1.1. Review by: M F E.
Studies: An Irish Quarterly Review 30 (117) (1941), 153-154.

Professor Hardy is a mathematician of the first rank, and a book such as this from his pen challenges comparison with 'La Science et l'Hypothèse' and 'La Valeur de la Science'. Frankly, it does not stand the test, (the severest one possible), too well. Henri Poincaré, in addition to his amazing mathematical gifts, had an exceptionally sympathetic mind, a wit that was for ever bubbling over, and a most engaging personality. It was a delight to him to tell his readers of his enthusiasms and his achievements; and their response was such that his works competed with 'les romans scandaleux' in popularity.

Professor Hardy, on the contrary, is at pains to let his audience know that it doesn't matter twopence what they think. "It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. ... There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain." And we ask ourselves, what must this great man think of us, poor folk who have to have things explained to us?

In the passage I have just cited, Professor Hardy is unfair to himself in more than one way. For one thing, the chief interest of a mathematician is not in the newness of the theorem that he has found, in the fact that no one else has managed to get in ahead of him. If it were, his science would be a sorry one. And that it is not so with Professor Hardy himself is obvious to anyone who has read any of his works, or who has even noticed the evident pleasure with which he recalls Euclid's theorem about prime numbers, the proof of which has been an unfailing source of delight for two thousand years.

Poincaré put the meaning of mathematics, the secret of the mathematician's devotion to his science, into memorable words. The highest of human functions and of human aims is the contemplation of truth; and the mathematician contemplates truths whose permanence and universality, whose independence of local and temporal bounds, bring him in touch with the eternal Order.

Professor Hardy's book is one to read and meditate upon. It is full of suggestion; for instance, his analysis of the idea of 'depth' in mathematical theorems opens up a whole field of thought.

1.2. Review by: I Bernard Cohen.
Isis 33 (6) (1942), 723-725.

G H Hardy, is concerned with the problem: Why pursue mathematics? This book begins by explaining why some men pursue mathematics, others literary studies, others chess or cricket. Mr Hardy reiterates his famous dictum that all good mathematics is useless - "the study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation." This thesis is extended in the form of an argument to prove that the difference between chess and group theory is not a question of application, but entirely one of seriousness and significance. "The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but the significance of the mathematical ideas which it connects." After a most interesting and stimulating discussion of the various fields of mathematics, including several examples to illustrate the aspects of the subject, we are given the following conclusion:

"I have never done anything "useful." No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. I have helped to train other mathematicians, but mathematicians of the same kind as myself, and their work has been, so far at any rate as I have helped them to it, as useless as my own. Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value."

"The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them."

I suspect that many mathematicians will disagree with Hardy's ideas, but none will find his book dull. All classes of readers will find it stimulating to read about the development of the mind of a great mathematician in his own terms, and the meaning of his great creativity according to his own standards. Not every one will agree with Hardy's statement that it is a delusion to "suppose that there is a great difference in the utility between 'pure' and 'applied' mathematics," and that the only dichotomy in mathematics is between what Hardy chooses to call real mathematics and trivial mathematics. Non-mathematicians will not get the significance of the statement:

"I still say to myself when I am depressed, and find myself forced to listen to pompous tiresome people, 'Well, I have done one thing you could never have done, and that is to collaborated with both Littlewood and Ramanujan on something like equal terms.'"

But all readers will find this book a very worth-while experience.

1.3. Review by: Eric Temple Bell.
The Scientific Monthly 54 (1) (1942), 81.

Hardy's "Apologia pro Vita Sua" is ludicrously reminiscent of Cardinal Newman's famous history of his personal religious opinions. Unlike the late cardinal's pious effusion, however, Hardy's somewhat defiant challenge to the impure among mathematicians has a stimulating dash of satire occasionally, broadening into uproarious farce comedy in the final section. Hardy's sardonic confession of how he ever came to be a professional pure mathematician may be specially commended to solemn young men who believe they have a call to preach the higher arithmetic to mathematical infidels.

With his usual clarity the author explains several simple examples of what he calls "real" mathematical theorems. All are within the comprehension of any one who has had a few days of elementary algebra. Contrasted with "real" or stainlessly "pure" mathematics, is the baser kind, "useful" or applied mathematics, and we are shown why "pure mathematics is on the whole distinctly more useful than applied.'' This of course is an immediate corollary of a classic paradox of G K Chesterton's. We learn that when the mathematical physicist wants to be useful, "he must work in a humdrum way. ... 'Imaginary' universes are so much more beautiful than this stupidly constructed real one. ..." Well, God, not the mathematical physicist, must take the blame. And this, brings us to what is perhaps the most remarkable passage in the book. It is a statement of Hardy's mathematical creed: I believe that mathematical reality lies outside us, and 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. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards.... Indeed it has. But not by all. The impregnable strength of this creed is that it can be neither proved nor disproved. We may take it or leave it as we please. Congenital believers will embrace it with joy, possibly as a compensation for the loss of the religious beliefs of their childhood. Some, like Bertrand Russell, who once clung to it, will "abandon it with regret." The majority will probably ignore it as a museum piece from an incredibly credulous past.

The mathematician's apology, though franker and less casuistical than the cardinal's, deserves a place on the shelf with the churchman's masterpiece of special pleading. Even those who dislike what the mathematician says may like the enthusiastic way he says it.

