A Mathematician's Apology (Canto Classics) Reissue Edition

"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fields which I find indefensible in an increasingly democratized information economy.Still other arguments are rather outdated.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.

There is a deep mystery about this book – it has been widely praised and remained in print for nearly 80 years over 22 editions since its publication in 1940. The question is why? An answer will be attempted at the end of this review, especially as it is not well organized and meanders across many topics.It has no clear structure or direction, bouncing lightly over many tough ideas while promising an uninformative autobiography. Fortunately, the esteemed novelist, C. P. Snow wrote a 60 page foreword that was added to the 4th edition in 1967. Snow wrote a sympathetic synopsis of the life of his friend Godfrey Harold, so we can see some of his better aspects of his character, like their mutual obsession with cricket and the privileged lives of Cambridge university ‘dons’. Otherwise, there would be almost no biographical details but we would be left with the impression of a very proud man (“good work is not done by humble men” [p.66]), who had few interests outside his professional research. Hardy was always a very conventional member of the English elite, defending ambition as a worthy motivation. Cambridge was dedicated to educating the best of the next generation of the ruling class, especially in science and mathematics. He wrote an “apologia” because he was obsessed with the question: “Is mathematics worth doing?” Hardy surprised himself by having to defend mathematics as he admits it is “generally recognized as profitable and praiseworthy” but acknowledges that he is really defending himself and his obsessive dedication, with only stellar astronomy and atomic physics, as sciences standing in higher esteem in the popular estimation.Much of the book is dedicated to convincing the non-mathematician that there are (at least in Hardy’s eyes) two types of mathematics: Pure (or ‘real’) mathematics and all the rest! Only a few real mathematicians can truly appreciate (or even recognize) pure mathematics. Most educated people will only learn “trivial” or simple ‘school’ mathematics and even those whose technical careers depend on mathematics (such as engineers and physicists) will only learn ‘applied’ mathematics, such as integral and differential calculus (but these topics are both “dull and lacking in aesthetic appeal”). Although admitting that a definition is difficult, he later lists modern geometry and algebra, number theory and the theory of functions as good examples of pure mathematics. This is confirmed in his eyes by declaring them “useless” – a critical feature for him, as such topics are usually ‘harmless’ (Hardy was a notorious pacifist along with Bertrand Russell). He believes that ‘real’ mathematics is “serious” because its theorems play significant roles in other, major mathematical areas. The book contains two sections where ‘real’ mathematical theorems are discussed: these include Euclid’s proof of the infinitude of prime numbers and Pythagoras’s proof that the square root of two cannot be expressed as a rational fraction (e.g. n/m); in other words, the shattering realization that there are quantities that are not related to integers (“counting numbers”). This fascination with prime numbers (not divisible by any other number) lies at the heart of Number Theory, which leads to the ‘fundamental’ theorem of arithmetic, where every integer number can be expressed as a unique multiplication of prime numbers. Most people visualize non-prime numbers as rectangular arrangements of rows of equal numbers of objects. Hardy makes no mention of Descartes’ radical invention of “real” numbers (like 2.7134) that allowed all physical quantities to be assigned to such numbers; vital to all modern physics with its arbitrary units of measure.Perhaps, the reason for this omission was that Hardy viewed physics as too closely linked to material reality; in fact, he makes the extreme claim that mathematicians are much more in direct contact with reality (they call it “mathematical reality”) than physicists (ignoring the key role of experimental physicists that make physics an empirical science). It is convenient for him that he counts famous physicists, such as Maxwell, Einstein, Eddington and Dirac as “real” mathematicians. He is more impressed with the properties of specific numbers, like the ‘fact’ that ‘317’ is a prime (whereas most people would respond: “So what?”). He is proud to be associated with Plato’s views of ‘deep’ reality; not surprisingly, as Plato was also a fanatic follower of the ancient religious mystic, Pythagoras. In fact, Catholic intellectuals have also pointed to mathematics as perfect examples of their own timeless world created by their God (ironically, Hardy was a harsh atheist). All these intellectual mystics deny that mathematics (and theology) were just mental constructions but lie objectively outside of all people; failing to see that artists too do not ‘discover’ their original creations but reflect the hard work of their communicable imaginations, reflecting socially evolving ideas. Most mathematicians allude to the generality of arithmetic, like 2+3=5. Once again, they fail to think deeply about their basic ideas, such that integers are the common results of the physical act of counting stable, distinguishable existents. In fact, a small number of mathematicians (the “constructivists”) believe that mathematics is just an extended intellectual analysis of abstract definitions; this certainly covers much of Euclidean geometry, which has become the classic example of what constitutes a logical proof.The true motivation for this book was Hardy’s psychological crisis brought on by his awareness that his creative abilities had deserted him. At first, he tried to commit suicide but took too many sleeping pills; he reconciled himself with the view that “mathematics is a young man’s game” and he was still quite creative in his forties. Although Hardy recognized that pure mathematics was much like great art “in promoting a lofty habit mind”, he believed that it was superior due to its greater demonstrable ability to endure through many centuries and civilizations. Like many well-educated intellectuals, he was a snob; indeed, he admits that he is only interested in mathematics as a “creative art”, so that pure mathematics does not have to be ‘useful’ (i.e. “increase humanity’s well-being or comfort”).So, why does this book keep finding readers? In my own case, it was to learn more about his famous relationship with the Indian mathematical genius, Ramanujan after seeing the movie “The Man who knew Infinity”. My own theory here is that generations of mathematics teachers and professors recommend it to their better students, hoping it will help commit them to this ancient profession. In fact, the abstract symbolism and definitions of mathematics have helped make it the ideal subject for secondary education across all societies. Mathematics is harmless and useful and easy to teach, and easy to mark: giving an ‘objective’ measure of the student’s intelligence (so many believe – but like most educational subjects: a good memory is important). In fact, as a manmade invention (I am an Aristotelian, not a Platonist) the finitude of mathematics actually makes it much easier (to learn and remember) than science, which is challenged by the complexity and open-ended characteristics of Nature. Indeed, the timeless nature of mathematics both preserves its structures and appeals to many intellectuals, who (like Plato) were threatened by the inevitability of their personal existence.

