一个数学家的辩白

《一个数学家的辩白》是数学大师 Godfrey Harold Hardy 在 1940 年左右的作品,可以称之为 Hardy 本人的自传和内心独白。Hardy 通过自己多年从事数学科研的经验,对数学,文学,哲学,美学等诸多学科的理解,将其写成了一本小册子。给其余的数学工作者,数学爱好者,以及不了解数学的人一个了解数学家内心的机会。

Hardy的照片
G.H.Hardy 的照片

G.H.Hardy(1877 年 2 月 7 日 – 1947 年 12 月 1 日)是一代数论大师,英国数学家,先后在牛津大学(Oxford)和剑桥大学(Cambridge)担任数学教授,与另一位英国数学家 Littlewood 共同研究数学,其研究领域包括解析数论,三角级数,不等式等诸多方向。对数论领域和分析学领域的贡献巨大,是二十世纪英国分析学派的代表人物之一。

Hardy的数学家族谱
G.H.Hardy 的数学家族谱

在卢昌海《黎曼猜想漫谈》这本书里面,提到了 G.H.Hardy 的一则小故事。当年,Hardy 在一次从丹麦回英国的途中,碰巧遇到暴风肆虐。他很担心小船会沉没,于是在上船前给朋友寄了一张明信片,上面写道:“我证明了黎曼猜想。” 如果哈代不幸遇难,因死无对证,后世将无法否认(当然也没法确认)他真的证明了黎曼猜想。然而,上帝不想让他有这样大的荣耀,所以没有让这条船沉没。

在《一个数学家的辩白》这本书中,Hardy 基于个人经历,向大家展示了一位数学家对数学这门学科的一些看法和观点。下面将会摘录其中的一些语句。

假如真的能把我的雕像塑在伦敦纪念碑上的话,我是希望这座碑高耸入云,以至人们见不到雕像呢,还是希望纪念碑矮得可以使人们对雕像一目了然呢?我会选择前一种,而斯诺博士可能会选择后一种。

如果一个数学家发现自己在写关于数学的东西,他会感到很忧伤的。因为数学家的工作是做实事,比如证明新定理,使数学有所发展,而不是谈论自己或别的数学家干了些什么。政治家蔑视时事评论家;画家蔑视艺术评论家;生理学家、物理学家或数学家一般都有类似的感觉。做事者对评论者的蔑视是最深刻的,总的来看也是最合理的。解释、评论、鉴赏是次等工作。

认为“谦卑”的人做不出优秀的工作。比方说,在任何一个学科里,教授的首要职责之一就是对自己这一学科的重要性以及自己本人在这一学科的重要性进行一点夸大。假如一个人总在问自己:“我所做的事是值得做的吗?”以及“我做这个合适吗?”这都会使自己永远无能而且也让别人泄气。这种人该把眼睛闭上一会儿,更多地考虑自己的学科和自己本人的情况,而不是更多地考虑学科与自己所应得的报酬。这不太困难,因为更加困难的是依靠紧闭眼睛来使自己的学科与自己本人不受他人所嘲笑。

我之所以做我的事,因为这事是,而且是惟一的一件我完全可以做好的事。我是个律师,或者是一个股票经纪人,或者是一个职业板球手,这都是因为我对这一特别的工作有些真正的才能。我做律师,是因为我伶牙俐齿,而且对法律之微妙感兴趣;我做股票经纪人,是因为我对股市行情的判断迅速而准确;我做职业板球手,是因为我挥拍非同一般地好。有人说,我做个诗人或数学家也许更好,但不幸的是,我并没有才能做这样的工作。

我并不认为大多数人能够做出上述那样的辩解,因为多数人什么工作也做不好。可是只要这种辩解说得振振有词,它就很难反驳,事实上只有少数人能进行这样的辩解:也许只有 5% 或 10% 的人可做得不错。而只有极少数人可做得真正好。而能做好两件事的人只有寥寥无几的了。假如一个人有真正的才能,他就应该乐于牺牲几乎所有的一切,以充分发挥自己的才能。

我会设想我是在为那些现在和过去都满怀雄心壮志的人写这本书的。一个人的首要任务,进一步说,一个年轻人的首要任务是能显示雄心壮志。雄心是一种可以合情合理地以许多形式表现出的一种宏大高尚的志向。阿提拉(Attila)和拿破仑的野心中就有某种高尚的志向,但最高尚的雄心壮志是在自己身后留下某种永存的价值。

  在这平坦的沙滩上,
  海洋与大地间,
  我该建起或写些什么,
  来阻止夜幕的降临?
  告诉我神秘的字符,
  去喝退那汹涌的波涛,
  告诉我时间的城堡,
  去规划那更久的白昼。

