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Calculus & Mathematical Analysis
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 Customer Reviews
No need to apologize for this, 27 Nov 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.
How maths used to be, 24 Nov 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.
Hardy's Apology, 23 Jul 2008
I bought Hardy's biography almost 50 years ago, just before going up to university to read Maths. The gentle almost humourous tone of the book convinced me that the life of a mathematician was a potential source of pleasure. In fact I later moved to read Philosophy, for my interests grew to centre on Logic and Mathematical Foundations, but my current work on Infinity still finds me using Hardy's "Pure Mathematics", and I still occasionally read his "Apology" for the pure pleasure that this outstanding book provides.
Disappointing, 26 Mar 2008
I'm not sure I read the same book as the other reviewers. I'm fascinated by mathematics and have, for a long time, been meaning to read this book. Finally I have, and I have to say I was somewhat disappointed. C.P. Snow's foreword was an interesting potted biography of one of Britain's greatest 20th century mathematicians; but, the "apology" itself is overshadowed by its own introduction.
The first nine chapters are a clear indication - if any was needed - that intellectual brilliance is no talisman against small-mindedness. Maybe I'm being unfair but in those nine short chapters Hardy manages to denigrate most of humanity - "most people can do nothing at all well" - by which he means, nothing that he personally considers worthy. I'm not convinced we should readily accept the social opinions of a man who was born to privilege and spent his entire life in rarefied cloisters shielded from what the "most" in question would consider reality. A man who thought we should not go to war with Germany because of the caliber of its mathematicians and scientists. (?) There are many reasons why we shouldn't be at war with Iraq, for example, but that the region is the cradle of civilisation is not one of them.
From chapter ten onwards the book makes a change of direction and deals with the actual subject of mathematics and being a mathematician. This latter section of the book is a much more interesting (less infuriating) read but I still felt he dealt rather ineptly with the attempt to differentiate between "common and garden" mathematics and the truly profound and beautiful creative mathematics that he so clearly loves.
When you compare this book to say Marcus du Sautoy's "Music of the Primes" or Simon Singh's "Fermat's Last Theorem", both books which truly promote the essential character of mathematics, "A Mathematician's Apology" seems very pale.
Read it. Seriously., 04 Feb 2008
This book is a facinating insight into the mind of one of the century's greatest mathematians. However, Hardy's ideas go far beyond this into the purpose of human endevour in general.
The best part of this book is the foreword by C.P Snow. This amounts to a lucid, if brief, biography of Hardy from a man who is brilliant in his own right.
If you have any kind of interest in the realities of mathematics, or the workings of genius, then this book is the place to start.
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Customer Reviews
No need to apologize for this, 27 Nov 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.
How maths used to be, 24 Nov 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.
Hardy's Apology, 23 Jul 2008
I bought Hardy's biography almost 50 years ago, just before going up to university to read Maths. The gentle almost humourous tone of the book convinced me that the life of a mathematician was a potential source of pleasure. In fact I later moved to read Philosophy, for my interests grew to centre on Logic and Mathematical Foundations, but my current work on Infinity still finds me using Hardy's "Pure Mathematics", and I still occasionally read his "Apology" for the pure pleasure that this outstanding book provides.
Disappointing, 26 Mar 2008
I'm not sure I read the same book as the other reviewers. I'm fascinated by mathematics and have, for a long time, been meaning to read this book. Finally I have, and I have to say I was somewhat disappointed. C.P. Snow's foreword was an interesting potted biography of one of Britain's greatest 20th century mathematicians; but, the "apology" itself is overshadowed by its own introduction.
The first nine chapters are a clear indication - if any was needed - that intellectual brilliance is no talisman against small-mindedness. Maybe I'm being unfair but in those nine short chapters Hardy manages to denigrate most of humanity - "most people can do nothing at all well" - by which he means, nothing that he personally considers worthy. I'm not convinced we should readily accept the social opinions of a man who was born to privilege and spent his entire life in rarefied cloisters shielded from what the "most" in question would consider reality. A man who thought we should not go to war with Germany because of the caliber of its mathematicians and scientists. (?) There are many reasons why we shouldn't be at war with Iraq, for example, but that the region is the cradle of civilisation is not one of them.
From chapter ten onwards the book makes a change of direction and deals with the actual subject of mathematics and being a mathematician. This latter section of the book is a much more interesting (less infuriating) read but I still felt he dealt rather ineptly with the attempt to differentiate between "common and garden" mathematics and the truly profound and beautiful creative mathematics that he so clearly loves.
When you compare this book to say Marcus du Sautoy's "Music of the Primes" or Simon Singh's "Fermat's Last Theorem", both books which truly promote the essential character of mathematics, "A Mathematician's Apology" seems very pale.
Read it. Seriously., 04 Feb 2008
This book is a facinating insight into the mind of one of the century's greatest mathematians. However, Hardy's ideas go far beyond this into the purpose of human endevour in general.
