A Mathematician's Apology

by G. H. Hardy

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Hardy's Text as an Artistic Treatise and Memoir

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Mathematics is an exclusive club that opens its doors to a small number of gifted and often misunderstood individuals. Those who remain outside only have a vague perception of what it means to be a mathematician, and the perception that they do hold is more often than not hindered by an inability to understand exactly what it is that a mathematician does. Conversely, the expert mathematician is almost as ill-equipped as the layperson in trying to convey the beauty and joy of pure mathematics to non-mathematicians; the mathematician’s reliance on abstractions and the specialized vocabulary that define him as a mathematician make him a poor choice to describe and verbalize his field to the layperson. One either grasps the inherent beauty of theorems and numbers, thereby earning entry into the club, or one cannot and remains a perplexed outsider unable to grasp the inscrutable formulas. At least this was the case until G. H. Hardy, one of the foremost mathematicians of the twentieth century, bridged the gap and allowed the non-mathematicians of the world a glimpse into the mind and values of a pure mathematician.

Past his intellectual prime and restricted physically by several years of failing health, Hardy decided to write A Mathematician’s Apology, a book that can be appreciated by the mathematician and non-mathematician alike. The theorems that he outlines are among the most basic in the entire field. They are chosen so that the reader can both readily comprehend the explanations and easily perceive their aesthetic qualities. Hardy writes with the flavor and passion of an art lover about Euclid’s proof of the existence of an infinity of prime numbers and Pythagoras’s proof of the irrationality of the square root of two. The theorems he describes are representative of works of art precisely because they are so simple, which also makes them convenient as perfect examples for the general reader.

Since Hardy writes for an audience in large part comprising non-mathematicians, one must classify his essay with literary rather than mathematical headings. While the cold narrative voice of a mathematician does come forth at times in the prose, so do tones of elitism, disdain, and artistic snobbery, all of which do not normally belong in a mathematical essay. How then should we classify the essay if not as ‘‘mathematical’’? A Mathematician’s Apology is so multifaceted that it seems to transcend pigeonholing or categorizing. Restricting it to any single genre is an error that would cause a very restricted interpretation. For a thorough understanding of Hardy’s intentions, one must read the work as a representative example of various literary genres including the apology, artistic manifesto, and memoir. In each of these genres, Hardy’s elitist, valueladen tone invariably either demands unconditional acceptance or provokes severe disagreement.

A literary apology is a defense or justification for a particular way of life. Hardy, in calling his essay an apology, feels compelled to defend his chosen discipline. The urgency which he brings to the task leads one to believe that he is, at any given moment, trying to convince himself of the arguments. In order to present his belief that mathematics is an art, Hardy returns again and again to the concept of ‘‘utility’’ or ‘‘usefulness.’’ Judging from the disproportionate amount of writing he dedicates to these definitions, the charge that real mathematics has no practical use must have truly bothered him over the years. The disdain Hardy reserves for the widely accepted notions of ‘‘utility’’ and ‘‘useA fulness’’ is a further indication that the core of Hardy’s philosophy resides in these definitions.

Hardy continually splits hairs in defining ‘‘utility’’ or ‘‘usefulness’’...

(This entire section contains 1844 words.)

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in order to refine a definition that contradicts common sense. After Hardy has finished, the conventional conceptions of ‘‘useful’’ and ‘‘useless’’ have been inverted from what is generally accepted: what we commonly hold to be ‘‘useful’’ applications, such as engineering, geometry, and calculus, are ‘‘trivial’’ and useless to the real mathematician, according to Hardy. Conversely, the ‘‘uselessness’’ of real mathematics is precisely the reason why it is immortal and why one can consider it an art form. Hardy cannot contain his contempt and scorn for applied mathematics, calling it ‘‘school’’ mathematics and referring to its various worldly applications—such as engineering feats, ballistics, and aerodynamics—as ‘‘repulsively ugly and intolerably dull.’’

