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A Mathematician's Apology
G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician the purest of the pure'. He was also, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, ready to talk about anything'. This 'apology', written in 1940, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it was like to be a creative artist'. C. P. Snow's Foreword gives sympathetic and witty insights into Hardy's life, with its rich store of anecdotes concerning his collaboration with the brilliant Indian mathematician Ramanujan, his idiosyncrasies, and his passion for cricket. This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times.
154 pages, Paperback
First published January 1, 1940
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About the author
G.H. Hardy
68books143followersGodfrey Harold Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis.
Non-mathematicians usually know him for A Mathematician's Apology, his essay from 1940 on the aesthetics of mathematics. The apology is often considered one of the best insights into the mind of a working mathematician written for the layman.
His relationship as mentor, from 1914 onwards, of the Indian mathematician Srinivasa Ramanujan has become celebrated. Hardy almost immediately recognized Ramanujan's extraordinary albeit untutored brilliance, and Hardy and Ramanujan became close collaborators. In an interview by Paul Erdős, when Hardy was asked what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan. He called their collaboration "the one romantic incident in my life."
Non-mathematicians usually know him for A Mathematician's Apology, his essay from 1940 on the aesthetics of mathematics. The apology is often considered one of the best insights into the mind of a working mathematician written for the layman.
His relationship as mentor, from 1914 onwards, of the Indian mathematician Srinivasa Ramanujan has become celebrated. Hardy almost immediately recognized Ramanujan's extraordinary albeit untutored brilliance, and Hardy and Ramanujan became close collaborators. In an interview by Paul Erdős, when Hardy was asked what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan. He called their collaboration "the one romantic incident in my life."
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August 14, 2019
A Serious Business of Taste
The dominant theme of A Mathematician’s Apology, established from the first page, is one of aesthetics. Aesthetics, the study of what is inherently important and valuable, is for Hardy the fundamental power of mathematics, not an incidental result of correct thought. Aesthetics, while not unique to mathematics, is arguably more single-mindedly applied in mathematics than in any other human activity, including art of all kinds.
Hardy, like many poets and artists as well as other mathematicians, is hesitant about his exposition of the aesthetics of mathematics. It seems to him vaguely disloyal and a possible waste of time. Mathematicians, after all, do mathematics, they don’t write about doing mathematics. And he has a point: either one gets the aesthetics of mathematics or one doesn’t. Mathematicians could care less about their reputations outside of mathematics. Hardy is therefore wary of going over the line from an ‘apologia�, that is, an explanation, to a defense, lest his should offend his colleagues. Most creative artists, I imagine, feel similarly.
Hardy is acutely aware that an aesthetic, that is, a specific criterion or set of criteria, is that which is valuable for itself, and for no other reason. The aesthetic aims at nothing practical, nothing beyond itself. It is an argument from first principles that cannot be gainsaid by any other argument. It proves itself by its own assertion and by its own internal logic. Paradoxically, this is what makes an aesthetic so powerful: it doesn’t care what is thought of it outside the rules of its own creation. It is the scale of its own value. Its attraction is precisely its special kind of absoluteness.
Aesthetics therefore is a dangerous business. It holds itself apart from criticism of any kind from those outside the circle which embraces it. It is an elitist activity. Its justification is merely the complete indifference about whether others subscribe to its views or not. There is no compulsion for others to ‘belong� nor even to recognise its existence. It does not even claim any right to exist, for that would imply a purpose beyond itself.
When a mathematician asserts that 1+1=2, there is no meaning beyond that assertion other than the expression of the aesthetic of mathematics. The assertion may have implications. Indeed this assertion has vast implications for the practice of mathematics. But that is the extent of its mathematical significance. It is a start in expressing the relevant aesthetic, but it is not intended to make the world better, or more intelligent, or more interested in mathematics.
Does that make all aesthetics equal? Hardy says that the highest ambition “is to leave behind something of permanent value.� Value cannot be permanent unless it is intrinsic, of value in and of itself. But such intrinsic value is not arbitrary. The aesthetics adhered to by the professions - law, medicine, science - for example are distinctly different and incommensurate; but they are not arbitrary. Rather they are arrived at through social processes and accepted for what they are - the way we do things.
Nonetheless a distinction can be made between fundamental criteria of action (I hesitate to call them aesthetics at this point) which are hidden, implicit, and unexpressed, from those that are made explicit, revealed, and given for consideration to others without the threat of compulsion. It is only the latter that are aesthetics. The former we can categorise as mere prejudice or, at best, unconsidered preferences.
An aesthetic then must be articulated and expressed to be considered as such. When it is, it develops an uncanny power. The greater the precision of the articulation, the greater its attractiveness. I can’t account for this except as an empirical observation of the way in which people respond to aesthetic propositions. ‘Justice�, for example, as an abstract criterion is far less captivating than ‘the just treatment of relationships among those with same sex preferences as a matter of law�. This latter aesthetic has become accepted throughout Western Europe and North America as it has become expressed.
An aesthetic is likely to attract those with a talent to employ it creatively. If for no otter reason than the social comfort of being among ‘like-minded� people, that is others who appreciate not just the same things but the ability itself to appreciate those things. Hardy identifies curiosity, professional pride and ambition for reputation within the profession as general aesthetic aspects, applicable to many others than mathematicians.
The principle aesthetic criteria of mathematics that Hardy identifies, however, is that of ‘pattern�, more specifically patterns of ideas. It has always struck me that it is precisely this aesthetic that is presented in Herman Hesse’s novel The Glass Bead Game, published almost contemporaneously with A Mathematician’s Apology.
The game in question is never described except to the extent that it involves the identification of patterns across otherwise discrete fields of human knowledge - mathematics, history, politics, painting, poetry, physics, etc. A mathematical aesthetic expanded universally in other words. My discussion with other readers suggests that indeed you either get this aesthetic, and find the book a treasure, or you don’t.
Pattern-seeing rather than pattern-making is the essential mathematical skill. The difference is crucial, and what makes mathematics an empirical science. Numbers are there to be explored and interrogated. Mathematicians don’t invent, they discover, patterns that numbers have always had. These patterns are as real, perhaps even more real, than the patterns proposed by, say, physicists. The latter involve themselves with ‘strings�, and ‘quarks�, and ‘dark energy�, for example. But these are mere hypotheses in comparison with the factual solidity of the number 2, or the logical necessity of more than one cardinal order of infinity.
There are ugly and beautiful patterns in mathematics. One might suppose that this distinction is also arbitrary. But as Hardy explains, it is not arbitrary at all, nor is it vague even if its details are obvious only within the profession. Hardy refers to this as the ‘seriousness� of a theorem or a proof. A serious proof, like ܳ’s proof of the infinity of prime numbers is short, un-showy, and (surprisingly) surprising. It has a seductive elegance that does not so much force as it does invite acceptance.
A component of seriousness is ‘significance�. This Hardy further divides into ‘generality� and ‘depth�. Without these characteristics, theorems, however ingenious, remain curiosities of interest only to puzzlers and hobbyists. On the other hand, too much generality and a theorem becomes abstractly insipid. Depth is even more subtle and has to do with the virtuosity involved in solving a problem that has just the right degree of generality and difficulty - again using innovative or unexpected ...“line of attack� to get to a solution. Such a proof “should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.�
Ultimately metaphors like this are inevitable, not because the criteria are vague but because they have been so internalized by practitioners that they are almost pre- (or perhaps post-) linguistic. They ‘know it when they see it� not because they are inexperienced and unsophisticated dilettantes, but they are able to ‘walk the talk� so well that it is unnoticed by consciousness. The aesthetic becomes truly a ‘law written on the heart� for better or worse.
Hardy makes several mentions of how the aesthetic becomes obsolete and how it might be modified. Cambridge, for example, “crippled its mathematics for a century� by insisting on examinations that were exclusively about technique at the expense of creativity. Only by having the personal courage to move against this established norm did the situation improve. The toleration of such ‘rebellion� is clearly, therefore, a necessary characteristic of the aesthetic-society, as it were.
A Mathematician’s Apology is a highly personal statement, as its title implies. But the fact that its contents are only usually revealed over late night brandies during international academic conferences, doesn’t make it idiosyncratic or merely anecdotal. Without the mathematical talent sufficient to participate in the community that shares the aesthetic, it is perhaps impossible to appreciate the power of the aesthetic Hardy outlines.
My summary of Hardy’s outline is undoubtedly inadequate. But it is, I think, sufficient to establish the almost miraculous way in which a professional discipline can create and sustain criteria of value that are not only independent of economic or commercial imperatives, but markedly antithetical to them. One might say that mathematics is serious business indeed.
Appendix: Aesthetics and Fake News
An aesthetic has no intentional meaning beyond itself but it does have an incidental effect in the sense that it eliminates any consideration of truth. Since an aesthetic is its own truth, it cannot be compared or verified by reference to any other truth. This might appear as an aesthetic defect until it is realized that an aesthetic has a great epistemological consequence. It eliminates what has come to be known as fake news.
Fake news isn’t fake because it is intentionally wrong or not (although it may be). It is fake because it is irrelevant in a given aesthetic. Fake ‘fake news� are purported facts which are presented for some reason other than their mere presentation. It exists when there is an ulterior motive that remains unexpressed, a purpose - political, economic, or otherwise pragmatic - which is beyond the simple factual assertion.
It takes some practice to know whether one is dealing with an aesthetic or some other instrumental or intermediate criterion of value. The phrase ‘this is important because...� is a giveaway that whatever is being discussed is not an aesthetic. Even what follows the ‘because� - it is right; it is expedient; it is effective; it is sensible - may not be an aesthetic. Often we hide our aesthetic under layers of rationalization so that we may not be aware ourselves what our aesthetic is.
When a politician claims that a news story is fake, it is because he has some underlying interest he wants to promote, some hidden aesthetic, possibly Power, possibly wealth, possibly reputation. But never ‘truth�, this being the underlying aesthetic whose revelation might be damaging to itself.
When a physicist or a social scientist makes a claim about reality, he or she is also making claim that is fake. They may use the term ‘truth� to defend such a claim, saying that it ‘fits the facts� better than alternatives, or that ‘it will cost us less to do X than to do Y� but these are statements that confirm the existence of some other criterion that constitutes the reason that their assertion should be accepted. Only when we reach the terminal response-point ‘just because� have we encountered what can be called the fundamental aesthetic that can’t be defended, only accepted or rejected, have these scientists approached the directness of the aesthetic of mathematics.
The dominant theme of A Mathematician’s Apology, established from the first page, is one of aesthetics. Aesthetics, the study of what is inherently important and valuable, is for Hardy the fundamental power of mathematics, not an incidental result of correct thought. Aesthetics, while not unique to mathematics, is arguably more single-mindedly applied in mathematics than in any other human activity, including art of all kinds.
Hardy, like many poets and artists as well as other mathematicians, is hesitant about his exposition of the aesthetics of mathematics. It seems to him vaguely disloyal and a possible waste of time. Mathematicians, after all, do mathematics, they don’t write about doing mathematics. And he has a point: either one gets the aesthetics of mathematics or one doesn’t. Mathematicians could care less about their reputations outside of mathematics. Hardy is therefore wary of going over the line from an ‘apologia�, that is, an explanation, to a defense, lest his should offend his colleagues. Most creative artists, I imagine, feel similarly.
Hardy is acutely aware that an aesthetic, that is, a specific criterion or set of criteria, is that which is valuable for itself, and for no other reason. The aesthetic aims at nothing practical, nothing beyond itself. It is an argument from first principles that cannot be gainsaid by any other argument. It proves itself by its own assertion and by its own internal logic. Paradoxically, this is what makes an aesthetic so powerful: it doesn’t care what is thought of it outside the rules of its own creation. It is the scale of its own value. Its attraction is precisely its special kind of absoluteness.
Aesthetics therefore is a dangerous business. It holds itself apart from criticism of any kind from those outside the circle which embraces it. It is an elitist activity. Its justification is merely the complete indifference about whether others subscribe to its views or not. There is no compulsion for others to ‘belong� nor even to recognise its existence. It does not even claim any right to exist, for that would imply a purpose beyond itself.