The best part of this book is the foreword by C.P Snow. This amounts to a lucid, if brief, biography of Hardy from a man who is brilliant in his own right.
If you have any kind of interest in the realities of mathematics, or the workings of genius, then this book is the place to start.
Great For A2 Physics, 15 Sep 2006
I have just started A2 Physics at my school. I didn't do maths to AS (just GCSE). I am the only one in my class without an AS in Maths. My teacher told me that I needed to know about Differentiation and Integration in order to progress my A grade in AS physics to an A in A2.
I always had the idea that Calculus was simply a very complicated area of Mathematics developed by Newton. I didn't know anything else (remember I only had a Maths GCSE).
I thought I'd buy this book to see if it could help me. The fantastic thing is that it explains every single concept in an extremely simple way (starting with the very basics). I believe that I am now up to the high standard of the mathematicians in my class and am able to understand the important mathematical concepts in physics that use calculus as a basis for their explanation. In particular, OCR's Advancing Physics A2 - Creating Models section.
You really don't need to be a math genius to understand this book fully!
Shock, horror, calculus is easy (well sort of!), 06 Nov 2003
If you`ve ever looked at a polynomial equation and thought that learning Martian would be easier, or even that knowing what a polynomial was would make a lot more sense, then this is the book for you. Granted, the book does assume that you have some prior knowledge of algebra and trig but I have only the most basic grasp of them and still managed to follow the book. The chapters are well set out and explain painful looking equations for differentiation and integration in simple,easy to follow terms. Calculus for Dummies acts like a kindly mentor; easing you through the hard bits, holding your hands when the numbers start to look like gibberish and celebrating with you at the end of the chapter that maths no longer seems like a strange and alien concept!
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Customer Reviews
No need to apologize for this, 27 Nov 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.
How maths used to be, 24 Nov 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.
Hardy's Apology, 23 Jul 2008
I bought Hardy's biography almost 50 years ago, just before going up to university to read Maths. The gentle almost humourous tone of the book convinced me that the life of a mathematician was a potential source of pleasure. In fact I later moved to read Philosophy, for my interests grew to centre on Logic and Mathematical Foundations, but my current work on Infinity still finds me using Hardy's "Pure Mathematics", and I still occasionally read his "Apology" for the pure pleasure that this outstanding book provides. Disappointing, 26 Mar 2008
I'm not sure I read the same book as the other reviewers. I'm fascinated by mathematics and have, for a long time, been meaning to read this book. Finally I have, and I have to say I was somewhat disappointed. C.P. Snow's foreword was an interesting potted biography of one of Britain's greatest 20th century mathematicians; but, the "apology" itself is overshadowed by its own introduction.
The first nine chapters are a clear indication - if any was needed - that intellectual brilliance is no talisman against small-mindedness. Maybe I'm being unfair but in those nine short chapters Hardy manages to denigrate most of humanity - "most people can do nothing at all well" - by which he means, nothing that he personally considers worthy. I'm not convinced we should readily accept the social opinions of a man who was born to privilege and spent his entire life in rarefied cloisters shielded from what the "most" in question would consider reality. A man who thought we should not go to war with Germany because of the caliber of its mathematicians and scientists. (?) There are many reasons why we shouldn't be at war with Iraq, for example, but that the region is the cradle of civilisation is not one of them.
From chapter ten onwards the book makes a change of direction and deals with the actual subject of mathematics and being a mathematician. This latter section of the book is a much more interesting (less infuriating) read but I still felt he dealt rather ineptly with the attempt to differentiate between "common and garden" mathematics and the truly profound and beautiful creative mathematics that he so clearly loves.
When you compare this book to say Marcus du Sautoy's "Music of the Primes" or Simon Singh's "Fermat's Last Theorem", both books which truly promote the essential character of mathematics, "A Mathematician's Apology" seems very pale. Read it. Seriously., 04 Feb 2008
This book is a facinating insight into the mind of one of the century's greatest mathematians. However, Hardy's ideas go far beyond this into the purpose of human endevour in general.
The best part of this book is the foreword by C.P Snow. This amounts to a lucid, if brief, biography of Hardy from a man who is brilliant in his own right.
If you have any kind of interest in the realities of mathematics, or the workings of genius, then this book is the place to start. Great For A2 Physics, 15 Sep 2006
I have just started A2 Physics at my school. I didn't do maths to AS (just GCSE). I am the only one in my class without an AS in Maths. My teacher told me that I needed to know about Differentiation and Integration in order to progress my A grade in AS physics to an A in A2.
I always had the idea that Calculus was simply a very complicated area of Mathematics developed by Newton. I didn't know anything else (remember I only had a Maths GCSE).
I thought I'd buy this book to see if it could help me. The fantastic thing is that it explains every single concept in an extremely simple way (starting with the very basics). I believe that I am now up to the high standard of the mathematicians in my class and am able to understand the important mathematical concepts in physics that use calculus as a basis for their explanation. In particular, OCR's Advancing Physics A2 - Creating Models section.
