A Mathematician's Apology

"It is a melancholy experience for a professional mathematician to find himself writing about mathematics." Thus begins Hardy's classic essay, laying bare the melancholy subtext of what is superficially a very positive book and simultaneously casting a verbal stone at those who busy themselves with critical exposition rather than creation. With that in mind, it seems a rather perverse exercise to write a review of such a book; nevertheless, Hardy's Apology merits some reflection.Essentially, this book explains its author's philosophy of mathematics in very brief terms. Proving only two simple and classic theorems from Ancient Greek mathematics in the entire text, it is written as an explanation of the mathematician's mind and directed to the non-mathematician. It paints a portrait of a man obsessed with his field and who wants to explain to the rest of the world why. Graham Greene called it "the best account of what it is like to be a creative artist."Indeed, I can think of no other book that more succinctly makes the case for viewing (I would also say, though Hardy does not, for teaching) mathematics as creative art. I can also think of no time when such an argument has been more needed. Though Hardy's essay was first published in 1940 (and C. P. Snow's lengthy foreword added in 1967), it is in the early twentieth century that I think the need for a widespread appreciation of mathematics has reached its peak at the same time that popular fear of mathematics has also reached an unprecedented level. Under such circumstances, it would behoove every mathematician to consider Hardy's philosophy as much as it would benefit every non-mathematician to understand the mathematician's perspective.To be sure, there are elements of Hardy's essay with which we may disagree. He has a general distrust of mathematics as applied to engineering (as might be expected from an essay written during World War II by a man who also saw World War I) which I cannot in good conscience endorse (though his point is well-argued) and a view that widespread knowledge of scientific subjects (chemistry, for instance) is largely useless outside of the communities of professionals trained and working in related fields which I find indefensible in an increasingly democratized information economy.Still other arguments are rather outdated. There is a deep and dark irony in the idea that a mathematician passionately concerned with the applications of mathematics to war would write something like the following: "There is one comforting conclusion which is easy for a real mathematician. 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 very unlikely that anyone will do so for many years." Of course, relatively birthed the atomic bomb just five years later and number theory came to be the foundation of modern cryptography within the following decades. Despite the essay's positive tone, there is a depressing thread throughout, but Hardy could never have known how false that paragraph would ring just a few short years later.Still, despite some historical incongruities and points of minor (if impassioned) disagreement, this work remains arguably the best explanation of mathematics as an aesthetic pursuit in addition to (and perhaps even above) an applied one. For that reason alone, it merits serious consideration.The inclusion of C. P. Snow's lengthy (50-page) foreword adds a great deal of benefit for the reader. While it seems odd that so short a book should merit so long an introduction, the fact of the matter is that Snow provides the essential biographical context that helps the reader understand the circumstances under which Hardy wrote. Such context transforms Hardy's essay from a mere defense of mathematics (though it is a triumph of that genre) into an examination of the human condition worthy of a novelist. In view of Hardy's life history, one cannot help but to be moved by the concluding words of section 28: "It is a pity that it should be necessary to make one very serious reservation--he must not be too old. Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal anyhow, and it would be silly to bother about him."Incidentally, despite his monumental mathematical achievements in his own right, one of Hardy's accomplishments was his "discovery" of the Indian mathematician Ramanujan. Mention of their collaboration is all but absent from Hardy's words but is given deft treatment in Snow's introduction. Those who enjoyed the recent film, "The Man Who Knew Infinity," or who are otherwise familiar with Ramanujan's work will find some of these anecdotes quite interesting.In sum, this is a book that can be easily read in a single sitting but which has resonated with mathematical and mathematically-curious audiences across nearly eight decades. It has done so for a very good reason and should be considered required reading even today.

I am an engineer with a little bit of mathematics knowledge. I expanded my view on mathematics through reading this book. Knew the differences between the pure mathematics and applied mathematics.

A testimony of a man that, who although is not dead, died mathematically. This book has a lot of depressing accents, especially in the beginning and at the end, but it is nonetheless also very beautiful. The author shares his views on mathematics: what it is, why do it, why it is beautiful, and why it is important, all of that in a very logical, elegant, clear and non-pretentious manner. It is sad that those views are shared only in order to justify the author's creative, mathematical life. Nonetheless, this additional context gives an additional emotional value to this book.

G. H. Hardy was an influential figure in British mathematics in the early 20th Century. C. P. Snow who was his friend in his later years wrote the foreword to this book. This is a short book with the foreword taking up 58 pages and Hardy's contribution about 90 pages. It is a bit pricey, but contains Hardy's own words, which to me do not come across as apologetic as its title might suggest. Hardy has explained his stance about his career as a pure mathematician compared to other professions. He defends his profession and thus himself not against any particular criticism, but in the larger arena of human activities and contributions to knowledge over recorded history. Hardy has portrayed himself in this book as having a firm and proud character. He was also a realist who recognized at the time of writing this book, when he was about sixty-three, that the era of his mathematical creativity was long past.

I have long believed in the power of enthusiasm to overcome apparently insuperable hurdles and this book is strong evidence.It comprises basically two essays. The first is CP Snow's (essentially biographical) account of his relationship with Hardy and the second is Hardy's exposition of how he views mathematics.The CP Snow essay is very readable. It gives profound insight into the character of the subject. This is a great amuse-bouche before the main course.Hardy's essay is a literary masterpiece from a man whose expertise was mathematical rather than literary. His enthusiasm for mathematics is infectious and his skill in conveying that is masterly. He is well aware that for Joe Public, mathematics is a yawn. Yet he overcomes by conveying the beauty of two (apparently abstruse) mathematical proofs: (i) Euclid's proof that the number of primes is infinite and (ii) the Pythagorean proof of the irrationality of the square root of two. Having seized the reader with these two gems, he exposes further wonders.This is a jewel of a book. I have read it many times over more than 40 years and recommended it (with varying degrees of success) to large numbers of friends.