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.