You really don't need to be a math genius to understand this book fully!
Shock, horror, calculus is easy (well sort of!), 06 Nov 2003
If you`ve ever looked at a polynomial equation and thought that learning Martian would be easier, or even that knowing what a polynomial was would make a lot more sense, then this is the book for you. Granted, the book does assume that you have some prior knowledge of algebra and trig but I have only the most basic grasp of them and still managed to follow the book. The chapters are well set out and explain painful looking equations for differentiation and integration in simple,easy to follow terms. Calculus for Dummies acts like a kindly mentor; easing you through the hard bits, holding your hands when the numbers start to look like gibberish and celebrating with you at the end of the chapter that maths no longer seems like a strange and alien concept! Compact knowledge, 15 Apr 2008
I found this book is very interesting. I used this book when doing my research. For me, it is a compact knowledge inside one book. I do not need to search for other book for reference.
This book provide easy to understand structure. A lot of example and proof make this book is very nice.
I agree that this book is good for undergraduate, but it is also good for high level education. Complete and effective, 10 Jul 2004
This is a very effective book for the student : - the knowledge is neatly summed up, and proofs are given just afterwards and don't interfere with the concepts, so that reading and understanding the lessons is easy, - hundreds of corrected exercises, very gradual, - very complete for undergrads (and enough for most grads who won't do Physics for example). This book and the S. Lipschutz one on linear algebra are among those I've kept for further use. Ce livre est hautement recommandable pour le premier cycle scientifique. Clair, complet et graduel.
A good reference, 02 Jan 2002
Thi sbook is not for everyone. It is pitched somewhere at the undergraduate level or to the high school student who would like to go beyond the syllabus. The book presents the axioms and assumptions clearly and concisely before moving quickly on to analysis and examples. Explanations may be too brief for some but all the information necessary to understand and to handle the exercises are there. This approach may not be suitable for readers who want a quick fix but for those whose passion is mathematics, this is a great primer.
Moves too fast. Does not spend enough time on each topic., 02 Sep 1999
My Advanced Calculus class covered only the first two chapters. While we were spending time on detailed analysis of different types of infinities, this book did not cover it at all. This book was almost no help. Make sure the style of the teacher matches the way this book is written, otherwise it may be a waste. My class would have been a lot easier and less satisfying if it had been conducted the way this is written.
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Customer Reviews
No need to apologize for this, 27 Nov 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.
How maths used to be, 24 Nov 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.
Hardy's Apology, 23 Jul 2008
I bought Hardy's biography almost 50 years ago, just before going up to university to read Maths. The gentle almost humourous tone of the book convinced me that the life of a mathematician was a potential source of pleasure. In fact I later moved to read Philosophy, for my interests grew to centre on Logic and Mathematical Foundations, but my current work on Infinity still finds me using Hardy's "Pure Mathematics", and I still occasionally read his "Apology" for the pure pleasure that this outstanding book provides. Disappointing, 26 Mar 2008
I'm not sure I read the same book as the other reviewers. I'm fascinated by mathematics and have, for a long time, been meaning to read this book. Finally I have, and I have to say I was somewhat disappointed. C.P. Snow's foreword was an interesting potted biography of one of Britain's greatest 20th century mathematicians; but, the "apology" itself is overshadowed by its own introduction.
The first nine chapters are a clear indication - if any was needed - that intellectual brilliance is no talisman against small-mindedness. Maybe I'm being unfair but in those nine short chapters Hardy manages to denigrate most of humanity - "most people can do nothing at all well" - by which he means, nothing that he personally considers worthy. I'm not convinced we should readily accept the social opinions of a man who was born to privilege and spent his entire life in rarefied cloisters shielded from what the "most" in question would consider reality. A man who thought we should not go to war with Germany because of the caliber of its mathematicians and scientists. (?) There are many reasons why we shouldn't be at war with Iraq, for example, but that the region is the cradle of civilisation is not one of them.
From chapter ten onwards the book makes a change of direction and deals with the actual subject of mathematics and being a mathematician. This latter section of the book is a much more interesting (less infuriating) read but I still felt he dealt rather ineptly with the attempt to differentiate between "common and garden" mathematics and the truly profound and beautiful creative mathematics that he so clearly loves.
When you compare this book to say Marcus du Sautoy's "Music of the Primes" or Simon Singh's "Fermat's Last Theorem", both books which truly promote the essential character of mathematics, "A Mathematician's Apology" seems very pale. Read it. Seriously., 04 Feb 2008
This book is a facinating insight into the mind of one of the century's greatest mathematians. However, Hardy's ideas go far beyond this into the purpose of human endevour in general.
The best part of this book is the foreword by C.P Snow. This amounts to a lucid, if brief, biography of Hardy from a man who is brilliant in his own right.
