Godfrey Harold “G. H.” Hardy FRS (7 February 1877 – 1 December 1947) was a prominent English
mathematicianMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, known for his achievements in
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and
mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
.
He is usually known by those outside the field of mathematics for his
essayAn essay is a piece of writing which is often written from an author's personal point of view. Essays can consist of a number of elements, including: literary criticism, political manifestos, learned arguments, observations of daily life, recollections, and reflections of the author. The definition...
from 1940 on the aesthetics of mathematics,
A Mathematician's ApologyA Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician.-Summary:...
, which is often considered one of the best insights into the mind of a working mathematician written for the
laymanA layperson or layman is a person who is not an expert in a given field of knowledge. The term originally meant a member of the laity, i.e. a non-clergymen, but over the centuries shifted in definition....
.
Starting in 1914, he was the mentor of the Indian mathematician
Srinivasa RamanujanSrīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...
, a relationship that has become celebrated. Hardy almost immediately recognized Ramanujan's extraordinary albeit untutored brilliance, and Hardy and Ramanujan became close collaborators. In an interview by
Paul ErdősPaul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
, 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 years
G.H. Hardy was born 7 February 1877, in
CranleighCranleigh is a large village, self-proclaimed the largest in England, and is situated 8 miles south east of Godalming in Surrey. It lies to the east of the A281 which links Guildford with Horsham; neighbouring villages include: Ewhurst, Alfold and Hascombe....
,
SurreySurrey is a county in the South East of England and is one of the Home Counties. The county borders Greater London, Kent, East Sussex, West Sussex, Hampshire and Berkshire. The historic county town is Guildford. Surrey County Council sits at Kingston upon Thames, although this has been part of...
, England, into a teaching family. His father was
BursarA bursar is a senior professional financial administrator in a school or university.Billing of student tuition accounts are the responsibility of the Office of the Bursar. This involves sending bills and making payment plans with the ultimate goal of getting the student accounts paid off...
and Art Master at
Cranleigh SchoolCranleigh School is an independent English boarding school in the village of Cranleigh, Surrey. It was founded in 1865 as a boys' school and started to admit girls in the early 1970s. It is now co-educational. The current headmaster is Guy de W...
; 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 a young age. When just two years old, he wrote numbers up to millions, and when taken to church he amused himself by factorizing the numbers of the hymns.
After schooling at
CranleighCranleigh School is an independent English boarding school in the village of Cranleigh, Surrey. It was founded in 1865 as a boys' school and started to admit girls in the early 1970s. It is now co-educational. The current headmaster is Guy de W...
, Hardy was awarded a scholarship to
Winchester CollegeWinchester College is an independent school for boys in the British public school tradition, situated in Winchester, Hampshire, the former capital of England. It has existed in its present location for over 600 years and claims the longest unbroken history of any school in England...
for his mathematical work. In 1896 he entered
Trinity CollegeTrinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
,
CambridgeThe University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
. After only two years of preparation he was fourth in the
Mathematics TriposThe Mathematical Tripos is the taught mathematics course at the University of Cambridge. It is the oldest Tripos that is examined in Cambridge.-Origin:...
examination. Years later, Hardy 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 ApostlesThe Cambridge Apostles, also known as the Cambridge Conversazione Society, is an intellectual secret society at the University of Cambridge founded in 1820 by George Tomlinson, a Cambridge student who went on to become the first Bishop of Gibraltar....
, 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 JordanMarie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...
, 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 IWorld War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
. He returned to Cambridge in 1931, where he was Sadleirian Professor until 1942.
The Indian ClerkThe Indian Clerk is a novel by David Leavitt, published in 2007. It is inspired by the career of the self-taught mathematical genius Srinivasa Ramanujan, as seen mainly through the eyes of his mentor and collaborator G.H. Hardy, a British mathematics professor at Cambridge University...
(2007) is a novel by
David LeavittDavid Leavitt is an American novelist.-Biography:Born in Pittsburgh, Pennsylvania, Leavitt is a graduate of Yale University. and a professor at the University of Florida...
based on Hardy's life at
CambridgeThe city of Cambridge is a university town and the administrative centre of the county of Cambridgeshire, England. It lies in East Anglia about north of London. Cambridge is at the heart of the high-technology centre known as Silicon Fen – a play on Silicon Valley and the fens surrounding the...
, including his discovery of and relationship with
Srinivasa RamanujanSrīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...
.
