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Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
, known for his achievements in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
and
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
. In biology, he is known for the
Hardy–Weinberg principle In population genetics, the Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will remain constant from generation to generation in t ...
, a basic principle of
population genetics Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and po ...
. G. H. Hardy is usually known by those outside the field of mathematics for his 1940 essay '' A Mathematician's Apology'', 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 Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ...
, a relationship that has become celebrated.THE MAN WHO KNEW INFINITY: A Life of the Genius Ramanujan
. Retrieved 2 December 2010.
Hardy almost immediately recognised Ramanujan's extraordinary albeit untutored brilliance, and Hardy and Ramanujan became close collaborators. In an interview by
Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
, when Hardy was asked what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan. In a lecture on Ramanujan, Hardy said that "my association with him is the one romantic incident in my life".


Early life and career

G. H. Hardy was born on 7 February 1877, in
Cranleigh Cranleigh is a village and civil parish, about southeast of Guildford in Surrey, England. It lies on a minor road east of the A281, which links Guildford with Horsham. It is in the north-west corner of the Weald, a large remnant forest, the m ...
, Surrey, England, into a teaching family. His father was
Bursar A bursar (derived from "bursa", Latin for '' purse'') is a professional administrator in a school or university often with a predominantly financial role. In the United States, bursars usually hold office only at the level of higher education ...
and Art Master at Cranleigh School; his mother had been a senior mistress at Lincoln Training College for teachers. Both of his parents were mathematically inclined, though neither had a university education. 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 Cranleigh is a village and civil parish, about southeast of Guildford in Surrey, England. It lies on a minor road east of the A281, which links Guildford with Horsham. It is in the north-west corner of the Weald, a large remnant forest, the m ...
, Hardy was awarded a scholarship to
Winchester College Winchester College is a public school (fee-charging independent day and boarding school) in Winchester, Hampshire, England. It was founded by William of Wykeham in 1382 and has existed in its present location ever since. It is the oldest of ...
for his mathematical work. In 1896, he entered
Trinity College, Cambridge Trinity College is a constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge or Oxford. ...
. 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 The Cambridge Apostles (also known as '' Conversazione Society'') is an intellectual society at the University of Cambridge founded in 1820 by George Tomlinson, a Cambridge student who became the first Bishop of Gibraltar.W. C. Lubenow, ''The ...
, 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 in the same year he was elected to a Prize Fellowship at Trinity College. In 1903 he earned his M.A., which was the highest academic degree at English universities at that time. When his Prize Fellowship expired in 1906 he was appointed to the Trinity staff as a lecturer in mathematics, 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) at
Oxford Oxford () is a city in England. It is the county town and only city of Oxfordshire. In 2020, its population was estimated at 151,584. It is north-west of London, south-east of Birmingham and north-east of Bristol. The city is home to the ...
in the aftermath of the Bertrand Russell affair during
World War I World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was List of wars and anthropogenic disasters by death toll, one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, ...
. Hardy spent the academic year 1928–1929 at
Princeton Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the nin ...
in an academic exchange with
Oswald Veblen Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this wa ...
, who spent the year at Oxford. Hardy gave the Josiah Willards Gibbs lecture for 1928. Hardy left Oxford and returned to Cambridge in 1931, becoming again a fellow of Trinity College and holding the Sadleirian Professorship until 1942. He was on the governing body of
Abingdon School Abingdon School is a day and boarding independent school for boys in Abingdon-on-Thames, Oxfordshire, England. The twentieth oldest independent British school, it celebrated its 750th anniversary in 2006. The school was described as "highly ...
from 1922 to 1935.


Work

Hardy is credited with reforming British mathematics by bringing
rigour Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as ma ...
into it, which was previously a characteristic of French, Swiss and German mathematics. British mathematicians had remained largely in the tradition of
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
, in thrall to the reputation of
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
(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 Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
, in particular against the
hydrodynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
that was an important part of Cambridge mathematics. From 1911, he collaborated with John Edensor Littlewood, in extensive work in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
and
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...
. 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 A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
theory, they proved results and some notable conditional results. This was a major factor in the development of number theory as a system of
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...
s; 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." Hardy is also known for formulating the
Hardy–Weinberg principle In population genetics, the Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will remain constant from generation to generation in t ...
, a basic principle of
population genetics Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and po ...
, independently from
Wilhelm Weinberg Wilhelm Weinberg (Stuttgart, 25 December 1862 – 27 November 1937, Tübingen) was a German obstetrician-gynecologist, practicing in Stuttgart, who in a 1908 paper, published in German in ''Jahresheft des Vereins für vaterländische Naturkund ...
in 1908. He played
cricket Cricket is a bat-and-ball game played between two teams of eleven players on a field at the centre of which is a pitch with a wicket at each end, each comprising two bails balanced on three stumps. The batting side scores runs by st ...
with the geneticist
Reginald Punnett Reginald Crundall Punnett FRS (; 20 June 1875 – 3 January 1967) was a British geneticist who co-founded, with William Bateson, the ''Journal of Genetics'' in 1910. Punnett is probably best remembered today as the creator of the Punnett ...
, who introduced 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 realised how important the result became. Hardy's collected papers have been published in seven volumes by
Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print book ...
.