If you have any kind of interest in the realities of mathematics, or the workings of genius, then this book is the place to start. Great For A2 Physics, 15 Sep 2006
I have just started A2 Physics at my school. I didn't do maths to AS (just GCSE). I am the only one in my class without an AS in Maths. My teacher told me that I needed to know about Differentiation and Integration in order to progress my A grade in AS physics to an A in A2.
I always had the idea that Calculus was simply a very complicated area of Mathematics developed by Newton. I didn't know anything else (remember I only had a Maths GCSE).
I thought I'd buy this book to see if it could help me. The fantastic thing is that it explains every single concept in an extremely simple way (starting with the very basics). I believe that I am now up to the high standard of the mathematicians in my class and am able to understand the important mathematical concepts in physics that use calculus as a basis for their explanation. In particular, OCR's Advancing Physics A2 - Creating Models section.
You really don't need to be a math genius to understand this book fully!
Shock, horror, calculus is easy (well sort of!), 06 Nov 2003
If you`ve ever looked at a polynomial equation and thought that learning Martian would be easier, or even that knowing what a polynomial was would make a lot more sense, then this is the book for you. Granted, the book does assume that you have some prior knowledge of algebra and trig but I have only the most basic grasp of them and still managed to follow the book. The chapters are well set out and explain painful looking equations for differentiation and integration in simple,easy to follow terms. Calculus for Dummies acts like a kindly mentor; easing you through the hard bits, holding your hands when the numbers start to look like gibberish and celebrating with you at the end of the chapter that maths no longer seems like a strange and alien concept! Compact knowledge, 15 Apr 2008
I found this book is very interesting. I used this book when doing my research. For me, it is a compact knowledge inside one book. I do not need to search for other book for reference.
This book provide easy to understand structure. A lot of example and proof make this book is very nice.
I agree that this book is good for undergraduate, but it is also good for high level education. Complete and effective, 10 Jul 2004
This is a very effective book for the student : - the knowledge is neatly summed up, and proofs are given just afterwards and don't interfere with the concepts, so that reading and understanding the lessons is easy, - hundreds of corrected exercises, very gradual, - very complete for undergrads (and enough for most grads who won't do Physics for example). This book and the S. Lipschutz one on linear algebra are among those I've kept for further use. Ce livre est hautement recommandable pour le premier cycle scientifique. Clair, complet et graduel.
A good reference, 02 Jan 2002
Thi sbook is not for everyone. It is pitched somewhere at the undergraduate level or to the high school student who would like to go beyond the syllabus. The book presents the axioms and assumptions clearly and concisely before moving quickly on to analysis and examples. Explanations may be too brief for some but all the information necessary to understand and to handle the exercises are there. This approach may not be suitable for readers who want a quick fix but for those whose passion is mathematics, this is a great primer.
Moves too fast. Does not spend enough time on each topic., 02 Sep 1999
My Advanced Calculus class covered only the first two chapters. While we were spending time on detailed analysis of different types of infinities, this book did not cover it at all. This book was almost no help. Make sure the style of the teacher matches the way this book is written, otherwise it may be a waste. My class would have been a lot easier and less satisfying if it had been conducted the way this is written.
Good Book, 12 Sep 2007
If you want to solve lots of exercises this is a good choice. Of course, exercises are important to master this topics, but I advice you to get a good calculus (or vector calculus) book to understand theory. I think shaum's vector analysis is a bit incomplete on that matter. Well... it's a shaum's outline series...
I found this book useful, especially for electromagnetics. I'm happy to own it.
I love these Schaum's books!, 17 Nov 2000
As with all the Schaum's books, it's packed with all the theory you could possibly need to know (for a basic vector analysis course). Add this to enough worked examples and supplementary problems to shake a big stick at and you've got one brilliant book. Put it this way, the first time I sat my vector course at uni I got 3%. I bought this book, resat, and got 87%. Enough said. The introduction to tensors is quite basic, but if you want a book on tensors then buy the Schaum's Tensor Calculus one 'cause it's great too! Buy now and be a witness to the Schaum's domination of the world...
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Calculus Made Easy
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Customer Reviews
No need to apologize for this, 27 Nov 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.
How maths used to be, 24 Nov 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.
Hardy's Apology, 23 Jul 2008
I bought Hardy's biography almost 50 years ago, just before going up to university to read Maths. The gentle almost humourous tone of the book convinced me that the life of a mathematician was a potential source of pleasure. In fact I later moved to read Philosophy, for my interests grew to centre on Logic and Mathematical Foundations, but my current work on Infinity still finds me using Hardy's "Pure Mathematics", and I still occasionally read his "Apology" for the pure pleasure that this outstanding book provides. Disappointing, 26 Mar 2008
I'm not sure I read the same book as the other reviewers. I'm fascinated by mathematics and have, for a long time, been meaning to read this book. Finally I have, and I have to say I was somewhat disappointed. C.P. Snow's foreword was an interesting potted biography of one of Britain's greatest 20th century mathematicians; but, the "apology" itself is overshadowed by its own introduction.