When a mathematician asserts that 1+1=2, there is no meaning beyond that assertion other than the expression of the aesthetic of mathematics. The assertion may have implications. Indeed this assertion has vast implications for the practice of mathematics. But that is the extent of its mathematical significance. It is a start in expressing the relevant aesthetic, but it is not intended to make the world better, or more intelligent, or more interested in mathematics.
Does that make all aesthetics equal? Hardy says that the highest ambition “is to leave behind something of permanent value.� Value cannot be permanent unless it is intrinsic, of value in and of itself. But such intrinsic value is not arbitrary. The aesthetics adhered to by the professions - law, medicine, science - for example are distinctly different and incommensurate; but they are not arbitrary. Rather they are arrived at through social processes and accepted for what they are - the way we do things.
Nonetheless a distinction can be made between fundamental criteria of action (I hesitate to call them aesthetics at this point) which are hidden, implicit, and unexpressed, from those that are made explicit, revealed, and given for consideration to others without the threat of compulsion. It is only the latter that are aesthetics. The former we can categorise as mere prejudice or, at best, unconsidered preferences.
An aesthetic then must be articulated and expressed to be considered as such. When it is, it develops an uncanny power. The greater the precision of the articulation, the greater its attractiveness. I can’t account for this except as an empirical observation of the way in which people respond to aesthetic propositions. ‘Justice�, for example, as an abstract criterion is far less captivating than ‘the just treatment of relationships among those with same sex preferences as a matter of law�. This latter aesthetic has become accepted throughout Western Europe and North America as it has become expressed.
An aesthetic is likely to attract those with a talent to employ it creatively. If for no otter reason than the social comfort of being among ‘like-minded� people, that is others who appreciate not just the same things but the ability itself to appreciate those things. Hardy identifies curiosity, professional pride and ambition for reputation within the profession as general aesthetic aspects, applicable to many others than mathematicians.
The principle aesthetic criteria of mathematics that Hardy identifies, however, is that of ‘pattern�, more specifically patterns of ideas. It has always struck me that it is precisely this aesthetic that is presented in Herman Hesse’s novel The Glass Bead Game, published almost contemporaneously with A Mathematician’s Apology.
The game in question is never described except to the extent that it involves the identification of patterns across otherwise discrete fields of human knowledge - mathematics, history, politics, painting, poetry, physics, etc. A mathematical aesthetic expanded universally in other words. My discussion with other readers suggests that indeed you either get this aesthetic, and find the book a treasure, or you don’t.
Pattern-seeing rather than pattern-making is the essential mathematical skill. The difference is crucial, and what makes mathematics an empirical science. Numbers are there to be explored and interrogated. Mathematicians don’t invent, they discover, patterns that numbers have always had. These patterns are as real, perhaps even more real, than the patterns proposed by, say, physicists. The latter involve themselves with ‘strings�, and ‘quarks�, and ‘dark energy�, for example. But these are mere hypotheses in comparison with the factual solidity of the number 2, or the logical necessity of more than one cardinal order of infinity.
There are ugly and beautiful patterns in mathematics. One might suppose that this distinction is also arbitrary. But as Hardy explains, it is not arbitrary at all, nor is it vague even if its details are obvious only within the profession. Hardy refers to this as the ‘seriousness� of a theorem or a proof. A serious proof, like ܳ’s proof of the infinity of prime numbers is short, un-showy, and (surprisingly) surprising. It has a seductive elegance that does not so much force as it does invite acceptance.
A component of seriousness is ‘significance�. This Hardy further divides into ‘generality� and ‘depth�. Without these characteristics, theorems, however ingenious, remain curiosities of interest only to puzzlers and hobbyists. On the other hand, too much generality and a theorem becomes abstractly insipid. Depth is even more subtle and has to do with the virtuosity involved in solving a problem that has just the right degree of generality and difficulty - again using innovative or unexpected ...“line of attack� to get to a solution. Such a proof “should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.�
Ultimately metaphors like this are inevitable, not because the criteria are vague but because they have been so internalized by practitioners that they are almost pre- (or perhaps post-) linguistic. They ‘know it when they see it� not because they are inexperienced and unsophisticated dilettantes, but they are able to ‘walk the talk� so well that it is unnoticed by consciousness. The aesthetic becomes truly a ‘law written on the heart� for better or worse.
Hardy makes several mentions of how the aesthetic becomes obsolete and how it might be modified. Cambridge, for example, “crippled its mathematics for a century� by insisting on examinations that were exclusively about technique at the expense of creativity. Only by having the personal courage to move against this established norm did the situation improve. The toleration of such ‘rebellion� is clearly, therefore, a necessary characteristic of the aesthetic-society, as it were.
A Mathematician’s Apology is a highly personal statement, as its title implies. But the fact that its contents are only usually revealed over late night brandies during international academic conferences, doesn’t make it idiosyncratic or merely anecdotal. Without the mathematical talent sufficient to participate in the community that shares the aesthetic, it is perhaps impossible to appreciate the power of the aesthetic Hardy outlines.
My summary of Hardy’s outline is undoubtedly inadequate. But it is, I think, sufficient to establish the almost miraculous way in which a professional discipline can create and sustain criteria of value that are not only independent of economic or commercial imperatives, but markedly antithetical to them. One might say that mathematics is serious business indeed.
Appendix: Aesthetics and Fake News
An aesthetic has no intentional meaning beyond itself but it does have an incidental effect in the sense that it eliminates any consideration of truth. Since an aesthetic is its own truth, it cannot be compared or verified by reference to any other truth. This might appear as an aesthetic defect until it is realized that an aesthetic has a great epistemological consequence. It eliminates what has come to be known as fake news.
Fake news isn’t fake because it is intentionally wrong or not (although it may be). It is fake because it is irrelevant in a given aesthetic. Fake ‘fake news� are purported facts which are presented for some reason other than their mere presentation. It exists when there is an ulterior motive that remains unexpressed, a purpose - political, economic, or otherwise pragmatic - which is beyond the simple factual assertion.
It takes some practice to know whether one is dealing with an aesthetic or some other instrumental or intermediate criterion of value. The phrase ‘this is important because...� is a giveaway that whatever is being discussed is not an aesthetic. Even what follows the ‘because� - it is right; it is expedient; it is effective; it is sensible - may not be an aesthetic. Often we hide our aesthetic under layers of rationalization so that we may not be aware ourselves what our aesthetic is.
When a politician claims that a news story is fake, it is because he has some underlying interest he wants to promote, some hidden aesthetic, possibly Power, possibly wealth, possibly reputation. But never ‘truth�, this being the underlying aesthetic whose revelation might be damaging to itself.
When a physicist or a social scientist makes a claim about reality, he or she is also making claim that is fake. They may use the term ‘truth� to defend such a claim, saying that it ‘fits the facts� better than alternatives, or that ‘it will cost us less to do X than to do Y� but these are statements that confirm the existence of some other criterion that constitutes the reason that their assertion should be accepted. Only when we reach the terminal response-point ‘just because� have we encountered what can be called the fundamental aesthetic that can’t be defended, only accepted or rejected, have these scientists approached the directness of the aesthetic of mathematics.
January 30, 2015
I nearly studied maths at university, because of this book.
When I was sixteen, I was scared of the grades and numbers end of academia, and I was determined that whatever I was going to study - and it was going to be something, and a lot of it - I was going to do it for the love of it. I was going to read around my subjects, follow tangents and pick whatever took my fancy. So, a few months into a Maths A-level, I took this out of Southampton Central Library, and I didn't give it back for nearly a year. The joy for numbers and patterns is infectious. The pure love for picking out order and seeing connections is one of my favourite things. I am convinced that you can talk about the most boring subject in the world, whatever it may be, but if you do it with love and that sparkle in your eye, people will want to listen, and ask questions, and find out more.
This is how GH Hardy made mathematics for me, and why sometimes I still look him up, and come back to this book, and remember why scholarship and finding out about things make me more excited about the world than anything else. They make me remember why I fall in love with people who get enthusiastic, and why I seek out friends who know about things that I don't, and why I read widely. This book sparked my love of recreational maths, word puzzles, cryptic crosswords, steganography.
The first time I read his proof for why the square root of two is an irrational number - short, quick, and with "Isn't this beautiful?" commentary - I followed it along with a pen and paper. I love his discussion of elegance, and how you can do things with numbers that are elegant, and why that is a legitimate thing to strive for. In the rest of my life, I want to make things that are elegant and clever and reach out to people like GH Hardy does. Every so often I come back to it and I get exactly the same feelings and they push me forward again. This is a book for when I am tired of everything else. It helps me go back to looking things in the eye.
The first half of A Mathematician's Apology is a foreword, and it is a friend and former student of Hardy's writing about his life. The context of a story is never as clean and self-contained as the story itself. Real life that has not been picked and chosen is not as clean as that single persuasive essay with the purpose of showing you how much fun a thing can be. The foreword makes me want to go out and read everything and write things down, because I have to do it all now, because one day I won't be able to.
I don't get to recommend this book to very many people, because apparently "it's about maths" is a turn-off. Somehow when I recommend this to people they have a very long to-read list that rears its head in the way that it doesn't for other books. That's okay. It's special to me, and a gem that I will quietly nurse for the times that I'm sad, and I won't force it on people with such phenomenally long other lists. But this book has amplified the person that I might have been into the person that I am, and every so often it does it again, and GH Hardy feels like one of my best friends even though this is all of his work that I know, and I am an non-Oxbridge-educated girl so if we'd met I would have bored him to tears. It's okay. I've got a lot from it anyway.
When I was sixteen, I was scared of the grades and numbers end of academia, and I was determined that whatever I was going to study - and it was going to be something, and a lot of it - I was going to do it for the love of it. I was going to read around my subjects, follow tangents and pick whatever took my fancy. So, a few months into a Maths A-level, I took this out of Southampton Central Library, and I didn't give it back for nearly a year. The joy for numbers and patterns is infectious. The pure love for picking out order and seeing connections is one of my favourite things. I am convinced that you can talk about the most boring subject in the world, whatever it may be, but if you do it with love and that sparkle in your eye, people will want to listen, and ask questions, and find out more.
This is how GH Hardy made mathematics for me, and why sometimes I still look him up, and come back to this book, and remember why scholarship and finding out about things make me more excited about the world than anything else. They make me remember why I fall in love with people who get enthusiastic, and why I seek out friends who know about things that I don't, and why I read widely. This book sparked my love of recreational maths, word puzzles, cryptic crosswords, steganography.
The first time I read his proof for why the square root of two is an irrational number - short, quick, and with "Isn't this beautiful?" commentary - I followed it along with a pen and paper. I love his discussion of elegance, and how you can do things with numbers that are elegant, and why that is a legitimate thing to strive for. In the rest of my life, I want to make things that are elegant and clever and reach out to people like GH Hardy does. Every so often I come back to it and I get exactly the same feelings and they push me forward again. This is a book for when I am tired of everything else. It helps me go back to looking things in the eye.
The first half of A Mathematician's Apology is a foreword, and it is a friend and former student of Hardy's writing about his life. The context of a story is never as clean and self-contained as the story itself. Real life that has not been picked and chosen is not as clean as that single persuasive essay with the purpose of showing you how much fun a thing can be. The foreword makes me want to go out and read everything and write things down, because I have to do it all now, because one day I won't be able to.
I don't get to recommend this book to very many people, because apparently "it's about maths" is a turn-off. Somehow when I recommend this to people they have a very long to-read list that rears its head in the way that it doesn't for other books. That's okay. It's special to me, and a gem that I will quietly nurse for the times that I'm sad, and I won't force it on people with such phenomenally long other lists. But this book has amplified the person that I might have been into the person that I am, and every so often it does it again, and GH Hardy feels like one of my best friends even though this is all of his work that I know, and I am an non-Oxbridge-educated girl so if we'd met I would have bored him to tears. It's okay. I've got a lot from it anyway.
September 2, 2021
“what are the uses of mathematics?�
This has been a pervasive question that might have occurred to everyone who encounters mathematics in school or elsewhere throughout their lives.
Euclid( a Greek mathematician who lived in 4th century BC) believed in the search of mathematical truth for its own sake. One story tells of a student who questioned him about the use of mathematics. Upon completing the lesson, Euclid turned to his slave and said : � give the boy a penny since he desires to profit from all that he learns. � the student was then expelled.
