A Mathematician's Apology

by G. H. Hardy

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Chapters 1–2
Hardy opens his apology by asserting his belief that in the mere act of ‘‘writing about mathematics’’ he has lowered himself to a level below that of a pure mathematician. He equates himself in this position to that of an art critic—a profession he considers to be for ‘‘second-rate minds’’—as opposed to the artist himself. Hardy describes a discussion he had on this subject with British poet A. E. Houseman. In chapter 2, Hardy introduces the questions he proposes to answer throughout the remainder of the book: Why is it worthwhile to make a career out of mathematics? And what is the proper justification of a mathematician’s life?

Chapters 3–4
Hardy states that most people choose their career path because ‘‘it is the one and only thing that [they] can do at all well.’’ Mathematics is a particularly specialized subject, and mathematicians themselves are not noted for their versatility. In chapter 4, he lists several mathematicians whom he considers immortal geniuses, and he points out that most of them reached their intellectual peaks or died before the age of forty. Those men who attempted new careers later in life were largely failures. Hardy uses these points to illustrate why he is now writing this memoir: simply put, he is too old to continue with theory, and he has no talent for any other career.

Chapters 5–9
Hardy concludes his responses to the questions he posed in chapter 3. As to why one would choose to become a mathematician, Hardy refers to a lecture he gave at Oxford twenty years earlier in which he posited that mathematics is chosen for three reasons. First, it is essentially a ‘‘harmless’’ profession; second, because the universe is so vast, if a few professors wasted their lives doing something at which they excelled, it would be ‘‘no overwhelming catastrophe’’; and third, there is a ‘‘permanence’’ of mathematics that is ‘‘beyond the powers of the vast majority of men.’’ It is here that Hardy adds what he believes are the three prime motivations that impel men to choose their professions: intellectual curiosity, professional pride, and ambition for reputation and the rewards it brings. To support these statements, Hardy lists several ancient civilizations that are long forgotten save for their mathematical discoveries. He concludes with a dream that mathematician and philosopher Bertrand Russell once related that expressed Russell’s deep-seated fear that he would one day be forgotten by future generations.

Chapters 10–11
Hardy posits that mathematics has an aesthetic quality like that of art or poetry—a position for which he and this book are best remembered. Hardy takes a swipe at one of his contemporaries, mathematician Lancelot Hogben, who was well-known for his opposition to Hardy’s theories. Hardy uses the example of chess to refute Hogben. Because chess is revered by the masses and is an exercise in pure mathematics, though admittedly of a ‘‘lowly kind,’’ when one appreciates the beauty of a particular chess move, one is in essence appreciating its mathematical beauty. However, since the best mathematics also demands ‘‘seriousness,’’ or ‘‘importance,’’ and since no chess player or problem ‘‘has ever affected the general development of scientific thought,’’ chess is ‘‘trivial’’ compared to pure mathematics.

Chapters 12–14
Hardy uses the examples of proofs by Euclid and Pythagoras to illustrate the beauty of mathematics and then explains why they are significant in spite of the fact that they are not practical. These proofs are presented concisely and demand only a rudimentary background in mathematics to follow them. It is the only instance in the memoir in which Hardy attempts to explain mathematical concepts or...

(This entire section contains 1737 words.)

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Chapters 15–18
Hardy continues to refine his concept of mathematical beauty by further defining the idea of ‘‘seriousness.’’ To do this, he introduces the concepts of ‘‘generality’’ and ‘‘depth.’’ Generality can be loosely defined as ‘‘abstractness,’’ while depth is comparable to ‘‘difficulty.’’ He discusses mathematician and philosopher Alfred North Whitehead’s assertions, which he quotes: ‘‘The certainty of mathematics rests entirely on abstract generality.’’ Hardy partially accepts this argument from a logician’s point of view but argues that theoreticians must look for the difference among generalities. With regard to ‘‘depth,’’ Hardy employs a metaphor that equates theorems with geologic strata: the more difficult the theorem, the more layers it has, with each layer holding ideas that link to other ideas above and below it. He concludes this discussion in chapter 18 by explaining why chess can never be ‘‘beautiful.’’ In short, the very nature of chess demands that any given move can be answered with multiple countermoves—what Hardy refers to as ‘‘proof by enumeration of cases,’’ which is the antithesis of beauty in a mathematician’s eye. Rather than being a beautiful collection of mathematical theorems competing with one another, a chess game is, at its heart, a psychological battle between two intelligent beings.