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.

Maybe it is because it is so frequently cited as the ultimate defence of mathematics, maybe its because I don't really need to be convinced that pure research is of interest and value: but while I enjoyed this I was left feeling slightly underwhelmed at the end.Famously Hardy cites examples of mathematics that he says will never be of practical use, only for them all to be of practical (or at least applied) use in the 80 years that have followed, but it's not that I think he is selling mathematics short - his argument stands regardless. It's more that I hoped for something in here I hadn't previously considered and I didn't really find it.

There's no doubt that Hardy was a brilliant mathematician. My problem is that I didn't agree with his conclusions about mathematics being useless because they were almost immediately proved wrong by the works of Turing and others in solving the Enigma code, and indeed the whole business of digital computing.In addition, his attitudes are very much of their time. A bit elitist and maybe even conceited.But at least he helped Ramanujan.

"Exposition, criticism, appreciation, is work for second-rate minds." Already, in the first paragraph, G. H. Hardy is deploring the task of writing about mathematics in his characteristically forthright fashion. It's just as well that by the time we begin the Apology we have been softened up by C. P. Snow's excellent introduction and potted biography. Clearly, this first-rate book has attracted an altogether better class of reviewer, the self-effacing type not given to hissy fits on being reminded of their place in the intellectual pantheon."I had of course found at school, as every future mathematician does, that I could often do things much better than my teachers". Hardy's later achievements and his matter-of-fact style ensure that this is neither preening vanity nor a pompous boast. A professional mathematician might also agree that the "function of a mathematician is to do something" and not to talk about it. Mathematics as an active pursuit, being cleverer than your maths teacher - these count as revelations to ordinary mortals, even those of us who weren't too bad at maths. Then, and before any unsuspecting non-mathematician can run for cover, Hardy sets about proving "two of the famous theorems of Greek mathematics". There is really nothing to be scared of, even for the most equation-phobic humanities graduate. It's the ideas and the arguments that link them that matter, and they are not difficult to follow. In tracing the steps of Euclid and Pythagoras we are tracing patterns of thought that have lasted two thousand years, and we too can directly appreciate their beauty, and see for ourselves in a small way that a "mathematician, like a painter or a poet, is a maker of patterns."Hardy does not take kindly to the commonplace idea "that an academic career is one sought mainly by cautious and unambitious persons who care primarily for comfort and security." While he is, unsurprisingly, motivated by "intellectual curiosity" and a "desire to know the truth", he also admits to "professional pride", "ambition" and a "desire for reputation". A lesser mind might have been tempted to feign guilt over such worldly traits, but Hardy has only a good word for ambition, the "noble passion". And his own noblest ambition? That "of leaving behind... something of permanent value". No dreams of heavenly bliss for this atheist.Just when you might be thinking that all this talk of reputation and ambition must arise from an insufferable self-centredness, he declares that much of his best work was done in collaboration with two other mathematicians, Littlewood and Ramanujan, from very different backgrounds. Hardy's recognition of the unknown Indian was not inevitable: two other eminent Englishmen had returned the manuscripts without comment, on the assumption that Ramanujan was a crank. That too was Hardy's first