History of mathematics has shown the same incentive for other great mathematicians such as Pythagoras, Leonhard Euler, The legendary Gauss, Riemann and so many other fertile minds.
In � A mathematicians Apology� G.H Hardy who was one of the greatest number theorists himself, justifies pursuing pure mathematics independent of any plausible application in real life or in other disciplines.
He argues that in every scientific field, scientist have three main incentive for research. First, they wanted to satisfy their insatiable curiosity for knowing the truth of any kind. The second is that they wanted to have some intellectual advantage among other people because they know something that other don’t or they are capable of doing some stuff that other people can’t. And finally the last incentive is indeed money and social status.
Then Hardy asserts that in mathematics all those three incentives mentioned above have been met because as a matter of curiosity for truth mathematics has the most profound and sacred kinds of truth in it self. Furthermore, a proficient mathematician has special powers that is terrifically rare among ordinary people. And at last, mathematicians have always had an eminent status and fame in their society and even sometimes in the world. So this fact alone justifies craving mathematics for its own sake.
Hardy also claims that in so many other scientific fields- in fact all of them- such as biology, quantum mechanics, physics, astronomy etc there are an enormous amount of useless discoveries. Although just approximately 10 percent of mathematics is useful and adds to human comfort, it is more useful than so many other disciplines.
The author mentions so many other philosophical points which was boring to me so if you are fond of philosophical arguments you can read the rest of the book.
This has been a pervasive question that might have occurred to everyone who encounters mathematics in school or elsewhere throughout their lives.
Euclid( a Greek mathematician who lived in 4th century BC) believed in the search of mathematical truth for its own sake. One story tells of a student who questioned him about the use of mathematics. Upon completing the lesson, Euclid turned to his slave and said : � give the boy a penny since he desires to profit from all that he learns. � the student was then expelled.
History of mathematics has shown the same incentive for other great mathematicians such as Pythagoras, Leonhard Euler, The legendary Gauss, Riemann and so many other fertile minds.
In � A mathematicians Apology� G.H Hardy who was one of the greatest number theorists himself, justifies pursuing pure mathematics independent of any plausible application in real life or in other disciplines.
He argues that in every scientific field, scientist have three main incentive for research. First, they wanted to satisfy their insatiable curiosity for knowing the truth of any kind. The second is that they wanted to have some intellectual advantage among other people because they know something that other don’t or they are capable of doing some stuff that other people can’t. And finally the last incentive is indeed money and social status.
Then Hardy asserts that in mathematics all those three incentives mentioned above have been met because as a matter of curiosity for truth mathematics has the most profound and sacred kinds of truth in it self. Furthermore, a proficient mathematician has special powers that is terrifically rare among ordinary people. And at last, mathematicians have always had an eminent status and fame in their society and even sometimes in the world. So this fact alone justifies craving mathematics for its own sake.
Hardy also claims that in so many other scientific fields- in fact all of them- such as biology, quantum mechanics, physics, astronomy etc there are an enormous amount of useless discoveries. Although just approximately 10 percent of mathematics is useful and adds to human comfort, it is more useful than so many other disciplines.
The author mentions so many other philosophical points which was boring to me so if you are fond of philosophical arguments you can read the rest of the book.
November 1, 2019
A pragmatist's review of how math fits into our knowledge, world and psyche.
Q:
The mass of mathematical truth is obvious and imposing; its practical applications, the bridges and steam-engines and dynamos, obtrude themselves on the dullest imagination. The public does not need to be convinced that there is something in mathematics. (c)
Q:
I am a lawyer, or a stockbroker, or a professional cricketer, because I have some real talent for that particular job. I am a lawyer because I have a fluent tongue, and am interested in legal subtleties; I am a stockbroker because my judgment of the markets is quick and sound; I am a professional
cricketer because I can bat unusually well. I agree that it might be better to be a poet or a mathematician, but unfortunately I have no talent for such pursuits. (c)
Q:
I am not suggesting that this is a defence which can be made by most people, since most people can do nothing at all well. But it is impregnable when it can be made without absurdity, as it can by a substantial minority: perhaps five or even ten percent of men can do something rather well. It is a tiny minority who can do something really well, and the number of men who can do two things well is negligible. If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full. (c)
Q:
I said that a mathematician was a maker of patterns of ideas, and that beauty and seriousness were the criteria by which his patterns should be judged. (с)
Q:
Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years. It is true that there are branches of applied mathematics, such as ballistics and aerodynamics, which have been developed deliberately for war and demand a quite elaborate technique: it is perhaps hard to call them ‘trivial�, but none of them has any claim to rank as ‘real�. They are indeed repulsively ugly and intolerably dull; even Littlewood could not make ballistics respectable, and if he could not who can? So a real mathematician has his conscience clear; there is nothing to be set against any value his work may have; mathematics is, as I said at Oxford, a ‘harmless and innocent� occupation.
The trivial mathematics, on the other hand, has many applications in war. The gunnery experts and aeroplane designersfor example, could not do their work without it. And the general effect of these applications is plain: mathematics facilitates (if not so obviously as physics or chemistry) modern, scientific, ‘total� war.
It is not so clear as it might seem that this is to be regretted... (c)
Q:
The mass of mathematical truth is obvious and imposing; its practical applications, the bridges and steam-engines and dynamos, obtrude themselves on the dullest imagination. The public does not need to be convinced that there is something in mathematics. (c)
Q:
I am a lawyer, or a stockbroker, or a professional cricketer, because I have some real talent for that particular job. I am a lawyer because I have a fluent tongue, and am interested in legal subtleties; I am a stockbroker because my judgment of the markets is quick and sound; I am a professional
cricketer because I can bat unusually well. I agree that it might be better to be a poet or a mathematician, but unfortunately I have no talent for such pursuits. (c)
Q:
I am not suggesting that this is a defence which can be made by most people, since most people can do nothing at all well. But it is impregnable when it can be made without absurdity, as it can by a substantial minority: perhaps five or even ten percent of men can do something rather well. It is a tiny minority who can do something really well, and the number of men who can do two things well is negligible. If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full. (c)
Q:
I said that a mathematician was a maker of patterns of ideas, and that beauty and seriousness were the criteria by which his patterns should be judged. (с)
Q:
Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years. It is true that there are branches of applied mathematics, such as ballistics and aerodynamics, which have been developed deliberately for war and demand a quite elaborate technique: it is perhaps hard to call them ‘trivial�, but none of them has any claim to rank as ‘real�. They are indeed repulsively ugly and intolerably dull; even Littlewood could not make ballistics respectable, and if he could not who can? So a real mathematician has his conscience clear; there is nothing to be set against any value his work may have; mathematics is, as I said at Oxford, a ‘harmless and innocent� occupation.
The trivial mathematics, on the other hand, has many applications in war. The gunnery experts and aeroplane designersfor example, could not do their work without it. And the general effect of these applications is plain: mathematics facilitates (if not so obviously as physics or chemistry) modern, scientific, ‘total� war.
It is not so clear as it might seem that this is to be regretted... (c)
June 1, 2011
Amusing, even if it was as sad as the introduction suggested. Read it in high school, but haven't since. Glad I took another crack at it. It just about made me want to crack open one of my math books! I enjoyed the style of exposition, as well as much of the message, though, admittedly, I probably lost track of an argument here or there.
I think avoided pulling out some of the more quoted passages, though I'm sure these aren't entirely original selections:
68: If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.
70: ... it is undeniable that a gift for mathematics is one of the most specialized talents, and that mathematicians as a class are not particularly distinguished for general ability of versatility. If a man is any sense a real mathematician, then it is a hundred to one that his mathematics will be far better than anything else he can do, and that he would be silly if he surrendered any decent opportunity of exercising his one talent in order to do undistinguished work in other fields.
73: It is quite true that most people can do nothing well. <-- Dang, Hardy. How do you really feel?
77: A man's first duty, a young man's at any rate, is to be ambitious. Ambition is a noble passion which may legitimately take many forms...
80: [A mathematician's] subject is the most curious of all -- there is none in which truth plays such odd pranks.
116: Mathematics may, like poetry or music, 'promote and sustain a lofty habit of mind'...
117-8: It is indeed rather astonishing how little practical value scientific knowledge has for ordinary men, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to its reputed utility. <-- ah, school
143: When the world is mad, a mathematician may find in mathematics an incomparable anodyne. For mathematics is, of all the arts and sciences, the most austere and the most remote, and a mathematician should be of all men the one how can easily take refuge where, as Bertrand Russell says, 'one at least of our nobler impulses can best escape from the dreary exile of the actual world.'
I think avoided pulling out some of the more quoted passages, though I'm sure these aren't entirely original selections:
68: If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.
70: ... it is undeniable that a gift for mathematics is one of the most specialized talents, and that mathematicians as a class are not particularly distinguished for general ability of versatility. If a man is any sense a real mathematician, then it is a hundred to one that his mathematics will be far better than anything else he can do, and that he would be silly if he surrendered any decent opportunity of exercising his one talent in order to do undistinguished work in other fields.
73: It is quite true that most people can do nothing well. <-- Dang, Hardy. How do you really feel?
77: A man's first duty, a young man's at any rate, is to be ambitious. Ambition is a noble passion which may legitimately take many forms...
80: [A mathematician's] subject is the most curious of all -- there is none in which truth plays such odd pranks.
116: Mathematics may, like poetry or music, 'promote and sustain a lofty habit of mind'...
117-8: It is indeed rather astonishing how little practical value scientific knowledge has for ordinary men, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to its reputed utility. <-- ah, school
143: When the world is mad, a mathematician may find in mathematics an incomparable anodyne. For mathematics is, of all the arts and sciences, the most austere and the most remote, and a mathematician should be of all men the one how can easily take refuge where, as Bertrand Russell says, 'one at least of our nobler impulses can best escape from the dreary exile of the actual world.'
April 2, 2012
I wonder how much my enjoyment of this book was hampered by my mathematical incompetence. Not too much, I hope. CP Snow’s introduction is as good as the book, but you can’t fault Hardy with not giving you something to chew on. Rather than try to summarize my feelings about Hardy’s little book, I’m going to take the lazy option here and simply repost from my blog:
-
In A Mathematician’s Apology G.H. Hardy estimates that only five or ten people in a hundred can do something “rather well.� Considerably fewer are really gifted. We do not each have a valuable talent waiting for discovery. We may dream of making names for ourselves, but most “talents� are talents only by inflation, and many true talents are never valued. The influence of the cult of achievement extends even beyond its membership. Those who renounce the pursuit of worldly accomplishment often do so with other (more comfortably nebulous) goals in mind: sainthood, perhaps, or self-realization. They’re chasing the same fox by another tail.
Hardy’s calculation is stark, and depressing. I’ve been working on a novel two nights each week for the past four years. I’m well into my third draft and hope to shop it around to agents this summer. Re-reading and re-writing it is a bruising, infuriating, ego-punishing business. What I’ve created is, I think, better than a lot of what gets published today, but that’s saying awfully little when 99% of what gets published is an unjustifiable waste of both writer’s and reader’s time. Almost every book ever written more than deserves its inevitable oblivion.
My book surely will too. Though I’m bold enough to say that it’s “better than many,� I’m not going to fool myself and say “better than 99%.� It may beat fifty or even sixty percent of the schlock printed these days, but I won’t bluff any higher than that. Even if I succeed in getting it published, it’s not something to be too ridiculously proud of. If writing it has taught me anything, it’s that I am no Herman Melville or Henry James. Tonic as it may be to fess up to that inadequacy, my sickness is such that I plod on anyway. I’m even making notes for a second book. Ambition isn’t going to let lack of genius stand in its way.
I may be neglecting avenues for achievement that are better suited to me. Hardy writes that “poetry is more valuable than cricket, but Bradman would be a fool if he sacrificed his cricket in order to write second-rate minor poetry.� I know nothing about cricket or Bradman, but I’ll agree that you don’t give up on a first-rate talent merely because it happens to be for a second-rate activity. I manage to make a living in the business world without much effort. What might I achieve if I focused my ambitions in that direction? But most days it’s a struggle even to fake a tepid enthusiasm.
