作者：（英） 戈弗雷哈代
出版社：人民邮电出版社有限公司
格式: AZW3, DOCX, EPUB, MOBI, PDF, TXT
一个数学家的辩白(双语版)试读：
Preface
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Note
注记
译后记
译者注
[1]献给约翰 • 洛马斯 ，本书受他之约而作PrefaceI am indebted for many valuable criticisms to Professor C. D. Broad and Dr C. P. Snow, each of whom read my original manuscript. I have incorporated the substance of nearly all of their suggestions in my text, and have so removed a good many crudities and obscurities.
In one case I have dealt with them differently. My §28 is based on a short article which I contributed to Eureka (the journal of the Cambridge Archimedean Society) early in the year, and I found it impossible to remodel what I had written so recently and with so much care. Also, if I had tried to meet such important criticisms seriously, I should have had to expand this section so much as to destroy the whole balance of my essay. I have therefore left it unaltered, but have added a short statement of the chief points made by my critics in a note at the end.G. H. H
18 July 1940序[2][3]
我非常感谢查理 • 布罗德 教授和查尔斯 • 斯诺 博士对本书提出了许多有价值的批评意见，他们都读过我的手稿。他们提出的几乎所有建议的精髓都已被我采纳，我借此改正了书中许多经不起推敲或是令人费解的地方。
在处理那些建议时，我对书中的第 28 节采取了不同的做法，它是基于我年初发表在《顿悟》杂志（剑桥阿基米德协会的期刊）上的[4]一篇短文 写成的。我发现自己几乎没有办法去改写这篇文章，它是我精心写就的新作。倘若我认真回复那些重要批评，就会把那一节无限扩写，乃至于会破坏随笔的总体平衡。因此，我保留了第 28 节的原貌，只在注记中补充了一小段文字，探讨他们提出的主要观点。
戈弗雷 • 哈代1940 年 7 月 18 日1
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. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings; 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.
I can remember arguing this point once in one of the few serious conversations that I ever had with Housman. Housman, in his Leslie Stephen lecture The Name and Nature of Poetry, had denied very emphatically that he was a ‘critic’; but he had denied it in what seemed to me a singularly perverse way, and had expressed an admiration for literary criticism which startled and scandalized me.
He had begun with a quotation from his inaugural lecture, delivered twenty-two years before—Whether the faculty of literary criticism is the best gift
that Heaven has in its treasures, I cannot say; but Heaven
seems to think so, for assuredly it is the gift most charily
bestowed. Orators and poets..., if rare in comparison with
blackberries, are commoner than returns of Halley's comet:
literary critics are less common... .
And he had continued—In these twenty-two years I have improved in some
respects and deteriorated in others, but I have not so much
improved as to become a literary critic, nor so much
deteriorated as to fancy that I have become one.
It had seemed to me deplorable that a great scholar and a fine poet should write like this, and, finding myself next to him in Hall a few weeks later, I plunged in and said so. Did he really mean what he had said to be taken very seriously? Would the life of the best of critics really have seemed to him comparable with that of a scholar and a poet? We argued these questions all through dinner, and I think that finally he agreed with me. I must not seem to claim a dialectical triumph over a man who can no longer contradict me; but ‘Perhaps not entirely’ was, in the end, his reply to the first question, and ‘Probably no’ to the second.
