出版社: Cambridge University Press
出版年: 1992131
页数: 153
定价: $17.99
装帧: Paperback
丛书: Canto Classics
ISBN: 9780521427067
内容简介 · · · · · ·
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 as his mathematical powers were declining, offers a brilliant and engaging ...
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 as his mathematical powers were declining, 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 aphorisms and 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.
作者简介 · · · · · ·
A Mathematician's Apology is a profoundly sad book, the memoir of a man who has reached the end of his ambition, who can no longer effectively practice the art that has consumed him since he was a boy. But at the same time, it is a joyful celebration of the subjectand a stern lecture to those who would sully it by dilettantism or attempts to make it merely useful. "The mathem...
A Mathematician's Apology is a profoundly sad book, the memoir of a man who has reached the end of his ambition, who can no longer effectively practice the art that has consumed him since he was a boy. But at the same time, it is a joyful celebration of the subjectand a stern lecture to those who would sully it by dilettantism or attempts to make it merely useful. "The mathematician's patterns," G.H. Hardy declares, "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."
Hardy was, in his own words, "for a short time the fifth best pure mathematician in the world" and knew full well that "no mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game." In a long biographical foreword to Apology, C.P. Snow (now best known for The Two Cultures) offers invaluable background and a context for his friend's occasionally brusque tone: "His life remained the life of a brilliant young man until he was old; so did his spirit: his games, his interests, kept the lightness of a young don's. And, like many men who keep a young man's interests into their sixties, his last years were the darker for it." Reading Snow's recollections of Hardy's Cambridge University years only makes Apology more poignant. Hardy was popular, a terrific conversationalist, and a notoriously good cricket player.
When summer came, it was taken for granted that we should meet at the cricket ground.... He used to walk round the cinderpath with a long, loping, clumpingfooted stride (he was a slight spare man, physically active even in his late fifties, still playing real tennis), head down, hair, tie, sweaters, papers all flowing, a figure that caught everyone's eyes. "There goes a Greek poet, I'll be bound," once said some cheerful farmer as Hardy passed the scoreboard.
G.H. Hardy's elegant 1940 memoir has provided generations of mathematicians with pithy quotes and examples for their office walls, and plenty of inspiration to either be great or find something else to do. He is a worthy mentor, a man who understood deeply and profoundly the rewards and losses of true devotion. Therese Littleton
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魏厚生 (生于子宫，死于脑洞。)
感谢刘晴同学提供1967年C. P. Snow序言本扫描版 1 I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job. 2 A metaphysician, says Bradley, will be told that 'metaphysical knowledge is wholly impossible', or that 'even if possible to a certain degree, it is ...20141029 22:31 1人收藏
感谢刘晴同学提供1967年C. P. Snow序言本扫描版1 I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job.2 A metaphysician, says Bradley, will be told that 'metaphysical knowledge is wholly impossible', or that 'even if possible to a certain degree, it is practically no knowledge worth the name'. 'The same problems,' he will hear, 'the same disputes, the same sheer failure. Why not abandon it and come out? Is there nothing else worth your labour?' There is no one so stupid as to use this sort of language about mathematics. The mass of mathematical truth is obvious and imposing; its practical applications, the bridges and steamengines and dynamos, obtrude themselves on the dullest imagination....All this is in its way very comforting to mathematicians, but it is hardly possible for a genuine mathematician to be content with it.Good work is not done by 'humble' men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking 'Is what I do worth while?' and 'Am I the right person to do it?' will always be ineffective himself and a discouragement to others. 3 'I do what I do because it is the one and only thing that I can do at all well....'4 No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game.5 Is mathematics 'unprofitable'? In some ways, plainly, it is not; for example, it gives great pleasure to quite a large number of people. I was using the word, however, in a narrower senseis mathematics 'useful', directly useful, as other sciences such as chemistry and physiology are? This is not an altogether easy or uncontroversial question, and I shall ultimately say No, though some mathematicians, and most outsiders, would no doubt say Yes. 7 A man's first duty, a young man's at any rate, is to be ambitious.A physiologist may indeed be glad to remember that his work will benefit mankind, but the motives which provide the force and the inspiration for it are indistinguishable from those of a classical scholar or a mathematician.There are many highly respectable motives which may lead men to prosecute research, but three which are much more important than the rest. The first (without which the rest must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satisfied with one's performance, the shame that overcomes any selfrespecting craftsman when his work is unworthy of his talent. Finally, ambition, desire for reputation, and the position, even the power or the money, which it brings.