有很多高尚的动机驱使人们进行某项研究。在这些动机中,最为重要的有三种。 首先(舍此必一事无成)是理智的好奇心,也就是对了解真理的渴望。其次是对自己专业工作的自豪感,只有工作才能使自己得以满足的那种渴望。任何自尊的数学家,当他的工作与其才能不相称时,耻辱感会压倒一切。最后一个就是雄心壮志,期望得到名声、地位甚至随之而来的权力和金钱。

假如理智的好奇心、对专业工作的自豪感和雄心壮志是在研究工作中占支配地位的动机的话,那么,毫无疑问,没有哪个人比一个数学家有更好的机会来满足这些条件了。数学家的研究学科是所有学科中最令人好奇的。没有哪门学科中的真理会像数学那样奇异。数学是最精细与最富有魅力的技艺,而且数学研究提供了展示真正的专业技能的机会。最后我还要说的是,正如历史所充分证明的那样,不论数学内在的本质价值何在,其成就是一切成就中最持久的。

数学家,就像画家、诗人一样,都是模式的创制者。要说数学家的模式比画家、诗人的模式更长久,那是因为数学家的模式由思想组成,而画家以形状和色彩创制模式,诗人则以言语和文字造型。一幅画或许蕴含着某种”意境”,但通常是平凡而无关紧要的;比较之下,诗意要重要得多,不过,像豪斯曼坚持认为的那样,人们习以为常地夸大了诗意的重要性。他说:“我难以确信存在诗意之类的东西⋯⋯诗歌并不在于表述了什么,而在于怎样表述。”

 倾江海之水,洗不净帝王身上的膏香御气。

一个有意义的数学概念,一条严肃的数学定理从下述意义上被认为是“普遍的”。数学概念应该是许多数学构造的要素,应能应用于许多不同种定理的证明。

一个有意义的定理必须具备的第二个特性就是“深刻性”。其概念也不易定义,它与“难度”有关,深刻的思想往往难以掌握,但二者也并不完全一样。毕氏定理及推广所蕴含的概念有一定的深度,但现代数学家绝不会认为它难懂。相反,一个定理可能极为肤浅,但却难以证明–如丢番图(Diophantus)的有关求方程整数解的定理。

纯数学和应用数学的这些差异对它们本身很重要,但与我们关于数学“实用性”的讨论毫无关系。我曾谈到过费马和其他一些伟大的数学家的“真正的”数学,具有永恒美学价值的数学,如最好的希腊数学。它们之所以永恒,是因为其中的精华就像文学中的精英部分,在几千年后还能引起千万人强烈的满足感。

对于我的一生,或者说任何一个与我类似的数学家的情况是:我所做的工作扩充了知识,并且帮助他人在这座知识的大厦上添砖加瓦;而这些添加部分与伟大的数学家们的创新,或任何其他大大小小艺术家们的作品的价值的不同仅仅在于程度而不在于种类。这些数学家和艺术家都在死后留下了某种纪念物。

伽罗瓦 21 岁去世,阿贝尔 27 岁去世,拉曼纽扬 33 岁去世,黎曼 40 岁去世。也有些人确实是在较晚时取得伟大成就的,高斯就是在 55 岁时才发表了他的微分几何学的重要论文(但在此十年前他就已经形成了他的基本思想)。我还不知道有哪一个重要的数学进展是由一个年过半百的人创始的。假如一个年长的人对数学不感兴趣而放弃了它,这种损失不论对数学本身还是他本人来说,都不十分严重。

阅读《一个数学家的辩白》这本书,可以通过字里行间看到作为一个数学家的 Hardy 对数学的热爱,对自己正值花甲之年而无法做出更好的科研而流露出一种淡淡的忧伤。

其实,二十世纪三十年代末以后,Hardy 的学术活动就开始逐渐减少。1939 年第二次世界大战爆发,让 Hardy 感到更加苦闷。四十年代以后,Hardy 很少参与学术活动。1947 年 12 月 1 日,Hardy 在剑桥去世,享年 70 岁。留给大家的除了数学上的各种定理,书籍之外,还留下了这一本小册子,《一个数学家的自白》。

参考书籍:

  1. 《一个数学家的辩白》;
  2. 《黎曼猜想漫谈》。

一代数学大师 Rota 的经验与忠告

意大利裔的美籍数学家 Gian-Carlo Rota(1932 年 4 月 27 日 – 1999 年 4 月 18 日)是一位杰出的组合学家。他曾是研究泛函分析(Functional Analysis)出身,后来由于个人兴趣的转移,成为了一位研究组合数学(Combinatorial Mathematics)的学者。Rota 的职业生涯大部分都在麻省理工学院(MIT)度过,曾担任 MIT 的数学教授与哲学教授。