In belittling an entire subfield of mathematics, Hardy puts himself in a difficult position from which he can only extricate himself by twisting conventional definitions to justify his own field. His disdain for practical and useful applications forces him to redefine ‘‘uselessness,’’ a word that usually evokes negative images, in a manner that brings forth positive connotations. Herein lies the second stage of his definition and the beauty of the essay. Hardy may appear, to the careless reader, to have painted himself into a corner by proclaiming that it is ‘‘not possible to justify the life of any genuine mathematician on the ground of the ‘utility’ of his work.’’ According to Hardy’s philosophy, ‘‘applied’’ mathematics is ‘‘trivial’’ because it is useful, while ‘‘real’’ mathematics is immortal and superior because it is useless. Furthermore, it is useless in the way that the highest art forms of humankind are useless. Much the way Taoist thought holds a certain type of uselessness as an outstanding character trait, Hardy compares the uselessness of ‘‘real’’ mathematics to the uselessness of art. In this sense, to be useless is the ultimate compliment, and ‘‘real’’ mathematics is the highest form of art. Hardy writes

For mathematics is, of all the arts and sciences, the most austere, and the most remote, and a mathematician should be of all men the one who can most easily take refuge where, as Bertrand Russell says, ‘‘one at least of our nobler impulses can best escape from the dreary exile of the actual world.’’ . . . Mathematics is not a contemplative but a creative subject.

In setting down this philosophy and carefully describing its terms, Hardy creates a manifesto that describes real mathematics as an artistic movement, in much the same way the surrealist André Breton clarified an artistic movement in his Manifesto of Surrealism. Hardy puts forth the argument that real mathematicians have since time immemorial been artists of the highest caliber. This mathematicianas- artist motif was noticed immediately in early reviews of the work. In his 1941 review of A Mathematician’s Apology in the Spectator, British author Graham Greene asserts that Hardy’s philosophy is akin to the philosophy of an artist. ‘‘The real mathematician . . . must justify himself as an artist,’’ Greene writes.

Putting all accolades aside, there are those who remain unconvinced of the basic theories in Hardy’s concept of mathematics as art and who take offense to his dismissive views of other artistic genres. Arthur Waley writes in an early review for the New Statesman and Nation that ‘‘Dr. Hardy in this book is very definitely on the defensive, and his defense of mathematics consists in asserting that it is an art, like painting or poetry.’’ However, a poet could easily take offense or pick apart the examples and arguments that Hardy puts forth in an attempt to show the inferiority of poetry as an art. Not only does Waley not buy into the logic, he writes, ‘‘All this sounds like the comment of one whose contact with poetry is somewhat superficial.’’ A similar argument can be proposed for many of the other disciplines and fields Hardy writes off as ‘‘trivial.’’ The most glaring example is chess. Chess grandmaster Alexander Alekhine is derisively described as a ‘‘conjuror’’ or ‘‘ventriloquist,’’ and chess is constantly belittled as ‘‘trivial.’’ Keeping Waley’s objection in mind, it is clear that Hardy knows no more about chess than he does about poetry. His analysis and dismissals are superficial in that they do not take into account, for instance, the countless variations of set openings and the economy and aesthetic beauty of eliminating inefficient continuations in an attempt to bring about a ‘‘winA ning’’ position. In short, much as with poetry, Hardy writes off an entire field or artistic genre as inferior without having approached it with the same passion and knowledge he retains for theoretical mathematics; a chess grandmaster could easily argue that some of José Capablanca’s games contain the same simplicity and beauty of a Euclidian theorem.

Hardy’s derisive tone does not in any way imply final authority. The mathematician Lancelot Hogben (Hardy hesitates to even confer the title of mathematician on him), for whom Hardy reserves a flagrant contempt, has also achieved a significant and enduring reputation. A reprinted version of Hogben’s book Mathematics for the Million was reviewed in tandem with the reprint of A Mathematician’s Apology in 1967. The continuing popularity of both works indicates a particular rift within the field of mathematics and clearly shows that Hardy, though universally accepted as brilliant, is not necessarily considered the final authority he claims to be.

Although Hardy’s artistic philosophy has provoked widespread disagreement, his work remains extremely compelling as a personal memoir. Snow, writing in his biographical portrait of Hardy that initially appeared in his Variety of Men and that is now included as the introduction in later editions of Hardy’s essay, believes it to be a work of ‘‘haunting sadness’’ precisely because it is a ‘‘passionate lament for creative powers that used to be and that will never come again.’’ Viewed as a memoir, the work, particularly towards its conclusion, describes the trials and tribulations that a creative genius must undergo to excel in a field that does not appear to have the approval of the non-initiated. Both Snow’s biographical portrait and Hardy’s concluding chapters, for example, mention the insufficiency and stifling quality of the English educational system to which Hardy was subjected during his formative years. The mathematics departments at Cambridge relied on a contemptible and severe exam system, the triposes, which rewarded diligence rather than creativity. Hardy became a mathematician in spite of his education and was never truly appreciated by that system for the creative thinker that he was. The artistic genius is bound to remain misunderstood and held back by a callous society of Philistines; such a theme appears in numerous artistic memoirs and biographies of creative thinkers.