1.4. Review by: K V.
Current Science 10 (11) (1941), 499.

This is a small book of about 100 pages written by one of the foremost mathematicians of England. The book gives stimulating discussions of some points about mathematics which appear very intriguing to a layman. One may not however agree entirely with the author about the necessity of an apology. For e.g., when he says, p. 78, "The mathematics which can be used 'for ordinary purposes' by ordinary men is negligible. ... "one can point out that in these times, e.g., electrical appliances are in such ordinary use by ordinary men and the mathematics behind their design is by no means negligible.

1.5. Review by: Charlie Dunbar Broad.
Philosophy 16 (63) (1941), 323-326.

The author of this little book is known to his professional colleagues throughout the world as one of the most eminent of living English pure mathematicians. To his many friends he is known as a man of strongly marked and highly original personality, which expresses itself very characteristically in his conversation. During the most anxious period of the dreadful summer of 1940 he spent some time in looking back, from the early evening of his life, on his own professional activities and in writing down his reflections on the nature and value of his subject and on his own attitude towards it, He took the manuscript somewhat diffidently to the University Press, thinking that they might perhaps consent to publish it as a pamphlet if he undertook to bear the expense. Mr Roberts of the Press knows a good thing when he sees it, and he jumped at the opportunity to publish a fascinating and agreeably written little book. I hope, and have good reason to believe, that his reward will be something more than a deferred annuity payable in the Kingdom of Heaven.

Professor Hardy distinguishes two questions: (1) What are the motives which lead certain persons to devote themselves to mathematics? And (2) What is the value of their activities?.

Professor Hardy's answer to his first question is as follows. The main motives which lead anyone of first-rate abilities to devote himself to any kind of research are intellectual curiosity, pleasure and pride in the successful exercise of his technical skill, and the desire for favourable recognition of his work by competent contemporaries and successors. An eminent mathematician has a particularly good chance of satisfying all these desires. Moreover, great mathematical gifts are so rarely accompanied by comparable ability in any other department that anyone who possesses them is under little temptation to aim at any alternative kind of achievement.

In this connection there are two small points in Professor Hardy's argument which a captious reader might be inclined to criticise. (i) In his attempt to show that mathematical fame tends to be more enduring than, e.g. political fame, he does not sufficiently distinguish between continued admiration for a theorem in fact discovered by X and continued admiration for X as discoverer of that theorem. What he shows is that a first-rate mathematical theorem is likely to be recognised and admired as such by experts for an indefinitely long period. But surely it must depend on many highly contingent circumstances whether the name of its discoverer remains associated with it. Who of us, in admiring some theorem discovered by some Babylonian mathematician, is in a position to admire its discoverer for discovering it? No doubt there is a satisfaction in knowing that what is in fact one's work will continue to call forth admiration even though its admirers will cease to associate it with oneself. But that satisfaction might be enjoyed by many politicians and civil servants whose names will be unknown within a hundred years. (ii) Professor Hardy insists, probably with truth, that supreme achievement in mathematics is possible only at a comparatively early age. The only relevant instance which he gives is that of Newton. To produce, as he does, a list of persons who did supreme creative work in mathematics and then died young - e.g. Galois, Abel, Ramanujan and Riemann - is surely irrelevant. I suppose that the suppressed premise is that the work which they did before their early deaths was so stupendously great that it is incredible that they should have equalled it if they had lived.

The discussion of the intrinsic value of mathematics is somewhat rambling, but the gist of it is as follows. A mathematical theorem is a pattern of ideas. (We are not told what special kind of ideas are the constituents of specifically mathematical patterns.) If a theorem is to be non-trivial, it must have two characteristics, viz. beauty and seriousness. These are not independent, for the beauty depends to a considerable extent on the seriousness. Before attempting to analyse these characteristics Professor Hardy illustrates them by contrasting chess problems, which are genuine bits of mathematics but are essentially trivial and not particularly beautiful, with two very simple theorems which are both serious and beautiful. The examples which he takes are the proof that there is no greatest prime-number and the proof that there is no rational fraction whose square is equal to 2. These examples seem to me to be very happily chosen for his purpose; they are easy enough for any sane person to follow, and they are quite obviously weighty and beautiful. I think it would have been an advantage if Professor Hardy had distinguished more sharply between the proposition proved in a mathematical theorem and the reasoning by which it is established. In these two examples I should be inclined to feel that the beauty resides mainly in the reasoning and the seriousness mainly in the conclusion; but, no doubt, in more complicated examples each part would have a considerable share in both properties.

1.6. Review by: Eric Harold Neville.
The Mathematical Gazette 25 (264) (1941), 119.

In 1907 there were no Archimedeans at Cambridge, nor was there a mathematical club in Trinity, but one Sunday evening in a philosophical essay society which owed its existence to the hospitality of the Professor of Arabic a junior lecturer defended his vocation. So when a third of a century later Professor Hardy begins his apology by bemoaning the melancholy experience of finding himself writing "about" mathematics, we are not all wholly deceived. He has always enjoyed arguing "about" mathematics, and surely such talk is as legitimate a relaxation for a professional mathematician even in his most fertile years as chess or cricket.

