A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy, which offers a defence of the pursuit of mathematics. Central to Hardy's "apology" – in the sense of a formal justification or defence (as in Plato's Apology of Socrates) – is an argument that mathematics has value independent of possible applications. Hardy located this value in the beauty of mathematics, and gave some examples of and criteria for mathematical beauty. The book also includes a brief autobiography, and gives the layman an insight into the mind of a working mathematician.
Background
Hardy felt the need to justify his life's work in mathematics at this time mainly for two reasons. 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 and skills. By devoting time to writing the Apology, Hardy was admitting 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". In Hardy's words, "Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done."
Secondly, at the start of World War II, Hardy, 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. He began writing on this subject when he was invited to contribute an article to Eureka, the journal of The Archimedeans (the Cambridge University student mathematical society). One of the topics the editor suggested was "something about mathematics and the war", and the result was the article "Mathematics in war-time". Hardy later incorporated this article into A Mathematician's Apology.
He wanted to write a book in which he would explain his mathematical philosophy to the next generation of mathematicians; that would defend mathematics by elaborating on the merits of pure mathematics solely, without having to resort to the attainments of applied mathematics in order to justify the overall importance of mathematics; and that would inspire the upcoming generations of pure mathematicians. Hardy was an atheist, and makes his justification not to God but to his fellow men.
Hardy initially submitted A Mathematician's Apology to Cambridge University Press with the intention of personally paying for its printing, but the Press decided to fund publication with an initial run of four thousand copies.
Summary
One of the main themes of the book is the beauty that mathematics possesses, which Hardy compares to painting and poetry. For Hardy, the most beautiful mathematics was that which had no practical applications in the outside world (pure mathematics) and, in particular, his own special field of number theory. Hardy contends that if useful knowledge is defined as knowledge which is likely to contribute to the material comfort of mankind in the near future (if not right now), so that mere intellectual satisfaction is irrelevant, then the great bulk of higher mathematics is useless. He justifies the pursuit of pure mathematics with the argument that its very "uselessness" on the whole meant that it could not be misused to cause harm. On the other hand, Hardy denigrates much of the applied mathematics as either being "trivial", "ugly", or "dull", and contrasts it with "real mathematics", which is how he ranks the higher, pure mathematics.
Hardy expounds by commenting about a phrase attributed to Carl Friedrich Gauss that "Mathematics is the queen of the sciences and number theory is the queen of mathematics." Some people believe that it is the extreme non-applicability of number theory that led Gauss to the above statement about number theory; however, Hardy points out that this is certainly not the reason. If an application of number theory were to be found, then certainly no one would try to dethrone the "queen of mathematics" because of that. What Gauss meant, according to Hardy, is that the underlying concepts that constitute number theory are deeper and more elegant compared to those of any other branch of mathematics.