According to Hardy, first-rate minds care only for creation. If second-rate minds care for it too, so much the worse for them. They would do better, he says, to restrict themselves to the very second-rate tasks of criticism and appreciation. “Appreciation.� The term, as he utters it, drips condescension. But I want to say that Hardy gets it wrong here. He shows his scheme of values to be debased. It may be that I’m too democratic-minded, or just plotting myself an escape from Hardy’s sentence, but I hate the idea that the worthiest of human endeavors is beyond the reach of most people. Surely it’s not only scarce things that can have ultimate value?
Appreciation, in the sense of pure enjoyment, seems to me a better candidate than creative accomplishment for the title of “man’s true work.� It may sound Jeffersonian (“pursuit of happiness�), or Epicurean, or bourgeois of me to say so. I don’t mean that people with leisure are morally superior to those without it. But though it’s not an idea that lends itself to proof by argument, I do believe that, other things being equal, there’s no nobler human aspiration than simply to enjoy and delight in things. To appreciate a particular face, a meal, a tree, a note, a book, a fact, an idea is something available to most of us. To enjoy something to the limit of one’s capacity is better than to create it.
-
In A Mathematician’s Apology G.H. Hardy estimates that only five or ten people in a hundred can do something “rather well.� Considerably fewer are really gifted. We do not each have a valuable talent waiting for discovery. We may dream of making names for ourselves, but most “talents� are talents only by inflation, and many true talents are never valued. The influence of the cult of achievement extends even beyond its membership. Those who renounce the pursuit of worldly accomplishment often do so with other (more comfortably nebulous) goals in mind: sainthood, perhaps, or self-realization. They’re chasing the same fox by another tail.
Hardy’s calculation is stark, and depressing. I’ve been working on a novel two nights each week for the past four years. I’m well into my third draft and hope to shop it around to agents this summer. Re-reading and re-writing it is a bruising, infuriating, ego-punishing business. What I’ve created is, I think, better than a lot of what gets published today, but that’s saying awfully little when 99% of what gets published is an unjustifiable waste of both writer’s and reader’s time. Almost every book ever written more than deserves its inevitable oblivion.
My book surely will too. Though I’m bold enough to say that it’s “better than many,� I’m not going to fool myself and say “better than 99%.� It may beat fifty or even sixty percent of the schlock printed these days, but I won’t bluff any higher than that. Even if I succeed in getting it published, it’s not something to be too ridiculously proud of. If writing it has taught me anything, it’s that I am no Herman Melville or Henry James. Tonic as it may be to fess up to that inadequacy, my sickness is such that I plod on anyway. I’m even making notes for a second book. Ambition isn’t going to let lack of genius stand in its way.
I may be neglecting avenues for achievement that are better suited to me. Hardy writes that “poetry is more valuable than cricket, but Bradman would be a fool if he sacrificed his cricket in order to write second-rate minor poetry.� I know nothing about cricket or Bradman, but I’ll agree that you don’t give up on a first-rate talent merely because it happens to be for a second-rate activity. I manage to make a living in the business world without much effort. What might I achieve if I focused my ambitions in that direction? But most days it’s a struggle even to fake a tepid enthusiasm.
According to Hardy, first-rate minds care only for creation. If second-rate minds care for it too, so much the worse for them. They would do better, he says, to restrict themselves to the very second-rate tasks of criticism and appreciation. “Appreciation.� The term, as he utters it, drips condescension. But I want to say that Hardy gets it wrong here. He shows his scheme of values to be debased. It may be that I’m too democratic-minded, or just plotting myself an escape from Hardy’s sentence, but I hate the idea that the worthiest of human endeavors is beyond the reach of most people. Surely it’s not only scarce things that can have ultimate value?
Appreciation, in the sense of pure enjoyment, seems to me a better candidate than creative accomplishment for the title of “man’s true work.� It may sound Jeffersonian (“pursuit of happiness�), or Epicurean, or bourgeois of me to say so. I don’t mean that people with leisure are morally superior to those without it. But though it’s not an idea that lends itself to proof by argument, I do believe that, other things being equal, there’s no nobler human aspiration than simply to enjoy and delight in things. To appreciate a particular face, a meal, a tree, a note, a book, a fact, an idea is something available to most of us. To enjoy something to the limit of one’s capacity is better than to create it.
June 15, 2016
The first thing the reader of this book will notice is that Hardy is an excellent writer. Although he repeatedly insists that his only talent lay in his mathematical ability, it is clear that he is a seasoned wordsmith.
The first mark of a good writer is their seemingly effortless ability to convey their personality through the written word, no matter the subject or format. The reader is immediately presented with Hardy the man, as if he is sitting in front of you giving a lecture.
One of the drawbacks of revealing your personality in such a direct way is that the reader is tempted to judge you, the author, rather than the content of your book. I certainly feel this temptation, as Hardy is not very likable. He immediately comes across as arrogant, rude, forceful, snobbish, opinionated, and gruff. He shoots off his views with the nonchalance of a college professor used to his students sitting in forced attention in crowded lecture halls. He does not hesitate to speculate on matters of which he knows little. In short, I don’t wish to have met him.
Hardy insists that mathematics is the vocation in which one comes most directly in touch with reality. This is a sort of escapism, also seen in Russell and Plato, where brilliant men (and Hardy is clearly brilliant) play ducks and drakes with ideas, skipping them across the surface of their intellects like smooth stones, rather than contemplating vile things like breakfast or the weather. Hardy also insists that mathematical fame is permanent, the closest thing that a person can ever come to immortality. These ideas, so appealing when stated by Plato or Descartes, are almost nauseating with Hardy; I shudder at the thought that he has reached some sort of immortality.
How ironic, then, that Hardy will most likely be remembered by the general mass of humankind for this little essay, rather than his mathematical achievements. And it is a fantastic essay, don’t get me wrong. To repeat, Hardy is an excellent writer. He is direct, clear, organized, and bold. Most of all, however, Hardy is honest, and this is a rare enough quality to almost redeem his whole arrogant personality. I would especially recommend it to those capable of the cold, aesthetic joy one experiences from beautiful proofs, like ܳ’s proof of the infinite number of primes. For those like myself, whose dull intellect shuts them off from these joys, this is a curious look into the psychology of such individuals.
The first mark of a good writer is their seemingly effortless ability to convey their personality through the written word, no matter the subject or format. The reader is immediately presented with Hardy the man, as if he is sitting in front of you giving a lecture.
One of the drawbacks of revealing your personality in such a direct way is that the reader is tempted to judge you, the author, rather than the content of your book. I certainly feel this temptation, as Hardy is not very likable. He immediately comes across as arrogant, rude, forceful, snobbish, opinionated, and gruff. He shoots off his views with the nonchalance of a college professor used to his students sitting in forced attention in crowded lecture halls. He does not hesitate to speculate on matters of which he knows little. In short, I don’t wish to have met him.
Hardy insists that mathematics is the vocation in which one comes most directly in touch with reality. This is a sort of escapism, also seen in Russell and Plato, where brilliant men (and Hardy is clearly brilliant) play ducks and drakes with ideas, skipping them across the surface of their intellects like smooth stones, rather than contemplating vile things like breakfast or the weather. Hardy also insists that mathematical fame is permanent, the closest thing that a person can ever come to immortality. These ideas, so appealing when stated by Plato or Descartes, are almost nauseating with Hardy; I shudder at the thought that he has reached some sort of immortality.
How ironic, then, that Hardy will most likely be remembered by the general mass of humankind for this little essay, rather than his mathematical achievements. And it is a fantastic essay, don’t get me wrong. To repeat, Hardy is an excellent writer. He is direct, clear, organized, and bold. Most of all, however, Hardy is honest, and this is a rare enough quality to almost redeem his whole arrogant personality. I would especially recommend it to those capable of the cold, aesthetic joy one experiences from beautiful proofs, like ܳ’s proof of the infinite number of primes. For those like myself, whose dull intellect shuts them off from these joys, this is a curious look into the psychology of such individuals.
May 29, 2014
I object not to the message, but rather its form. Essentially, GH argues that mathematics is worth the world's time and effort--that it is a beautiful, creative, and noble pursuit. I'm already convinced of this, so maybe I'm not his target audience and should therefore shut up. I've spent a non-trivial amount of around mathematicians. They are almost a different species, and I envy their passion and analytic abilities. While I'm glad GH tried to be their advocate--which must've been more necessary when he wrote this than it is today--I don't understand why some hold this book/essay in such high esteem.
GH mingles mathturbation with emperor's-new-clothes rhetoric. He asserts (I'm paraphrasing) that a true intellect appreciates math's beauty, thus implying that if one fails to appreciate the promised beauty, she must not be an intellect (this is further paraphrased as, "if you can't do the math on that then I can't help ya'" [Or perhaps by describing it as a preemptive "No true Scotsman" fallacy]). He spends a large part of the essay/book itemizing mathematics nice qualities.
GH has a real inferiority complex with respect to his love, mathematics, and his writing ability about it. The former expresses itself via a prim smugness, the later via his confessions of his inability to express concepts neatly.
I wish GH had tried to write something less universal and more personal. I wanted more insight into what it's like to be a capital-M Mathematician. I wanted to understand his thought process, how he experiences mathematical joy, how he touches Truth, how math has shaped his person and life, etc. The last chapter, in which he drifts in that direction, is among the book/essay's most compelling. But it's far too short and superficial to be sufficient.
I suppose I will remain a math philistine.
And while they end up only tangentially relating to his thesis, the first few chapters, in which GH discusses the nature of ambition, ability, growth, etc., are ultimately what justify the price of admission.
GH mingles mathturbation with emperor's-new-clothes rhetoric. He asserts (I'm paraphrasing) that a true intellect appreciates math's beauty, thus implying that if one fails to appreciate the promised beauty, she must not be an intellect (this is further paraphrased as, "if you can't do the math on that then I can't help ya'" [Or perhaps by describing it as a preemptive "No true Scotsman" fallacy]). He spends a large part of the essay/book itemizing mathematics nice qualities.
GH has a real inferiority complex with respect to his love, mathematics, and his writing ability about it. The former expresses itself via a prim smugness, the later via his confessions of his inability to express concepts neatly.
I wish GH had tried to write something less universal and more personal. I wanted more insight into what it's like to be a capital-M Mathematician. I wanted to understand his thought process, how he experiences mathematical joy, how he touches Truth, how math has shaped his person and life, etc. The last chapter, in which he drifts in that direction, is among the book/essay's most compelling. But it's far too short and superficial to be sufficient.
I suppose I will remain a math philistine.
And while they end up only tangentially relating to his thesis, the first few chapters, in which GH discusses the nature of ambition, ability, growth, etc., are ultimately what justify the price of admission.
December 10, 2017
"Ogni tanto è necessario dire delle cose difficili, ma bisognerebbe dirle nel modo più semplice di cui si è capaci�
“La maggior parte della della gente è così spaventata al solo nome della matematica che è portata in tutta sincerità a esagerare la propria stupidità matematica�
G. H. Hardy (Cranleigh, 7 febbraio1877 � Cambridge, 1º dicembre 1947) è stato un matematico britannico. Fellow della Royal Society è noto per i suoi contributi in teoria dei numeri e analisi matematica, scrive questo libro alla fine della sua vita come matematico creativo, nonché giocatore di tennis e di cricket…esso decreta la fine della sua vita attiva. La ricerca matematica così come lo sport è cosa per giovani.
Apologia di un matematico, scritto intorno alla 60ina, è una lettura permeata di grande tristezza e rimpianto nonostante alcune folgoranti battute che contraddistinguono l'uomo di straordinario acume dotato di ironia e sense of humor molto ma molto british, proprio perché decreta ciò che Hardy non potrà più esercitare. L'apologia è spesso considerata una delle migliori introspezioni nella mente di un matematico ed è una delle più riuscite descrizioni di cosa significhi essere un'artista creativo.
Nel libello G.H. (di pochi il privilegio chiamarlo Harold) prova a spiegare in cosa consista il lavoro del matematico creativo, prova a spiegare in cosa consista la bellezza e il fascino della matematica pura. Inizia pertanto facendo un distinguo tra essa e quella che lui definisce matematica 'banale', ovvero quella scolastica o quella al servizio delle scienze, che lui ritiene terribilmente noiosa.
