A Mathematician's Apology

The late chapters form an extended argument that amplifies Hardy’s beliefs about the superiority of abstract math to the useful variety, underscoring the book’s theme on The Purity of Mathematics. Thus, Hardy delves into the philosophy, contemplating such deep topics as the ultimate nature of reality, aesthetics as a virtue, and math’s place in civilization. The book as a whole focuses on the nature of math and what it means to mathematicians and to society. Hardy reflects very little on his creative process, which he apparently considers too technical to be widely useful to others. However, one can glean information about his methods by examining his writing style and reading between the lines. Hardy often expresses his ideas as a series of nested thoughts. In written work, nested thoughts appear as a sequence of dependent phrases or clauses. For example, in Chapter 21, Hardy writes:

It is undeniable that a good deal of elementary mathematics—and I use the word ‘elementary’ in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus—has considerable practical utility (119).

He expresses the central idea, that basic math is valuable, in the opening subject phrase and the closing predicate phrase, but in the middle of the sentence, within the dashes, Hardy drops down one thought level to limit the term “elementary” to its use by math professionals and then drops another level to define “elementary” as inclusive of calculus, delimiting this even further by declaring calculus as merely one example of “elementary.” (This suggests that other fields regarded as fairly simple by top-tier mathematicians include such banes of high school students as trigonometry and upper-level algebra.) Hardy must keep all the sentence’s concepts active at once and not lose track of them, so that he can express them clearly to readers. For those who seek to know what it’s like to be a working mathematician, Hardy’s sentence about elementary math neatly encapsulates a fundamental part of how such professionals think.

Throughout the text, Hardy compares math to chess, pointing out similarities between their routine practices and the joys of making discoveries within them. This hints at how he perceives his own thought processes. Readers who play chess regularly, then, may have a small additional sense of Hardy’s thinking.

Hardy lauds the uselessness of his field of pure mathematics, arguing that its unsullied nature gives it an aesthetic beauty that proves its virtue. Critics might counter that Hardy is simply stricken with a bad case of “sour grapes”: Since his specialty has no utility, it’s therefore somehow superior to the more practical departments of math. Hardy’s attitude, however, connects to a deeper source. He had a specific moral objection to applied mathematics: its use in warfare.

Hardy’s worry proved prophetic shortly after he published his book. While he was writing it, English and American scientists and mathematicians were working frantically to construct an atomic weapon meant to win World War II. The weapon’s success later became the source of much stress to its inventors over its world-ending possibilities. Today, civilization faces an ongoing risk of self-annihilation from nuclear and other high-tech perils. Hardy’s concerns thus seem more justified than ever.

Hardy aligns with Plato’s view that math exists as a separate, perfect realm. It must somehow do so, he declares, because its principles persist entirely apart from the observed physical universe. This viewpoint took a hit several years before Hardy’s book was published, when mathematician Kurt Gödel published his Incompleteness Theorems: These proved that, in math at least, and probably in all logical systems, every system of axioms contains an inconsistency. (The famous example, which Gödel cited, is the sentence “This statement is false,” which is both true and untrue.)

As a practical matter, mathematicians can simply add a new axiom that accounts for a contradiction. Euclidean geometry ran into such a dilemma: A triangle on a sphere, for example, contains inner angles that don’t add up to 180 degrees, as in a flat triangle. One solution replaces the standard axiom about parallel lines, which, in turn, generates a new, non-Euclidean form of geometry. The problem is that even new axioms themselves contain contradictions, and this process continues indefinitely. Thus, even mathematics itself can never be proven to absolute certainty. Whether Gödel’s discovery unhinges Hardy’s concept of an eternally perfect math reality remains an open question.

Part of Hardy’s argument in favor of pure mathematics is its general uselessness. Of relativity and quantum mechanics, two of the grand achievements of mathematics and physics, he writes that “these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers” (131-32). Hardy allows that perhaps one day they’ll become useful despite their beauty. Indeed, in the decades since Hardy’s book was published, both of these sciences have become vital to modern technology. Certain forms of chemistry, as well as the manufacture of modern computers, lasers, and medical scanners, rely on quantum mechanics computations. Without the calculations made possible by relativity theory, the GPS trackers that help our smartphones determine our exact location would lose accuracy by several miles per day. One of the wonders of math, and of science in general, is that even their most arcane discoveries eventually become useful: “It is the dull and elementary parts of applied mathematics, as it is the dull and elementary parts of pure mathematics, that work for good or ill” (132). By this argument, though, all of math must eventually become dull.

Hardy thus turns away from the sheer power of his own field of study. For him, the everyday application of knowledge is at best a boring necessity that enables the true calling of the human mind, the contemplation of eternal truths. For Hardy, as the poet John Keats put it, “Beauty is truth, truth beauty,—that is all / Ye know on earth, and all ye need to know” (Keats, John. “Ode on a Grecian Urn.” 1819. Poetry Foundation. Lines 49-50). Perhaps someday civilization will advance to the point that all people may contemplate the beauty of mathematics at their leisure; in the meantime, math will continue to manifest its irritating habit of being highly useful.