There may have been some doubt about Housman's feelings, and I do not wish to claim him as on my side; but there is no doubt at all about the feelings of men of science, and I share them fully. If then I find myself writing, not mathematics but ‘about’ mathematics, it is a confession of weakness, for which I may rightly be scorned or pitied by younger and more vigorous mathematicians. 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.1
让职业数学家去写一本关于数学的书，他一定会很发愁。数学家的工作应该是去证明新定理、发现新数学，不该谈论自己或其他数学家做了什么。政治家看不起时事评论家，画家轻视艺术评论家，生理学家、物理学家、数学家们通常也有类似的感觉：这是实干家对评论家的藐视，没有比这种藐视更深刻，或总体来说更无可非议的了。解说、评论、品鉴，都是二等人才从事的工作。[5]
我记得，在和豪斯曼 为数不多的几次认真谈话里，就有一次[6]对这个话题展开过辩论。豪斯曼在他的莱斯利 • 斯蒂芬 讲座《诗歌的名与实》上，坚决不承认自己是一个“批评家”。在我看来，他表达的方式很荒谬，其对文学批评表示的赞赏，也让我非常震惊。
他以 22 年前就职演说中的一段话作为开头：我不能说，文学批评能力是否是上天赐予我们的最好礼
物。但上天似乎是这样认为的，毫无疑问，它是一份最谨慎
的馈赠。演说家和诗人……虽然不像随处可见的黑莓，但他
们可比哈雷彗星的回归来得常见，而文学批评家则更稀
缺……
他继续说道：在这 22 年里，我在某些方面有所进步，不过在另一些
方面退步了。但我还没有进步到足以成为一名文学批评家；
同样，我也没有退步到幻想自己已经是一名文学批评家。
一位伟大的学者和优秀的诗人竟然这样认为，在我看来是很可悲的。几个星期后，当我在大厅里发现旁边坐的是豪斯曼时，便单刀直入地和他聊起了以下话题：他的话当真吗？在他看来，最好的评论家真的能与学者和诗人相提并论吗？整个晚餐，我们都在辩论这些问[7]题，我想他最终同意了我的观点。对一个再也无法反驳我的人 ，我似乎并不能宣布这次辩论取得了胜利。不过，最终他对第一个问题的回答是“也许不能完全当真”，对第二个问题的回答是“或许不能相提并论”。
人们对豪斯曼的感受可能还有些不解，我也不指望他和我的想法是一致的。但科学家的感受是毋庸置疑的，我和他们有完全相同的体会。当我发现自己的创作只不过与数学“有关”，而并不是数学本身时，那就是在承认自己不行了，我很可能会因此而遭受更年轻、更有活力的数学家的轻视或怜悯。就像其他任何一位年逾花甲的数学家一样，我围绕着数学写作，是因为头脑已经老化，不再有足够的精力和耐心去有效地从事数学本职工作了。2
I propose to put forward an apology for mathematics; and I may be told that it needs none, since there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy. This may be true; indeed it is probable, since the sensational triumphs of Einstein, that stellar astronomy and atomic physics are the only sciences which stand higher in popular estimation. A mathematician need not now consider himself on the defensive. He does not have to meet the sort of opposition described by Bradley in the admirable defence of metaphysics which forms the introduction to Appearance and Reality.
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 more 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 steam-engines and dynamos, obtrude themselves on the dullest imagination. The public does not need to be convinced that there is something in mathematics.
All this is in its way very comforting to mathematicians, but it is hardly possible for a genuine mathematician to be content with it. Any genuine mathematician must feel that it is not on these crude achievements that the real case for mathematics rests, that the popular reputation of mathematics is based largely on ignorance and confusion, and that there is room for a more rational defence. At any rate, I am disposed to try to make one. It should be a simpler task than Bradley's difficult apology.
I shall ask, then, why is it really worth while to make a serious study of mathematics? What is the proper justification of a mathematician's life? And my answers will be, for the most part, such as are to be expected from a mathematician: I think that it is worth while, that there is ample justification. But I should say at once that my defence of mathematics will be a defence of myself, and that my apology is bound to be to some extent egotistical. I should not think it worth while to apologize for my subject if I regarded myself as one of its failures.