Per fare comprendere cosa sia la matematica pura fornisce esempi, ovvero spiega come si passi dall'esposizione di una intuizione (es. i numeri primi sono infiniti) e come questo venga dimostrato, per il tramite di un procedimento logico, non necessariamente univoco, che porta alla tesi. E una volta che un assunto viene provato, questo diventa immortale, una verità inconfutabile che si tramanderà nei secoli dei secoli.
In maniera speculativa disserta inoltre sulla utilità della matematica definendo in maniera rigorosa (se non è rigoroso lui...) cosa intenda per utilità, ovvero:
"Una scienza, o un’arte, può essere definita “utile� se il suo sviluppo accresce, anche indirettamente, il benessere materiale e fisico degli uomini, si favorisce la felicità, nel senso più semplice e banale della parola."
In questi termini la matematica banale è utile, mentre la matematica pura, quella accessibile a pochi, non lo è: nella vita concreta a che serve l'infinito? Ma per chi cerca il fascino e il mistero, la purezza e l'assoluto, è la dimensione dell'infinito quella da indagare.
"Per me e per la maggior parte dei matematici, esiste un’altra realtà che chiamerò “realtà matematica�, sulla cui natura né i matematici né i filosofi sono assolutamente d’accordo. Alcuni sostengono che è mentale che, in un certo senso, siamo noi che la costruiamo, altri che è esterna e indipendente da noi�
La matematica pura è una costruzione dell’intelletto dell’uomo o è una realtà parallela cui solo alcune menti hanno accesso? Le formule affascinanti e misteriose che regolano il comportamento dei numeri che spesso trovano rispondenze parallele nella descrizione dei fenomeni della natura, hanno sotteso qualcosa di soprannaturale? Le formule esistono di per sé stesse nella realtà matematica oppure sono solo frutto della mente dell'uomo?
La prefazione di Apologia di un Matematico pare la base per la sceneggiatura del film “L’uomo che vide l’infinito�. Appena terminato il film non ho potuto evitare di leggere il libro (scaricato ebook immediatamente giunta a casa).
Il film tratta della relazione lavorativa tra Hardy e "l’unico incidente romantico della sua vita",ovvero Ramanujan un giovane indiano che, non avendo conoscenze formali di matematica, la intuiva, la conosceva senza per altro possedere gli strumenti per descrivere ai colleghi perché ciò che affermava fosse corretto. E Hardy, si inchina davanti al talento. Al dono. E cerca di dare gli strumenti a Ramanujan per affermarsi e rendere partecipe il mondo accademico delle sue intuizioni provandole.
E d’altra parte esistono persone come il giovane indiano i cui amici intimi (privi di segreti, cioè) sono gli interi positivi (1, 2, 3...).
E solo loro ti sanno rivelare perché 1729 è un numero bellissimo.
“La maggior parte della della gente è così spaventata al solo nome della matematica che è portata in tutta sincerità a esagerare la propria stupidità matematica�
G. H. Hardy (Cranleigh, 7 febbraio1877 � Cambridge, 1º dicembre 1947) è stato un matematico britannico. Fellow della Royal Society è noto per i suoi contributi in teoria dei numeri e analisi matematica, scrive questo libro alla fine della sua vita come matematico creativo, nonché giocatore di tennis e di cricket…esso decreta la fine della sua vita attiva. La ricerca matematica così come lo sport è cosa per giovani.
Apologia di un matematico, scritto intorno alla 60ina, è una lettura permeata di grande tristezza e rimpianto nonostante alcune folgoranti battute che contraddistinguono l'uomo di straordinario acume dotato di ironia e sense of humor molto ma molto british, proprio perché decreta ciò che Hardy non potrà più esercitare. L'apologia è spesso considerata una delle migliori introspezioni nella mente di un matematico ed è una delle più riuscite descrizioni di cosa significhi essere un'artista creativo.
Nel libello G.H. (di pochi il privilegio chiamarlo Harold) prova a spiegare in cosa consista il lavoro del matematico creativo, prova a spiegare in cosa consista la bellezza e il fascino della matematica pura. Inizia pertanto facendo un distinguo tra essa e quella che lui definisce matematica 'banale', ovvero quella scolastica o quella al servizio delle scienze, che lui ritiene terribilmente noiosa.
Per fare comprendere cosa sia la matematica pura fornisce esempi, ovvero spiega come si passi dall'esposizione di una intuizione (es. i numeri primi sono infiniti) e come questo venga dimostrato, per il tramite di un procedimento logico, non necessariamente univoco, che porta alla tesi. E una volta che un assunto viene provato, questo diventa immortale, una verità inconfutabile che si tramanderà nei secoli dei secoli.
In maniera speculativa disserta inoltre sulla utilità della matematica definendo in maniera rigorosa (se non è rigoroso lui...) cosa intenda per utilità, ovvero:
"Una scienza, o un’arte, può essere definita “utile� se il suo sviluppo accresce, anche indirettamente, il benessere materiale e fisico degli uomini, si favorisce la felicità, nel senso più semplice e banale della parola."
In questi termini la matematica banale è utile, mentre la matematica pura, quella accessibile a pochi, non lo è: nella vita concreta a che serve l'infinito? Ma per chi cerca il fascino e il mistero, la purezza e l'assoluto, è la dimensione dell'infinito quella da indagare.
"Per me e per la maggior parte dei matematici, esiste un’altra realtà che chiamerò “realtà matematica�, sulla cui natura né i matematici né i filosofi sono assolutamente d’accordo. Alcuni sostengono che è mentale che, in un certo senso, siamo noi che la costruiamo, altri che è esterna e indipendente da noi�
La matematica pura è una costruzione dell’intelletto dell’uomo o è una realtà parallela cui solo alcune menti hanno accesso? Le formule affascinanti e misteriose che regolano il comportamento dei numeri che spesso trovano rispondenze parallele nella descrizione dei fenomeni della natura, hanno sotteso qualcosa di soprannaturale? Le formule esistono di per sé stesse nella realtà matematica oppure sono solo frutto della mente dell'uomo?
La prefazione di Apologia di un Matematico pare la base per la sceneggiatura del film “L’uomo che vide l’infinito�. Appena terminato il film non ho potuto evitare di leggere il libro (scaricato ebook immediatamente giunta a casa).
Il film tratta della relazione lavorativa tra Hardy e "l’unico incidente romantico della sua vita",ovvero Ramanujan un giovane indiano che, non avendo conoscenze formali di matematica, la intuiva, la conosceva senza per altro possedere gli strumenti per descrivere ai colleghi perché ciò che affermava fosse corretto. E Hardy, si inchina davanti al talento. Al dono. E cerca di dare gli strumenti a Ramanujan per affermarsi e rendere partecipe il mondo accademico delle sue intuizioni provandole.
E d’altra parte esistono persone come il giovane indiano i cui amici intimi (privi di segreti, cioè) sono gli interi positivi (1, 2, 3...).
E solo loro ti sanno rivelare perché 1729 è un numero bellissimo.
January 27, 2012
Recently I started teaching myself to program. An article I read recommended Project Euler, which is a set of math exercises intended to be completed with computer code. So for the last few months I've been doing more writing than reading, as I puzzled through these math problems. Research on various problems led to me to other math websites, and often G.H. Hardy's short book "A Mathematician's Apology" was mentioned in various contexts. I picked it up, and found a lot of what Hardy wrote applied to other fields, including the computer programming I was attempting to learn.
Boiled down to its essence, "A Mathematician's Apology" is a book about beauty. Hardy writes: "the mathematician's patterns, like the painter's or the poet's, must be beautiful... there is no permanent place in the world for ugly mathematics." The nature of this beauty, in all its subtlety, is what Hardy spends most of the book writing about. To him, beautiful math combines simplicity with depth- connecting previously unconnected branches of mathematics and revealing new solutions to unsolvable problems- as well as new problems to be solved. You can apply the same ideas to science, or to art- there is something deeply zen about Hardy's formulation.
This is strange, because Hardy himself is deeply unzen. As C.P. Snow writes in his long introduction, "it is very rare for a writer to realize, with the finality of truth, that he is absolutely finished." While beautiful, the book is very sad, because Hardy was deeply depressed when he wrote it. At the end of his career, with his health failing, he seems to have abandoned all hope. Even writing seems to bring him no satisfaction: "There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds."
For a man who saw so far into such undiscovered philosophical territory, it's sad that Hardy didn't acquire a sense of perspective about himself, or his place in the universe. He hung his entire self-worth on his sense of mathematical accomplishment, and as his powers faded, so did his ability to experience joy. This kind of unzen attitude is found throughout the book- here's just one small taste: "I shall assume that I am writing for readers who are full, or have in the past been full, of a proper spirit of ambition. A man's first duty, a young man's at any rate, is to be ambitious."
Hardy's unhappiness has been explained in a number of ways- dissatisfaction with his work, a horror at the oncoming war (this book was written in 1940), and perhaps problems of the heart (Snow hints that Hardy was gay, but it seems that he lived asexually.) But I think it's really his world-view that ultimately defeated him- this idea that a man is to be judged by what he creates and nothing else. No man who truly believes that could ever be lastingly happy- especially not in old age.
Boiled down to its essence, "A Mathematician's Apology" is a book about beauty. Hardy writes: "the mathematician's patterns, like the painter's or the poet's, must be beautiful... there is no permanent place in the world for ugly mathematics." The nature of this beauty, in all its subtlety, is what Hardy spends most of the book writing about. To him, beautiful math combines simplicity with depth- connecting previously unconnected branches of mathematics and revealing new solutions to unsolvable problems- as well as new problems to be solved. You can apply the same ideas to science, or to art- there is something deeply zen about Hardy's formulation.
This is strange, because Hardy himself is deeply unzen. As C.P. Snow writes in his long introduction, "it is very rare for a writer to realize, with the finality of truth, that he is absolutely finished." While beautiful, the book is very sad, because Hardy was deeply depressed when he wrote it. At the end of his career, with his health failing, he seems to have abandoned all hope. Even writing seems to bring him no satisfaction: "There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds."
For a man who saw so far into such undiscovered philosophical territory, it's sad that Hardy didn't acquire a sense of perspective about himself, or his place in the universe. He hung his entire self-worth on his sense of mathematical accomplishment, and as his powers faded, so did his ability to experience joy. This kind of unzen attitude is found throughout the book- here's just one small taste: "I shall assume that I am writing for readers who are full, or have in the past been full, of a proper spirit of ambition. A man's first duty, a young man's at any rate, is to be ambitious."
Hardy's unhappiness has been explained in a number of ways- dissatisfaction with his work, a horror at the oncoming war (this book was written in 1940), and perhaps problems of the heart (Snow hints that Hardy was gay, but it seems that he lived asexually.) But I think it's really his world-view that ultimately defeated him- this idea that a man is to be judged by what he creates and nothing else. No man who truly believes that could ever be lastingly happy- especially not in old age.
February 5, 2013
The text may be found at
I had the good fortune to come across this title just as I was finally beginning to see the glimmers of beauty in mathematics thanks to the efforts of some wonderful instructors on the subject during my later school years. It called upon me for a deeper reflection on my chosen pursuit, which at that point appealed to me for its fundamental importance to the other sciences and for the simple pleasure that can be gleaned of it. I was confronted with a fleeting, tricksome nature even in mathematical truth, which had till then been sacrosanct and taken for granted. It became apparent that there was more artistic excitement in mathematics than the safety of empirical sciences and it has been my great love ever since, however modestly ability dresses me.
Hardy's is an essay of paradoxes and duals, with examples littered through the text. These may seem contradictory at a glance, but close scrutiny reveals them to be mutually reliant points that hint to the complexities of Hardy's pursuit. One instance is in his claim that mathematical patterns are more enduring than any artist's or poet's. While this may leave many an artist or poet bristling, this endurance is manifest in the progression of a mathematician's significant ideas, even though these are all in a language of ideas that does not rely on being evocative (so unlike the artist or poet). This is at odds however, with the aesthetic appreciation to be had in nontrivial mathematics that Hardy keeps returning to. The prospect of leaving some mark so enduring is certainly catering to ambition (which Hardy encourages), but this is incompatible with failing to capture beauty, and so it is that one cannot stand without the other. It has been said that this is a tragic account, but here is a thread of hope- that one should remain steadfast in the area of one's talent and atop that cultivate ambitions and seek too what is beautiful. There is a terrible, indubitable rationale here which could strike some as arrogant (though even if it were taken to be so, not something diminishing from the work), but is ultimately a course of efficiency (something Hardy valued in his mathematical aesthetic) devoid of any lingering melancholy.
