G. H. Hardy

English mathematician

• Born: February 7, 1877; Cranleigh, England
• Died: December 1, 1947; Cambridge, England

English mathematician G. H. Hardy made advances in number theory, mathematical analysis, and statistical analysis, though he focused primarily on pure mathematics. Hardy also wrote mathematics textbooks and is sometimes credited as the founder of population genetics analysis.

Also known as: Godfrey Harold Hardy

Primary field: Mathematics

Specialties: Mathematical analysis; set theory; decision theory

Early Life

Godfrey Harold Hardy was born in Cranleigh, a village in England’s southeastern Surrey County. His parents, Isaac and Sophia Hardy, were both employed in the educational system. Harvey attended local schools in Cranleigh, where he excelled in all subjects and performed well enough in mathematics to receive a scholarship to Winchester College in 1889, then considered the top school in England for mathematics education.

Hardy graduated from Winchester College at the top of his class and received an open scholarship to continue his education. In 1896, he began studying at Cambridge University’s Trinity College, where he first developed an interest in theoretical mathematics research. In his personal memoirs, Hardy largely attributes his growing interest in mathematics to the mentorship of mathematics professor A. E. H. Love. Love introduced Hardy to the work of French mathematicians of the previous generations, and it was in these works that Hardy first discovered his interest in theoretical mathematics.

Hardy was elected a fellow at Trinity College in 1900 and earned his master’s degree, then the highest academic degree offered in Britain, in 1903. He remained at the university as a lecturer in the mathematics department and authored a number of papers describing integrals and convergence series.

Life’s Work

Though Hardy published a variety of research papers during his early years of research, his first major contribution to mathematics came with the publication of his textbook A Course of Pure Mathematics (1908), which was the first undergraduate-level text to cover number theory and pure mathematics research. Hardy’s text set a new standard and direction for undergraduate mathematics education, and some historians credit this text with helping to inspire an overall change in mathematics education in Britain.

Hardy began what would become a long-term collaboration with British mathematician J. E. Littlewood in 1911. Hardy and Littlewood worked together for more than thirty years, becoming close friends. By the end of their careers, the two were considered to be the leading mathematicians in Britain, largely due to their joint research. Hardy and Littlewood worked primarily on number theory and produced two famous conjectures, a conjecture being a scientific proposition largely believed to be true because of sufficient evidence but not yet proved or disproved. The Hardy-Littlewood conjectures focus on the distribution of prime numbers in complex functions, based on the work of German mathematician Bernhard Riemann.

In 1913, Hardy received a letter from Indian mathematician Srinivasa Aiyangar Ramanujan, a largely untrained mathematician who had been working on elliptical functions and other complex equations in relative isolation. Ramanujan wrote to several prominent Western mathematicians in an effort to collaborate and discuss his various theorems and conjectures. Hardy recognized the strength of his work and convinced Ramanujan to travel to Britain for a visiting scholarship in 1914. Ramanujan spent several years working in Britain, with Hardy acting as his tutor and helping him to better disseminate his work.

Before Ramanujan’s early death in 1920, he produced thousands of notable mathematical calculations, many of which would not be fully investigated until after his death. Hardy and Ramanujan collaborated to produce the Hardy-Ramanujan asymptotic formula, which was later used in calculations involving quantum mechanics, though the formula was not developed with any specific applications in mind. Hardy said later in biographical accounts that he considered his mentoring of Ramanujan to be his most important contribution to mathematics. After his work on asymptotics with Ramanujan, Hardy and Littlewood formulated the Hardy-Littlewood circle method, which became a commonly used technique in number theory to prove the asymptotic—meaning a curve infinitely approaching a straight line (the asymptote)—behavior of a series.

During World War I, Hardy found himself at odds with many prominent English intellectuals, as he was committed to pacifism and did not approve of Britain entering the war. This partially motivated his decision to move from Cambridge to Oxford University in 1919, where he was appointed Savilian Professor of Geometry. At Oxford, Hardy completed more mathematics work in conjunction with Littlewood, who remained at Cambridge. Hardy returned to Cambridge in 1931, where he was appointed Sadleirian Chair of Pure Mathematics, at the time considered the most prominent position in British mathematics.

