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BySphexon 27 November 2008

"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.

"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.

ByKindle Customeron 5 October 2017

Hardy was a great mathematician but it is unfortunate that he should have written a book like this. The results of many theorems have been applied in finance and commerce in the 21st century that it makes the predictions in this book dull and unimaginative. Wiles proved Fermat theorem in his 40s; Thankfully, he did not feel the same about age as Hardy. Let us hope that he is proven wrong again if and when Riemann hypothesis is proven by an AI machine or an elderly mathematician way past his prime (in age that is).

BySphexon 27 November 2008

"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.

"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.

ByKindle Customeron 5 October 2017

Hardy was a great mathematician but it is unfortunate that he should have written a book like this. The results of many theorems have been applied in finance and commerce in the 21st century that it makes the predictions in this book dull and unimaginative. Wiles proved Fermat theorem in his 40s; Thankfully, he did not feel the same about age as Hardy. Let us hope that he is proven wrong again if and when Riemann hypothesis is proven by an AI machine or an elderly mathematician way past his prime (in age that is).

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ByAmazon Customeron 7 August 2017

an very interesting philosophy

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ByJ. E. Finchon 27 December 2005

This is a delightful read. The foreword by C.P. Snow takes up approximately one-third of the book, and is effectively a short biography of Hardy. It follows his life from late Victorian public school, to Trinity at Cambridge, then to New College Oxford, and then back to Cambridge. His initial decision to go to Cambridge came after reading “A Fellow of Trinity” by “Alan St Aubyn” – this is apparently not one of the world’s greatest works of literature, but I just have to read it now to see what was in it that could inspire him so strongly!

CP Snow paints a delightful picture of the life of an honest, eccentric, and intellectually gifted man – a life revolving around academia in general, mathematics, cricket, radical ideas and some superb eccentricities. Hardy was suspicious of all things mechanical – “If you fancy yourself at the telephone, there is one in the other room”. This book is worth reading for the foreword alone.

Hardy’s work then follows, written in a series of short, pithy chapters, a bit too long to be called aphorisms, but each almost stands alone in placing an argument, crafted in step-by-step fashion, as you would expect of a mathematician. Now, maybe my interpretation of Hardy’s words is different to others, but for me, although he concentrates on the rights or wrongs of devoting one’s life to pure mathematics, discussing how “worthwhile” mathematics is as a profession, I think you can read this as an argument on the merits or otherwise of any human endeavour. He basically concludes that it is far better to exercise to the full whatever talent one has, than do undistinguished work in other fields. There’s more depth to it than that of course, all very readable, and an interesting set of views for those faced with an awkward crossroads in life!

CP Snow paints a delightful picture of the life of an honest, eccentric, and intellectually gifted man – a life revolving around academia in general, mathematics, cricket, radical ideas and some superb eccentricities. Hardy was suspicious of all things mechanical – “If you fancy yourself at the telephone, there is one in the other room”. This book is worth reading for the foreword alone.

Hardy’s work then follows, written in a series of short, pithy chapters, a bit too long to be called aphorisms, but each almost stands alone in placing an argument, crafted in step-by-step fashion, as you would expect of a mathematician. Now, maybe my interpretation of Hardy’s words is different to others, but for me, although he concentrates on the rights or wrongs of devoting one’s life to pure mathematics, discussing how “worthwhile” mathematics is as a profession, I think you can read this as an argument on the merits or otherwise of any human endeavour. He basically concludes that it is far better to exercise to the full whatever talent one has, than do undistinguished work in other fields. There’s more depth to it than that of course, all very readable, and an interesting set of views for those faced with an awkward crossroads in life!

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Some excellent passages and the discussion of maths as beauty is illuminating, but his elitist distinctions between those with outstanding ability and those without no longer resonates with a reader and some of the discussion of maths' innocuousness in comparison to other disciplines just lacks any great insight: certainly Physics and Engineering makes atom bombs, but Feynmann, Oppenheimer, etc. and Einstein were using some fairly tricky maths to realise them.