Some egotism of this sort is inevitable, and I do not feel that it really needs justification. 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. He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.2
我打算替数学做一次辩白。也许有人会和我说，数学根本不需要这些，因为当下很少有研究工作能像数学一样，无论出于什么原因，都能被公认为是有益的，并且也值得称道。这也许是真的。事实上，[8]由于爱因斯坦 激动人心的成果，在大众眼里，可能只有恒星天文学和原子物理学的地位会比数学高。数学家不必认为自己正处于守势，[9]也不需要面对像布拉德利 在维护形而上学时所做的辩白里描述的那种敌意，那份令人钦佩的辩白就是《现象与实在》的引言。
据布拉德利说，人们会对形而上学家说，“形而上学的知识是根本不存在的”，或是“即便在某种情况下它们是存在的，但实际上它们还是没有什么名副其实的内容”。还有人会说：“同样的问题，同样的争论，同样的彻底溃败。为什么不另起炉灶呢？难道没有别的事情值得去做了吗？”没有人会愚蠢到对数学说这种话。大量数学真理的权威性是明摆着的。它的实际应用随处可见，桥梁、蒸汽机和发电机都是例子。不用唠叨，人们就知道数学很有用。
在某种程度上，所有这些都能让数学家感到欣慰，但真正的数学家几乎不可能会对此感到满意。任何一位真正的数学家一定会认为，数学的口碑所仰仗的并不是这些朴素的实际应用成果，它在很大程度上是出于人们的无知和不解，所以还有更合理的辩词。无论如何，我打算试一试。相较于布拉德利艰难地为形而上学辩白，这应该会简单些。
那么我得问，为什么认真研究数学的确是值得的呢？数学家存在的意义又是什么呢？在很大程度上，我的答案就是数学家的答案：我认为数学研究是值得的，数学家的存在也是有充分理由的。但同时我还要说明，我为数学的辩白也是在为自己说话，这份辩白在某种程度上必然会很本位。如果我认为自己在数学上很失败，那就不会认为有必要为它辩白。
这种本位主义是不可避免的，我不认为真的需要为此辩解。优秀的成果不是由那些“谦虚”的人做出来的。无论什么学科，教授的首要职责之一，便是把他教的课程以及自己在其中的重要性稍作夸大。一个人若总是问自己“我做的事值得吗？”“我是研究这个的合适人选吗？”，那就永远做不好自己，也会让别人情绪低落。他必须不要太在意，稍微拔高一下学科和自身。这点并不难做到，不盲目把它们吹嘘得荒唐可笑才是更难的。3
A man who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be. The first question is often very difficult, and the answer very discouraging, but most people will find the second easy enough even then. Their answers, if they are honest, will usually take one or other of two forms; and the second form is merely a humbler variation of the first, which is the only answer which we need consider seriously.
(1) ‘I do what I do because it is the one and only thing that I can do at all well. 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 judgement 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.’
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 per cent of men can do something rather well. It is a tiny minority who can do anything 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.
This view was endorsed by Dr Johnson—When I told him that I had been to see [his namesake]
Johnson ride upon three horses, he said ‘Such a man, sir,
should be encouraged, for his performances show the extent
of the human powers...’—
and similarly he would have applauded mountain climbers, channel swimmers, and blindfold chess-players. For my own part, I am entirely in sympathy with all such attempts at remarkable achievement. I feel some sympathy even with conjurors and ventriloquists; and when Alekhine and Bradman set out to beat records, I am quite bitterly disappointed if they fail. And here both Dr Johnson and I find ourselves in agreement with the public. As W. J. Turner has said so truly, it is only the ‘highbrows’ (in the unpleasant sense) who do not admire the ‘real swells’.
We have of course to take account of the differences in value between different activities. I would rather be a novelist or a painter than a statesman of similar rank; and there are many roads to fame which most of us would reject as actively pernicious. Yet it is seldom that such differences of value will turn the scale in a man's choice of a career, which will almost always be dictated by the limitations of his natural abilities. 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 (and I suppose that it is unlikely that he could do better). If the cricket were a little less supreme, and the poetry better, then the choice might be more difficult: I do not know whether I would rather have been Victor Trumper or Rupert Brooke. It is fortunate that such dilemmas occur so seldom.
