A Mathematician's Apology

The principles of mathematics, Hardy believes, exist on a high plane of existence, a realm of perfection, beauty, and a certain sublime uselessness. As a formalized version of logic, mathematics expresses eternal truths. The laws of nature are conditional, subject to the uncertainties of real life, and liable, now and then, to be overturned by new evidence. Math, conversely, is proved by logic, not by outside experience and, in that sense, is exempt from doubt. Hardy believes that mathematics’ perfection suggests that it dwells in its own realm outside the natural world. This is somewhat akin to Plato’s ideal Forms, the perfect expressions of imperfect nature that Plato believed exist outside space and time. Hardy therefore argues that the principles of math, and any new such axioms or proposals still being developed by mathematicians, aren’t inventions but discoveries.

However, Hardy held a distinctly negative view of applied mathematics, and its use in technology, industry, and everyday life—for example, its use by engineers and chemists to calculate the sizes, shapes, and power of the products they invent or the statistical methods that enable businesses, transit systems, governments, media, and other institutions to function properly. He considered applied math dull and crass despite its usefulness. Unlike pure math, with its pristinely perfect verities, “there is no permanent place in the world for ugly mathematics” (85). Hardy held that though the dullness of applied math lies in its ordinary, everyday applications, much of its ugliness stems from its use in destructive pursuits such as creating the instruments of warfare. The vast casualties of World War I resulted from advances in aeronautics, ballistics, and armaments, all of which relied on math. Hardy believed “that it is not on these crude achievements that the real case for mathematics rests” (65). That case relies on pure math, a realm in which the mind moves freely, discovering everlasting truths that have no connection to the dryness and frequent cruelty of everyday life.

Pure mathematics, or what Hardy called “real” mathematics, has no practical use and is therefore exempt from the moral culpability inherent in applied math. Thus, Hardy considered pure mathematics the most beautiful form of math. It’s so elegantly abstract that it has no obvious use in the real world, which Hardy saw as central to its value. His specialties were in number theory and math analysis, two fields with vast areas that are useless to science, business, government, or culture. He respected pure math for its elegance, aesthetic qualities, and serene isolation from the crudities of ordinary life. The sheer beauty of pure math’s empire of thought separates it and makes it valuable.