The first nine chapters are a clear indication - if any was needed - that intellectual brilliance is no talisman against small-mindedness. Maybe I'm being unfair but in those nine short chapters Hardy manages to denigrate most of humanity - "most people can do nothing at all well" - by which he means, nothing that he personally considers worthy. I'm not convinced we should readily accept the social opinions of a man who was born to privilege and spent his entire life in rarefied cloisters shielded from what the "most" in question would consider reality. A man who thought we should not go to war with Germany because of the caliber of its mathematicians and scientists. (?) There are many reasons why we shouldn't be at war with Iraq, for example, but that the region is the cradle of civilisation is not one of them.
From chapter ten onwards the book makes a change of direction and deals with the actual subject of mathematics and being a mathematician. This latter section of the book is a much more interesting (less infuriating) read but I still felt he dealt rather ineptly with the attempt to differentiate between "common and garden" mathematics and the truly profound and beautiful creative mathematics that he so clearly loves.
When you compare this book to say Marcus du Sautoy's "Music of the Primes" or Simon Singh's "Fermat's Last Theorem", both books which truly promote the essential character of mathematics, "A Mathematician's Apology" seems very pale. Read it. Seriously., 04 Feb 2008
This book is a facinating insight into the mind of one of the century's greatest mathematians. However, Hardy's ideas go far beyond this into the purpose of human endevour in general.
The best part of this book is the foreword by C.P Snow. This amounts to a lucid, if brief, biography of Hardy from a man who is brilliant in his own right.
If you have any kind of interest in the realities of mathematics, or the workings of genius, then this book is the place to start. Great For A2 Physics, 15 Sep 2006
I have just started A2 Physics at my school. I didn't do maths to AS (just GCSE). I am the only one in my class without an AS in Maths. My teacher told me that I needed to know about Differentiation and Integration in order to progress my A grade in AS physics to an A in A2.
I always had the idea that Calculus was simply a very complicated area of Mathematics developed by Newton. I didn't know anything else (remember I only had a Maths GCSE).
I thought I'd buy this book to see if it could help me. The fantastic thing is that it explains every single concept in an extremely simple way (starting with the very basics). I believe that I am now up to the high standard of the mathematicians in my class and am able to understand the important mathematical concepts in physics that use calculus as a basis for their explanation. In particular, OCR's Advancing Physics A2 - Creating Models section.
You really don't need to be a math genius to understand this book fully!
Shock, horror, calculus is easy (well sort of!), 06 Nov 2003
If you`ve ever looked at a polynomial equation and thought that learning Martian would be easier, or even that knowing what a polynomial was would make a lot more sense, then this is the book for you. Granted, the book does assume that you have some prior knowledge of algebra and trig but I have only the most basic grasp of them and still managed to follow the book. The chapters are well set out and explain painful looking equations for differentiation and integration in simple,easy to follow terms. Calculus for Dummies acts like a kindly mentor; easing you through the hard bits, holding your hands when the numbers start to look like gibberish and celebrating with you at the end of the chapter that maths no longer seems like a strange and alien concept! Compact knowledge, 15 Apr 2008
I found this book is very interesting. I used this book when doing my research. For me, it is a compact knowledge inside one book. I do not need to search for other book for reference.
This book provide easy to understand structure. A lot of example and proof make this book is very nice.
I agree that this book is good for undergraduate, but it is also good for high level education. Complete and effective, 10 Jul 2004
This is a very effective book for the student : - the knowledge is neatly summed up, and proofs are given just afterwards and don't interfere with the concepts, so that reading and understanding the lessons is easy, - hundreds of corrected exercises, very gradual, - very complete for undergrads (and enough for most grads who won't do Physics for example). This book and the S. Lipschutz one on linear algebra are among those I've kept for further use. Ce livre est hautement recommandable pour le premier cycle scientifique. Clair, complet et graduel.
A good reference, 02 Jan 2002
Thi sbook is not for everyone. It is pitched somewhere at the undergraduate level or to the high school student who would like to go beyond the syllabus. The book presents the axioms and assumptions clearly and concisely before moving quickly on to analysis and examples. Explanations may be too brief for some but all the information necessary to understand and to handle the exercises are there. This approach may not be suitable for readers who want a quick fix but for those whose passion is mathematics, this is a great primer.
Moves too fast. Does not spend enough time on each topic., 02 Sep 1999
My Advanced Calculus class covered only the first two chapters. While we were spending time on detailed analysis of different types of infinities, this book did not cover it at all. This book was almost no help. Make sure the style of the teacher matches the way this book is written, otherwise it may be a waste. My class would have been a lot easier and less satisfying if it had been conducted the way this is written.
