A Mathematician's Apology

by G. H. Hardy
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Historical Context

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Last Updated on May 6, 2015, by eNotes Editorial. Word Count: 726

Since Hardy elucidated a philosophy that stresses the timelessness and immortality of pure mathematics, it is very difficult to contextualize A Mathematician’s Apology with respect to a single historical period. Viewed as an autobiographical memoir, A Mathematician’s Apology is a product of a genius who came of age towards the end of the Victorian era and who died as the world entered the nuclear age. However, viewed as a philosophical treatise and justification of mathematics, the book begins with the ancient Greeks and extends to the eve of World War II.

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Hardy was a product of the English educational system that retained intellectually mediocre clergymen as the main instructors until well into the nineteenth century. Although this system had been largely reformed by the turn of the century, mathematics was one of the last disciplines to be affected. Cambridge mathematician Norbert Wiener described the level of mathematics at Oxford as ‘‘contemptible.’’ The training available at Cambridge was not much better, consisting of a severe exam system, the triposes, which relied on rote memorization rather than any degree of unique creativity; it was not a system that inspired a mathematical genius. The insufficiency of the English system meant that the English lagged behind other European countries in producing mathematicians and modern mathematical theories.

In his memoir, Hardy emphasizes that he became a mathematician in spite of this early training. Much of A Mathematician’s Apology can be read as a subtle jab at the stifling environment of the English educational system. As Hardy’s generation of mathematicians gained international recognition, they quickly instituted changes that treated mathematics more as a creative art than as an endless series of exercises in rote learning. Hardy himself, for instance, was instrumental in opposing the continuation of the rigid tripos exam system.

During the early part of the twentieth century, Britain was still very much an empire with territories spanning the globe, including India. The great Indian mathematician Srinivasa Ramanujan learned English as a result of the English colonial system. It was as a result of Britain’s close relationship with India that Ramanujan and Hardy began a correspondence which was to result in one of the great collaborations in mathematics history. Hardy was able to procure a position for Ramanujan at Trinity College, Cambridge, which allowed the collaboration to flourish.

But Hardy’s other collaborations at the time did not fare so well. By the time World War I broke out, Hardy was in his prime and had already begun working with several other mathematicians outside of England, who would ultimately have a lasting effect on both his own career and on mathematics as a whole. But with the outbreak of the war and the virulent nationalism that accompanied it, international collaboration proved exceedingly difficult, if not outright impossible.

Hardy became well known throughout his life for his outspoken views outside of the field of mathematics. The narrow-mindedness of many scientists within England, for instance, greatly concerned Hardy, and he let that be known. Unlike most intellects of his day, Hardy had a great reverence for the German mathematical school and was greatly distressed by the anti-German sentiment that proliferated throughout England and particularly at Cambridge. His ability to separate German intellectual achievement from the exaggerated ‘‘inhuman’’ traits of the enemy which were spoken of throughout England made him somewhat of a pariah figure in this regard. He even went so far as to carry on an extensive correspondence with Swedish mathematician Gösta Mittag-Leffler, in which the two worked towards a reconciliation between German and Allied mathematicians with the war still raging.

Hardy wrote A Mathematician’s Apology under the threat of another world war. Although he could not ignore the threat of that war, it is almost as if he includes the relationship between war and mathematics as an afterthought. He admits this in a brief note that follows the last chapter. Viewed with the hindsight of today, his views concerning the improbability that a theory like relativity would have an effect on war in Hardy’s lifetime appear to be grossly miscalculated and anachronistic. These views are the last gasp of an age of innocence and naiveté, ignoring or not fully recognizing the devastating effects that science—even the science of pure mathematics which Hardy considered to be ‘‘gentle and clean’’—could have on humanity.

Literary Style

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A Mathematician’s Apology is, as the title implies, written in the form of an ‘‘apology,’’ or defense. In this case, the author sets out to defend his chosen career: namely, theoretical, or what he calls ‘‘pure,’’ mathematics. Although he was generally accepted for his brilliant theoretical insights, which resulted in many remarkable works and collaborations, Hardy’s view that theoretical mathematics is an art form, while its counterpart, applied mathematics, is at best an application of trivial exercises, caused great disagreement among his contemporaries and thus spurred the need for this defense.

With this book, Hardy set out to address a general audience of both mathematicians and nonmathematicians alike, and as a result he employs a narrative style that could best explain in simple terms his profound and complex array of ideas. To that end, his tone, while often conveying a derogatory and elitist attitude toward his subject matter, never condescends to the reader with lofty diction; anyone with a rudimentary knowledge of mathematics would feel at home and comfortable with Hardy’s style. At the same time, the ideas he expresses are of a depth that would satisfy his colleagues.

Hardy himself is an archetype of the misunderstood artist; a creative genius who was either far ahead of his times or hopelessly behind. As history has proven, he was a little of both. Hardy’s own refusal to bow to the conventions of the time in regards to any subject matter, and his irrepressible need to offer his opinions and ideas regardless of the potential social or professional consequences, placed him in this lonely position. As a result, A Mathematician’s Apology is anything but objective. While Hardy’s argument is generally well defended, many of his subjective views, especially in regards to applied mathematics and chess, have been harshly criticized, and it is clear that as his life was drawing to a close, Hardly had achieved a melancholic acceptance of this position. Nevertheless, even to the end he refused to retreat on any of the views that defined his life and career.

Compare and Contrast

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Last Updated on May 6, 2015, by eNotes Editorial. Word Count: 292

1930s: Technology still has limited, though powerful, uses in warfare; a war must be won largely from the strength of armed forces, with technology playing a secondary role.