按：此处可与George Orwell《我为何写作》参照10A 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.It may be very hard to define mathematical beauty, but that is just as true of beauty of any kindwe may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.15A significant mathematical idea, a serious mathematical theorem, should be 'general' in some such sense as this. The idea should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds. The theorem should be one which, even if stated originally (like Pythagoras's theorem) in a quite special form, is capable of considerable extension and is typical of a whole class of theorems of its kind. The relations revealed by the proof should be such as connect many different mathematical ideas. All this is very vague, and subject to many reservations.16It is quite common, for example, for an astronomer or a physicist to claim that he has found a 'mathematical proof' that the physical universe must behave in a particular way. All such claims, if interpreted literally, are strictly nonsense. It cannot be possible to prove mathematically that there will be an eclipse tomorrow,because eclipses, and other physical phenomena, do not form part of the abstract world of mathematics; and this, I suppose, all astronomers would admit when pressed, however many eclipses they may have predicted correctly.We do not choose our friends because they embody all the pleasant qualities of humanity, but because they are the people that they are. And so in mathematics; a property common to too many objects can hardly be very excitng, and mathematical ideas also become dim unless they have plenty of individuality.21But here I must deal with a misconception. It is sometimes suggested that pure mathematicians glory in the usefulness of their work, and make it a boast that it has no practical applications. The imputation is usually based on an incautious saying attributed to Gauss, to the effect that, if mathematics is the queen of the sciences, then the theory of numbers is, because of its supreme usefulness, the queen of mathematicsI have never been able to find an exact quotation. I am sure that Gauss's saying (if indeed it be his) has been rather crudely misinterpreted. If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and lesser mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.按：许多人把Hardy对应用（在非常窄的意义下）的漠不关心误解为鄙视。比如下面这段话（Reuben Hersh, What is Mathematics, Really?, Oxford University Press, Criteria for a philosophy of mathematics in Part One）就带着打Hardy脸的口气说道：G. H. Hardy "famously" boasted: "(按出自《自白》28节)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." Nevertheless, the HardyWeinberg law of genetics is better known than his profound contributions to analytic number theory. What's worse, cryptology is making number theory applicable. Hardy's contribution to that pure field may yet be useful.但说Hardy以纯数学为傲倒是不假。23The room in which I am lecturing is part of the physical world, and has itself a certain pattern....Suppose now that a violent dynamo, or a massive gravitating body, is introduced into the room. Then the physicists tell us that the geometry of the room is changed, its whole physcial pattern slightly but definitely distorted. Do the theorems which I have proved become false? Surely it would be nonsense to suppose that the proofs of them which I have given are affected in any way.The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics. I may add that even a pure mathematician may find his appreciation of this geometry quickened, since there is no mathematician so pure that he feels no interest at all in the physical world; but, in so far as he succumbs to this tempatation, he will be abandoning his purely mathematical position.27It may be objected that my concept of 'utility' has been too narrow, that I have define it in terms of 'happiness' or 'comfort' only, and have ignored the general 'social' effects of mathematics on which recent writers, with very different sympathies, have laid so much stress. Thus Whitehead (who has been a mathematician) speaks of 'the tremendous effort of mathematical knowledge on the lives of men, on their daily avocations, on the organization of society';...I cannot really believe that all this eloquence will do much to comfort mathematicians. The language of both writers is violently exaggerated, and both of them ignore very obvious distinctions....It is not lack of understanding or of sympathy which is the trouble in Whitehead's cases; but he forgets, in his enthusiasm, distinctions with which he is quite familiar. The mathematics which has this 'tremendous effect' on the 'daily avocations of men' and on 'the organization of society' is not the Whitehead but the Hogben mathematics. The mathematics which can be used 'for ordinary purposes by ordinary men' is negligible, and that which can be used by economists or sociologists hardly rises to 'scholarship standard'. The Whitehead mathematics may affect astronomy or physics profoundly, philosophy very appreciablyhigh thinking of one kind is always likely to affect high thinking of anotherbut it has extremely little effect on anything else. Its 'tremendous effects' have been, not on men generally, but on men like Whitehead himself.按：Whitehead的观点以及Kline在Mathematics in Western Culture里表达的观点，正是Hardy之前所云的，对学科重要性的必要的夸张。回应 20141029 22:31 
C.P. Snow Nevertheless, he had to live with an overdelicate nature. He seems to have been born with three skins too few. Unlike Einstein, who had to subjugate his powerful ego in the study of the external world before he could attain his moral stature, Hardy had to strengthen an ego which wasn't much protected. This at times in later life made him selfassertive (as Einstein never was) when...