Rota's Photo 1970
1970 年的 Rota

从数学家族谱(Mathematics Genealogy Project)上面可以看到:Gian-Carlo Rota 的导师是 Jacob T. Schwartz,Rota 于 1956 年在耶鲁大学获得数学博士学位,其博士论文的题目是 Extension Theory of Differential Operators。

Rota的数学家族谱_1
Rota 的数学族谱

在 1997 年,Rota 发表了两篇关于人生经验和忠告的文章,分别是 “Ten Lessons I wish I Had Been Taught” 和 “Ten Lessons for the Survival of a Mathematics Department“。下面就来逐一分享这两篇文章中的一些观点。

Ten Lessons I wish I Had Been Taught

Springer Link Ten Lessons I wish I Had Been Taught
Ten Lessons I wish I Had Been Taught

讲座(Lecturing)

每次讲座或者分享的时候都有几个需要注意的事情。

(a)每次讲座都应该只有一个重点。(Every lecture should make only one main point.)

Every lecture should state one main point and repeat it over and over, like a theme with variations. An audience is like a herd of cows, moving slowly in the direction they are being driven towards. If we make one point, we have a good chance that the audience will take the right direction; if we make several points, then the cows will scatter all over the field. The audience will lose interest and everyone will go back to the thoughts they interrupted in order to come to our lecture.

(b)不要超时。(Never run overtime.)

Running overtime is the one unforgivable error a lecturer can make. After fifty minutes (one micro-century as von Neumann used to say) everybody’s attention will turn elsewhere even if we are trying to prove the Riemann hypothesis. One minute overtime can destroy the best of lectures.

(c)提及听众的成果。(Relate to your audience.)

As you enter the lecture hall, try to spot someone in the audience with whose work you have some familiarity. Quickly rearrange your presentation so as to manage to mention some of that person’s work. In this way, you will guarantee that at least one person will follow with rapt attention, and you will make a friend to boot.

Everyone in the audience has come to listen to your lecture with the secret hope of hearing their work mentioned.

(d)给听众一些值得回忆的东西。(Give them something to take home.)

Most of the time they admit that they have forgotten the subject of the course and all the mathematics I thought I had taught them. However, they will gladly recall some joke, some anecdote, some quirk, some side remark, or some mistake I made.

板书技巧(Blackboard Technique)

(a)开讲前保持黑板干净(Make sure the blackboard is spotless.)

By starting with a spotless blackboard you will subtly convey the impression that the lecture they are about to hear is equally spotless.

(b)从黑板的左上角开始书写(Start writing on the top left-hand corner.

What we write on the blackboard should correspond to what we want an attentive listener to take down in his notebook. It is preferable to write slowly and in a large handwriting, with no abbreviations.

When slides are used instead of the blackboard, the speaker should spend some time explaining each slide, preferably by adding sentences that are inessential, repetitive, or superfluous, so as to allow any member of the audience time to copy our slide. We all fall prey to the illusion that a listener will find the time to read the copy of the slides we hand them after the lecture. This is wishful thinking.

多次公布同样的结果(Publish the Same Result Several Times)

The mathematical community is split into small groups, each one with its own customs, notation, and terminology. It may soon be indispensable to present the same result in several versions, each one accessible to a specific group; the price one might have to pay otherwise is to have our work rediscovered by someone who uses a different language and notation and who will rightly claim it as his own.

说明性的工作反而更有可能被记得(You Are More Likely to Be Remembered by Your Expository Work)

When we think of Hilbert, we think of a few of his great theorems, like his basis theorem. But Hilbert’s name is more often remembered for his work in number theory, his Zahlbericht, his book Foundations of Geometry, and for his text on integral equations.

每个数学家只有少数的招数(Every Mathematician Has Only a Few Tricks)

You admire Erdös’s contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdös’s work could be “reduced” to a few tricks which Erdös repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory, it was surprising to verify that Hilbert’s proofs relied on the same few tricks. Even Hilbert had only a few tricks!

别害怕犯错(Do Not Worry about Your Mistakes)

There are two kinds of mistakes. There are fatal mistakes that destroy a theory, but there are also contingent ones, which are useful in testing the stability of a theory.

使用费曼的方法(Use the Feynman Method)

You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, “How did he do it? He must be a genius!”

不要吝啬你的赞美(Give Lavish Acknowledgments)

I have always felt miffed after reading a paper in which I felt I was not being given proper credit, and it is safe to conjecture that the same happens to everyone else.

写好摘要(Write Informative Introductions)

If we wish our paper to be read, we had better provide our prospective readers with strong motivation to do so. A lengthy introduction, summarizing the history of the subject, giving everybody his due, and perhaps enticingly outlining the content of the paper in a discursive manner, will go some of the way towards getting us a couple of readers.

