“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. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings; there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.”

Hardy condemns critics as people who judge what they can’t do themselves. Often, critics assume that the work they’re reviewing must be simple to achieve, and that efforts that don’t completely triumph must be the products of inferior talents. This rankles professionals who actually understand the difficulties they face. Unlike some other critics of the processes of mathematics, however, Hardy had one of the finest minds ever to face its challenges; his observations about his own thought processes are therefore highly worthwhile.

“Good work is not done by ‘humble’ men.”

As a historically prominent mathematician, Hardy took pride in his achievements—triumphs that, he believed, wouldn’t have been possible had he been filled with self-doubt. Success requires a certain cockiness of attitude about one’s abilities; Hardy freely admits to a large dose of positive self-regard, at least with respect to his professional work.

“It is a tiny minority who can do anything really well, and the number of men who can do two things well is negligible. If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.”

Today, Hardy would have written, “If any person has any genuine talent”—indeed, he supported the feminists of his era, and he mentored Mary Cartwright, one of the 20th century’s most prominent female mathematicians—but his main point here is that talent is valuable, and that, wherever it’s found, it should be nurtured. Hardy was skeptical that most people could be world-class in more than one area, but he knew it wasn’t out of the question and that such rarities too should be encouraged.