The greater part of this booklet deals with the nature of mathematics and with the relation of mathematics to the life of the community. The "real" mathematics of the great mathematicians is contrasted with the "trivial" mathematics which is developed for the sake of applications. Had he not retired behind the impervious cover of the admission that "the most practical of mathematicians may be seriously handicapped if his knowledge is the bare minimum which is essential to him", Professor Hardy might be charged with underestimating the range of "real" mathematics which has been found "useful": conformal transformations and the properties of geodesics are far above the datum line of his academic contempt. But when every allowance is made, the demands of "trivial" mathematics are wholly insufficient to justify the study of "real" mathematics. What is more, the creative mathematician disclaims passionately, resents bitterly, the suggestion that a justification of his lifework can be found in the ugliest and most despicable outgrowths from the vulgar roots of his subject. If pictures and statues have no intrinsic value, the painter will not accept patronage as an interior decorator or the sculptor as an architect's assistant, and if we loathe music, we must not hope to placate the musician by our gratitude when we find that it is profitable to milk cows to a seductive melody.

The apology for a life devoted to "real" mathematics hangs therefore on the aesthetic value of the mathematics itself. Every mathematician believes this in his heart, and Professor Hardy, in the perfect prose of which he is master, defends and elaborates our common conviction. He even succeeds in describing in simple language some of the constituents of mathematical beauty.

But Professor Hardy's creed is vitiated by an irreconcilable contradiction, for it includes one article which would be disputed by many artists and which is, I hope, peculiar to himself among mathematicians. "Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune" (p. 26); "The best [mathematics] may, like the best literature, continue to cause intense emotional satisfaction to thousands of people after thousands of years" (p. 71). On the other hand, "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" (p. 83). That is to say, when Littlewood or Ramanujan showed him a theorem, the only reaction possible to Professor Hardy was "What can I make out of this myself? How does it help me in my own researches?" The spontaneous enjoyment of a beautiful result for its own sake, the generous congratulation of a friendly investigator, the ready appreciation of achievement in a branch of mathematics which he has not made his own - these, if we are to believe him, are outside Professor Hardy's experience. I for one do not believe him "(nor should I think the better of him if I did)." (p. 19).

1.7. Review by: Ananda K Coomaraswamy.
The Art Bulletin 23 (4) (1941), 339.

Everyone knows that mathematicians sometimes speak of perfectly formulated equations as "beautiful" and are excited by them as the connoisseur is excited by works of art. The present volume will be of the greatest interest and value to "aestheticians," since it is here for the first time that the "beauty" of mathematics has been discussed by a mathematician. Professor Hardy's analysis of this beauty is penetrating and illuminating, and in welcome contrast to the vagueness that is so characteristic of most modern writings on the criteria of beauty in other kinds of art.

"A mathematician, like a painter or a poet, is a maker of patterns. ... 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" (pp. 24, 25). "The best mathematics is serious as well as beautiful. ... The beauty of a mathematical theorem depends a great deal on its seriousness. ... A 'serious' theorem is a theorem that contains 'significant' ideas ... [for which]. ... There are two things at any rate that seem essential, a certain generality and a certain depth" (pp. 29-43). By generality it is meant "That the relations revealed by the proof should be such as to connect many different mathematical ideas ... [not one of] the isolated curiosities in which arithmetic abounds" (p. 44). Depth "has something to do with difficulty; the deeper ideas are usually the harder to grasp" (p. 49). In such beautiful theorems as those propounded by Euclid and Pythagoras "there is a very high degree of unexpectedness, combined with inevitability and economy ... the weapons used seem so childishly simple compared with the far-reaching results; but there is no escape from the conclusions" (p. 53). And thus Professor Hardy is "interested in mathematics only as a creative art" (p. 55).

Having so well defined what are in fact the essentials in any art, the author, who seems to be acquainted only with modern ("aesthetic") conceptions of art, naturally rates the beauty of mathematics above that of "art." He quotes without protest Housman's "Poetry is not the thing said but a way of saying it" - a pronouncement fit to make Dante or Asvaghosa turn in their graves. He takes an example from Shakespeare:

Not all the water in the rough rude sea
Can wash the balm from an anointed King

and asks "Could lines be better, and could ideas be at once more trite and more false? The poverty of the ideas seems hardly to affect the beauty of the verbal pattern." What the example really proves is, not that beauty can be independent of validity, but that beauty and validity are relative. There is nothing made that can be either beautiful or apt in all contexts. "Nothing is beautiful for any other purpose than that for which that thing is adapted" (Socrates in Xenophon Mem. iv. 6, 9). The example also shows that no pronouncement can be true except for those to whom its truth is apparent. To any Platonist or other traditionalist, and to the reviewer Shakespeare's words are beautiful and true, but they are not true for Professor Hardy or in any democratic context. And where they are not true, the mere fact that the sounds of the words is liked does not make them beautiful in the sense of the tradition that maintains that "Beauty pertains to cognition"; but only "beautiful" (or rather, "lovely") to those whom Plato calls "lovers of fine colours and sounds." Professor Hardy is not one of these; he confesses ignorance of aesthetics, but all he needs to do is to apply his own mathematical standards of intelligibility and economy to other works of art, and let the Housmans say what they will. "Ideas do matter to the pattern" (p. 31).

As an "Apology," Professor Hardy's book is a defence of real or higher mathematics against those who raise objection to their uselessness (in the crude sense of the word). All he need have said is that mathematics as a whole serves needs both of the soul and of the body, like the arts of primitive man and those which Plato would have admitted to his Republic. That the higher mathematics have served his own soul well is shown by his concluding statement that, if he had a statue on a column in London, and were able to choose whether the column should be so high that the features of the statue would be invisible, or so low that they could be clearly seen, he would choose the first alternative (p. 93); and since it is man's first duty to work out his own salvation (from himself), no further defence is needed. He makes it perfectly clear that he could not have "done better" in any other field; mathematics was his vocation. He was right to be a mathematician, not because he succeeded (p. 90), but rather, he succeeded because he did "what it was his to do, by nature," which is Plato's type of "justice" and in the Bhagavad Gita the way that leads to perfection.