Pure mathematics

Hardy preferred his work to be considered ''
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
'', 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'': However, aside from formulating the
Hardy–Weinberg principle In population genetics, the Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will remain constant from generation to generation in t ...
in
population genetics Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and po ...
, 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 Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
) 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." 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 Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell, mathematician and physicist * Justice Maxwell (disambiguation) * Maxwell baronets, in the Baronetage of ...
and
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
, 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, Hardy was associated with the
Bloomsbury group The Bloomsbury Group—or Bloomsbury Set—was a group of associated English writers, intellectuals, philosophers and artists in the first half of the 20th century, including Virginia Woolf, John Maynard Keynes, E. M. Forster and Lytton St ...
and the
Cambridge Apostles The Cambridge Apostles (also known as '' Conversazione Society'') is an intellectual society at the University of Cambridge founded in 1820 by George Tomlinson, a Cambridge student who became the first Bishop of Gibraltar.W. C. Lubenow, ''The ...
; G. E. Moore,
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
and
J. M. Keynes John Maynard Keynes, 1st Baron Keynes, ( ; 5 June 1883 – 21 April 1946), was an English economist whose ideas fundamentally changed the theory and practice of macroeconomics and the economic policies of governments. Originally trained in m ...
were friends. He was an avid cricket fan. Maynard Keynes observed that if Hardy had read the
stock exchange A stock exchange, securities exchange, or bourse is an exchange where stockbrokers and traders can buy and sell securities, such as shares of stock, bonds and other financial instruments. Stock exchanges may also provide facilities for t ...
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 The Union of Democratic Control was a British advocacy group, pressure group formed in 1914 to press for a more responsive foreign policy. While not a pacifism, pacifist organisation, it was opposed to military influence in government. World War ...
during World War I, and For Intellectual Liberty in the late 1930s. Apart from close friendships, he had a few platonic relationships with young men who shared his sensibilities, and often his love of cricket. A mutual interest in cricket led him to befriend the young C. P. Snow. 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.


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. * 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''.Hardy, G. H. ''A Mathematician's Apology'', 1992
940 Year 940 ( CMXL) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. Events By place Europe * The tribe of the Polans begins the construction of the following fortified settlements (Gi ...
/ref> * We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not. * Galois died at twenty-one,
Abel Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...
at twenty-seven, Ramanujan at thirty-three, 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. * Hardy once told
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
"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". * A chess problem is genuine mathematics, but it is in some way 'trivial' mathematics. However ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are ''unimportant''. The best mathematics is ''serious'' as well as beautiful - 'important.'


Cultural references

Hardy is a key character, played by
Jeremy Irons Jeremy John Irons (; born 19 September 1948) is an English actor and activist. After receiving classical training at the Bristol Old Vic Theatre School, Irons began his acting career on stage in 1969 and has appeared in many West End theatre ...
, in the 2015 film ''
The Man Who Knew Infinity ''The Man Who Knew Infinity'' is a 2015 British biographical drama film about the Indian mathematician Srinivasa Ramanujan, based on the 1991 book of the same name by Robert Kanigel. The film stars Dev Patel as Srinivasa Ramanujan, a real-life ...
'', based on the biography of Ramanujan with the same title. Hardy is a major character in
David Leavitt David Leavitt (; born June 23, 1961) is an American novelist, short story writer, and biographer. Biography Leavitt was born in Pittsburgh, Pennsylvania to Harold and Gloria Leavitt. Harold was a professor who taught at Stanford University and G ...
's fictive biography, '' The Indian Clerk'' (2007), which depicts his Cambridge years and his relationship with John Edensor Littlewood and Ramanujan. Hardy is a secondary character in '' Uncle Petros and Goldbach's Conjecture'' (1992), a mathematics novel by
Apostolos Doxiadis Apostolos K. Doxiadis ( el, Απόστολος Κ. Δοξιάδης; born 1953) is a Greek writer. He is best known for his international bestsellers '' Uncle Petros and Goldbach's Conjecture'' (2000) and ''Logicomix'' (2009). Early life Doxiad ...
. Hardy is also a character in the 2014 Indian film, '' Ramanujan'', played by Kevin McGowan.


Bibliography

*
Full text
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. * * * *
Full text
*
Vol.1Vol.3Vol.6Vol.7
* *


See also

* Critical line theorem * Campbell–Hardy theorem *
Hardy hierarchy In computability theory, computational complexity theory and proof theory, the Hardy hierarchy, named after G. H. Hardy, is a hierarchy of sets of numerical functions generated from an ordinal-indexed family of functions ''h''α: N →&nbs ...
* Hardy notation *
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . I ...
* Hardy–Hille formula * Hardy–Littlewood definition * Hardy–Littlewood inequality *
Hardy–Littlewood maximal function In mathematics, the Hardy–Littlewood maximal operator ''M'' is a significant non-linear operator used in real analysis and harmonic analysis. Definition The operator takes a locally integrable function ''f'' : R''d'' → C and returns another ...
* Hardy–Littlewood tauberian theorem * Hardy–Littlewood zeta-function conjectures * '' Hardy–Ramanujan Journal'' *
Hardy–Ramanujan number 1729 is the natural number following 1728 and preceding 1730. It is a taxicab number, and is variously known as Ramanujan's number and the Ramanujan-Hardy number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian ...
* Hardy–Ramanujan theorem * Hardy's inequality * Hardy's theorem * Hardy field * Hardy Z function * Pisot–Vijayaraghavan number * Ulam spiral


Notes


References


Further reading

* * Reprinted as *


External links

* * * *
Quotations of G. H. HardyHardy's work on Number Theory
* {{DEFAULTSORT:Hardy, G. H. 1877 births 1947 deaths Mathematical analysts Number theorists Population geneticists 19th-century English mathematicians 20th-century English mathematicians Savilian Professors of Geometry Fellows of the Royal Society Members of the French Academy of Sciences Foreign associates of the National Academy of Sciences Fellows of Trinity College, Cambridge Alumni of Trinity College, Cambridge Cambridge University Moral Sciences Club English atheists People educated at Cranleigh School People educated at Winchester College Royal Medal winners Recipients of the Copley Medal People from Cranleigh Fellows of New College, Oxford De Morgan Medallists Mathematics writers Governors of Abingdon School British textbook writers Sadleirian Professors of Pure Mathematics