20130327 11:16
C.P. Snow Nevertheless, he had to live with an overdelicate nature. He seems to have been born with three skins too few. Unlike Einstein, who had to subjugate his powerful ego in the study of the external world before he could attain his moral stature, Hardy had to strengthen an ego which wasn't much protected. This at times in later life made him selfassertive (as Einstein never was) when he had to take a moral stand. On the other hand, it gave him his introspective insight and beautiful candour, so that he could speak of himself with absolute simplicity (as Einstein never could). His behaviour was often different, bizarrely so, from ours: but it came to seem a kind of superstructure set upon a nature which wasn't all that different from our own, except that it was more delicate, less paddd, finernerved. Hardy: ``Cricket is the only game where you are playing against eleven of the other side and then of your own.'' That is why A Mathematician's Apology is, read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits; yes ,the crystalline clarity and candour are still there; yes, it is the testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again. I know nothing like it in the language: partly because most people with the literary gift to express such a lament don't come to feel it: it is very rare for a writer to realize, with the finality of truth, that he is absolutely finished. G.H. Hardy 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 of secondrate minds. Good work is not done by ``humble'' men. It is one of the first duties of professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking ``Is what I do worth while?'' will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. A.E. Housman: Smooth Between Sea and Land Here on the level sand, Between the sea and land, What shall I build or write Against the fall of night? Tell me of runes to grave That hold the bursting wave, Or bastions to design For longer date than mine. The first (without which the rest must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satisfied with one's performance, the shame that overcomes any selfrespecting craftsman when his work is unworthy of his talent. Finally, ambition, desire for reputation, and the position, even the power or the money, which it brings. As Housman insisted, the importance of ideas in poetry is habitually exaggerated: ``I cannot satisfy myself that there are any such things as poetical ideas. Poetry is not the thing said but a way of saying it.'' Chess problems are the hymntunes of mathematics. It (reductio ad absurdum) is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of apawn or even a piece, but a mathematician offers the game. Mathematics may, like poetry or music, ``promote and sustain a lofty habit of mind'', and so increase the happiness of mathematicians and even of other people; but to defend it on that ground would be merely to elaborate what I have said already. I may add that even a pure mathematician may find his appreciation of this geometry quickened, since there is no mathematician so pure that he feels no interestat all in the physical world; but, in so far as he succumbs to this temptation, he will be abandoning his purely mathemtical position. One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. ``Imaginary'' universes are so much more beautiful than this stupidly constructed ``real'' one; and most of the finest products of an applied mathematician's fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.
回应 20130327 11:16 
窜 (You ignorant fuck//)
There are many highly respected motives which may lead men to prosecute research, but three which are much more important to prosecute research, but three which are much more important than the rest. The first (without which the reat must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satified with one's performance, the shame that o...20121105 22:02
There are many highly respected motives which may lead men to prosecute research, but three which are much more important to prosecute research, but three which are much more important than the rest. The first (without which the reat must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satified with one's performance, the shame that overcomes any selfrespecting craftsman when his work is unworthy of his talent. Finally, ambition, deire for reputation, and the position,even the power or the money, which it brings. It may be fine to feel, when you have done your work, that you have added to the happiness or alleviated the sufferings of others, but that will not be why you did it. So if a mathematician, or a chemist, or even a physiologist, were to tell me that the driving force in his work had been the desire to benefit humanity, then I should not believe him (nor should I think the better of him if I did). His dominant motives have been those which I have stated, and in which, surely, there is nothing of which any decent man need be ashamed.