Chapters 19–21
Hardy returns to his Oxford lecture in order to address the question of the usefulness of mathematics. In short, Hardy states emphatically that although some ‘‘elementary’’ mathematics such as calculus have some utility, the ‘‘pure’’ mathematics with which he concerns himself cannot be justified on utilitarian grounds.

Chapters 22–24
Hardy returns to his comparison of ‘‘applied’’ and ‘‘theoretical’’ mathematics and states that it is a gross oversimplification to say that one has utility while the other does not. Hardy supports this statement by setting out to argue that pure mathematics is closer to ‘‘reality’’ than is applied mathematics. His assumption here is that there is a ‘‘mathematical reality’’ that exists that is no different from the ‘‘physical reality’’ to which most of us can relate. Mathematical reality is not a mental construct but rather an objective reality that exists in the world that can be discovered and described. Mathematicians who ‘‘create’’ proofs are actually doing little more than taking notes on their observations.

To illustrate this point, Hardy draws on the field of geology, which sets out to draw a ‘‘picture’’ of a part of mathematical reality. However, because geometry does not account for changes in spatiotemporal reality, such as those created by eclipses and earthquakes (since these are not mathematical concepts), the ‘‘drawing’’ a geometer creates in his theorems may suddenly have little to do with the physical reality surrounding him. However, the truths of the theorem remain unaffected. Or, to put it in even simpler terms, while spilling coffee on the pages of a Shakespeare play may make certain pages unreadable, the spill does not affect the play itself. By analogy, pure mathematicians concern themselves with the play, while applied mathematicians concern themselves with the pages on which the play is written.

In chapter 24, Hardy makes the seemingly paradoxical claim that despite these relationships, pure mathematicians are in fact the closer of the two to reality. Hardy’s argument is as follows: an applied mathematician must work with a physical reality over which there is ample disagreement as to what comprises it. There is confusion as to what constitutes a chair, for instance: it may be a mass of whirling electrons, or it may be an ‘‘idea of God.’’ The pure mathematician, however, works with a mathematical reality about which there is no ambiguity. No one disagrees as to what ‘‘2’’ or ‘‘317’’ is, and ‘‘317’’ is a prime number not because we ‘‘think it so’’ but rather because ‘‘it is so.’’

Chapter 25
Continuing the comparison of pure and applied mathematics, Hardy claims that pure mathematics is timeless, has a permanent aesthetic value, and its eternal qualities bring about a lasting sense of emotional satisfaction. The achievements of applied mathematicians, on the other hand, are more modern and temporal. Hardy leans towards calling the applied mathematical theories ‘‘useless.’’

Chapter 26
Hardy continues to delve into the idea of utility in mathematics, asking, ‘‘What part of mathematics are useful?’’ He goes on to list branches according to utility. He comes to the general conclusion that the more useful a type of mathematics is to an engineer or physicist, the less aesthetic value it has. Hardy prefers the world of imagination and art to the ‘‘humdrum’’ reality of applied mathematics. Hardy writes

‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest productions 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.

Chapters 27–28
Hardy addresses some of the objections of his critics, especially applied mathematician Lancelot Hogben. Hardy’s tone is snide and superior as he sums up his arguments regarding the differentiation between real and applied mathematics. He repeatedly uses the word ‘‘trivial’’ in reference to applied mathematics. According to Hardy, real mathematicians are artists. Hardy does not offer any justification of applied mathematics, saying only that it would appeal to Hogben. It is here that Hardy finally broaches the subject of utility and harm. Writing under the threat of an impending world war, Hardy feels that it is necessary to lead his discussion towards the relationship between mathematics and war. He comes to the conclusion that ‘‘real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.’’ Hardy wrote these words only five years before the theories of relativity helped the United States develop the first atomic bomb.

Chapter 29
Hardy concludes his memoir by returning to a more personal narrative voice. This final chapter is far more autobiographical than the rest of the memoir. Having already stated his theories, Hardy feels justified in summing up his life. It is the summation of a man knowingly in his declining years. Although the tone is sad and melancholic, he seems to convince himself that his life has had meaning. In lines often quoted by critics of the work, Hardy writes, ‘‘Well, I have done one thing you could never have done, and that is to have collaborated with both [mathematicians John Edensor] Littlewood and [Srinivasa] Ramanujan on something like equal terms.’’ One quotes these lines so often because while so much of the work paints Hardy as pompous, this quotation is a clear illustration of his humility as well. He is able to recognize genius and also admit to his own limitations. In a work that carries a subtle sadness throughout it, these line spring forth as a positive affirmation of a genius’ existence.