1.8. Review by: Virginia Modesitt.
National Mathematics Magazine 16 (6) (1942), 311.

Although A Mathematician's Apology is addressed to non-mathematicians, it should appeal to any thoughtful man, mathematician or not. Its purpose is to answer the questions, "Why is it really worth while to study mathematics? What is the proper justification of a mathematician's life?" The answer to such questions can be given only by a man who is worthy of the name of "mathematician" and who is willing to give a frank appraisal of his own life. Professor Hardy can speak with authority and he has spoken frankly.

To justify a man's choice of mathematics as a career, Professor Hardy shows that such a choice offers an exceptional opportunity to gratify ambition, the driving force behind nearly all the best work of the world. He is speaking of men of ability and dismisses the problems of the ordinary man as unimportant.

To show the value of mathematics, Professor Hardy discusses the permanence of the ideas from which mathematical patterns are formed. The appeal of "serious" mathematics is compared with the appeal of the genuine but "trivial" mathematics of chess. Examples of "real" mathematical theorems are given to show the qualities of generality and depth which characterise "serious" mathematics.

A real mathematician, and Hardy is one, is interested in mathematics as a creative art, but no discussion of the worthwhileness of mathematics would be complete without considering its utility. This requires a statement of the distinction between pure and applied mathematics, between physical reality and the reality of ideas. Hardy distinguishes two mathematics, that which is useful, comprising the large bulk of elementary mathematics, and that which he calls "real" mathematics, most of which cannot be justified on the basis of its utility at all but rather as art is justified.

This is a small volume but a very compact one. It is a challenge to every reader to examine the justification of his own life as frankly as Hardy has done, as well as a personal and direct answer to the controversial questions which the author has proposed for himself in the beginning. A review of this particular book would not be complete without mention of one of its most outstanding characteristics. It is written in a perfect essay form and in a very simple, direct, and pleasing style.

1.9. Review by: John F Randolph.
The American Mathematical Monthly 49 (6) (1942), 396-397.

Here is a little book every mathematician should have in his circulating library. This book should be read and then loaned; first, to every young person with mathematical leaning, then to non-mathematicians, and finally to any mathematician who inadvertently has failed to but a copy.

The young mathematician will learn, probably to his surprise, that his very youth with its freshness and lack of restraint, is a much greater asset in his chosen field than the greater knowledge and experience that he hopes to accumulate over the years to come. He should realise after reading this book how important it is for him to pour what he has into an all out effort at the very beginning of his career. At that time, according to Hardy, he is more likely to win fame and place (but not fortune) for himself than at any later date. An old mathematician, as an old tennis player, may be a fairly young man.

A non-mathematician may be well repaid for the two of his hours spent with this book, but he would profit relatively more id he invested five or six. Even the person capable of the remark "Who cares if there is an infinity of prime numbers?" may feel a sense of satisfaction after Hardy has led him to this trough and made him drink. Of course the non-mathematician will puzzle over the sentence "Real mathematics has no effect on war," but it will be good for his soul to be jarred from his notion that school mathematics is the ultimate in mathematics. He will, of course, not see what lies beyond but must realise that there is something to be seen.

A mathematician needs no apology for mathematics, but he can read this book and after every section feel a deep sense of gratitude to one of his greatest contemporaries for putting the cause so clearly and with such force. He may not have had the nerve to say for himself, but he will be moved when he reads "and there are probably more people really interested in mathematics than in music."

It is not to be expected that all mathematicians will agree with Hardy on every point. There may, for example, be mathematicians who object to Hardy's use of the term "trivial mathematics," and certainly there are "real mathematicians" who would-not feel they were criticising the bible if they threw a little mud at some "real mathematics."

This book is not only about mathematics, it is about ideals, art, beauty, importance, significance, seriousness, generality, depth, young men, old men and G H Hardy. It is a book to be read, thought about, talked about, criticised, and read again.

2.1. Review by: Carl Douglas Olds.
Mathematics Magazine 41 (3) (1968), 155-156.

This is the long awaited reissue of the Apology which first appeared in 1940. Hardy sets the theme of this exciting, beautifully written essay on the firth page of the Apology. He writes: "I shall ask, then, why is it really worth while to make a serious study of mathematics? What is the proper justification of a mathematician's life? And my answer will be, for the most part, such as are to be expected from a mathematician. ... But I shall say at once that, in defending mathematics, I shall be defending myself, and that my apology is bound to be to some extent egotistical.

Hardy's defence of mathematics can be read by anyone; except for a theorem of two there is nothing here that the layman cannot understand. However, in order to fully appreciate the Apology one should know something about Hardy as a man, and also have some knowledge of his mathematical achievements.

Thus the reviewer was extremely pleased to see a forward included in this edition which gives an excellent sketch of Hardy's life. C P Snow traces Hardy's life from his birth in Surrey in 1877 until his tragic death in 1947. We learn about his parents and about his early education. We follow his career as a Fellow at Trinity College, Cambridge (1898-1919); his stay at Oxford (1919-1931); and his return to Cambridge. Snow tells of the one romantic incident in Hardy's life, his discovery of Ramanujan, the poor Indian clerk born with a genius for mathematics.