Although his primary interest lay in pure mathematics and number theory, Hardy also made several well-known contributions to applied mathematics, such as when he derived an equation for the frequency distribution of alleles (variant forms of a gene) in a certain genotype (genetic sequence), a calculation that describes the overall genetic variance within a population. He determined that genotype and allele frequencies are constant within an idealized population and only fluctuate when some outside force, such as small population size or sexual selection, influences the gene pool. Hardy’s calculations of genotype and allele frequency mirrored the independent discovery of the same principle by German physician Wilhelm Weinberg, who delivered a speech on his calculations in 1908, about six months prior to Hardy’s publication of a paper explaining the same concept. This idealized gene pool representation was later named the Hardy-Weinberg principle of population genetics to honor both men’s contributions to the idea.

Though the Hardy-Weinberg principle and Hardy’s work with Ramanujan indicated a wide range of potential applications to be derived from his calculations, Hardy was outspoken regarding his desire to focus on the study of pure mathematics, independent of possible applications of his work. During his career, he became one of the most effective spokesmen for the benefits and the value of number theory and mathematics research, helping to counteract the widespread focus on applied mathematics that dominated early twentieth-century British academia.

In 1940, Hardy wrote an essay entitled A Mathematician’s Apology, in which he attempted to explain his approach to mathematics for a general audience. Throughout the book, Hardy extols the virtues of pure mathematics, comparing mathematicians to artists in the sense that their calculations and methods of thinking must take into account beauty and aesthetics in the same way an artist would. Additionally, Hardy used his essay to justify his belief that mathematics research should be pursued in its own right, outside of any reference to potential or realized applications. In part, Hardy believed that pure mathematics research was in need of defense because the First and Second World Wars saw the international mathematics community moving toward applied, rather than theoretical, research programs. Hardy believed that a focus on research for the pure beauty and mystery of mathematics was a peaceful and harmonious pursuit that enriched human culture, rather than assisting in endangering it.

Impact

During his life, Hardy held the most prominent positions in British mathematics and mentored numerous students who went on to make significant contributions to both pure and applied mathematics. His 1908 book, A Course of Pure Mathematics, was a major step in mathematics education, providing a concise treatment of complex subjects for undergraduate students. The text and many derivative works based on his example became standard tools in the field of mathematics education, leading historians and biographers to credit Hardy with bringing his approach to mathematics research to the British mathematics community.

The various principles, conjectures, and formulas that Hardy worked on became influential in number theory and other areas of mathematics, as well as in quantum mechanics, nuclear physics, and genetics. His calculations regarding the proportions of gene types within populations helped to inspire a new field of biological study called population genetics. Population geneticists have utilized Hardy and Weinberg’s calculations as a theoretical baseline state from which the actual state of a population’s genetic structure can be derived. Population genetics grew from these early beginnings in the 1910s to become a dominant field in biology in the 1970s and 1980s.

Bibliography

Borwein, Peter B. The Riemann Hypothesis: A Resource for the Aficionado and Virtuoso Alike. New York: Springer, 2008. Print. Provides a historical account of the Riemann hypothesis, known as one of the most challenging mathematical problems in history, and describes Hardy and Littlewood’s attempts to address certain issues concerning it.

Fitzgerald, Michael, and Ioan James. The Mind of the Mathematician. Baltimore: Johns Hopkins UP, 2007. Print. Explores the lives and thought processes of great mathematicians, looking for significant commonalities between them. Contains a discussion of Hardy with biographical information.

Hardy, Godfrey Harold. A Mathematician’s Apology. London: Cambridge UP, 2012. Print. A semiautobiographical essay written in 1940, explaining Hardy’s approach to mathematics and his experience with academic research and pure mathematics. Contains detailed descriptions of some of Hardy’s notable cooperative work, as well as his personal feelings about the benefits of pure mathematics research. Foreword includes biographical information.