impression, but he soon changed his mind and saw in Ramanujan a brilliant if untutored mathematical mind. It is a remarkable story by any standards, and has been recently staged as "A Disappearing Number" - a brilliant production in which a rather battered copy of this very edition gets a turn in the limelight.What was a "melancholy experience" for Hardy (writing about mathematics) provides a rewarding experience for us. Graham Greene considered this the best account of what it is like to be a creative artist. I don't know if Greene is right, or if Hardy is right in his belief "that mathematical reality lies outside us," waiting to be discovered. I defer to their judgements but can better appreciate their conclusions after reading this book. C. P. Snow describes Hardy's "mocking horror of pretentiousness, self-righteous indignation, and the whole stately pantechnicon of the hypocritical virtues." More intriguing, given Hardy's hatred of God and all the pious nonsense carried out in God's name, and given that the spiritual side of human nature has been unthinkingly yoked to religious mumbo jumbo for far too long, is Snow's description of Hardy "as spiritually delicate" and "spiritually candid as few men are".The dominant sense of "apology" implies a fault for which contrition is being expressed. Hardy's Apology is no craven exercise in self-abasement but a serious and vigorous justification of the intellectual and creative life, whether led by a mathematician or anyone else.

This essay by one of the great pure mathematicians is rightly famous, but not for the right reasons. The author's central thesis - that real mathematics is, like the other forms of art, wholly useless - was shown to be wrong shortly after his death. The "wholly useless" theory of numbers, in which Hardy spent most of his professional life, is in fact of paramount importance these days. When you buy this book from Amazon the only reason you can be assured that naughty people won't steal your credit card number in transit is because of work done by pure mathematicians, and Hardy's own work has proven to be important in physics.Hardy is writing for the non-mathematical layman here, so the book is very approachable, with only a minimum of elementary mathematics in it, which he provides as examples, and all of which should be accessible to anyone, including small children and Media Studies students. His intention is to provide a view into the mind of "real" mathematicians and explain the fascination that some people have with his "wholly useless" subject. And I suppose he does a decent job of that.But in my opinion, the best bit is the foreword by C. P. Snow, which first appeared in the 1967 edition, 20 years after Hardy's death. That is a clear, touching - but critical in parts - portrait, and would be worth reading on its own. Hardy's essay is just a bonus.

If you have already encountered A Mathematician's Apology, I very much doubt you will demur from my review title. If not - read at once. Little need be said except the basics: this is an insight into mathematics, and why mathematicians persue it, by a leading professional mathematician of the early 20th century. Hardy was, however, a very unusual personality in many ways by the standards of mathematicians. His strange personality and, oddly for such an avowed atheist, his very strong spirituality, make for a poetic yet precise approach to his topic. The preface by C.P. Snow is a masterpiece of character study, and essential reading so as best to appreciate Hardy's own thoughts. The only drawback of the Kindle edition is the lack of the splendid cover picture used for so many years on the Cambridge University Press print editions. By the way, no experience or ability in mathematics is needed to enjoy, and benefit from, this book. Quite the contrary, both mathematicians and those to whom mathematics is a closed book will relish Hardy's work in different ways. That is the remarkable achievement of this niche classic.