From Snow's sketch Hardy emerges as a strange man, an eccentric who hated telephones and watches, and wouldn't have a looking-glass in his room; a man so shy that he dreaded introductions, but brave enough to stand up for his beliefs; a man with two strong passions, mathematics and cricket.

Hardy hated war and this perhaps explains to some extent why he regarded applied mathematics (ballistics, for example) as "repulsively ugly and intolerably dull." He writes: "I have never done anything 'useful'. No discovery of mine has made or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." This, of course, is not true, for it has often been pointed out that some of Hardy's own pure mathematical discoveries have found applications in applied fields (see Newman, The World of Mathematics, p. 2026). This is one of the attractions of the Apology; the reader quickly finds himself in agreement with, or quite opposed to some of Hardy's statements.

Hardy was a pure mathematician whose main work was in analysis and the theory of numbers (or the higher arithmetic). His great reputation, of course, rests on his very original and advanced research papers, many of the most famous of which were written in collaboration with J E Littlewood, and with Ramanujan. The first volume of his Collected Papers has been published by the Oxford University Press. This is a volume of 700 pages; eventually there will be seven volumes.

Snow's moving, beautifully written story of Hardy's last years explains clearly why many regard the Apology as he does as a "book of haunting sadness." Hardy died December 1, 1947, the day the Copley Medal of the Royal Society, its highest award, was to have been presented to him.

There is a striking photograph of Hardy on the jacket of the book, and a sample of his writing style on the back. Since jackets get torn and lost, it is a pity the photo does not face the title page. Since he hated to be photographed there are few pictures of him.

We conclude this review by quoting from the Scientific Monthly: "Hardy's sardonic confession of how he ever came to be a professional mathematician may be specially recommended to solemn young men who believe they have the call to preach the higher arithmetic to mathematical infidels ..."

2.2. Review by: Howard Eves.
The Mathematics Teacher 62 (5) (1969), 423.

It is rare for a top-flight mathematician to write an account of what mathematics has personally meant to him. We have some such excursions made by Hadamard, Poincaré, Russell, Weiner, Mordell, and a few others, and the discussions have always constituted extremely interesting reading. A very valuable and more lengthy addition to this scant literature is 'A Mathematician's Apology', by G H Hardy, one of England's foremost mathematicians of the early half of the present century, and in his prime (by his own honest evaluation) probably the fourth- or fifth-best mathematician in the world at the time.

This moving little book is written with disarming candour and honesty, exquisite beauty and insight, and a touch of the sardonic and the sad. It attempts to answer such questions as these: Why is it really worthwhile to make a serious study of mathematics? What is the proper justification of a mathematician's life? In particular, how can a mathematician who has spent his entire life on "useless" mathematics justify his existence in this world? What about the evil and harm that some mathematics can do, especially in connection with war efforts? What is the ultimate test of "good" mathematics? How does nontrivial mathematics differ from, say, a game of chess? What makes one theorem of mathematics more significant or more "serious" than another? How does "real" mathematics differ from merely "useful" mathematics? These are all questions which must, at one time or another, bother a conscientious devotee and creator of mathematics.

While dealing in his book with deep and some times disturbing questions like those cited above, Hardy drops many pearls by the way. Thus he gives simple examples of those elusive qualities of "beauty" and "elegance" so desired by a mathematician in his mathematics. And consider what Hardy insightfully says about the 'reductio ad absurdum', or indirect, method of reasoning, so frequently employed by mathematicians: "It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game."

The present edition of Hardy's little work, which is a reprint of the original 1940 edition, is augmented and enriched by a lengthy foreword (almost a third of the entire book) from the pen of Hardy's good friend, C P Snow. This foreword is charmingly written and tells us much about the life and character of Hardy. Here we read about Hardy's childhood, his introduction to mathematics, his early years at Cambridge, his great love of cricket, his remarkable collaboration with Littlewood, his discovery of Ramanujan, and the very sad circumstances attendant upon his last days and his attempted suicide. Those whose space-time graphs at one time or another intersected that of Professor Hardy will be particularly grateful to Dr Snow for his warm, understanding, and sympathetic foreword.

2.3. Review by: Viggo Brun.
Nordisk Matematisk Tidskrift 16 (1-2) (1968), 35-36.

The first edition of G H Hardy's book was published in 1940, the second and third in 1941 and 1948. The fourth edition was published in 1967, augmented by a foreword by C P Snow of over 50 pages.

Since Hardy's "Apology" already belongs to the classic and highly esteemed works, I shall settle for a brief mention.

Hardy seems to be greatly gripped by the problem of his age: "I write about mathematics because, like any other mathematician who has passed sixty, I no longer have the freshness of mind, the energy, or the patience to carry on effectively with my proper job. ... No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game."

I think it is dangerous to generalise as Hardy does here, although one should also beware of the misjudgement that J Sylvester should have been guilty of, always claiming that the last paper was the best he had written. The truth lies well in between. At least I can attest that "the energy and the patience to carry on" can survive the 80 years.

Hardy's book contains many funny remarks. Let me be careful with two: "Good work is not done by 'humble' men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. " - "There is no instance, as far as I know, of a first-rate mathematician abandoning mathematics and attaining first-rate distinction in any other field."