I may add that they are particularly unlikely to present themselves to a mathematician. It is usual to exaggerate rather grossly the differences between the mental processes of mathematicians and other people, but 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 or versatility. If a man is in 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. Such a sacrifice could be justified only by economic necessity or age.3
一个人想要证明自己的存在和行为是有意义的，就必须辨别两个不同的问题。第一个问题是，他的工作是否值得去做；第二个则是，无论其价值如何，他为什么要去做。前者通常很难回答，答案也常令人十分沮丧。然而，多数人会觉得回答后一个问题很容易。如果这些人是诚实的，那么答案通常会符合两种形式，由于第二种只是比第一种更谦卑，于是第一种形式便成了我们唯一需要认真讨论的答案。（1）“我做我所做的，因为这是唯一一件我能做好的事。我做律师，或股票经纪人，或职业板球运动员，是因为我真的有天赋来胜任这份工作。我是一名律师，因为我口齿伶俐，并且有志于把控法律的微妙之处；我是一名股票经纪人，因为我对行情的判断既快又准；我是一名职业板球运动员，因为我的击球技术出类拔萃。我同意，成为诗人或数学家也许会更好，但不幸的是我并没有从事这些职业的天赋。”
我并不是说多数人都能这样替自己辩解，其实多数人什么都做不好。也许只有 5%、最多也就 10% 的人可以在他的行当里干得相当出色。倘若是这一小部分人如此辩解，那么他们的说法一点儿也不荒谬，它是无懈可击的。能真正做好一件事的人非常少，能做好两件事的人更是寥寥无几。如果某人真有天赋，那么为了把这份天赋发挥到极致，他应该做好牺牲一切的准备。[10]
约翰逊博士 赞同这个观点：当我告诉他，我曾见过一个和他一样也叫约翰逊的人骑
着三匹马时，他说：“先生，这样的人应该受到鼓励，因为[11]
他的表演展示了人类力量的极限……”
同样，他也会赞美登山者、横渡海峡的泳将，以及盲棋手。就我个人而言，我完全支持这种为了取得杰出成就而做出的全部努力。即[12]使是魔术师和口技演员，我也能表示理解。在阿廖欣 和布拉德[13]曼 即将打破纪录的那一刻，倘若他们失败了，我会感到非常失望。就这点而言，约翰逊博士和我都觉得，我们和公众的观点是一致的。[14]正如沃尔特 • 特纳 所说，只有那些“趣味高雅的人”（带有贬义）才不欣赏“真正的名家”。
当然，我们也必须考虑不同活动之间的价值差异。我宁愿当小说家或画家，也不愿做相同级别的政治家。还有许多出名的办法，多数人都会因其有问题而加以拒绝。然而，这种价值差异几乎不会改变一个人对职业的选择，择业几乎总是由个人天赋的局限性决定的。诗歌比板球更有价值，但如果布拉德曼为了写二流小诗（我料想他不太可能写出更好的作品）而放弃板球，那他就是个傻瓜。倘若他的板球技巧不那么高超，而诗歌还写得稍好一些，那么可能会更难以抉择：我[15]不知道自己会更愿意成为维克托 • 特兰佩 还是鲁珀特 • 布鲁克 [16]。幸运的是，这样的困境几乎没有出现。
我还可以补充一点，这些人绝不可能想要当一名数学家。尽管数学家与其他人在思维过程上的差异常常被过分夸大，但不可否认的是，数学才能是一种最专业的天赋，而数学家是这样一类人，他们的常规能力或通才能力并不特别突出。如果一个人不管以什么标准衡量，都能算得上是真正的数学家，那么几乎可以肯定，相较他能做的其他工作，从事数学会好得多。倘若他为了能在其他领域有一份普普通通的工作，而放弃可以发挥自己数学才能的良机，那他就是愚蠢的。只有出于经济或年龄的考虑，这种牺牲才说得过去。4
I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics.
We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time that he was forty that his great creative days were over. His greatest ideas of all, fluxions and the law of gravitation, came to him about 1666, when he was twenty-four—‘in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since’. He made big discoveries until he was nearly forty (the ‘elliptic orbit’ at thirty-seven), but after that he did little but polish and perfect.
Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.
On the other hand the gain is no more likely to be substantial; the later records of mathematicians who have left mathematics are not particularly encouraging. Newton made a quite competent Master of the Mint (when he was not quarrelling with anybody). Painlevé was a not very successful Premier of France. Laplace's political career was highly discreditable, but he is hardly a fair instance, since he was dishonest rather than incompetent, and never really ‘gave up’ mathematics. It is very hard to find an instance of a first-rate mathematician who has abandoned mathematics and attained first-rate distinction in any other field.1 There may have been young men who would have been first-rate mathematicians if they had stuck to mathematics, but I have never heard of a really plausible example. And all this is fully borne out by my own very limited experience. Every young mathematician of real talent whom I have known has been faithful to mathematics, and not from lack of ambition but from abundance of it; they have all recognized that there, if anywhere, lay the road to a life of any distinction.