回应 20121105 22:02 
窜 (You ignorant fuck//)
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. Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambitions of Attila or Napoleon; but the noblest ambition is that of leaving behind something of permanent val... (5回应)20121105 21:38
King Gillette 发明了现代刮胡刀。William Willett，彩色玻璃设计师。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. Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambitions of Attila or Napoleon; but the noblest ambition is that of leaving behind something of permanent value Here, on the level sand, Between the sea and land, What shall I build or write Against the fall of night? Tell me of runes to grave That hold the bursting wave, Or bastions to design, For longer date than mine. Ambition has been the driving force behind nearly all the best work of the world. In particular, practically all substantial contributions to human happiness have been made by ambitious men. To make two famous examples, were not Lister and Pasteur ambitous?Or, on a humbler level, King Gillette and William Willet; and who in recent times have contributed more to human comfort than they?
Married Couple, 1915 Leaded Stained Glass Window designed and fabricated by William WillePhysiology provides particularly good examples, just because it is so obviously a 'beneficial' study. We must guard against a fallacy common among aplogist of science, the fallacy of supposing that the men whose work most benefits humanity are thinking much of that while they do it, that physiologists, for example, have particularly noble souls. A physiologist may indeed be glad to remember that his work will benefit mankind, but the motives which provide the force and the inspiration for it are indistinguishable form those of a classical scholar or mathematician.
5回应 20121105 21:38

C.P. Snow Nevertheless, he had to live with an overdelicate nature. He seems to have been born with three skins too few. Unlike Einstein, who had to subjugate his powerful ego in the study of the external world before he could attain his moral stature, Hardy had to strengthen an ego which wasn't much protected. This at times in later life made him selfassertive (as Einstein never was) when...
20130327 11:16
C.P. Snow Nevertheless, he had to live with an overdelicate nature. He seems to have been born with three skins too few. Unlike Einstein, who had to subjugate his powerful ego in the study of the external world before he could attain his moral stature, Hardy had to strengthen an ego which wasn't much protected. This at times in later life made him selfassertive (as Einstein never was) when he had to take a moral stand. On the other hand, it gave him his introspective insight and beautiful candour, so that he could speak of himself with absolute simplicity (as Einstein never could). His behaviour was often different, bizarrely so, from ours: but it came to seem a kind of superstructure set upon a nature which wasn't all that different from our own, except that it was more delicate, less paddd, finernerved. Hardy: ``Cricket is the only game where you are playing against eleven of the other side and then of your own.'' That is why A Mathematician's Apology is, read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits; yes ,the crystalline clarity and candour are still there; yes, it is the testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again. I know nothing like it in the language: partly because most people with the literary gift to express such a lament don't come to feel it: it is very rare for a writer to realize, with the finality of truth, that he is absolutely finished. G.H. Hardy 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 of secondrate minds. Good work is not done by ``humble'' men. It is one of the first duties of professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking ``Is what I do worth while?'' will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. A.E. Housman: Smooth Between Sea and Land Here on the level sand, Between the sea and land, What shall I build or write Against the fall of night? Tell me of runes to grave That hold the bursting wave, Or bastions to design For longer date than mine. The first (without which the rest must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satisfied with one's performance, the shame that overcomes any selfrespecting craftsman when his work is unworthy of his talent. Finally, ambition, desire for reputation, and the position, even the power or the money, which it brings. As Housman insisted, the importance of ideas in poetry is habitually exaggerated: ``I cannot satisfy myself that there are any such things as poetical ideas. Poetry is not the thing said but a way of saying it.'' Chess problems are the hymntunes of mathematics. It (reductio ad absurdum) is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of apawn or even a piece, but a mathematician offers the game. Mathematics may, like poetry or music, ``promote and sustain a lofty habit of mind'', and so increase the happiness of mathematicians and even of other people; but to defend it on that ground would be merely to elaborate what I have said already. I may add that even a pure mathematician may find his appreciation of this geometry quickened, since there is no mathematician so pure that he feels no interestat all in the physical world; but, in so far as he succumbs to this temptation, he will be abandoning his purely mathemtical position. One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. ``Imaginary'' universes are so much more beautiful than this stupidly constructed ``real'' one; and most of the finest products of an applied mathematician's fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.
