The most beautiful mathematics had no applications in the real world, according to the prominent British mathematician G. H. Hardy. He called them “pure mathematics,” and for him the purest of the pure was number theory, a branch of math that studies the properties of numbers.
Hardy’s A Mathematician’s Apology is a layman-accessible personal essay on theoretical mathematics. It’s hardly an “apology” in the traditional sense of the word. Hardy’s essay is a justification of his life and work.
At the age of 62, Hardy is a mathematician who’s aware of his waning intellectual abilities.
“Exposition, criticism, appreciation, is work for second-rate minds…It is a melancholy experience for a professional mathematician to find himself writing about mathematics.”
As a committed pacifist at the outset of the Second World War, Hardy also writes about his repulsion of war and his justification of math for its own sake. Number theory, like other mathematical fields, is not a foreboding realm for reclusive egg-heads but a sublime calling akin to creative professions like literature and art. Even readers who shudder at the thought of basic calculus will see the elegance of mathematical theorems and sympathize with Hardy’s conviction in mathematicians’ quest for truth.
Hardy praised number theory for lacking warfare applications unlike branches of applied mathematics such as ballistics and aerodynamics. Ironically, the Internet has applied number theory to applications ranging from buying goods on e-Bay to sending instant messages. Nonetheless, “Apology” remains an insightful peek into the mind of a prominent mathematician devoted to his craft regardless of money or application.
The turning point in number theory’s change from a pure, non-applicable science to one of highly practical use was the development of public-key cryptography in the 1970s. The method works like a locked mailbox with a key. The mailboxes address can be shared publicly so anybody who knows the address can send a message to that mailbox. Only individuals with a key to the mailbox, however, can open it to read the messages. Number theory has been applied to technological, financial, and, ironically for Hardy, military uses.