A Mathematician's Apology

Summary

A Mathematician's Apology is a short book of twenty-nine numbered sections in which the Cambridge mathematician G. H. Hardy, at sixty-two and conscious that his creative powers had faded, sets out to justify a life spent on pure mathematics. Cambridge University Press published it in 1940. The word apology is used in its old sense of a defence, as in Plato's Apology of Socrates. Hardy opens by calling it a melancholy experience for a working mathematician to write about mathematics rather than do it; exposition and criticism are for second-rate minds, and he writes only because he can no longer do better. He then insists that mathematics is a young man's game. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty, and he knows of no major advance begun by a man past fifty. The defence proper begins with the question of why anyone should do mathematics. Most people, Hardy says, do what they do because it is the one thing they can do well, and ambition, the desire to leave something of permanent value, is a noble passion. Mathematics offers the most lasting fame of all. Greek mathematics is still read as the work of colleagues, and Archimedes, he says, will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. In the central sections Hardy describes the mathematician as a maker of patterns made of ideas. Such patterns must be beautiful, and he argues that real mathematics is judged by beauty and by seriousness, meaning the significance of its ideas and their connection with other ideas. To show what he means he gives two proofs from Greek mathematics that any patient reader can follow: Euclid's proof that the prime numbers never come to an end, and the proof attributed to the Pythagoreans that the square root of two is irrational. He contrasts these with chess problems, which are genuine mathematics but trivial, because nothing important follows from them. Hardy then asks whether mathematics is useful. He distinguishes real mathematics from the trivial elementary parts that serve engineers and accountants, and argues that the real mathematics of Fermat, Euler, Gauss and Riemann is almost wholly useless in practice. He finds comfort in this: real mathematics, he claims, has no effects on war. In the final sections he judges his own life. He has never done anything useful, but he has added something to knowledge, and his collaboration with Littlewood and Ramanujan was the decisive event of his life. A short note at the end acknowledges criticisms from C. D. Broad and C. P. Snow. Snow's long foreword, added to the 1967 edition, gives a portrait of Hardy that many readers now take as part of the book. Because the work is still under copyright in the United States, the library offers summary and guide only, with two short quotations.

Top Lessons

• Beauty as a test of truth's worth

  Hardy holds that mathematical ideas must fit together harmoniously and that ugly mathematics does not last. He shows rather than asserts this by walking the reader through Euclid's proof about the primes. The lesson is that beauty in an argument can be described with precision: depth, surprise, inevitability and economy.

• Usefulness is not the only measure

  Hardy defends knowledge pursued for its own sake and even takes pride in its uselessness. His claim that number theory would never serve war was overturned within a generation by cryptography and nuclear physics, a fact that sharpens rather than settles his question about why such knowledge matters.

• An honest accounting of a life

  In his last sections Hardy assesses his own work without false modesty or boasting. He ranks himself below the greatest mathematicians, names the collaborations that made his best work possible, and asks whether his life was futile. The book models a sober self-examination near the end of a career.

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Medical Lens

The Apology is an essay on mathematics, but it is also a document of ageing and lost creative power. Hardy begins by calling it a melancholy experience to write about mathematics, insists that mathematical creativity belongs to youth, and in section 29 mentions what he says to himself when depressed. He wrote after a heart attack in 1939. C. P. Snow's later foreword describes Hardy's decline in his last years and reports a suicide attempt in 1947, shortly before his death; those details belong to Snow's memoir, not to Hardy's text. The claim that mathematical ability peaks early has been debated by later researchers on age and creativity. Separately, the one applied result bearing Hardy's name, the Hardy-Weinberg principle of 1908, became basic to medical and population genetics, an irony given his praise of useless mathematics.

• Sections 1-5

  Hardy treats writing about mathematics as a sign of exhausted creative power and argues that mathematics is a young man's game.

• Section 29

  Hardy's final self-assessment, including what he tells himself when depressed and his sense of steady deterioration since his forties.