回应 20130327 11:16 
窜 (You ignorant fuck//)
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. Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambitions of Attila or Napoleon; but the noblest ambition is that of leaving behind something of permanent val... (5回应)20121105 21:38
King Gillette 发明了现代刮胡刀。William Willett，彩色玻璃设计师。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. Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambitions of Attila or Napoleon; but the noblest ambition is that of leaving behind something of permanent value Here, on the level sand, Between the sea and land, What shall I build or write Against the fall of night? Tell me of runes to grave That hold the bursting wave, Or bastions to design, For longer date than mine. Ambition has been the driving force behind nearly all the best work of the world. In particular, practically all substantial contributions to human happiness have been made by ambitious men. To make two famous examples, were not Lister and Pasteur ambitous?Or, on a humbler level, King Gillette and William Willet; and who in recent times have contributed more to human comfort than they?
Married Couple, 1915 Leaded Stained Glass Window designed and fabricated by William WillePhysiology provides particularly good examples, just because it is so obviously a 'beneficial' study. We must guard against a fallacy common among aplogist of science, the fallacy of supposing that the men whose work most benefits humanity are thinking much of that while they do it, that physiologists, for example, have particularly noble souls. A physiologist may indeed be glad to remember that his work will benefit mankind, but the motives which provide the force and the inspiration for it are indistinguishable form those of a classical scholar or mathematician.
5回应 20121105 21:38 
窜 (You ignorant fuck//)
There are many highly respected motives which may lead men to prosecute research, but three which are much more important to prosecute research, but three which are much more important than the rest. The first (without which the reat must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satified with one's performance, the shame that o...20121105 22:02
There are many highly respected motives which may lead men to prosecute research, but three which are much more important to prosecute research, but three which are much more important than the rest. The first (without which the reat must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satified with one's performance, the shame that overcomes any selfrespecting craftsman when his work is unworthy of his talent. Finally, ambition, deire for reputation, and the position,even the power or the money, which it brings. It may be fine to feel, when you have done your work, that you have added to the happiness or alleviated the sufferings of others, but that will not be why you did it. So if a mathematician, or a chemist, or even a physiologist, were to tell me that the driving force in his work had been the desire to benefit humanity, then I should not believe him (nor should I think the better of him if I did). His dominant motives have been those which I have stated, and in which, surely, there is nothing of which any decent man need be ashamed.
回应 20121105 22:02

魏厚生 (生于子宫，死于脑洞。)
感谢刘晴同学提供1967年C. P. Snow序言本扫描版 1 I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job. 2 A metaphysician, says Bradley, will be told that 'metaphysical knowledge is wholly impossible', or that 'even if possible to a certain degree, it is ...20141029 22:31 1人收藏
感谢刘晴同学提供1967年C. P. Snow序言本扫描版1 I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job.2 A metaphysician, says Bradley, will be told that 'metaphysical knowledge is wholly impossible', or that 'even if possible to a certain degree, it is practically no knowledge worth the name'. 'The same problems,' he will hear, 'the same disputes, the same sheer failure. Why not abandon it and come out? Is there nothing else worth your labour?' There is no one so stupid as to use this sort of language about mathematics. The mass of mathematical truth is obvious and imposing; its practical applications, the bridges and steamengines and dynamos, obtrude themselves on the dullest imagination....All this is in its way very comforting to mathematicians, but it is hardly possible for a genuine mathematician to be content with it.Good work is not done by 'humble' men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking 'Is what I do worth while?' and 'Am I the right person to do it?' will always be ineffective himself and a discouragement to others. 