A Mathematician's Apology

“Good work is not done by ‘humble’ men.”

As a historically prominent mathematician, Hardy took pride in his achievements—triumphs that, he believed, wouldn’t have been possible had he been filled with self-doubt. Success requires a certain cockiness of attitude about one’s abilities; Hardy freely admits to a large dose of positive self-regard, at least with respect to his professional work.

“It is a tiny minority who can do anything really well, and the number of men who can do two things well is negligible. If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.”

Today, Hardy would have written, “If any person has any genuine talent”—indeed, he supported the feminists of his era, and he mentored Mary Cartwright, one of the 20th century’s most prominent female mathematicians—but his main point here is that talent is valuable, and that, wherever it’s found, it should be nurtured. Hardy was skeptical that most people could be world-class in more than one area, but he knew it wasn’t out of the question and that such rarities too should be encouraged.

“[M]athematics, more than any other art or science, is a young man’s game.”

Mathematics is a vast domain whose size expands with every new discovery. At its frontier, where the creative work gets done, it remains challenging, and Hardy notes that this calls for a very specific type of mind. It requires a relentlessly rigorous mental process—almost mechanical perfection in thinking—yet also demands the ability to create completely original ideas. That ability is at its peak from late adolescence until the beginnings of middle age. This applies to other professions as well—physics, chemistry, and chess, for example—but is especially true for mathematicians. Hardy watched helplessly as aging slowed his highly refined mental apparatus. His book is in part a commentary on this loss.

“What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men.”

Hardy argues that math itself is essentially harmless, though it can be used for destructive ends. Most of the preoccupations of pure math point nowhere but to itself; it’s an exercise in abstract thought whose products may be astonishing and elegant but usually have no application. This, for Hardy, is the essential wonder of mathematics: It’s beautifully useless yet requires great minds to produce.

“It may be fine to feel, when you have done your work, that you have added to the happiness or alleviated the sufferings of others, but that will not be why you did it.”

Doing good is a laudable motive, but, for Hardy, more important motivators, especially for mathematicians, are intellectual curiosity, professional pride, and the desire for acclaim, power, and money. The work must satisfy one’s personal needs or it will fail, and with that failure will be lost the benefits to humanity.

“Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.”

Playwright Aeschylus made great contributions to the development of stage dramas, while mathematician, scientist, and engineer Archimedes laid many of the foundations for modern math, especially calculus. Mathematics transcends languages and cultures, it’s always available and often useful, and its principles work anytime and anywhere. Hardy notes that most of Aeschylus’s plays have been lost, and the remainder must be translated or explained, while Archimedean principles remain infinitely clear to any mathematician.

“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

Hardy notes that a math theorem’s truth inspires wonder, and its elegance gives it beauty. The idea that it encapsulates may stun an awestruck mind, but when rendered simply and clearly, that awe changes to an appreciation of great artistry. If a theorem is too complicated or poorly expressed, this may signal that the theorem is false or at least not as elemental or fundamental as it could be.

“Chess problems are the hymn-tunes of mathematics.”

Hardy considered the chief appeal of math its beauty, which stems from the elegance of its ideas. Any puzzle includes mathematical elements, which provide the thrill and aesthetic joy of solving them. The pleasure that people derive from such hobbies is essentially the same satisfaction that mathematicians obtain from their work. The complexity of chess thus provides a kind of music for the intellect.

“I will say only that if a chess problem is, in the crude sense, ‘useless’, then that is equally true of most of the best mathematics; that very little of mathematics is useful practically, and that that little is comparatively dull.”

According to Hardy, math is a sublime pursuit for its own sake. Talk of its usefulness irritated him. During World War I, to his great dismay, he witnessed its use to amplify the relentlessly destructive power of industrialized combat. Although aware of its potential peacetime benefits, Hardy preferred to justify mathematics as an exercise of the highest mental faculties. For him, math is best seen as a type of art as well as a science.

“The chess problem is the product of an ingenious but very limited complex of ideas, which do not differ from one another very fundamentally and have no external repercussions. We should think in the same way if chess had never been invented, whereas the theorems of Euclid and Pythagoras have influenced thought profoundly, even outside mathematics.”

Although Hardy regarded pure mathematics as fundamentally useless, he noted that its profound discoveries affect human thought and influence the sciences. Whereas chess puzzles basically have application only in chess, mathematical problems influence how people think about everything. The logic of math is, in a sense, the logic of the mind; its rigor clarifies everyday thought.

“We do not choose our friends because they embody all the pleasant qualities of humanity, but because they are the people that they are. And so in mathematics; a property common to too many objects can hardly be very exciting, and mathematical ideas also become dim unless they have plenty of individuality.”

An overly general theory, such as the idea that all mathematics involves quantities, may be true but not inherently interesting to its practitioners. If it’s true of all math, it doesn’t suggest differences that lead to further discoveries. The power of a theorem—or, as Hardy noted, its seriousness—lies in its ability to affect much, but not all, of mathematics.

“It seems that mathematical ideas are arranged somehow in strata […] The lower the stratum, the deeper (and in general the more difficult) the idea.”

