A Mathematician's Apology

The opening chapters explain some of Hardy’s thoughts on how mathematicians choose their career. In his view, the choice usually occurs when young people discover they have a gift for math, and, if they’re ambitious, decide to apply themselves and concentrate on it.

Each of the book’s chapters is short, most less than 1,000 words. This pattern persists throughout the book. Hardy’s thoughts are often complex, but he expresses them concisely.

Hardy opens with a statement of self-reproach: He finds himself reduced, in old age, to writing about his process rather than doing it. It’s also an apology of sorts for stepping outside the bounds of mathematics to take up the pen of an author. Should this effort prove unsatisfactory, he claims, it’s because his mind is no longer at its best. In effect, he’s deflecting in advance snickers of reproach from his fellow scientists.

One reason that Hardy has contempt for those who write about how math gets done is that mathematics is arguably one of the sciences and thus is predefined by a rigorous process. It must conform to the rules of logic, as math in the modern world does, partly because of Hardy’s own efforts; and logic has a merciless quality that strictly limits the results it finds valid. The arts are different: They’re open-ended, and every new creation alters the rules. Math is a creative enterprise in its own right, but each discovery must submit itself to a tight filter that automatically rejects anything less than logically perfect.

Hardy’s writing conforms to an older standard of gender equality. The year 1940, when the book was published, was decades before the time when women were understood as men’s intellectual equals. In the literary tradition of that era, a practice that marginalized women, Hardy refers to all people as “men.” It’s a habit that today he’d probably find embarrassing, as he was sympathetic to women and their efforts to rise above their status in what he called a “depressed class.”

Hardy was the PhD advisor to mathematics student Mary Cartwright, who became a prominent English mathematician and multi-award-winning member of the Royal Society. Cartwright’s work, along with that of Hardy’s colleague John Littlewood, formed the beginnings of what would become Chaos Theory, known for its contribution to the understanding of complex systems and for its description of the “butterfly effect,” whereby an insect’s flapping wings can change the course of a storm half a world away.

Chapter 4 presents Hardy’s central argument that mathematics is A Field for the Young and Ambitious, which is one of the book’s primary themes. At first, it may seem strange that a mathematician’s career is at its best only when the practitioner is young, while in many other careers the required skills seem to improve with age. Those professions usually demand a greater width of experience, which accumulates over time, adds wisdom and perspective, and more than makes up for any loss of mental speed or exactitude. Thus, writers, musicians, actors, teachers, business leaders, clerics, chefs, administrators, and political leaders often reach the heights of their careers in middle age and beyond.

Fortunately, Hardy was an excellent communicator, and his book speaks eloquently about the challenges that he, other mathematicians, and anyone involved in creative efforts confront as the years go by. Hardy’s mind may have flagged in mathematics—enough, at least, for him to notice—but it grew in wisdom, perspective, and literary ability, as is vividly clear in A Mathematician’s Apology.

Hardy’s comments on the importance of leaving something for posterity bring to mind Steve Jobs’s belief that what matters is to “make a dent in the universe” (Linzmayer, Owen W. Apple Confidential 2.0: The Definitive History of the World’s Most Colorful Company. No Starch Press, 2004). Both Jobs and Hardy were extremely competitive people whose overriding goal was to matter. Hardy admits that his perspective is “egotistical” (65) but argues that only people with great self-confidence can do the work that changes civilization.

Hardy believed that mathematicians can achieve an immortality that even great playwrights can’t attain. He cites Aeschylus, one of the most important ancient Greek playwrights, as more liable to historical oblivion than Archimedes because the latter’s mathematical discoveries are universally understandable and so, therefore, is the mathematician. This alludes to another of the book’s main themes, The Purity of Mathematics. Of course, whether future generations will remember all mathematical principles thus far discovered, much less their discoverers, remains to be seen.