3 'I do what I do because it is the one and only thing that I can do at all well....'4 No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game.5 Is mathematics 'unprofitable'? In some ways, plainly, it is not; for example, it gives great pleasure to quite a large number of people. I was using the word, however, in a narrower senseis mathematics 'useful', directly useful, as other sciences such as chemistry and physiology are? This is not an altogether easy or uncontroversial question, and I shall ultimately say No, though some mathematicians, and most outsiders, would no doubt say Yes. 7 A man's first duty, a young man's at any rate, is to be ambitious.A physiologist may indeed be glad to remember that his work will benefit mankind, but the motives which provide the force and the inspiration for it are indistinguishable from those of a classical scholar or a mathematician.There are many highly respectable motives which may lead men to prosecute research, but three which are much more important than the rest. The first (without which the rest must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satisfied with one's performance, the shame that overcomes any selfrespecting craftsman when his work is unworthy of his talent. Finally, ambition, desire for reputation, and the position, even the power or the money, which it brings.按：此处可与George Orwell《我为何写作》参照10A 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.It may be very hard to define mathematical beauty, but that is just as true of beauty of any kindwe may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.15A significant mathematical idea, a serious mathematical theorem, should be 'general' in some such sense as this. The idea should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds. The theorem should be one which, even if stated originally (like Pythagoras's theorem) in a quite special form, is capable of considerable extension and is typical of a whole class of theorems of its kind. The relations revealed by the proof should be such as connect many different mathematical ideas. All this is very vague, and subject to many reservations.16It is quite common, for example, for an astronomer or a physicist to claim that he has found a 'mathematical proof' that the physical universe must behave in a particular way. All such claims, if interpreted literally, are strictly nonsense. It cannot be possible to prove mathematically that there will be an eclipse tomorrow,because eclipses, and other physical phenomena, do not form part of the abstract world of mathematics; and this, I suppose, all astronomers would admit when pressed, however many eclipses they may have predicted correctly.We do not choose our friends because they embody all the pleasant qualities of humanity, but because they are the people that they are. And so in mathematics; a property common to too many objects can hardly be very excitng, and mathematical ideas also become dim unless they have plenty of individuality.21But here I must deal with a misconception. It is sometimes suggested that pure mathematicians glory in the usefulness of their work, and make it a boast that it has no practical applications. The imputation is usually based on an incautious saying attributed to Gauss, to the effect that, if mathematics is the queen of the sciences, then the theory of numbers is, because of its supreme usefulness, the queen of mathematicsI have never been able to find an exact quotation. I am sure that Gauss's saying (if indeed it be his) has been rather crudely misinterpreted. If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and lesser mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.按：许多人把Hardy对应用（在非常窄的意义下）的漠不关心误解为鄙视。比如下面这段话（Reuben Hersh, What is Mathematics, Really?, Oxford University Press, Criteria for a philosophy of mathematics in Part One）就带着打Hardy脸的口气说道：G. H. Hardy "famously" boasted: "(按出自《自白》28节)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." Nevertheless, the HardyWeinberg law of genetics is better known than his profound contributions to analytic number theory. What's worse, cryptology is making number theory applicable. Hardy's contribution to that pure field may yet be useful.但说Hardy以纯数学为傲倒是不假。23The room in which I am lecturing is part of the physical world, and has itself a certain pattern....Suppose now that a violent dynamo, or a massive gravitating body, is introduced into the room. Then the physicists tell us that the geometry of the room is changed, its whole physcial pattern slightly but definitely distorted. Do the theorems which I have proved become false? Surely it would be nonsense to suppose that the proofs of them which I have given are affected in any way.The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics. I may add that even a pure mathematician may find his appreciation of this geometry quickened, since there is no mathematician so pure that he feels no interest at all in the physical world; but, in so far as he succumbs to this tempatation, he will be abandoning his purely mathematical position.27It may be objected that my concept of 'utility' has been too narrow, that I have define it in terms of 'happiness' or 'comfort' only, and have ignored the general 'social' effects of mathematics on which recent writers, with very different sympathies, have laid so much stress. Thus Whitehead (who has been a mathematician) speaks of 'the tremendous effort of mathematical knowledge on the lives of men, on their daily avocations, on the organization of society';...I