Co., 1991. LCCN 91-75377. ISBN 0828403341.^{} - 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

## References

## 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.