为老年做好心理准备(Be Prepared for Old Age)

You must realize that after reaching a certain age you are no longer viewed as a person. You become an institution, and you are treated the way institutions are treated. You are expected to behave like a piece of period furniture, an architectural landmark, or an incunabulum.

 

Ten Lessons for the Survival of a Mathematics Department

Springer Link Ten Lessons for the Survival of a Mathematics Department
Ten Lessons for the Survival of a Mathematics Department

不要在其他系讲自己系同事的坏话(Never wash your dirty linen in public)

Departments of a university are like sovereign states: there is no such thing as charity towards one another.

别越级打报告(Never go above the head of your department)

Your letter will be viewed as evidence of disunity in the rank and file of mathematicians. Human nature being what it is, such a dean or provost is likely to remember an unsolicited letter at budget time, and not very kindly at that.

不要进行领域评价(Never Compare Fields)

You are not alone in believing that your own field is better and more promising than those of your colleagues. We all believe the same about our own fields. But our beliefs cancel each other out. Better keep your mouth shut rather than make yourself obnoxious. And remember, when talking to outsiders, have nothing but praise for your colleagues in all fields, even for those in combinatorics. All public shows of disunity are ultimately harmful to the well-being of mathematics.

别看不起别人使用的数学(Remember that the grocery bill is a piece of mathematics too)

The grocery bill, a computer program, and class field theory are three instances of mathematics. Your opinion that some instances may be better than others is most effectively verbalized when you are asked to vote on a tenure decision. At other times, a careless statement of relative values is more likely to turn potential friends of mathematics into enemies of our field. Believe me, we are going to need all the friends we can get.

善待擅长教学的老师(Do not look down on good teachers)

Mathematics is the greatest undertaking of mankind. All mathematicians know this. Yet many people do not share this view. Consequently, mathematics is not as self-supporting a profession in our society as the exercise of poetry was in medieval Ireland. Most of our income will have to come from teaching, and the more students we teach, the more of our friends we can appoint to our department. Those few colleagues who are successful at teaching undergraduate courses should earn our thanks as well as our respect. It is counterproductive to turn up our noses at those who bring home the dough.

学会推销自己的数学成果(Write expository papers)

When I was in graduate school, one of my teachers told me, “When you write a research paper, you are afraid that your result might already be known; but when you write an expository paper, you discover that nothing is known.”

It is not enough for you (or anyone) to have a good product to sell; you must package it right and advertise it properly. Otherwise you will go out of business.

When an engineer knocks at your door with a mathematical question, you should not try to get rid of him or her as quickly as possible.

不要把提问者拒之门外(Do not show your questioners to the door)

What the engineer wants is to be treated with respect and consideration, like the human being he is, and most of all to be listened to with rapt attention. If you do this, he will be likely to hit upon a clever new idea as he explains the problem to you, and you will get some of the credit.

Listening to engineers and other scientists is our duty. You may even learn some interesting new mathematics while doing so.

联合阵线(View the mathematical community as a United Front)

Grade school teachers, high school teachers, administrators and lobbyists are as much mathematicians as you or Hilbert. It is not up to us to make invidious distinctions. They contribute to the well-being of mathematics as much as or more than you or other mathematicians. They are right in feeling left out by snobbish research mathematicians who do not know on which side their bread is buttered. It is our best interest, as well as the interest of justice, to treat all who deal with mathematics in whatever way as equals. By being united we will increase the probability of our survival.

把科学从不可靠中拯救出来(Attack Flakiness)

Flakiness is nowadays creeping into the sciences like a virus through a computer, and it may be the present threat to our civilization. Mathematics can save the world from the invasion of the flakes by unmasking them and by contributing some hard thinking. You and I know that mathematics is not and will never be flaky, by definition.

This is the biggest chance we have had in a long while to make a lasting contribution to the well-being of Science. Let us not botch it as we did with the few other chances we have had in the past.

善待所有人(Learn when to withdraw)

Let me confess to you something I have told very few others (after all, this message will not get around much): I have written some of the papers I like the most while hiding in a closet. When the going gets rough, we have recourse to a way of salvation that is not available to ordinary mortals: we have that Mighty Fortress that is our Mathematics. This is what makes us mathematicians into very special people. The danger is envy from the rest of the world.

When you meet someone who does not know how to differentiate and integrate, be kind, gentle, understanding. Remember, there are lots of people like that out there, and if we are not careful, they will do away with us, as has happened many times before in history to other Very Special People.

参考资料:

  1. Rota, Gian-Carlo. “Ten lessons I wish I had been taught.” Indiscrete thoughts. Birkhäuser, Boston, MA, 1997. 195-203.
  2. Rota, Gian-Carlo. “Ten Lessons for the Survival of a Mathematics Department.” Indiscrete Thoughts. Birkhäuser, Boston, MA, 1997. 204-208.