Study Guide

A Mathematician's Apology

by G. H. Hardy

A Mathematician's Apology Summary

Introduction

Godfrey Harold (G. H.) Hardy’s A Mathematician’s Apology, first published in 1940 in England, is the memoir of the world-renowned mathematician, written in the last few years of his life while he was in failing health. The work is written in the form of an apology, which in literary terms means a defense. In this case, Hardy is defending his career as a theoretical mathematician. To make the defense comprehensible to the layperson, Hardy discards the language he would use in an academic paper and instead adopts a succinct and simple writing style aimed at a general audience. The book is not mathematical; rather, it is an affirmation of a career that happens to be mathematical and purely speculative.

It should be noted that Hardy speaks exclusively of men in his writing, which reflects the secondary role women of his era played in the British university system in general and in the field of mathematics in particular. Hardy does not mention or refer to a single woman intellectual or a work by a woman.

A Mathematician’s Apology is a lasting testament to Hardy’s passion for intellectual pursuits. Hardy likens mathematics to art and explains math in much the same way a critic explains art. He elaborates on the qualities of mathematical genius and the logical reasons for pursuing a career in mathematics, and he briefly outlines three of the most basic and timeless theorems in order to illustrate the inherent beauty of mathematics for the layperson. Many of the chapters also address the differences between theoretical or ‘‘pure’’ mathematics— to which Hardy dedicated his life—and several types of ‘‘applied math,’’ which he regards as largely inferior. The work also reveals the grave doubts Hardy harbored about the overall usefulness of his work and life. While A Mathematician’s Apology has had an enormous influence on generations of mathematicians, it has also been viewed by many as a psychological document of a genius with depressive tendencies. As Hardy contemporary C. P. Snow acknowledges in the book’s introduction, A Mathematician’s Apology ‘‘is a book of haunting sadness.’’

A Mathematician's Apology Summary

Chapters 1–2
Hardy opens his apology by asserting his belief that in the mere act of ‘‘writing about mathematics’’ he has lowered himself to a level below that of a pure mathematician. He equates himself in this position to that of an art critic—a profession he considers to be for ‘‘second-rate minds’’—as opposed to the artist himself. Hardy describes a discussion he had on this subject with British poet A. E. Houseman. In chapter 2, Hardy introduces the questions he proposes to answer throughout the remainder of the book: Why is it worthwhile to make a career out of mathematics? And what is the proper justification of a mathematician’s life?

Chapters 3–4
Hardy states that most people choose their career path because ‘‘it is the one and only thing that [they] can do at all well.’’ Mathematics is a particularly specialized subject, and mathematicians themselves are not noted for their versatility. In chapter 4, he lists several mathematicians whom he considers immortal geniuses, and he points out that most of them reached their intellectual peaks or died before the age of forty. Those men who attempted new careers later in life were largely failures. Hardy uses these points to illustrate why he is now writing this memoir: simply put, he is too old to continue with theory, and he has no talent for any other career.

Chapters 5–9
Hardy concludes his responses to the questions he posed in chapter 3. As to why one would choose to become a mathematician, Hardy refers to a lecture he gave at Oxford twenty years earlier in which he posited that mathematics is chosen for three reasons. First, it is essentially a ‘‘harmless’’ profession; second, because the universe is so vast, if a few professors wasted their lives doing something at which they excelled, it would be ‘‘no overwhelming catastrophe’’; and third, there is a ‘‘permanence’’ of mathematics that is ‘‘beyond the powers of the vast majority of men.’’ It is here that Hardy adds what he believes are the three prime motivations that impel men to choose their professions: intellectual curiosity, professional pride, and ambition for reputation and the rewards it brings. To support these statements, Hardy lists several ancient civilizations that are long forgotten save for their mathematical discoveries. He concludes with a dream that mathematician and philosopher Bertrand Russell once related that expressed Russell’s deep-seated fear that he would one day be forgotten by future generations.

Chapters 10–11
Hardy posits that mathematics has an aesthetic quality like that of art or poetry—a position for which he and this book are best remembered. Hardy takes a swipe at one of his contemporaries, mathematician Lancelot Hogben, who was well-known for his opposition to Hardy’s theories. Hardy uses the example of chess to refute Hogben. Because chess is revered by the masses and is an exercise in pure mathematics, though admittedly of a ‘‘lowly kind,’’ when one appreciates the beauty of a particular chess move, one is in essence appreciating its mathematical beauty. However, since the best mathematics also demands ‘‘seriousness,’’ or ‘‘importance,’’ and since no chess player or problem ‘‘has ever affected the general development of scientific thought,’’ chess is ‘‘trivial’’ compared to pure mathematics.

Chapters 12–14
Hardy uses the examples of proofs by Euclid and Pythagoras to illustrate the beauty of mathematics and then explains why they are significant in spite of the fact that they are not practical. These proofs are presented concisely and demand only a rudimentary background in mathematics to follow them. It is the only instance in the memoir in which Hardy attempts to explain mathematical concepts or logic.

Chapters 15–18
Hardy continues to refine his concept of mathematical beauty by further defining the idea of ‘‘seriousness.’’ To do this, he introduces the concepts of ‘‘generality’’ and ‘‘depth.’’ Generality can be loosely defined as ‘‘abstractness,’’ while depth is comparable to ‘‘difficulty.’’ He discusses...

(The entire section is 1737 words.)