Good Book, 12 Sep 2007
If you want to solve lots of exercises this is a good choice. Of course, exercises are important to master this topics, but I advice you to get a good calculus (or vector calculus) book to understand theory. I think shaum's vector analysis is a bit incomplete on that matter. Well... it's a shaum's outline series...
I found this book useful, especially for electromagnetics. I'm happy to own it.
I love these Schaum's books!, 17 Nov 2000
As with all the Schaum's books, it's packed with all the theory you could possibly need to know (for a basic vector analysis course). Add this to enough worked examples and supplementary problems to shake a big stick at and you've got one brilliant book. Put it this way, the first time I sat my vector course at uni I got 3%. I bought this book, resat, and got 87%. Enough said. The introduction to tensors is quite basic, but if you want a book on tensors then buy the Schaum's Tensor Calculus one 'cause it's great too! Buy now and be a witness to the Schaum's domination of the world...
Magnificant work., 11 Jan 2004
I have a number of texts on the calculus andgeneral maths and this book, written originally in 1910, but recently edited by Martin Gardener) stands head and shoulders above all the introductory texts, for introducing calculus in an understandable way, slice by slice. Also, the book being a small paperback fits into one's pocket unlike many/most texts on calculus!!!
A review of great praise towards this magnificent work..., 29 Oct 2000
This book was written way back in 1910 by a Fellow of the Royal Society. However, unlike Newton's works (he was also a member) this is extremely lucid. The essence of this work is that anyone can do calculus. Moreover, since the fools that are university professors can do it, so can you or I. The book begins with an amusing prologue about the stupidity of the mathematical teaching establishment and how it likes to show off with its amazing ability by portraying calculus as a difficult art. The author had to 'unteach himself the difficulties' and undertakes to explain them as clearly as possible. And that he does. From personal experience, before reading this book my maths grades were in the toilet - almost immediately after I read and understood it, my grades trebled. This is because the book is the best explanation of calculus I have ever seen or heard of. This was also the favourite of an eminent American physicist who read this book himself, and went on to win the Nobel Prize in 1965. The concepts are reduced to their bare (understandable) bones, and built up again leading to some great understanding of the calculus, and a confidence to approach mathematics. This book really is a MUST for all A-level students of maths - you may as well throw your textbooks in the bin. Good work Mr. Thompson!!!
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Customer Reviews
No need to apologize for this, 27 Nov 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.
How maths used to be, 24 Nov 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.
Hardy's Apology, 23 Jul 2008
I bought Hardy's biography almost 50 years ago, just before going up to university to read Maths. The gentle almost humourous tone of the book convinced me that the life of a mathematician was a potential source of pleasure. In fact I later moved to read Philosophy, for my interests grew to centre on Logic and Mathematical Foundations, but my current work on Infinity still finds me using Hardy's "Pure Mathematics", and I still occasionally read his "Apology" for the pure pleasure that this outstanding book provides. Disappointing, 26 Mar 2008
I'm not sure I read the same book as the other reviewers. I'm fascinated by mathematics and have, for a long time, been meaning to read this book. Finally I have, and I have to say I was somewhat disappointed. C.P. Snow's foreword was an interesting potted biography of one of Britain's greatest 20th century mathematicians; but, the "apology" itself is overshadowed by its own introduction.
The first nine chapters are a clear indication - if any was needed - that intellectual brilliance is no talisman against small-mindedness. Maybe I'm being unfair but in those nine short chapters Hardy manages to denigrate most of humanity - "most people can do nothing at all well" - by which he means, nothing that he personally considers worthy. I'm not convinced we should readily accept the social opinions of a man who was born to privilege and spent his entire life in rarefied cloisters shielded from what the "most" in question would consider reality. A man who thought we should not go to war with Germany because of the caliber of its mathematicians and scientists. (?) There are many reasons why we shouldn't be at war with Iraq, for example, but that the region is the cradle of civilisation is not one of them.
From chapter ten onwards the book makes a change of direction and deals with the actual subject of mathematics and being a mathematician. This latter section of the book is a much more interesting (less infuriating) read but I still felt he dealt rather ineptly with the attempt to differentiate between "common and garden" mathematics and the truly profound and beautiful creative mathematics that he so clearly loves.
When you compare this book to say Marcus du Sautoy's "Music of the Primes" or Simon Singh's "Fermat's Last Theorem", both books which truly promote the essential character of mathematics, "A Mathematician's Apology" seems very pale. Read it. Seriously., 04 Feb 2008
This book is a facinating insight into the mind of one of the century's greatest mathematians. However, Hardy's ideas go far beyond this into the purpose of human endevour in general.
The best part of this book is the foreword by C.P Snow. This amounts to a lucid, if brief, biography of Hardy from a man who is brilliant in his own right.
If you have any kind of interest in the realities of mathematics, or the workings of genius, then this book is the place to start. Great For A2 Physics, 15 Sep 2006
I have just started A2 Physics at my school. I didn't do maths to AS (just GCSE). I am the only one in my class without an AS in Maths. My teacher told me that I needed to know about Differentiation and Integration in order to progress my A grade in AS physics to an A in A2.
