A Mathematician's Apology

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A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy (ISBN 0521427061). It concerns the aesthetics of mathematics with some personal content, and gives the layman an excellent insight into the mind of a working mathematician. It is, however, a very individual view, and Hardy's opinions are not universally held by mathematicians.

Summary

In the book's title, Hardy uses the word "apology" in the sense of a formal justification or defense (as in Plato's Apology of Socrates), not in the sense of a plea for forgiveness. There were two main reasons why Hardy felt the need to justify his life's work in mathematics at this time. Firstly, at age 62, Hardy felt the approach of old age (he had survived a heart attack in 1939) and the decline of his mathematical creativity. He wanted to explain his mathematical philosophy to the next generation of mathematicians. Secondly, at the start of the Second World War, Hardy, who was a committed pacifist, wanted to justify his belief that mathematics should be pursued for its own sake, rather than for the sake of its applications. As Hardy was an atheist, he makes his justification not to God but to his fellow man.

One of the main themes of the book is the beauty of mathematics, which Hardy compares to painting and poetry. For Hardy, the most beautiful mathematics was that which had no applications in the outside world, by which he meant pure mathematics, and, in particular, his own special field of number theory. He justifies the pursuit of pure mathematics with the argument that its very "uselessness" meant that it could not be misused to cause harm. On the other hand, Hardy denigrates applied mathematics, describing it as "ugly", "trivial" and "dull".

Another theme is that mathematics is a "young man's game", so anyone with a talent for mathematics should develop and use that talent while they are young, before their ability to create original mathematics starts to decline in middle age. This view reflects Hardy's increasing depression at the wane of his own mathematical powers. For Hardy, real mathematics was essentially a creative activity, rather than an explanatory or expository one.

Ironically, by devoting time to writing the Apology, Hardy was admitting to himself that his own time as a creative mathematician was finished. In his foreword to the 1967 edition of the book, C. P. Snow describes the Apology as "a passionate lament for creative powers that used to be and that will never come again".

Critiques

The Apology is a book of its time, and Hardy's opinions were heavily influenced by the academic culture of the universities of Cambridge and Oxford between World War I and World War II. His assumption that only the very best original work in any field has any lasting value can sound elitist to the modern ear.

The sharp distinction that Hardy draws between pure mathematics and applied mathematics has been overtaken by events, as the separation is much less clear-cut today. For example, Hardy wrote: No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years. Since then, the application of relativity was part of the development of nuclear weapons, while number theory figures prominently in public-key cryptography.

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