Today: The United States uses advanced technologies in its bombing campaigns against Iraq and Afghanistan, thus severely limiting the need for ground troops directly engaging in battle.

1930s: The world has still not been exposed to the threat of nuclear annihilation. Nuclear fission is viewed as impractical, and Einstein’s theory of relativity is still a concept remote from everyday life.

Today: With the help of Einstein’s theories, many nations have nuclear capabilities and can cause the destruction of mankind. Atomic testing in certain desert and ocean regions has had a lasting and adverse affect on the environment, and the threat of nuclear war between states continues to exist.

1930s: Alexander Alekine, Mikhail Botvinnik, and José Capablanca are celebrated for their mastery of chess. Chess is viewed as a game with an infinite number of continuations that can only be mastered by a particular kind of genius.

Today: Chess computers have been developed that can beat some of the best players in the world. IBM has developed a computer that defeats the reigning champion grandmaster, Gary Kasparov. Computers are capable of calculating an immense number of various chess continuations.

1930s: The profession of mathematics is an exclusive club, with a nearly all-male membership. In A Mathematician’s Apology, Hardy does not mention a single female colleague or refer to a single female author.

Today: Although less than one-fifth of all mathematicians and scientists are female, the Association of Women in Mathematics, founded in 1971, has over 4100 members, and there is wider recognition that gender disparities in the field are an issue to be addressed.

Bibliography and Further Reading

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Dauben, Joseph W., ‘‘Mathematics and World War I: The Internal Diplomacy of G. H. Hardy and Gösta Mittag-Leffler as Reflected in Their Personal Correspondence,’’ in Historia Mathematica, Vol. 7, 1980, pp. 261–88.

Greene, Graham, ‘‘The Austere Art,’’ in Spectator, Vol. 165, December 20, 1940, p. 682.

‘‘People Who Count,’’ in Times Literary Supplement, December 28, 1967, p. 1266.

Snow, C. P., ‘‘Foreword,’’ in A Mathematician’s Apology, by G. H. Hardy, Cambridge University Press, 1967, originally published in Variety of Men, by C. P. Snow, Scribner’s, 1967.

Waley, Arthur, ‘‘The Pattern of Mathematics,’’ in New Statesman and Nation, Vol. 21, February 15, 1941, p. 169.

Wiener, Norbert, ‘‘Obituary: Godfrey Harold Hardy (1877–1947),’’ in Bulletin of American Mathematics, Vol. 55, 1949, pp. 72–77.

Further Reading
Berndt, Bruce C., and Robert A. Rankin, eds., Ramanujan: Essays and Surveys, American Mathematical Society, 2001. This collection of largely non-technical, highly accessible essays on the Indian mathematician, is the first of three books covering Srinivasa Ramanujan’s life and includes several articles on his wife, his Indian colleagues, and his long illness.

Chan, L. H., ‘‘Godfrey Harold Hardy (1877–1947)—the Man and the Mathematician,’’ in Menemui Matematik, Vol. 1, 1979, pp. 1–13. Chan provides a biographical portrait of Hardy that can be compared to that by C. P. Snow.

Golomb, Solomon W., ‘‘Mathematics after Forty Years of the Space Age,’’ in The Mathematical Intelligencer, Fall 1999, p. 38. Examining Hardy’s assertion that pure mathematics has no relationship to issues of everyday life due to its inapplicability, Golomb argues that technological advances in the forty years since the publication of A Mathematician’s Apology have largely proved his assertion to be false. Prime number theory, for instance, an area Hardy had a special claim to, has contributed to advances in cryptology. Golomb explains how several other mathematical fields that were also once considered ‘‘pure’’ are now clearly ‘‘applied,’’ and he recounts his own experiences working in space programs during the fifties to further argue his case.

Hardy, G. H., Bertrand Russell’s Trinity, Arno Press, 1977. This book was originally published privately for Hardy by the University Press of Cambridge in 1942. In 1916, the philosopher Bertrand Russell was expelled from Trinity College, where he was lecturing, due to his objection to World War I. Hardy, who defended Russell and helped get him reinstated to the college after the war, sets out in this book to provide a full account of that incident and further helps to elucidate the lesser known history of conscientious objection during World War I.

Hoffman, Paul, The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth, Hyperion, 1998. Hoffman provides a popular account of the life of the Hungarian mathematician Paul Erdös, who died in 1996 and was widely revered as one of the most prolific, if not the most bizarre, mathematicians who ever lived. He was known as much for his obsessive nature, nomadic existence, and boundless energy for the search for mathematical proofs as he was for his actual contributions to the science. Hoffman’s book achieved widespread popular recognition at its publication and is accessible for the lay reader.

Kanigel, Robert, The Man Who Knew Infinity: A Life of the Genius Ramanujan, Scribner’s, 1991. Regarded as the definitive biography of Ramanujan, this book covers the mathematician’s life from his early childhood to his death in 1920 with a strong emphasis on his years in England, where he collaborated with Hardy.

‘‘A Professor’s Ideals,’’ in Times Literary Supplement, January 18, 1941, p. 33. This article reiterates Hardy’s philosophy that mathematics is a quest for beauty and truth.

Snow, C. P., Variety of Men, Scribner’s, 1967. C. P. Snow, a writer and a scientist who was a contemporary of Hardy, writes essays about several key early twentieth-century figures, including Hardy. The biographical sketch on Hardy has come to be included as the introduction in most modern editions of A Mathematician’s Apology.

Wiener, Norbert, ‘‘Obituary: Godfrey Harold Hardy (1877–1947),’’ in Bulletin of the American Mathematical Society, Vol. 55, 1949, pp. 72–77. This obituary gives an overview of Hardy’s life and also details the problems he faced as a young man in the stifling English educational system.

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