That said the book is to be read for the discussion of maths as beauty and for the pathos of a great mathematician past his prime. I enjoyed Snow's introduction, which discussed Ramunjan etc., at least as much as the main text

That said the book is to be read for the discussion of maths as beauty and for the pathos of a great mathematician past his prime. I enjoyed Snow's introduction, which discussed Ramunjan etc., at least as much as the main text

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ByA. Froston 12 June 2015

First class, almost a must read as books describing the work of first class pure mathematicians are rare. Hardy with his collaborators Littlewood and Ramanujan brought British Mathematics back as a force in European Mathematics after a sterile century following Newton's death. Hardy's prose is as precise and crisp as presumably his mathematical work was. C.P Snow' s preface is equally enjoyable , very perceptive , and fills in much of the background of Hardy's early life. The C.P Snow introduction takes up at least half of the volume , but I for one, would have liked both parts to have been even longer.As a bonus both parts of the book give an insight into the life of Cambridge dons in the 1930s. Excellent.

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ByM. R. Hudsonon 10 March 2012

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.

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ByA customeron 10 May 2001

This is a good book for people intersted in mathematics, no prior knowledge needed! It discusses how mathematics is important in the everyday world. I would recommend this book especially to mathematics students (or prospective ones) as wider reading. It is interesting to see how in Hardy's time, less than sixty years ago, areas of mathematics that had no obvious use and were studied purely for their beauty have become the centre of importance in the computer and internet technology of today and the future with important applications and areas of research.

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ByD. R. Cantrellon 1 April 2010

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.

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.

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ByDr. Mike Goldsmithon 24 November 2008

Reading this book is like opening a time capsule - which is fun, up to a point. The academic world Hardy inhabited was largely swept away by the first world war, and this book is full of the feeling of that threatened world - one of Hardy's main aims is to show that a. "proper" mathematics is of no practical use, and that b. this is good news, partly because this makes it "purer" and hence better (in some unexplained way) partly because then it can't be blamed for killing people.

The idea that mathematicians should justify their incomes through the value of their work (either through actual income generation or through explaining how the universe works) seems not to occur to Hardy; it's hard to imagine any mathematician today making the uselessness of their subject the major theme of a book.

Unfortunately for Hardy, the very areas he singles out as being especially satisfyingly useless are key to modern code-breaking and secure financial transactions - in fact it's hard to think of any substantial area of mathematics that is "useless" in Hardy's sense.

Hardy's style is also very antiquated, involving a great deal of preparatory throat-clearing, unsupported value-judgments, meandering around the subject, and lengthy vagueness. To a modern reader it all seems a bit smug and fake.

On the other hand, the actual points that Hardy makes about the way mathematicians develop ideas, and the examples of explanations of the amazing way mathematical proof works, are good ones. And, as a look back at a vanished period of academic history, it's great. But as an introduction to how mathematicians work now, or the role of mathematical research, it has become a very weak book.

The idea that mathematicians should justify their incomes through the value of their work (either through actual income generation or through explaining how the universe works) seems not to occur to Hardy; it's hard to imagine any mathematician today making the uselessness of their subject the major theme of a book.

Unfortunately for Hardy, the very areas he singles out as being especially satisfyingly useless are key to modern code-breaking and secure financial transactions - in fact it's hard to think of any substantial area of mathematics that is "useless" in Hardy's sense.

Hardy's style is also very antiquated, involving a great deal of preparatory throat-clearing, unsupported value-judgments, meandering around the subject, and lengthy vagueness. To a modern reader it all seems a bit smug and fake.

On the other hand, the actual points that Hardy makes about the way mathematicians develop ideas, and the examples of explanations of the amazing way mathematical proof works, are good ones. And, as a look back at a vanished period of academic history, it's great. But as an introduction to how mathematicians work now, or the role of mathematical research, it has become a very weak book.

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