A Mathematician's Apology

Published in 1940, A Mathematician’s Apology, by G. H. Hardy, is an extended essay on why people study mathematics and how its logical purity, much more than its usefulness in daily life, makes it a worthy pursuit.

Hardy was one of the 20th century’s most important mathematicians. His work in math analysis and number theory, coupled with his high standards of logical proof, helped advance mathematics and increase its rigor. Hardy’s discovery of—and work with—untutored math genius Srinivasa Ramanujan is his best-known contribution to the field.

The book contains a biographical Foreword by Hardy’s friend, celebrated English scientist and author C. P. Snow.

This study guide references the e-book version of the 2012 edition.

Content Warning: The book’s Foreword briefly discusses suicide.

Summary

In the book’s Foreword, English literary giant C. P. Snow describes Hardy’s life and their friendship. Hardy’s parents were both teachers; they nurtured his education, channeling him toward a career in math, a field in which the precocious boy showed great aptitude. Young Hardy was generally first in his class at school and at, or near, the top when he attended Trinity College at Cambridge University.

Hardy discovered a French textbook on calculus, Cours d’Analyse, that used the rigorous proof system of geometry; this inspired him to campaign for a deeper logical foundation for math studies in England. As a professor at Cambridge—and, for a time, at Oxford—Hardy made remarkable mathematical discoveries, first with Indian mathematician Srinivasa Ramanujan and later with fellow math professor John Littlewood. His influence on mathematics is profound.

Hardy was shy, though warmly talkative with close friends, most of whom were among the top minds at Cambridge. He loved cricket and indoor tennis, hated war and lobbied against it, and favored liberal social reforms. Strictly honest in his thinking, he was skeptical of the Cambridge social elite but favored anyone with a brilliant mind. He and Snow initially became friends over their common interest in sports but later developed an even closer bond as intellectual peers.

In his sixties, Hardy struggled with health problems and a mind slowed by age. During this period, he penned a long essay on Bertrand Russell’s World War I anti-war efforts, and he completed his book about the creative process in math, A Mathematician’s Apology. Depressed and bored, Hardy attempted death by suicide in 1947 but survived. During his final months, Snow visited him weekly, and they talked mostly about cricket. Hardy is remembered as a remarkable person who, at the height of his powers, was one of the greatest mathematical minds of the 20th century.

Writing at the end of his career, Hardy felt awkward about describing his thought processes as a mathematician. It seemed somewhat like being a critic, which he regarded as a lowly profession. Hardy wrote about his career partly because he was no longer young enough to do mathematics work well and instead felt the need to offer perspective on his experiences in the field.

Mathematics is famous for its ability to guide the advance of modern technology, but it needs no such justification. It’s worthwhile in itself and is so specialized that only a few great minds are able fully to answer its call. According to Hardy, those people must be ambitious and work hard right away, because they will do their best work during their young adulthood, when their minds are sharpest. Ambitious minds produce great works and may affect civilization positively, but their real motives are curiosity, pride, and desire for prestige and money. The reward is a kind of immortality, since mathematics has a universal language and its principles are retained across centuries, independent of culture or human history. It can be understood anywhere, at any point in time. Unlike fine arts, whose rules shift with the latest schools of practice, math is eternal, its principles unchanging.

The best math theorems have a beauty all their own, an elegance that bespeaks serious significance. This beauty is all the greater for its influence on human thought, especially in the sciences. To be significant, a math theorem must have generality and depth. Generality is the wide application of a theorem to other math theories; depth derives from the ability to force mathematicians to think in new ways about their own work. A serious theorem must be surprising, inevitable, and economical.

Math has a logical purity that exists apart from the physical world. Science often uses mathematical concepts that happen to fit the structures of reality, but these are conditional applications, while the underlying math remains unchanged. The most beautiful parts of math generally have no practical use, while the mundane, if useful, fields—calculus, for example—lack the elegance of pristine regions like number theory.

Applied math tends to be dull, crude, and ugly. Although useful, it’s uninspiring, and it can be put to bad ends such as warfare. Abstract math, on the other hand, pulses with significance and beauty. Its purity signals its perfection, and its uselessness makes it ever an art form and never a cause of harm.