1Pascal seems the best.4
关于年龄问题，我最好补充几句，因为它对数学家特别重要。任何一位数学家都不应该让自己忘记，比起任何其他艺术或科学，数学更是年轻人的游戏。举一个相对简单的例子，在英国皇家学会的入选者中，数学家的平均年龄是最小的。
我们还可以很轻松地找到更多引人注目的例证。比如，我们可以看看下面这个人的职业生涯，他无疑是世界上最伟大的三位数学家之[17]一。牛顿 在 50 岁时放弃了数学研究，他在很久以前就失去了对数学的热情；毫无疑问，他在 40 岁时就意识到他那最富有创造力的数学生涯已经结束。牛顿最伟大的思想——流数术和万有引力定律——是在 1666 年左右产生的，那时他才 24 岁。“在那些日子里，我正处于发明创造的黄金时期，我比任何时候都更专注于数学和哲学。”他不断地取得重大发现，一直到将近 40 岁（他在 37 岁时算出了“椭圆轨道”），但在此之后，他除了修正和完善之前的成果，几乎再也没有做出什么新的东西了。[18][19][20]
伽罗瓦 21 岁就死了，阿贝尔 27 岁，拉马努金 33 岁，[21]黎曼 也只活到 40 岁。也有人在上了年纪之后做出过了不起的成[22]就，高斯 关于微分几何的著名论文是在他 50 岁时发表的（尽管 10 年前他就有这方面的基本思想）。据我所知，在数学上没有一项重大的进步是由超过 50 岁的人提出的。如果一把年纪的人丧失了对数学的兴趣并将它抛弃，由此造成的损失对数学和他个人而言都不会很严重。
另一方面，数学家们在离开数学领域之后的状况也并不那么振奋人心，他们也都没什么实质性的建树。牛顿（在不和别人争吵的时候）[23]是一个相当能干的铸币厂厂长。班勒卫 是一位不太成功的法国总[24]理。拉普拉斯 的政治生涯极不光彩，但他几乎不能算是一个合适的例子，因为他不是无能，而是不诚实，而且他从来没有真正“放弃”过数学。很难找到第一流的数学家在放弃数学之后，在其他领域取得[25]卓越成就的例子 1 。也许有一些年轻人，倘若他们专攻数学，就会成为一流的数学家，但我从未听说过一个确实可信的例子。我自己有限的经历反复证明了这一切。我所认识的每一位真正才华横溢的年轻数学家都对数学忠心耿耿，他们志存高远，充满雄心壮志。他们都意识到，如果存在一条可以通往卓越人生的道路，那这条道路就是数学。
1似乎帕斯卡已经是做得最好的了。5
There is also what I called the ‘humbler variation’ of the standard apology; but I may dismiss this in a very few words.
(2) ‘There is nothing that I can do particularly well. I do what I do because it came my way. I really never had a chance of doing anything else.’ And this apology too I accept as conclusive. It is quite true that most people can do nothing well. If so, it matters very little what career they choose, and there is really nothing more to say about it. It is a conclusive reply, but hardly one likely to be made by a man with any pride; and I may assume that none of us would be content with it.5
还有一种答案，我把它称为第一种“更谦卑的变体”，我只用几句话一带而过。
(2)“没有什么事我能做得特别出色。我做我现在所做的，因为它刚好就在我面前。我的确从来没有机会去从事别的工作。”我同样认为这个辩解是无法反驳的。的确，大多数人什么也做不好。如果是这样，那么他们选择什么职业是无关紧要的，也确实没什么好说的。这是一个不容置疑的回答，但有自尊心的人不太可能说得出口，我想我们都不会满足于此。6
It is time to begin thinking about the first question which I put in §3, and which is so much more difficult than the second. Is mathematics, what I and other mathematicians mean by mathematics, worth doing; and if so, why?
I have been looking again at the first pages of the inaugural lecture which I gave at Oxford in 1920, where there is an outline of an apology for mathematics. It is very inadequate (less than a couple of pages), and it is written in a style (a first essay, I suppose, in what I then imagined to be the ‘Oxford manner’) of which I am not now particularly proud; but I still feel that, however much development it may need, it contains the essentials of the matter. I will resume what I said then, as a preface to a fuller discussion.
(1) I began by laying stress on the harmlessness of mathematics—‘the study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation’. I shall stick to that, but obviously it will need a good deal of expansion and explanation.
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 thinking of ‘profit’, however, in a narrower sense. Is mathematics
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