G. H. (Godfrey
Harold) Hardy FRS (February 7, 1877 Cranleigh, Surrey, England – December
1, 1947 Cambridge, Cambridgeshire,
England ) was a prominent English mathematician, known for his achievements in
number theory and mathematical analysis.
Non-mathematicians usually know him for
A Mathematician's Apology,
his
essay from 1940 on the
aesthetics of mathematics. The
Apology is often considered one of the best insights into
the mind of a working mathematician written for the
layman.
His relationship as mentor, from 1914 onwards, of the Indian
mathematician
Srinivasa
Ramanujan 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 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."
Life
G.H.
Hardy
was born 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 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 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 he was
fourth in the
Mathematics
Tripos 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 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. He 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
J.
E. Littlewood, in extensive work in
mathematical analysis and
analytic number theory. 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
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."
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."[13100]
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.
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 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 "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".
Books
- Hardy, G. H. (1940) Ramanujan, Cambridge University Press: London
(1940). Ams Chelsea Pub. (November 25, 1999) ISBN
0-8218-2023-0.
See also
Notes
References
External links