Depth is chief among Hardy’s requirements for a seriously significant math theorem. He doesn’t fully explain what this means, holding that the idea of depth in math is ultimately indefinable but mathematicians know it when they see it. A theorem that adds to another one usually requires more complex thought. For example, Pythagoras’s discovery of irrational numbers subtly advanced Euclid’s theory of numbers and thus was deeper, in some sense, than Euclid’s.

“A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.”

Solid proofs must be cleanly straightforward, lest their arguments become obscured by unnecessary complexity. Determining whether something is true is difficult when it’s garbled. Hardy preferred economy of logic in which an idea goes straight to the heart of a problem and attaches it firmly to the great edifice of mathematical truth.

“But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and lesser mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.”

Hardy vehemently objected to the use of science in warfare, famously opposing English participation in World War I, and his A Mathematician’s Apology was published just after the outbreak of World War II. His argument that mathematics is morally the cleanest science may be a shield against condemnation by other anti-war activists, but it’s also a cudgel that he uses on scientists tempted to build ever-more-destructive weapons. He had no moral objection to peaceful uses of science and math (though he generally found applied math far less intellectually interesting than pure math), but the wartime context added urgency to his plaint.

“I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.”

Hardy’s view of math followed that of Plato, who maintained that basic ideas such as a line or a sphere exist separately from reality, and that reality is merely an imperfect copy of the realm of perfect ideas. This isn’t the only viewpoint among philosophers but remains popular to this day: The concepts behind math are so compelling that they generate a strongly attractive argument that math somehow is an essential part of an ultimate truth independent of human reality.

“317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.”

Hardy notes that mathematics contains immutable principles. Nothing in math is conditional, whereas the physical sciences depend on the gathering of data to determine their truths. Math’s theorems, discovered through reasoning, require no evidence beyond the proofs that demonstrate them. Math thus has an absolute clarity and perfection that speak for its eternal universality. This is central to Hardy’s belief that math exists in its own right, separately from the external world or even the human mind.

“[T]he best Greek mathematics […] is eternal because the best of it may, like the best literature, continue to cause intense emotional satisfaction to thousands of people after thousands of years.”

The timeless, classic elegance and precision of the great discoveries in mathematics remain constant over time, throughout every age, so that Einstein’s equations take their place alongside those of the ancients. Hardy believed that what matters most about math is its beauty rather than its utility. The beauty of an equation is a sign that it is both useless and timeless.

“[I]t is what is commonplace and dull that counts for practical life.”

Hardy declares not merely that the best of math and science is beautiful but also that the most useful parts are colorless and dreary. Applied techniques have a mundane quality that, in his view, prevent their being attractive: Not every useless corner of math is beautiful, but nearly all useful aspects lack charm.

“But is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights.”

Applied math is a tool; as such, it’s rarely modified. Creativity isn’t much of an option when using a hammer or screwdriver, nor is it an issue when mathematicians apply their skills to practical matters. Hardy’s assertion is that applied math requires expertise but not much in the way of inventiveness.

“‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.”

For Hardy, math existed on its own in a pure realm unsullied by the messiness of the physical universe. As applied to real-life problems, the best math often must struggle to match the untidy realities of life. That impurity denigrates the mathematicians who attempt to tie math to scientific evidence.

“Modern geometry and algebra, the theory of numbers, the theory of aggregates and functions, relativity, quantum mechanics—no one of them stands the test much better than another, and there is no real mathematician whose life can be justified on this ground. If this be the test, then Abel, Riemann, and Poincaré wasted their lives; their contribution to human comfort was negligible, and the world would have been as happy a place without them.”

Hardy cites some of the most celebrated mathematicians as ironic examples of wasted effort but only if one regards pure mathematics as unimportant. He believed that applied math is vastly overrated and that the truly worthwhile effort takes place in the rarified air of abstract math. Had he known in 1940 that one of the fields he regarded as “pure,” quantum mechanics, would literally explode onto the world stage via the invention of the atomic bomb, he might have reassessed his view of what constitutes pure mathematics.

“So a real mathematician has his conscience clear; there is nothing to be set against any value his work may have; mathematics is, as I said at Oxford, a ‘harmless and innocent’ occupation.”

Hardy believed that the “applied” mathematics developed by great physicists such as Einstein actually rise to the level of pure math in both beauty and uselessness. After he published his book, Hardy lived long enough to witness the application of quantum mechanics—which he regarded as equal to relativity in quality—to the design of nuclear weapons. This challenged the notion that abstract scientific math, at least, is innocent of destructive potential.

“Many people, of course, use ‘sentimentalism’ as a term of abuse for other people’s decent feelings, and ‘realism’ as a disguise for their own brutality.”

Some critics argued during Hardy’s life that modern, high-tech warfare was more merciful than the old slash-and-stab methods, and that those who wished for a return to the glories of the past were overly sentimental. Struggling to discover a qualitative difference between old and new methods of slaughter, Hardy refused to join that argument other than to point out that war continues to be a plague that needs a cure.