Work
Hardy is credited with reforming British mathematics by bringing rigour into it, which was previously a characteristic of French,
SwissSwitzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
and German mathematics. British mathematicians had remained largely in the tradition of
applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
, in thrall to the reputation of
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
(see
Cambridge Mathematical TriposThe Mathematical Tripos is the taught mathematics course at the University of Cambridge. It is the oldest Tripos that is examined in Cambridge.-Origin:...
). Hardy was more in tune with the
cours d'analyse methods dominant in France, and aggressively promoted his conception of
pure mathematicsBroadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
, in particular against the hydrodynamics which was an important part of Cambridge mathematics.
From 1911 he collaborated with J. E. Littlewood, in extensive work in
mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
and
analytic number theoryIn 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 Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
. This (along with much else) led to quantitative progress on the Waring problem, as part of the Hardy–Littlewood circle method, as it became known. In
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
theory, they proved results and some notable conditional results. This was a major factor in the development of number theory as a system of
conjectureA conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
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 BohrHarald August Bohr was a Danish mathematician and football player. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the Nobel Prize-winning physicist Niels 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, a basic principle of
population geneticsPopulation genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...
, independently from
Wilhelm WeinbergDr Wilhelm Weinberg was a German half-Jewish physician and obstetrician-gynecologist, practicing in Stuttgart, who in a 1908 paper Dr Wilhelm Weinberg (Stuttgart, December 25, 1862 – Tübingen, November 27, 1937) was a German half-Jewish physician and obstetrician-gynecologist, practicing in...
in 1908. He played
cricketCricket is a bat-and-ball game played between two teams of 11 players on an oval-shaped field, at the centre of which is a rectangular 22-yard long pitch. One team bats, trying to score as many runs as possible while the other team bowls and fields, trying to dismiss the batsmen and thus limit the...
with the geneticist
Reginald PunnettProfessor Reginald Crundall Punnett FRS 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 square, a tool still used by biologists to predict the probability of possible genotypes...
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 mathematicsBroadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
, perhaps because of his detestation of war and the military uses to which mathematics had been
appliedApplied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
. He made several statements similar to that in his
ApologyA Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician.-Summary:...
:
- "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."http://www.numbertheory.org/obituaries/LMS/hardy/page83.html
However, aside from formulating the Hardy–Weinberg principle in
population geneticsPopulation genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...
, 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-EinsteinIn statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...
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
MaxwellJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
and
EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
, 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 groupThe Bloomsbury Group or Bloomsbury Set was a group of writers, intellectuals, philosophers and artists who held informal discussions in Bloomsbury throughout the 20th century. This English collective of friends and relatives lived, worked or studied near Bloomsbury in London during the first half...
and the
Cambridge ApostlesThe Cambridge Apostles, also known as the Cambridge Conversazione Society, is an intellectual secret society at the University of Cambridge founded in 1820 by George Tomlinson, a Cambridge student who went on to become the first Bishop of Gibraltar....
; G. E. Moore,
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
and J. M. Keynes were friends. He was an avid cricket fan and befriended the young
C. P. SnowCharles Percy Snow, Baron Snow of the City of Leicester CBE was an English physicist and novelist who also served in several important positions with the UK government...
who was one also.
He was at times politically involved, if not an activist. He took part in the
Union of Democratic ControlThe Union of Democratic Control was a British pressure group formed in 1914 to press for a more responsive foreign policy. While not a pacifist organization, it was opposed to military influence in government.-World War I:...
during World War I, and For Intellectual Liberty in the late 1930s.
Hardy was an
atheistAtheism is, in a broad sense, the rejection of belief in the existence of deities. In a narrower sense, atheism is specifically the position that there are no deities...
. Apart from close friendships, he had a few platonic relationships with young men who shared his sensibilities. He was a life-long 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.
In his
obituary, a former student reports:
"He was an extremely kind-hearted man, who could not bear any of his pupils to fail in their researches." — E. C. Titchmarsh (1950)
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.
- Nothing I have ever done is of the slightest practical use.
- Hardy once told Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
"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.
See also
- Critical line theorem
- Hardy notation
- Hardy space
In complex analysis, the Hardy spaces 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...
- Hardy–Littlewood tauberian theorem
In mathematical analysis, the Hardy–Littlewood tauberian theorem is a tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation...
- Hardy–Littlewood zeta-function conjectures
In mathematics, the Hardy–Littlewood zeta-function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density of zeros of the Riemann zeta function....
- Hardy's inequality
- Hardy's theorem
In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions.Let f be a holomorphic function on the open ball centered at zero and radius R in the complex plane, and assume that f is not a constant function...
- Pisot–Vijayaraghavan number
External links