Snow's foreword deepens the impression one gets through Hardy's portrayal of this pioneering mathematician and the original human being. Snow met Hardy for the first time in Cambridge in 1930: "You are supposed to know something about cricket, aren't you?" Hardy said. And so the friendship was formed. Snow says of Hardy: "He was unorthodox, eccentric, radical, ready to talk about anything." - "This was intellectually the most valuable friendship of my life. His mind was brilliant and concentrated: so much so that by his side anyone else's seemed a little muddy, a little pedestrian and confused."

2.4. Review by: George Daniel Mostow.
American Scientist 58 (4) (1970), 448.

"It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art critics, and physiologists, physicists, or mathematicians have usually similar feelings; there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds."

With these words, Hardy begins the greatest mathematical exposition in the English tongue. This monograph was first published in 1940 as England entered World War II; it was reprinted in 1969 with a 50-page foreword by C P Snow, who had Hardy in mind when he wrote 'The Masters'.

This book is a basic contribution to the question of "relevance" which publicly concerns our young today and which Hardy here confronts in his old age.

2.5. Review by: Thomas Arthur Alan Broadbent.
The Mathematical Gazette 52 (379) (1968), 59.

No reader of the Gazette should need to be told that Hardy's 'Apology' describes, with intense charm, and justifies, with lucid candour, the mathematician as creative artist. If at times it is near enough to proclaiming "art for art's sake" to have been derided as "cloistral clowning", its brilliance and integrity have won it admirers whose aesthetic perceptions are not primarily mathematical. The publishers quote Graham Greene: "I know of no writing-except perhaps Henry James's introductory essays - which conveys so clearly and with such an absence of fuss the excitement of the creative artist". High praise, but few readers will disagree with it.

Hardy has been dead for twenty years; there must now be many mathematicians who never heard him lecture, many to whom Pure Mathematics is an outmoded text, not the doorway it was, fifty or sixty years ago, through which the young English student entered the realm of modern analysis. For them, C P Snow's foreword, which has also appeared in his book, Varieties of men, will do something to help them to see Hardy the man. "Unorthodox, eccentric, radical, ready to talk about anything"; loving cats and cricket, hating dogs, telephones and looking-glasses. Some of this may have sprung from his shyness, his dislike of pompous, pretentious people, his contempt for shams. Snow speaks of Hardy's "introspective insight and beautiful candour", qualities which are clearly shown in the Apology. One other quality Snow finds there, a "haunting sadness"; when it was written, Hardy was a little over 60, and the creative power which he had exercised with so much delight was on the wane. This was hard to bear: he had been in "the Bradman class" and shrank from the substitute of Saturday afternoon cricket on the village green. But even in his sadness he could write "I have added something to knowledge, and helped others to add more"; what nobler epitaph could a mathematician crave?

2.6. Review by: Victor F Weisskopf.
Scientific American 218 (3) (1968), 148.

This touching, candid and sharp self-portrait by an ageing artist - one can describe Hardy in no other way - appears after 10 years out of print, with a brief new account of the man by an old friend.

3.1. Review by: John A Dossey.
The Mathematics Teacher 85 (9) (1992), 766-767.

A Mathematician's Apology is perhaps the most concise piece every written describing the creative process of doing mathematics. Its author, G H Hardy, a world-class theorist of the period between the great wars, describes his attraction to mathematics, defending the role of the pure mathematician. In this apology for his life spent chasing theorems of little application Hardy states that "a mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas ... and so his patterns are likely to last longer, since ideas wear less with time than words."

Hardy's retrospective analysis of his career and his conception of mathematics dwells on the distinctions he would hold for pure and applied mathematics, of beautiful and ugly mathematics, of nontrivial and trivial mathematics. His journey through developing these comparisons tells much about his inner view of the creative act that world-class mathematics requires. It also paints a picture of mathematics as a discipline at Cambridge in the 1930s and the 1940s. But, most of all, it tells the story of a mathematician who recognises that his creative powers are beginning to slip away and comes to the stark realisation of living life without his one great love.

3.2. Review by: Rajendra Prasad.
Social Scientist 21 (3-4) (1993), 98-101.

Scientists working in the frontier areas of their disciplines quite naturally become involved in questions that go far beyond their own disciplines. Boundaries became hazy, nature is found in tension, dynamic and energised (Goethe). The response to the infinite inexhaustibility of cognition depends on a whole gamut of factors including personal predilections and familiarity with contemporary thought.

This is best illustrated by G H Hardy's A Mathematician's Apology .... Hardy was one of this century's finest mathematical thinkers. And in the biographical notes on Hardy, C P Snow notes another characteristic of Hardy that marks him apart from any other great genius: 'That is at turning any work of the intellect, major or minor or sheer play, into a work of art.' ...

Hardy did most of his creative work in collaboration. In 1911 he began a collaboration with Littlewood which lasted 35 years and they produced around 100 papers which dominated pure mathematics for a generation. His second and perhaps more well known (at least in India) collaboration was with Ramanujan. Between 1914-18 Hardy worked in collaboration with this great Indian genius, in discovering whom he had played a singular role. Of these two men Hardy himself says in the book: 'I still say to myself when I am depressed and find myself forced to listen to pompous and tiresome people, "Well, I have done one thing you could never have done, and that is to have collaborated with Littlewood and Ramanujan on something like equal terms".'

In poetic, precise and lucid style, Hardy describes how he came to choose the career that he did and what it is to do great mathematics. He observes, 'If intellectual curiosity, professional pride and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of gratifying them than a mathematician. His subject is the most curious of all-there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivalled openings for the display of sheer professional skill. Finally, as history proves abundantly, mathematical achievement, whatever it intrinsic worth, is the most enduring of all.'