Another instance is what he has to say about the worthwhile mathematics and trivial. Frequently, Hardy is misrepresented as disdaining applied mathematics upon the haughty perch of the pure. This is unfortunate since special mention is made that mathematicians do not tend to revel in the uselessness of their work. Hardy is no fool to try untangling the mathematics that will come to some use and what stands detached and austere. He draws the crucial distinction instead based on aesthetic value once again, setting aside those abundant trifles of mathematics so often connected with the mundane world of everyday and advocating the deep framework (of rather few significant contributions) of what might be dubbed the 'true mathematics'. Hardy is rather half-hearted about elaborating on his standards of beauty, but there is no real surprise in this for anyone acquainted with higher mathematics, because it really is a difficult thing to define. There is efficiency and sweeping elegance in a great result, but too a contribution to that progression of ideas which makes mathematics so enduring. Whether these criteria capture beauty is contestable and so I make no claims. But this difficulty in explaining a thing like beauty highlights the crucial problem in Hardy's endeavour to explain to the public what a mathematician does and why. He justifies the pursuit of the true mathematics exactly because it is so removed from mundane being and so comes of purer intentions than much else that can be directed to harm as well as good, but it is a difficult thing to impart the joy of actually doing mathematics to the uninitiated, which is perhaps why so many reviews have picked upon the thread of personal tragedy here (partly owing to the foreword I suppose, thought Hardy's circumstances are the least of the work's significance, I thought, both on my first reading and now) instead of the 'apology' itself. There is nowadays a great scuffle to get some manner of emotional fix upon an author one is reading. This sort of thing is seldom the author's intent (else they would simply save themselves the trouble of writing a reasoned and coherent account with objective value and instead send their diaries to a publisher), especially in a work as this where the author is attempting to convey a position and defend it. The arguments do not require lenses coloured with the author's circumstances. I imagine Hardy the pure mathematician would rather have avoided this, as he remarks the following in the present work (something of a spoiler warning is due here)-
"Dr Snow had also made an interesting point about §8. Even if we grant that ‘Archimedes will be remembered when Aeschylus is forgotten�, is not mathematical fame a little too ‘anonymous� to be wholly satisfying? We could form a fairly coherent picture of the personality of Aeschylus (still more, of course, of Shakespeare or Tolstoi) from their works alone, while Archimedes and Eudoxus would remain mere names. Mr J M Lomas put this point more picturesquely when we were passing the Nelson column in Trafalgar square. If I had a statue on a column in London, would I prefer the columns to be so high that the statue was invisible, or low enough for the features to be recognizable? I would choose the first alternative, Dr Snow, presumably, the second."
(Dr Snow here is C P Snow, who wrote the foreword to this edition. It's worth reading, but perhaps after Hardy's own words, since its biographical tint somewhat along the line I've discussed above may detract from the potent general ideas Hardy is forwarding here.)
Few other works have attempted the exposition of what creative enterprise demands and entails quite so well as this one, and this from a master who has seen ability and passions come and go. I recommend it highly.
I had the good fortune to come across this title just as I was finally beginning to see the glimmers of beauty in mathematics thanks to the efforts of some wonderful instructors on the subject during my later school years. It called upon me for a deeper reflection on my chosen pursuit, which at that point appealed to me for its fundamental importance to the other sciences and for the simple pleasure that can be gleaned of it. I was confronted with a fleeting, tricksome nature even in mathematical truth, which had till then been sacrosanct and taken for granted. It became apparent that there was more artistic excitement in mathematics than the safety of empirical sciences and it has been my great love ever since, however modestly ability dresses me.
Hardy's is an essay of paradoxes and duals, with examples littered through the text. These may seem contradictory at a glance, but close scrutiny reveals them to be mutually reliant points that hint to the complexities of Hardy's pursuit. One instance is in his claim that mathematical patterns are more enduring than any artist's or poet's. While this may leave many an artist or poet bristling, this endurance is manifest in the progression of a mathematician's significant ideas, even though these are all in a language of ideas that does not rely on being evocative (so unlike the artist or poet). This is at odds however, with the aesthetic appreciation to be had in nontrivial mathematics that Hardy keeps returning to. The prospect of leaving some mark so enduring is certainly catering to ambition (which Hardy encourages), but this is incompatible with failing to capture beauty, and so it is that one cannot stand without the other. It has been said that this is a tragic account, but here is a thread of hope- that one should remain steadfast in the area of one's talent and atop that cultivate ambitions and seek too what is beautiful. There is a terrible, indubitable rationale here which could strike some as arrogant (though even if it were taken to be so, not something diminishing from the work), but is ultimately a course of efficiency (something Hardy valued in his mathematical aesthetic) devoid of any lingering melancholy.
Another instance is what he has to say about the worthwhile mathematics and trivial. Frequently, Hardy is misrepresented as disdaining applied mathematics upon the haughty perch of the pure. This is unfortunate since special mention is made that mathematicians do not tend to revel in the uselessness of their work. Hardy is no fool to try untangling the mathematics that will come to some use and what stands detached and austere. He draws the crucial distinction instead based on aesthetic value once again, setting aside those abundant trifles of mathematics so often connected with the mundane world of everyday and advocating the deep framework (of rather few significant contributions) of what might be dubbed the 'true mathematics'. Hardy is rather half-hearted about elaborating on his standards of beauty, but there is no real surprise in this for anyone acquainted with higher mathematics, because it really is a difficult thing to define. There is efficiency and sweeping elegance in a great result, but too a contribution to that progression of ideas which makes mathematics so enduring. Whether these criteria capture beauty is contestable and so I make no claims. But this difficulty in explaining a thing like beauty highlights the crucial problem in Hardy's endeavour to explain to the public what a mathematician does and why. He justifies the pursuit of the true mathematics exactly because it is so removed from mundane being and so comes of purer intentions than much else that can be directed to harm as well as good, but it is a difficult thing to impart the joy of actually doing mathematics to the uninitiated, which is perhaps why so many reviews have picked upon the thread of personal tragedy here (partly owing to the foreword I suppose, thought Hardy's circumstances are the least of the work's significance, I thought, both on my first reading and now) instead of the 'apology' itself. There is nowadays a great scuffle to get some manner of emotional fix upon an author one is reading. This sort of thing is seldom the author's intent (else they would simply save themselves the trouble of writing a reasoned and coherent account with objective value and instead send their diaries to a publisher), especially in a work as this where the author is attempting to convey a position and defend it. The arguments do not require lenses coloured with the author's circumstances. I imagine Hardy the pure mathematician would rather have avoided this, as he remarks the following in the present work (something of a spoiler warning is due here)-
"Dr Snow had also made an interesting point about §8. Even if we grant that ‘Archimedes will be remembered when Aeschylus is forgotten�, is not mathematical fame a little too ‘anonymous� to be wholly satisfying? We could form a fairly coherent picture of the personality of Aeschylus (still more, of course, of Shakespeare or Tolstoi) from their works alone, while Archimedes and Eudoxus would remain mere names. Mr J M Lomas put this point more picturesquely when we were passing the Nelson column in Trafalgar square. If I had a statue on a column in London, would I prefer the columns to be so high that the statue was invisible, or low enough for the features to be recognizable? I would choose the first alternative, Dr Snow, presumably, the second."
(Dr Snow here is C P Snow, who wrote the foreword to this edition. It's worth reading, but perhaps after Hardy's own words, since its biographical tint somewhat along the line I've discussed above may detract from the potent general ideas Hardy is forwarding here.)
Few other works have attempted the exposition of what creative enterprise demands and entails quite so well as this one, and this from a master who has seen ability and passions come and go. I recommend it highly.
August 3, 2020
What a wonderful time I had with this book. GH Hardy, one of last century’s towering mathematical figures now known to a wider audience through the film “The Man Who Knew Infinity�, looks back on his life at a time when, by his own testimony, his mathematical genius was fading.
The opening lines establish a sense of melancholy that I was never quite able to banish when reading his memoir:
It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done.
And yet, throughout the rest of his account, we encounter an intellect as sharp and penetrating as it is possible to meet. Hardy shares his thoughts about the aesthetics and truth value of mathematics, and his thoughts are profound, but they are offered with the charming nonchalance of somebody who does not seem to be truly aware of his own intellectual calibre.
The thought that resonated with me most is the idea that mathematics is, to speak with Hardy’s own diction, ‘useless�, or to put it into less colourful language reserved for somebody who is not a mathematician (me...), free of purpose. I believe what Hardy had in mind was to convey the idea that the study of mathematical objects is itself its purpose, that discovery of new mathematics “pushes back the darkness by a millimetre�, as Greg Chaitin, another genius mathematician, put it, that the purpose of mathematics is not measurable by any practical application that may not have been made without it. In this interpretation, the study of mathematics can be an example of pure thought, of pure intellectual pursuit unencumbered, constrained or possibly diminished, by the demands of the practical world. And that, to me at least, is a beautiful and encouraging thought.
When he wrote his account, Hardy, in the words of Jethro Tull, may have felt “too old to rock and roll, too young to die�, but I want to finish by letting the man himself speak. I started my comment with the first lines of Hardy’s book, allow me to finish it with some of the last:
I have never done anything ‘useful�. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating.
The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them.
The opening lines establish a sense of melancholy that I was never quite able to banish when reading his memoir:
It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done.
And yet, throughout the rest of his account, we encounter an intellect as sharp and penetrating as it is possible to meet. Hardy shares his thoughts about the aesthetics and truth value of mathematics, and his thoughts are profound, but they are offered with the charming nonchalance of somebody who does not seem to be truly aware of his own intellectual calibre.
The thought that resonated with me most is the idea that mathematics is, to speak with Hardy’s own diction, ‘useless�, or to put it into less colourful language reserved for somebody who is not a mathematician (me...), free of purpose. I believe what Hardy had in mind was to convey the idea that the study of mathematical objects is itself its purpose, that discovery of new mathematics “pushes back the darkness by a millimetre�, as Greg Chaitin, another genius mathematician, put it, that the purpose of mathematics is not measurable by any practical application that may not have been made without it. In this interpretation, the study of mathematics can be an example of pure thought, of pure intellectual pursuit unencumbered, constrained or possibly diminished, by the demands of the practical world. And that, to me at least, is a beautiful and encouraging thought.
When he wrote his account, Hardy, in the words of Jethro Tull, may have felt “too old to rock and roll, too young to die�, but I want to finish by letting the man himself speak. I started my comment with the first lines of Hardy’s book, allow me to finish it with some of the last:
I have never done anything ‘useful�. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating.
The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them.
November 13, 2011
This memoir from G.H Hardy has truly changed my perception of mathematics and mathematicians. Hardy is a remarkable man, though unusual (he likes cricket!) and with collaborations with Littlewood and Ramanujan he made astonishing breakthroughs in the mathematical field. The one thing which struck me in this novel was Hardy's sorrow caused by old age, he seemed in mourning for the creativity and drive for mathematics that he had once held. Some of his deep emotions are layed bare in this novel, and describes his woes and triumphs through his life, in particular the book which describe drinking in a common room at Trinity College inspiring him to become the reputable mathematician that he became.
Truly wonderful, I would recommend this to anyone with a heart, or an appreciation for the art of mathematics.
Truly wonderful, I would recommend this to anyone with a heart, or an appreciation for the art of mathematics.