cannot really believe that all this eloquence will do much to comfort mathematicians. The language of both writers is violently exaggerated, and both of them ignore very obvious distinctions....It is not lack of understanding or of sympathy which is the trouble in Whitehead's cases; but he forgets, in his enthusiasm, distinctions with which he is quite familiar. The mathematics which has this 'tremendous effect' on the 'daily avocations of men' and on 'the organization of society' is not the Whitehead but the Hogben mathematics. The mathematics which can be used 'for ordinary purposes by ordinary men' is negligible, and that which can be used by economists or sociologists hardly rises to 'scholarship standard'. The Whitehead mathematics may affect astronomy or physics profoundly, philosophy very appreciablyhigh thinking of one kind is always likely to affect high thinking of anotherbut it has extremely little effect on anything else. Its 'tremendous effects' have been, not on men generally, but on men like Whitehead himself.按：Whitehead的观点以及Kline在Mathematics in Western Culture里表达的观点，正是Hardy之前所云的，对学科重要性的必要的夸张。回应 20141029 22:31 
C.P. Snow Nevertheless, he had to live with an overdelicate nature. He seems to have been born with three skins too few. Unlike Einstein, who had to subjugate his powerful ego in the study of the external world before he could attain his moral stature, Hardy had to strengthen an ego which wasn't much protected. This at times in later life made him selfassertive (as Einstein never was) when...
20130327 11:16
C.P. Snow Nevertheless, he had to live with an overdelicate nature. He seems to have been born with three skins too few. Unlike Einstein, who had to subjugate his powerful ego in the study of the external world before he could attain his moral stature, Hardy had to strengthen an ego which wasn't much protected. This at times in later life made him selfassertive (as Einstein never was) when he had to take a moral stand. On the other hand, it gave him his introspective insight and beautiful candour, so that he could speak of himself with absolute simplicity (as Einstein never could). His behaviour was often different, bizarrely so, from ours: but it came to seem a kind of superstructure set upon a nature which wasn't all that different from our own, except that it was more delicate, less paddd, finernerved. Hardy: ``Cricket is the only game where you are playing against eleven of the other side and then of your own.'' That is why A Mathematician's Apology is, read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits; yes ,the crystalline clarity and candour are still there; yes, it is the testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again. I know nothing like it in the language: partly because most people with the literary gift to express such a lament don't come to feel it: it is very rare for a writer to realize, with the finality of truth, that he is absolutely finished. G.H. Hardy 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 of secondrate minds. Good work is not done by ``humble'' men. It is one of the first duties of professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking ``Is what I do worth while?'' will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. A.E. Housman: Smooth Between Sea and Land Here on the level sand, Between the sea and land, What shall I build or write Against the fall of night? Tell me of runes to grave That hold the bursting wave, Or bastions to design For longer date than mine. The first (without which the rest must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satisfied with one's performance, the shame that overcomes any selfrespecting craftsman when his work is unworthy of his talent. Finally, ambition, desire for reputation, and the position, even the power or the money, which it brings. As Housman insisted, the importance of ideas in poetry is habitually exaggerated: ``I cannot satisfy myself that there are any such things as poetical ideas. Poetry is not the thing said but a way of saying it.'' Chess problems are the hymntunes of mathematics. It (reductio ad absurdum) is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of apawn or even a piece, but a mathematician offers the game. Mathematics may, like poetry or music, ``promote and sustain a lofty habit of mind'', and so increase the happiness of mathematicians and even of other people; but to defend it on that ground would be merely to elaborate what I have said already. I may add that even a pure mathematician may find his appreciation of this geometry quickened, since there is no mathematician so pure that he feels no interestat all in the physical world; but, in so far as he succumbs to this temptation, he will be abandoning his purely mathemtical position. One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. ``Imaginary'' universes are so much more beautiful than this stupidly constructed ``real'' one; and most of the finest products of an applied mathematician's fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.