Hardy's position on the nature of 'mathematical reality' is quite unambiguous. '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.' The same clarity of mind is reflected in his total rejection of idealism. 'It may be that modern physics fits best into some framework of idealistic philosophy - I don't believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 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.' Characterising it as a 'testament of a creative artist', C P Snow also finds Apology a book of 'haunting sadness' because of a 'passionate lament for creative powers that used to be and that will never come again'.

3.3. Review by: Jessica Sekhon.
Math Horizons 1 (1) (1993), 20-21.

Staring at the ceiling of the Sistine Chapel, the majority of us are left awestruck. We see past the cracks and imperfections to the grand figures and majestic scenery. We appreciate the work and admire the ingenuity. Yet, at the same time, we think of mathematics as a dull subject used in building bridges, launching rockets, and designing buildings. We seldom look past the grade-school equations and formulas. We fail to recognise the beauty and power of the proofs presented by Archimedes and Euclid centuries ago. True mathematics is about elegant proofs that invoke serious reasoning and stir strong emotions. G H Hardy's A Mathematician's Apology embodies this theme. ...

Anyone planning on studying mathematics should read 'A Mathematician's Apology'. The book is written with extraordinary clarity, interest and enthusiasm. G H Hardy demonstrates that the proofs provided by Pythagoras and Euclid have stood the test of time. They are just as beautiful and logical today as they were more than two thousand years ago. He not only explains why one should study mathematics, but also shows its aesthetic value and where the breadth of the subject matter lies. The story is riveting and the proofs provided continue to invoke passion and fascinate the reader.

3.4. Review by: Walter Warwick Sawyer.
The Mathematical Gazette 78 (483) (1994), 348-351.

I first read this book about 50 years ago. I was forcibly struck by a passage in the opening paragraph:

There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.

At the time I thought "He seems to have lost concentration here; let's hope it will be better later on". Since then I have met some mathematicians who agree with this view of Hardy's, and I am now convinced that this statement was not a temporary feeling, caused by Hardy's distress that illness had deprived him of the power to do the research in which he delighted, but - as is clear from statements that come a little later - was an opinion he had held firmly throughout.

That Hardy was a very great mathematician is beyond question, and I do not intend to question it in any way. However, when any person eminent in some field makes statements outside that field, it is legitimate to consider the validity of these statements, as any teacher must certainly wish to do in relation to the passage quoted above.

Teaching

Of his own preferences Hardy writes:

I hate 'teaching'. ... I love lecturing, and have lectured a great deal to extremely able classes.

Here lecturing means imparting mathematical knowledge to those able to understand it with little or no difficulty; teaching means giving time and effort to make it accessible to those who require assistance. There is nothing wrong with Hardy's preference in this matter. I knew two teachers at a school. One was a mild clergyman who taught Greek successfully to the Classical Sixth, the other a good footballer who kept order and conveyed some learning to the boisterous spirits of the First Form. Both were providing a service; neither could have done the other's job. Good administration consists in appreciating the merits of a wide variety of individuals and combining them into an effective team. Now it is precisely this appreciation that Hardy lacks. He makes the extraordinary statement Most people can do nothing at all well.

One gets the impression that he regards you as doing well only if you are one of the ten best in the world at this particular activity. The theorem that very few people do anything well is an immediate logical consequence. However in life we continually depend on the co-operation of men and women far below this exacting standard. Even the publication of mathematical research depends on publishers, editors, printers, postmen and others, who need to be conscientious and competent, but rarely need to have truly exceptional qualities. Indeed this is true even of the social process that links the great mathematicians of one generation to those of the next. There may of course be direct contact, as when Riemann was a student at Gottingen University under Gauss. But the fact that Gauss was able to reach university at all was due to two teachers, Buttner, who recognised Gauss' extraordinary ability when Gauss was only 10 years old, and Bartels, who not only worked at mathematics with Gauss but also told influential citizens about him, as a result of which support was provided for Gauss' further education. In science the importance of the expositor is perhaps as great as that of the discoverer. Mendel's work in genetics remained unknown for many years because there was no one to publicise it and fight for it as Huxley did for Darwin. In the arts, critics play a similar role. Once, when I was a student, Leavis spoke to a college society on recent poetry. Prior to this meeting I had regarded modern poetry as pretentious nonsense, much as I still do for - not all, but most - modern art. He explained the allusions in some obscure poems and showed that these conveyed, very powerfully, intense emotion. This is what good criticism does - it makes available to a wider audience artistic work that would be meaningless without such intervention.

It is interesting that Hardy had such a poor opinion of teachers when he himself possessed an important quality of a good teacher - the ability to recognise a correct idea behind a very imperfect statement of it. Some writers deal very scornfully with the work of analysts before the age of rigour. Hardy however finds essentially correct, even very modern, ideas in the work of Euler, even though these are sometimes expressed in a way that would not find favour today.

Ambition

In Hardy's scheme of thought, ambition plays an important role. Now ambition has two sides. There is the intrinsic aspect, the desire to solve some significant problem, and the personal reflection of this - the hope that by so doing you will become famous and be remembered, perhaps for centuries. The second aspect is certainly important for Hardy. He makes this curiously objective division of mankind into minds that are first-class, second-class and so on; the ambitious person is concerned about where he fits into this scheme. There is no part of this that should be accepted as sound advice. If is something you think worth doing, that you are able to do, that you opportunity to do, and that you enjoy doing, wisdom lies in getting on with not giving a second's thought to what ordinal number attaches to you system of intellectual snobbery. As for concern with the self, you happiest and most effective when you are so absorbed in what you are doing for a while you forget the limited being that is actually performing it.

