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...