I always had the idea that Calculus was simply a very complicated area of Mathematics developed by Newton. I didn't know anything else (remember I only had a Maths GCSE).
I thought I'd buy this book to see if it could help me. The fantastic thing is that it explains every single concept in an extremely simple way (starting with the very basics). I believe that I am now up to the high standard of the mathematicians in my class and am able to understand the important mathematical concepts in physics that use calculus as a basis for their explanation. In particular, OCR's Advancing Physics A2 - Creating Models section.
You really don't need to be a math genius to understand this book fully!
Shock, horror, calculus is easy (well sort of!), 06 Nov 2003
If you`ve ever looked at a polynomial equation and thought that learning Martian would be easier, or even that knowing what a polynomial was would make a lot more sense, then this is the book for you. Granted, the book does assume that you have some prior knowledge of algebra and trig but I have only the most basic grasp of them and still managed to follow the book. The chapters are well set out and explain painful looking equations for differentiation and integration in simple,easy to follow terms. Calculus for Dummies acts like a kindly mentor; easing you through the hard bits, holding your hands when the numbers start to look like gibberish and celebrating with you at the end of the chapter that maths no longer seems like a strange and alien concept! Compact knowledge, 15 Apr 2008
I found this book is very interesting. I used this book when doing my research. For me, it is a compact knowledge inside one book. I do not need to search for other book for reference.
This book provide easy to understand structure. A lot of example and proof make this book is very nice.
I agree that this book is good for undergraduate, but it is also good for high level education. Complete and effective, 10 Jul 2004
This is a very effective book for the student : - the knowledge is neatly summed up, and proofs are given just afterwards and don't interfere with the concepts, so that reading and understanding the lessons is easy, - hundreds of corrected exercises, very gradual, - very complete for undergrads (and enough for most grads who won't do Physics for example). This book and the S. Lipschutz one on linear algebra are among those I've kept for further use. Ce livre est hautement recommandable pour le premier cycle scientifique. Clair, complet et graduel.
A good reference, 02 Jan 2002
Thi sbook is not for everyone. It is pitched somewhere at the undergraduate level or to the high school student who would like to go beyond the syllabus. The book presents the axioms and assumptions clearly and concisely before moving quickly on to analysis and examples. Explanations may be too brief for some but all the information necessary to understand and to handle the exercises are there. This approach may not be suitable for readers who want a quick fix but for those whose passion is mathematics, this is a great primer.
Moves too fast. Does not spend enough time on each topic., 02 Sep 1999
My Advanced Calculus class covered only the first two chapters. While we were spending time on detailed analysis of different types of infinities, this book did not cover it at all. This book was almost no help. Make sure the style of the teacher matches the way this book is written, otherwise it may be a waste. My class would have been a lot easier and less satisfying if it had been conducted the way this is written.
Good Book, 12 Sep 2007
If you want to solve lots of exercises this is a good choice. Of course, exercises are important to master this topics, but I advice you to get a good calculus (or vector calculus) book to understand theory. I think shaum's vector analysis is a bit incomplete on that matter. Well... it's a shaum's outline series...
I found this book useful, especially for electromagnetics. I'm happy to own it.
I love these Schaum's books!, 17 Nov 2000
As with all the Schaum's books, it's packed with all the theory you could possibly need to know (for a basic vector analysis course). Add this to enough worked examples and supplementary problems to shake a big stick at and you've got one brilliant book. Put it this way, the first time I sat my vector course at uni I got 3%. I bought this book, resat, and got 87%. Enough said. The introduction to tensors is quite basic, but if you want a book on tensors then buy the Schaum's Tensor Calculus one 'cause it's great too! Buy now and be a witness to the Schaum's domination of the world...
Magnificant work., 11 Jan 2004
I have a number of texts on the calculus andgeneral maths and this book, written originally in 1910, but recently edited by Martin Gardener) stands head and shoulders above all the introductory texts, for introducing calculus in an understandable way, slice by slice. Also, the book being a small paperback fits into one's pocket unlike many/most texts on calculus!!!
A review of great praise towards this magnificent work..., 29 Oct 2000
This book was written way back in 1910 by a Fellow of the Royal Society. However, unlike Newton's works (he was also a member) this is extremely lucid. The essence of this work is that anyone can do calculus. Moreover, since the fools that are university professors can do it, so can you or I. The book begins with an amusing prologue about the stupidity of the mathematical teaching establishment and how it likes to show off with its amazing ability by portraying calculus as a difficult art. The author had to 'unteach himself the difficulties' and undertakes to explain them as clearly as possible. And that he does. From personal experience, before reading this book my maths grades were in the toilet - almost immediately after I read and understood it, my grades trebled. This is because the book is the best explanation of calculus I have ever seen or heard of. This was also the favourite of an eminent American physicist who read this book himself, and went on to win the Nobel Prize in 1965. The concepts are reduced to their bare (understandable) bones, and built up again leading to some great understanding of the calculus, and a confidence to approach mathematics. This book really is a MUST for all A-level students of maths - you may as well throw your textbooks in the bin. Good work Mr. Thompson!!!