回应 20130327 11:16 
窜 (You ignorant fuck//)
There are many highly respected motives which may lead men to prosecute research, but three which are much more important to prosecute research, but three which are much more important than the rest. The first (without which the reat must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satified with one's performance, the shame that o...20121105 22:02
There are many highly respected motives which may lead men to prosecute research, but three which are much more important to prosecute research, but three which are much more important than the rest. The first (without which the reat must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satified with one's performance, the shame that overcomes any selfrespecting craftsman when his work is unworthy of his talent. Finally, ambition, deire for reputation, and the position,even the power or the money, which it brings. It may be fine to feel, when you have done your work, that you have added to the happiness or alleviated the sufferings of others, but that will not be why you did it. So if a mathematician, or a chemist, or even a physiologist, were to tell me that the driving force in his work had been the desire to benefit humanity, then I should not believe him (nor should I think the better of him if I did). His dominant motives have been those which I have stated, and in which, surely, there is nothing of which any decent man need be ashamed.
回应 20121105 22:02 
窜 (You ignorant fuck//)
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. Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambitions of Attila or Napoleon; but the noblest ambition is that of leaving behind something of permanent val... (5回应)20121105 21:38
King Gillette 发明了现代刮胡刀。William Willett，彩色玻璃设计师。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. Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambitions of Attila or Napoleon; but the noblest ambition is that of leaving behind something of permanent value Here, on the level sand, Between the sea and land, What shall I build or write Against the fall of night? Tell me of runes to grave That hold the bursting wave, Or bastions to design, For longer date than mine. Ambition has been the driving force behind nearly all the best work of the world. In particular, practically all substantial contributions to human happiness have been made by ambitious men. To make two famous examples, were not Lister and Pasteur ambitous?Or, on a humbler level, King Gillette and William Willet; and who in recent times have contributed more to human comfort than they?
Married Couple, 1915 Leaded Stained Glass Window designed and fabricated by William WillePhysiology provides particularly good examples, just because it is so obviously a 'beneficial' study. We must guard against a fallacy common among aplogist of science, the fallacy of supposing that the men whose work most benefits humanity are thinking much of that while they do it, that physiologists, for example, have particularly noble souls. A physiologist may indeed be glad to remember that his work will benefit mankind, but the motives which provide the force and the inspiration for it are indistinguishable form those of a classical scholar or mathematician.
5回应 20121105 21:38
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订阅关于A Mathematician's Apology的评论:
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0 有用 Alkis_Tangelis 20130624
选问题的品味很重要，尤其是当在生活和兴趣之间做trade off的时候，既要兼顾应用，又要深刻
0 有用 rhine 20110323
为什么要学数学？数学的实际意义在哪里？数学的永恒不朽和普遍意义（permanence, immortality,generality）区别于其他科学学科。应用数学或可有所实用，但纯数学的价值在哪里？
0 有用 lovepooh 20110910
来生，请赐予我数学的天赋，让我安心自在的驰骋在“无用”的数学想象里
0 有用 ninainaid 20110404
"Ambition is a noble passion..." 英伦摇滚风啊。
0 有用 壞人 20111010
差不多每句話都是我想說的啊，太偉大了。以後需要談論有關的東西，引用就好了，因為我不可能比Hardy說得更好。
0 有用 小狮子 20161026
在central和教授看的ramanujan话剧 还和poonen大神握上了手～～
0 有用 Sparkling 20161006
略奇怪哈代作为一个二十世纪的数学家讲起话来还是十九世纪那一副派头，有股维护数学独立尊严的倔强劲
0 有用 Seaweed Udon 20150704
对数学家与数学的探讨深入浅出，不涉及任何艰深的数学知识。一个伟大数学家晚年直抒胸臆的告白萦绕着淡泊而又动人心魄的悲伤，其深厚内涵已超越数学，任何创造性工作者都能从中找到共鸣。文笔冷静精准，逻辑如数学证明般简约明晰，全篇有希腊雕塑般的纯净高冷。
0 有用 Serendipity 20150101
书中明显的矛盾是一个数学伯拉图主义者却将数学视为Creative Arts,细想一下却也可以解释......
1 有用 刘德维希 20141006
连夜看完了，哈代文笔相当不错，读起来舒服又快活。