Although past the prime of his ability to contribute to the field of theoretical mathematics, Hardy retained the ability to describe to the layperson why the field of real mathematics is so lofty and noble. His eloquence and reluctant acceptance of his declining abilities allowed him to bridge the gap between genius and the common person, leaving a unique memoir to accompany his more creative artistic and mathematical endeavors. And although his derogatory statements and biased appraisal of real mathematics as the loftiest art form make him appear irrepressibly elitist, an undertone of humility caused by the realization of his declining physical and intellectual abilities balances Hardy’s writing and has rendered A Mathematician’s Apology an enduring classic.

Source: David Partikian, Critical Essay on A Mathematician’s Apology, in Nonfiction Classics for Students, Gale, 2003.

Pure Mathematician and the Artist

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Mathematics may not be the first pursuit that comes to mind when we speak of the creative process. The artist and the mathematician may seem to be on different ends of the spectrum. Storytelling, painting, literature, dance—these appear to be the realm of creative artists. Math, on the other hand, is an ‘‘austere’’ profession, little understood and sometimes feared. In A Mathematician’s Apology, G. H. Hardy distinguishes between pure and applied mathematics and compares the pursuit of pure mathematics to the creative process. For the most part, the comparison works.

According to its definition, the word ‘‘create’’ means to bring into being, to make, or to make by giving a new character function or status. Creation is the formulation of the new. Writers create stories that have never been told or have never been told with that author’s particular slant. Musical composers create original works or variations on existing works. Visual artists bring their visions into being using a number of media. And so, claims Hardy, pure mathematicians create new thought and new direction with their medium—numbers. In the introduction to A Mathematician’s Apology, C. P. Snow refers to a review of Hardy’s book by Graham Greene; Greene called A Mathematician’s Apology one of the best accounts of ‘‘what it was like to be a creative artist.’’

Snow, who knew Hardy personally, claims that Hardy had little ‘‘ego’’ and thus had to make a great effort in later life to assert his opinions. According to Snow, this aspect of Hardy’s personality contributed to the ‘‘introspective insight and beautiful candour’’ of Hardy’s thought process and writing. Throughout the book, Hardy clearly displays an artist’s passion for his work, with candor that can be direct and unflinching. Hardy discusses the creative artist’s potential to do something exceptional, and with characteristic bluntness and high standards claims that ‘‘if a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.’’

While Hardy promotes these standards for those with talent, he writes off much of the human population by claiming that ‘‘most people can do nothing well at all.’’ For the reader who never knew Hardy personally, it is hard to tell whether this remark is indicative of excessive ego or of a creative person’s high demands of himself and what he wishes to accomplish in life. To Snow, at least, Hardy’s purpose in life and in the field of mathematics was ‘‘to bring rigour into English mathematical analysis.’’ Hardy’s purpose is so integral to his existence that he admits this and calls it ‘‘inevitable egotism.’’ Good work, says Hardy, ‘‘is not done by ‘humble’ men.’’ Like a creative artist, Hardy believes that for a human, ‘‘the noblest ambition is that of leaving behind one something of value.’’

Throughout A Mathematician’s Apology, Hardy does not deny that he accomplished his life goal of bringing rigor into his field. However, he bemoans the fact (or the perception) that mathematicians do their best, most groundbreaking work at a relatively young age. The cause of Hardy’s sadness is one aspect of a mathematician’s life that seems to deviate from that of some creative artists. Hardy never tells the reader why older mathematicians do less than cutting-edge work. We might assume that as mathematicians age, their mental faculties decrease. However, creative artists in certain other fields may paint, write, or perform well into old age. Given the creative artist’s passion for creating, the reader might correctly assume that being unable to continue to create is akin to personal catastrophe or unbearable sadness.