March 11, 2008
Here's a reason one might want to read this book. In his introduction, C.P. Snow points out that Hardy's capacity for dissimulation "was always minimial." And he goes on to illustrate this with a passage in the Apology where Hardy says, "I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships; I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively." Besides revealing a lucid self-knowledge, it provides some of the tragedy to this book. For we know it is said by a person who's decisive abilities were decisively layed low by a heart attack which occured in Hardy's 60th year; reminding us that we do not own our gifts and that they can be taken from us at any time. It is a beautiful story of a man trying to hold himself up with dignity and survey a beautiful territory that is now inaccessible to him. He sees that not all of his struggles and triumphs in it were noble. Thus he manages to transcend the competition that once carried him to fame and tells us the beauty of his subject. He asks us to accompany him whereas in former times we would have been rigorously examined by his "brillant and concentrated mind."
February 24, 2013
I read this book because it was quoted in a coursera class about Genetics and Evolution. The quote was: "I have never done anything ‘useful�. No discovery of mine has
made, or is likely to make, directly or indirectly, for good or ill,
the least difference to the amenity of the world." Interestingly, one of Hardy's equation called the "Hardy-Weinberg's equation" is used in genetics, in population allele analysis and in fact it has become very useful for geneticians that ultimately benefit medicine and our well being.
I wanted to read this also because I often feel very guilty and apologetic about studying a basic science, and I wanted to see his point.
What I keep thinking about is a quote from Leonardo Da Vinci: "Everything connects with everything else". It sounds maybe a little bit mystic but I realised that any knowledge, even though we cannot forsee it's application clearly or at all, is information that can be useful in any other field. And why not consider the mere joy of finding things out? I often have the feeling that scientists should get involved with humanistics more and then they would appreciate more their work outside of their own fields, plus, better decisions can be taken.
made, or is likely to make, directly or indirectly, for good or ill,
the least difference to the amenity of the world." Interestingly, one of Hardy's equation called the "Hardy-Weinberg's equation" is used in genetics, in population allele analysis and in fact it has become very useful for geneticians that ultimately benefit medicine and our well being.
I wanted to read this also because I often feel very guilty and apologetic about studying a basic science, and I wanted to see his point.
What I keep thinking about is a quote from Leonardo Da Vinci: "Everything connects with everything else". It sounds maybe a little bit mystic but I realised that any knowledge, even though we cannot forsee it's application clearly or at all, is information that can be useful in any other field. And why not consider the mere joy of finding things out? I often have the feeling that scientists should get involved with humanistics more and then they would appreciate more their work outside of their own fields, plus, better decisions can be taken.
September 22, 2024
"The best account of what it was like to be a creative artist"
First, a word about the title. "Apology" is not here used in the most common modern sense of "an admission of error or discourtesy accompanied by an expression of regret", bur rather in the older sense of "a formal explanation or defense of a belief or system, especially one that is unpopular". is in no way expressing regret or admitting error for being a mathematician. In fact, he makes it clear that it is the thing in his life that he is most proud of. His purpose is to explain for non-mathematicians what a mathematician does and why it is valuable. Briefly, this is what he has to say
Mathematics is, to my knowledge, the only profession whose first concerns are explicitly truth and beauty. (I do not except the arts. I have never met an artist who would agree that beauty was a goal of his or her work.)
knew, of course, that it was not adequate to simply assert the importance of beauty. He attempts therefore to demonstrate it, and it is here, I fear, that fails. To this purpose he presents two classical theorems, Euclid's proof that the number of primes is infinite, and the proof that there is no rational number whose square is two. believes that the capacity to perceive and appreciate mathematical beauty is widespread, and that these two theorems will be perceived as beautiful by his readers, allowing him to demonstrate its nature.
I fear he is too optimistic. I suspect the vast majority of people will not put in the effort to understand these proofs, simple though they are, and even if they do, will not perceive any beauty therein. is not, in my opinion, a book you can give to non-mathematicians to explain math to them, except for those who are already more than halfway there.
also spends some time on the "usefulness" of math. The money quote is "very little of mathematics is useful practically, and that that little is comparatively dull." This argument has not aged well. For instance, he writes
Still, 's main point, that math is not to be justified by its practical usefulness, but by its beauty, holds.
This edition of begins with a biographical introduction by , a good friend of 's and a successful writer. This introduction does perhaps a better job of presenting 's arguments than himself does.
.
First, a word about the title. "Apology" is not here used in the most common modern sense of "an admission of error or discourtesy accompanied by an expression of regret", bur rather in the older sense of "a formal explanation or defense of a belief or system, especially one that is unpopular". is in no way expressing regret or admitting error for being a mathematician. In fact, he makes it clear that it is the thing in his life that he is most proud of. His purpose is to explain for non-mathematicians what a mathematician does and why it is valuable. Briefly, this is what he has to say
A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas...If you take that and nothing else away from , you will have learned the main lesson. described as one of "the best accounts of what it is like to be a creative artist."
The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.
Mathematics is, to my knowledge, the only profession whose first concerns are explicitly truth and beauty. (I do not except the arts. I have never met an artist who would agree that beauty was a goal of his or her work.)
knew, of course, that it was not adequate to simply assert the importance of beauty. He attempts therefore to demonstrate it, and it is here, I fear, that fails. To this purpose he presents two classical theorems, Euclid's proof that the number of primes is infinite, and the proof that there is no rational number whose square is two. believes that the capacity to perceive and appreciate mathematical beauty is widespread, and that these two theorems will be perceived as beautiful by his readers, allowing him to demonstrate its nature.
I fear he is too optimistic. I suspect the vast majority of people will not put in the effort to understand these proofs, simple though they are, and even if they do, will not perceive any beauty therein. is not, in my opinion, a book you can give to non-mathematicians to explain math to them, except for those who are already more than halfway there.
also spends some time on the "usefulness" of math. The money quote is "very little of mathematics is useful practically, and that that little is comparatively dull." This argument has not aged well. For instance, he writes
If useful knowledge is, as we agreed provisionally to say, knowledge which is likely, now or in the comparatively near future, to contribute to the material comfort of mankind, so that mere intellectual satisfaction is irrelevant, then the great bulk of higher mathematics is useless. Modern geometry and algebra, the theory of numbers, the theory of aggregates and functions, relativity, quantum mechanics—no one of them stands the test much better than another.In fact, in the 84 years since he published those sentences, quantum mechanics gave us light-emitting-diodes and transistors, transistors gave us high-speed computers, and number theory gave us public-key encryption standards. Reading this, you are using results of these mathematical disciplines.
Still, 's main point, that math is not to be justified by its practical usefulness, but by its beauty, holds.
This edition of begins with a biographical introduction by , a good friend of 's and a successful writer. This introduction does perhaps a better job of presenting 's arguments than himself does.
.
March 23, 2021
A paixão de G.H.Hardy pela Matemática é lendária. Este é um breve livro (demasiado breve!) sobre o amor por aquilo que se faz. Hardy tinha duas paixões: o críquete e a matemática. Metódico, com um humor muito peculiar e apaixonado pela partilha do conhecimento.
O filme «The man who knew infinity» aborda, ainda que lateralmente, esta personalidade britânica tão peculiar.
Mas, neste livrinho, a história é contada na primeira pessoa e vale mesmo muito a pena a sua leitura. A visão de Hardy é acutilante e muito crítica. Uma breve reflexão sobre a utilidade do que fazemos e o amor que emprestamos às ideias.
Bonito.
O filme «The man who knew infinity» aborda, ainda que lateralmente, esta personalidade britânica tão peculiar.
Mas, neste livrinho, a história é contada na primeira pessoa e vale mesmo muito a pena a sua leitura. A visão de Hardy é acutilante e muito crítica. Uma breve reflexão sobre a utilidade do que fazemos e o amor que emprestamos às ideias.
Bonito.
October 14, 2009
As Hardy pointed out himself, criticisms are work of second-rate mind. This book is awesome, it sheds so much light on what is going on inside mathematician's mind. I particularly like the part when he argues why mathematics is beautiful, and what constitutes a beautiful mathematical theorem.
December 11, 2016
Mr. Hardy explains the difference between pure and applied mathematics. Math history thrown in and some stories about Bertrand Russell and others. There's also a hint of defensiveness as he tries to explain the beauty of spending a life on math that has no real world applications.
Shelved as 'dnf'
December 11, 2015PDF available here:
June 14, 2024
Learned more from Hardy's character and influence on others (foreword) than his justification of mathematics. (Ironically, his friends proved the worthiness of his life far more than his dialectics.)
Unlike engineering, math is useful *because* it is abstract; unlike physics, realistic because it doesn't depend too much on reality; unlike art/poetry, creative because isn’t colorful. This is not to say physics, engineering, and art are not useful; but that just because they are useful and math is different in nature, doesn't undermine math.
only 3 real motivations: Truth, Ambition, Pride. Anyone who claims otherwise, e.g. Impact, is false. Do I agree? Ambition and Pride can be explained by selfishness, but what makes Truth different from Impact? Hardy is clearly biased towards the abstract, the universe and ideas as more inborn than speciesism (which I think is arguably more innate). Replace Truth with Curiosity, Impact with Speciesism, now both are evolvable traits, and I don't see how one is more real than the other. I agree that all real motivations are intrinsic/selfish, but for every external action there's an internal force, so it's hard to separate the two.
We are concerned with finding the deeply fundamental. In math we generalize to the theory of mathematical reality, in physics we zoom out out to the scale of the universe, bio zoom in to the secret of life: what is the fundamental substrate, what is the connection between connections, what are the laws of the "real" world. To Hardy, the mathematical world is more real than the physical one, because it is invariant where physical reality is not (2=2 but if we attach it to reality it may be 2 black balls one day and 2 white balls the next). At some level we're holding all these different maps of worlds but talking about the same thing, he's living in the "lift" and we're in the pre-image, but the point is to simplify reasoning so in different cases we may want to work in one or the other. I think most people benefit from jumping between maps, good for Hardy he sees clarity and completeness in one.
Hardy as a character: crossexamine convos. Only say controversial things: "It is not worth an intelligent man's time to be in the majority. By definition, there are already enough people to do that." he kept a list of 100 adjectives to mark off people. Loved cricket, the game and the theory, but even more to play it with others who understood it in the way he did
Unlike engineering, math is useful *because* it is abstract; unlike physics, realistic because it doesn't depend too much on reality; unlike art/poetry, creative because isn’t colorful. This is not to say physics, engineering, and art are not useful; but that just because they are useful and math is different in nature, doesn't undermine math.
only 3 real motivations: Truth, Ambition, Pride. Anyone who claims otherwise, e.g. Impact, is false. Do I agree? Ambition and Pride can be explained by selfishness, but what makes Truth different from Impact? Hardy is clearly biased towards the abstract, the universe and ideas as more inborn than speciesism (which I think is arguably more innate). Replace Truth with Curiosity, Impact with Speciesism, now both are evolvable traits, and I don't see how one is more real than the other. I agree that all real motivations are intrinsic/selfish, but for every external action there's an internal force, so it's hard to separate the two.
We are concerned with finding the deeply fundamental. In math we generalize to the theory of mathematical reality, in physics we zoom out out to the scale of the universe, bio zoom in to the secret of life: what is the fundamental substrate, what is the connection between connections, what are the laws of the "real" world. To Hardy, the mathematical world is more real than the physical one, because it is invariant where physical reality is not (2=2 but if we attach it to reality it may be 2 black balls one day and 2 white balls the next). At some level we're holding all these different maps of worlds but talking about the same thing, he's living in the "lift" and we're in the pre-image, but the point is to simplify reasoning so in different cases we may want to work in one or the other. I think most people benefit from jumping between maps, good for Hardy he sees clarity and completeness in one.
Hardy as a character: crossexamine convos. Only say controversial things: "It is not worth an intelligent man's time to be in the majority. By definition, there are already enough people to do that." he kept a list of 100 adjectives to mark off people. Loved cricket, the game and the theory, but even more to play it with others who understood it in the way he did
February 5, 2023
Hardy haalt enkele zeer fundamentele, en redelijk eenvoudige karakteristieken der wiskunde aan. De meest verfrissende is het beschouwen van wiskunde als scheppende kunst. Niets leidt echter af van de duidelijke bitterheid en competitieve geest van Hardy als wiskundige, ongeacht het feit dat Snow ons hiertoe lijkt te behoeden in zijn voorwoord per waarschuwing dat een sluier van somberheid Hardy leek te bevangen. Uitspraken zoals dat eenieders drijfveer de competitie of zelfs geld is halen Hardy's naam door het slijk.