A godsend, 15 Sep 2006
In the first year of my maths degree I was lost... until I found this book. It's unbelievable!! It makes sense, it has nice little historic interest bits and most importantly it'll answer all the exam questions. You won't need another analysis book. I actually love it... yes, I do realise it's just a textbook but trust me, you'll love it too.
Most useful analysis book ever, 05 Feb 2003
This book was the best value for money out of all the books I have ever bought. It covers everything you will need to know to start off your analysis coverage. Proofs are both concise but fully acceptable under examinable conditions. Definately worth buying. I used this in my first year of my Mathematics degree course.
Excellent theoretical text, 08 Jan 2002
An excellent book for learning about Analysis - although more about the general theory rather than how to actually perform the methods. A traditional analysis text, as opposed to something such as Burn's Numbers and Functions which consists largely of exercises. Sometimes, the solutions in this book can be a little scant, but for looking up proofs and theorems it is very useful. Recommended by everybody on my degree course, and very useful to me.
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Customer Reviews
No need to apologize for this, 27 Nov 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.
How maths used to be, 24 Nov 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.
Hardy's Apology, 23 Jul 2008
I bought Hardy's biography almost 50 years ago, just before going up to university to read Maths. The gentle almost humourous tone of the book convinced me that the life of a mathematician was a potential source of pleasure. In fact I later moved to read Philosophy, for my interests grew to centre on Logic and Mathematical Foundations, but my current work on Infinity still finds me using Hardy's "Pure Mathematics", and I still occasionally read his "Apology" for the pure pleasure that this outstanding book provides.
Disappointing, 26 Mar 2008
I'm not sure I read the same book as the other reviewers. I'm fascinated by mathematics and have, for a long time, been meaning to read this book. Finally I have, and I have to say I was somewhat disappointed. C.P. Snow's foreword was an interesting potted biography of one of Britain's greatest 20th century mathematicians; but, the "apology" itself is overshadowed by its own introduction.
The first nine chapters are a clear indication - if any was needed - that intellectual brilliance is no talisman against small-mindedness. Maybe I'm being unfair but in those nine short chapters Hardy manages to denigrate most of humanity - "most people can do nothing at all well" - by which he means, nothing that he personally considers worthy. I'm not convinced we should readily accept the social opinions of a man who was born to privilege and spent his entire life in rarefied cloisters shielded from what the "most" in question would consider reality. A man who thought we should not go to war with Germany because of the caliber of its mathematicians and scientists. (?) There are many reasons why we shouldn't be at war with Iraq, for example, but that the region is the cradle of civilisation is not one of them.
From chapter ten onwards the book makes a change of direction and deals with the actual subject of mathematics and being a mathematician. This latter section of the book is a much more interesting (less infuriating) read but I still felt he dealt rather ineptly with the attempt to differentiate between "common and garden" mathematics and the truly profound and beautiful creative mathematics that he so clearly loves.
When you compare this book to say Marcus du Sautoy's "Music of the Primes" or Simon Singh's "Fermat's Last Theorem", both books which truly promote the essential character of mathematics, "A Mathematician's Apology" seems very pale.
Read it. Seriously., 04 Feb 2008
This book is a facinating insight into the mind of one of the century's greatest mathematians. However, Hardy's ideas go far beyond this into the purpose of human endevour in general.
The best part of this book is the foreword by C.P Snow. This amounts to a lucid, if brief, biography of Hardy from a man who is brilliant in his own right.
If you have any kind of interest in the realities of mathematics, or the workings of genius, then this book is the place to start.
Great For A2 Physics, 15 Sep 2006
I have just started A2 Physics at my school. I didn't do maths to AS (just GCSE). I am the only one in my class without an AS in Maths. My teacher told me that I needed to know about Differentiation and Integration in order to progress my A grade in AS physics to an A in A2.
I always had the idea that Calculus was simply a very complicated area of Mathematics developed by Newton. I didn't know anything else (remember I only had a Maths GCSE).
I thought I'd buy this book to see if it could help me. The fantastic thing is that it explains every single concept in an extremely simple way (starting with the very basics). I believe that I am now up to the high standard of the mathematicians in my class and am able to understand the important mathematical concepts in physics that use calculus as a basis for their explanation. In particular, OCR's Advancing Physics A2 - Creating Models section.
You really don't need to be a math genius to understand this book fully!
Shock, horror, calculus is easy (well sort of!), 06 Nov 2003
If you`ve ever looked at a polynomial equation and thought that learning Martian would be easier, or even that knowing w | | |