It would be interesting to know how much satisfaction Hardy gleaned from continuing to work in mathematics into older age. Was his work truly inferior to what he had produced at a younger age? Was he able to feel as passionately about his later work in the field? In some creative endeavors such as literature, age, maturity, and experience may enrich the final product. An author’s first novel published when she is twenty-five is likely to be vastly different than a novel the same author publishes when she is forty-five. Snow concurs with this view of literature, stating that ‘‘it is very rare for a writer to realize, with the finality of truth, that he is absolutely finished.’’ But Snow, like Hardy, never explains why the work or the art of aging mathematicians diminishes.

Hardy’s world of pure mathematics in this respect more resembles the career of an athlete or a dancer. Why is pure mathematics such an ‘‘all or nothing’’ proposition? Why does Hardy believe that when a creative man has lost the power or desire to create, ‘‘it is a pity but in that case he does not matter a great deal anyway, and it would be silly to bother about him?’’ Has Hardy really lost the ability to continue to create or is he feeling the pinch of competition from up-and-coming, younger mathematicians? While competition may enter into an artist’s life, it does not need to affect the ability to continue to produce.

Regardless, it is Hardy’s exposition of the mathematical process as a creative process that makes A Mathematician’s Apology so accessible to the non-mathematical reader. Certainly, readers who are involved in some form of artistic creation or readers who have passionately and single-mindedly pursued the creation of something new in their lives are shown mathematics in a new light. And this is a good thing. Hardy admits that many people have an irrational fear of basic, applied mathematics. How could such people, therefore, be expected to understand the esoteric realms of pure mathematics, a field which Hardy calls ‘‘the most austere and most remote of all the arts and sciences’’?

To Hardy, artists as well as mathematicians create patterns. Like the patterns that a poet or painter creates, the patterns that the mathematician creates must be beautiful. Hardy claims that the ideas in any of these forms need to flow well together; ‘‘there is no permanent place in the world for ugly mathematics.’’ On the other hand, Hardy readily admits to the difficulty of defining beauty, a dilemma shared by artists and mathematicians alike. Worthwhile mathematics, according to Hardy, should be ‘‘serious as well as beautiful—‘important.’’’

One parallel between the creative process and the study of pure mathematics that Hardy does not elaborate upon is the role of the unknown. During the process of creativity, depending upon the particular artist and his or her style or mode of work, the end result may be completely unknown. For some creative artists, this is part of the thrill of creation.

Hardy never alludes directly to any personal fascination, distaste, or indifference to this aspect of the creative process. We can probably assume that most mathematicians would be thrilled, in the course of their work, to discover something previously unknown; something so cutting-edge that it changed the direction of the field of mathematics and had profound implications. But the role of the unknown in the creative process can take on subtler aspects.

In the area of novel writing, for example, some authors outline a novel completely before they start to write. Other novelists refuse to outline, writing the novel and figuring out the story, plot, and ending as they go. Many authors fall somewhere in between on this spectrum; outlining and preplanning to some degree but becoming fluid if needed to change course. While it may not be fair to compare pure mathematics research to novel writing, it might be interesting to know how comfortable or uncomfortable Hardy was with the unknown during his own research processes.

Hardy certainly has the purity of vision of a creative artist, or of anyone who knows what he wants to do and is doing what he loves. Immediately, he makes it clear that he prefers to do mathematics rather than engage in ‘‘exposition, criticism, appreciation-work for second-rate minds.’’ Critics, to Hardy, rank lower than scholars or poets, and he admits that it is a confession of weakness on his part to write about mathematics rather than actually writing mathematics. Like a creative artist, Hardy is so sure of his passion for his subject that ‘‘a defence of mathematics will be a defence of myself.’’ The artist and the art seem to be one and the same.

Finally, Hardy makes clear the difference between applied and pure (or real) mathematics, and it is clear that his heart and work are in the latter. To Hardy, the position of an applied mathematician is

in some ways a little pathetic . . . 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. ‘‘Imaginary’’ universes are so much more beautiful than this stupidly constructed real one.’’

As a creative artist might, Hardy sees value in transcending the ‘‘real’’ in pursuit of creativity, beauty, and significance. He can see no other way to justify real mathematics, other than justifying it as art, a view he claims is common among mathematicians.

Source: Catherine Dybiec Holm, Critical Essay on A Mathematician’s Apology, in Nonfiction Classics for Students, Gale, 2003.


Critical Overview