Utility

Hardy is very anxious to show that the value of mathematics lies in its beauty, not in its practical consequences. Real mathematics is that "which has permanent aesthetic value". On the other hand, "It is what is commonplace and dull that counts for practical life." In connection with 'real' mathematics he writes:

I was not thinking only of pure mathematics. I count Maxwell and Einstein, Eddington and Dirac among 'real' mathematicians.

It is not surprising to find Einstein, Eddington and Dirac classified as remote from practical consequences; when this was written, atomic theory had produced neither bombs nor medical applications. The surprising inclusion is that of Maxwell. When, after a delay of 25 years, his at first derided prediction of radio was confirmed by the physicists, this had immediate and important practical consequences - for example, aiding rescues at sea - and later profound social consequences, brought by the coming of wireless. The thesis that only dull mathematics has practical consequences can surely not be maintained.

Mathematical beauty

Of great interest are Hardy's discussion of mathematical beauty, and the characteristics of a good mathematical theorem. Essentially he says that a good theorem is simple, surprising and fruitful. He contrasts very effectively mathematical theorems "that connect many different mathematical ideas" with the special puzzles of mathematical recreations. He believes that nearly everyone is capable of appreciating mathematical beauty, and cites the popularity of mathematical puzzles in newspapers;

What the public wants is a little intellectual 'kick', and nothing else has quite the kick of mathematics.

It is interesting that the term, 'kick', used here is often heard in relation to drugs, and indeed there can be a danger of addiction to mathematics, the temptation to indulge in mathematics and neglect important, but much less exciting, mundane activities.

3.5. Review by: Devendra A Kapadia.
The Mathematical Gazette 77 (478) (1993), 104-105.

Hardy's apology for the pursuit of mathematics offers an excellent insight into the mind of a great mathematician. It is also a literary masterpiece and, though certain portions of the book appear to be dated, the clarity and precision that characterise Hardy's exposition make this reprint worth reading.

C P Snow's lengthy introduction complements the rest of the book very well. In particular it highlights Hardy's despair on being old and, therefore, useless for doing any original mathematics.

Hardy candidly admits that his defence of mathematics is a defence of his own life, namely, the life of a pure mathematician. His views on pure mathematics are unlikely to be challenged in the near future:

a) The "useful" parts of mathematics (e.g. Arithmetic, Statistics, Elementary Differential and Integral calculus, etc.) are the parts with no real aesthetic value. The life of a real mathematician cannot be justified on the grounds of the utility of his work alone.

b) The seriousness of a mathematical idea is intimately connected with its beauty.

c) A theorem should be judged by the depth of ideas contained in it as well as the generality of its applications. He cites the example of the Pythagorean proof of the irrationality of √2 which satisfies both these criteria in an admirable manner.

d) Mathematical ideas have a permanence and "reality" which no physical theory can hope to achieve.

Hardy is on less firm ground when he discusses applied mathematics. He counts Maxwell, Dirac and Einstein among "real" mathematicians on the basis that their mathematical work has permanent aesthetic value. This is certainly true, but, contrary to Hardy's conjecture, relativity and quantum mechanics have proved to be extremely useful in the modern technological era. Thus, there may be a role for the real mathematician to play in shaping advanced physical theories. Roger Penrose is an excellent example of a mathematical physicist who believes in a close link between the correctness of a theory (physical or mathematical) and its aesthetic content.

A mathematician's apology is bound to remain a popular choice for teachers who wish to initiate their students into mathematical thought.

4.1. Preface by annotator Alan J Cain.

Although G H Hardy, in his mathematical writing, was 'above the average in his care to cite others and provide bibliographies in his books', A Mathematician's Apology is filled with quotations, allusions, and references that are often unsourced.

This annotated edition aims to give sources for all quotations and clarify allusions to works, people, or events, as well as adding background information. Hardy made a number of minor misquotations, suggesting that he quoted from memory or used paraphrased notes of his own; the annotations point these out. This edition also includes an annotated version of Hardy's essay 'Mathematics in war-time', which formed the kernel around which Hardy shaped the Apology. The annotations point out how parts of this essay were incorporated into the Apology.

In both the Apology and 'Mathematics in war-time', Hardy's original footnotes are preserved and marked with an asterisk * or a dagger †. The annotations are in numbered footnotes.

Also included is a list of editions, excerptings, and translations of the Apology and 'Mathematics in war-time', and three essays by the annotator: the first sets the Apology in context in the debate about the justification for mathematics, particularly as an aesthetic pursuit; the second attempts to survey comprehensively contemporary reviews of the Apology, the third examines the legacy and ongoing influence of the Apology. This edition includes a unified bibliography for the Apology, 'Mathematics in war-time', the annotations, and the essays. Also included is an index, which has been lacking in previous editions.

The annotator thanks Yumi Murayama for reading and commenting on this edition, and Erkko Lehtonen for supplying details of the Finnish translation and for pointing out typos.

Finally, the annotator feels obliged to point out that he is fully aware of the irony of producing annotations and commentary on a work whose author wrote that 'exposition, criticism, appreciation, is work for second-rate minds'.

Alan J Cain
Lisbon,
21 January 2019