Snows voorwoord is echter zeer mooi en, zoals Hardy zelf afsluit, lijkt hij me een veel minder faamhongering persoon die zijn leven niet leidt met als motivatie het beter zijn dan anderen.
Snows voorwoord is echter zeer mooi en, zoals Hardy zelf afsluit, lijkt hij me een veel minder faamhongering persoon die zijn leven niet leidt met als motivatie het beter zijn dan anderen.
May 9, 2014
Since I have recommended this book to some friends, I'd better review it for them. It isn't going to be easy, just like the book wasn't easy to read, so here goes nothing:
1. This is an enlightening book, especially if you are looking to create some form of original work in academics. Are you planning on writing a doctorate thesis at some point? Read this, because it puts forth another academician's ideas about what a good contribution is. It helps that Hardy managed very well for himself.
Even if you're not looking to pursue an academic career, it is still an insightful read in case at some point you would like to create something original.
It sheds light on the nature and characteristics of good work, and I think the principles may be extended to any field with modifications. It even warns well about addressing the audience with some respect, and gives proof of what happens when an author doesn't do that (Point 2).
2. This book was difficult to read, not so much due to the subject, or the words used to describe them, but because it is:
a. Addressed constantly and only to men-
This was a perennial distraction for me. I've never read anything else that excludes women so thoroughly. I was ridiculously relieved when he used the word "women" once in the last few pages of the books, which backhandedly described why he felt men were more valuable to society than women. Once charming sentence says something about "... parents clamouring for a useful education for their sons."
I want to stress here that those of use who cannot travel in time are all moulded by the time we're embedded in, and so please don't put this book down in a fit of feminist outrage. Reading this also helped me appreciate how much more difficult lives must have been for women in those times, and before that. I appreciate where we are academically now with this insight. It boggled my mind, and it is what I will remember most about this very short book. Wow.
b. Addressed to people (men) already in the know-
This is particularly baffling as he makes it a point to say several times that the book was written for laypersons (laymen), and goes to some lengths to explain some math in easy terms. He refers to his peers by their names very often, and agrees with or refutes their work or arguments without explaining what they were. Perhaps people of his era then, or his profession now are aware of these gentlemen he refers to, and of what they said, but I could only read one side of the discussion- the author's, and so again, this book isn't written for a general audience.
I may read this book again in the future, but simply due to point one, and as a text. I don't expect to derive any pleasure from that exercise.
I do still recommend this book. Read it at least once, if just to understand how not to write about your work for laypeople.
1. This is an enlightening book, especially if you are looking to create some form of original work in academics. Are you planning on writing a doctorate thesis at some point? Read this, because it puts forth another academician's ideas about what a good contribution is. It helps that Hardy managed very well for himself.
Even if you're not looking to pursue an academic career, it is still an insightful read in case at some point you would like to create something original.
It sheds light on the nature and characteristics of good work, and I think the principles may be extended to any field with modifications. It even warns well about addressing the audience with some respect, and gives proof of what happens when an author doesn't do that (Point 2).
2. This book was difficult to read, not so much due to the subject, or the words used to describe them, but because it is:
a. Addressed constantly and only to men-
This was a perennial distraction for me. I've never read anything else that excludes women so thoroughly. I was ridiculously relieved when he used the word "women" once in the last few pages of the books, which backhandedly described why he felt men were more valuable to society than women. Once charming sentence says something about "... parents clamouring for a useful education for their sons."
I want to stress here that those of use who cannot travel in time are all moulded by the time we're embedded in, and so please don't put this book down in a fit of feminist outrage. Reading this also helped me appreciate how much more difficult lives must have been for women in those times, and before that. I appreciate where we are academically now with this insight. It boggled my mind, and it is what I will remember most about this very short book. Wow.
b. Addressed to people (men) already in the know-
This is particularly baffling as he makes it a point to say several times that the book was written for laypersons (laymen), and goes to some lengths to explain some math in easy terms. He refers to his peers by their names very often, and agrees with or refutes their work or arguments without explaining what they were. Perhaps people of his era then, or his profession now are aware of these gentlemen he refers to, and of what they said, but I could only read one side of the discussion- the author's, and so again, this book isn't written for a general audience.
I may read this book again in the future, but simply due to point one, and as a text. I don't expect to derive any pleasure from that exercise.
I do still recommend this book. Read it at least once, if just to understand how not to write about your work for laypeople.
April 3, 2017
How wonderful it is to contact with passionate people! Or, in this case, someone who claims to no longer be passionate, but still transpires passion in his speech. I have always had a penchant for what makes people tick.
How delightful to read about the beauty of mathematical patterns, the harmonious way in which ideas ought to fit together, about the seriousness of a theorem and the significance of the ideas which it connects, about the permanence of mathematical achievement and what makes a good problem. How whetting to follow the reasoning behind theorems of the highest class, and consider what the 'purely aesthetical' qualities of such theorems are (unexpectedness, inevitability, economy).
To read such well structured ideas, such a crystalline and sound reasoning, is truly refreshing.
The way Hardy defends his field, even though he does it in a strictly rational and unsentimental way which may not please some, his thoughts on why mathematics is worth it and in which ways it is superior to other fields, his passion, for lack of better word, made me truly enjoy this book. Notwithstanding, I disagree with many of Hardy's points. Two examples: the idea that most people can't do anything well strikes me as gratuitious; the same for the idea that men who make scorn the men who explain, as I know men who excell at both, and value the explaining as well as the making.
Mathematics is indeed a matchless field, one of the most praised, feared and misunderstood. That intellectual 'kick' provided best by a problem - mathematicians know it better than anyone else. I cannot aspire to transpose the field of 'trivial' mathematics and reach the 'real' mathematics of the 'real' mathematicians; my specific abilities do not lie that way. I am not given to properly explore the mathematical reality, yet it fascinates me. To read an apology where my fascination is validated and logically justified is invigorating.
How delightful to read about the beauty of mathematical patterns, the harmonious way in which ideas ought to fit together, about the seriousness of a theorem and the significance of the ideas which it connects, about the permanence of mathematical achievement and what makes a good problem. How whetting to follow the reasoning behind theorems of the highest class, and consider what the 'purely aesthetical' qualities of such theorems are (unexpectedness, inevitability, economy).
To read such well structured ideas, such a crystalline and sound reasoning, is truly refreshing.
The way Hardy defends his field, even though he does it in a strictly rational and unsentimental way which may not please some, his thoughts on why mathematics is worth it and in which ways it is superior to other fields, his passion, for lack of better word, made me truly enjoy this book. Notwithstanding, I disagree with many of Hardy's points. Two examples: the idea that most people can't do anything well strikes me as gratuitious; the same for the idea that men who make scorn the men who explain, as I know men who excell at both, and value the explaining as well as the making.
Mathematics is indeed a matchless field, one of the most praised, feared and misunderstood. That intellectual 'kick' provided best by a problem - mathematicians know it better than anyone else. I cannot aspire to transpose the field of 'trivial' mathematics and reach the 'real' mathematics of the 'real' mathematicians; my specific abilities do not lie that way. I am not given to properly explore the mathematical reality, yet it fascinates me. To read an apology where my fascination is validated and logically justified is invigorating.
December 9, 2020
Due to the reading assignment from the "Ten Year Reading Plan" I came across this short essay written by the mathematician G.H. Hardy. The introduction to the essay presents briefly his life and work in the domain of pure mathematics and for someone like me who had not known him before was very informative.
The essay itself was written at maturity and Mr. Hardy tries to explain to us and to himself his life choice, which was mathematics. The story is not difficult to read and the mathematical parts are very easy to understand.
The "apology" is not about trying to teach mathematics to the reader or trying to show the reader how wonderful this subject is. I think it is mainly an attempt from Mr. Hardy to make us see mathematics through his eyes. The most important aspect is the fact that this book makes one wonder oneself about his/her life choices. Would you do it again? What do you like about your actual job/calling? Is it what you do now your true calling? Do you regret not having done something else?
These were questions I got asking myself while reading it.
I have never done anything "useful". No discovery of mine has made, or is likely to make[...]the least difference to the amenity of the world [...]I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about value.
The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them
The essay itself was written at maturity and Mr. Hardy tries to explain to us and to himself his life choice, which was mathematics. The story is not difficult to read and the mathematical parts are very easy to understand.
The "apology" is not about trying to teach mathematics to the reader or trying to show the reader how wonderful this subject is. I think it is mainly an attempt from Mr. Hardy to make us see mathematics through his eyes. The most important aspect is the fact that this book makes one wonder oneself about his/her life choices. Would you do it again? What do you like about your actual job/calling? Is it what you do now your true calling? Do you regret not having done something else?
These were questions I got asking myself while reading it.
I have never done anything "useful". No discovery of mine has made, or is likely to make[...]the least difference to the amenity of the world [...]I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about value.
The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them
April 19, 2011
I'm finally actually getting to it. I just read the wonderful 50+ page intro my C.P. Snow (one of my heros). The intro is almost as long as the actual Hardy part. Apparently Graham Greene, in a review, said that along with Henry James' notebooks, Hardy's book was the best description of what it's like to be a creative artist. Despite much googling, I sadly cannot find a copy of the original Greene review. Perhaps I'll finish the rest this evening, if the wonderful Indian food we're off to eat with friends in celebration of J's birthday does not put me right to sleep;-)
I've been meaning to read this for years. I only gave it a 4 instead of 5 because some of the discussion of pure or real Math was over my head. I absolutely loved the intro by C.P. Snow and greatly enjoyed the play by the Nat'l Theatre in UK about Hardy and Ramanujan: A Disappearing Number. I think I would have loved to have heard Hardy lecture.
I've been meaning to read this for years. I only gave it a 4 instead of 5 because some of the discussion of pure or real Math was over my head. I absolutely loved the intro by C.P. Snow and greatly enjoyed the play by the Nat'l Theatre in UK about Hardy and Ramanujan: A Disappearing Number. I think I would have loved to have heard Hardy lecture.
February 6, 2018
I give five stars relative to my expectation - I was totally surprised how much I liked this book. I was taking an after-lunch walk in the library and suddenly the name came to mind, so I pick it up from the shelf. I was expecting some self-important manifesto with little information. I was wrong. Though often sounds like he has no interaction with the world outside of mathematics, he's pretty honest. He started off by saying that serious mathematicians do math, don't write stuff about math. I was like "hmmm". Then he went on: but I'm too old to be a serious mathematician, so now I'm gonna write this book about math. That cracked me up. Anyways, I was pleased to see his perspective on what constitutes "real" math, and how the math's "importance" depends on how it connects ideas and the aesthetics. It's an interesting book that's too short to not read.
May 27, 2023
En önemli matematiksel edebî eser olarak kabul edilen Bir Matematikçinin Savunması’nı ilk kez liseye geçtiğim yıl okumuştum. Şimdi, başka deneyimlerle, yaklaşık 20 senelik profesyonel matematik hayatımın gözlemleriyle bir kere daha okudum. Matematiksel gerçeği, matematiğin yararlılık tartışmasını ele alan Hardy, matematikle uğraşmayan insanların da rahatlıkla anlayabileceği bir dille matematiği diğer bilim dallarıyla, sanatla ve edebiyatla kıyaslayarak tartışıyor. Aslında, temel hedefi matematiği matematikçi olmayanlara anlatmak. Cevabı zor verilebilecek felsefi soruları da tartışmaya açıyor. Bazı matematiksel detayları ve gerçekleri, genel okuyucuya anlatmakta oldukça başarılı Hardy. Sanırım bunu ondan daha iyi yapabilecek çok az matematikçi vardır, hatta belki de yoktur�
November 19, 2009
G.H. Hardy argues that mathematics should be studied, not primarily for its utility, but for its aesthetics. He gives the best description that I have read of mathematical beauty, including the qualities of surprise, generality, and depth. These qualities give the study of mathematics permanent value.
May 8, 2024
Yazar, matematiğin gerçek hayatta hiçbir işe yaramadığını deneyimleriyle beraber okuyucuya göstermeye çalışmış. Matematiğe ilgisi olan arkadaşlara yol gösterebilecek güzel çerezlik bir kitap.
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