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学习数学的唯一方法是做数学

有谁能告诉别人怎样去做研究,怎样去创造,怎样去发现新东西?几乎肯定这是不可能的。在很长一段时间里,我始终努力学习数学,理解数学,寻求真理,证明一个定理,解决一个问题——现在我要努力说清楚我是怎样去做这些工作的,整个工作过程中重要部分是脑力劳动,那可是难以讲清楚的——但我至少可以试着讲一讲体力劳动的那一部分。

Can anyone tell others how to do research, how to create, and how to discover new things? It is almost certain that this is impossible. For a long time, I have been working hard to learn mathematics, understand mathematics, seek truth, prove a theorem, and solve a problem. Now I have to work hard to make clear how I do this work. The important part of the whole work process is Mental work, that is hard to explain - but I can at least try to talk about the part of manual labor.

数学并非是一门演绎科学——那已是老生常谈了。当你试图去证明一个定理时,你不仅只是罗列假设,然后开始推理,你所要做的工作应是反复试验,不断摸索,猜测。你要想弄清楚事实真相,在这点上你做的就像实验室里的技师,只是在其精确性和信息量上有些区别罢了。如果哲学家有胆量,他们也可能像看技师一样地看我们。

Mathematics is not a deductive science - it is a cliché. When you try to prove a theorem, you not only list the hypotheses, but then start the reasoning. The work you have to do should be trial and error, constant exploration, and guesswork. You have to figure out the truth. At this point, you are doing the same as the technicians in the lab, but there are some differences in their accuracy and amount of information. If philosophers have the courage, they may also look at us like a technician.

我喜欢做研究,我想做研究,我也得做研究,我却不愿坐下来开始做研究——我是能拖则拖迟迟不肯动手。尽管我对工作无限眷恋,我仍是不愿意着手去做它;每做一项工作都像是一场打仗格斗。难道就没有什么事我能(或必须?)先行干好吗?难道我就不能先将铅笔削好吗?事实上我从来不用铅笔,但“削铅笔”已成为一切有助于延迟集中创造精力带来的痛苦的手法的代名词。它的意思可以是在图书馆查阅资料,可以是整理旧笔记,甚至可以视为明天要讲的课作准备,干这些事的理由是:一旦这些事了结了,我就真正能做到一心一意而不受干扰了。

I like to do research, I want to do research, I have to do research, but I don't want to sit down and start doing research - I can delay it and delay it. Even though I am infinitely attached to my work, I am still reluctant to start doing it; every job is like a fight. Is there nothing I can (or must?) do it first? Can't I cut the pencil first? In fact, I never used a pencil, but "sharp pencil" has become synonymous with all the techniques that help to delay the concentration of energy. It can mean reading materials in the library, sorting out old notes, or even preparing for the class to be told tomorrow. The reason for doing these things is: once these things are over, I can really do it. Undisturbed.

当卡米查埃(Carmichael)抱怨说他当研究生院主任每周可用于研究工作的时间不超过20小时的时候,我感到很奇怪,我现在仍觉得很奇怪。在我大出成果的那些年代里,我每周也许平均用20小时作全神贯注的数学思考,但大大超过20小时的情况是极少的。这极少的例外,在我的一生中只有两三次,他们都是在我长长的思想阶梯接近顶点时来到的。尽管我从来未当过研究生院主任,我似乎每天只有干三,四个小时工作的精力,这是真正的“工作”;剩下的时间我用于写作,教书,作评论,与人交换意见,作鉴定,作讲座,干编辑活,旅行。一般地说,我总是想出各种办法来“削铅笔”。每个做研究工作的人都陷入过休闲期。在我的休闲期中,其他的职业活动,低到并包括教教课,成了我生活的一种借口。是的,是的,我也许今天没有证明出任何新定理,但至少我今天将正弦定理解释得十分透彻,我没白吃一天饭。

When Carmichael complained that he was a graduate school director who could spend less than 20 hours a week on research work, I was surprised that I still feel very strange. In those years when I was a big achievement, I might spend an average of 20 hours a week thinking about mathematics, but the situation is much less than 20 hours. These few exceptions, only two or three times in my life, came when my long thought ladder approached the apex. Although I have never been a graduate school director, I seem to have only three or four hours of work per day. This is the real "work"; the rest of the time I use for writing, teaching, commenting, and exchanging opinions. For identification, lectures, editorial work, travel. Generally speaking, I always come up with various ways to "sharp pencils." Everyone who does research work has fallen into a leisure period. During my leisure period, other professional activities, as low as and including teaching lessons, became an excuse for my life. Yes, yes, I may not have proved any new theorem today, but at least I explained the sine theorem very thoroughly today. I didn't eat a day.

数学家们为什么要研究?这问题有好几个回答。我喜爱的回答是:有好奇心——我们需要知道.这几乎等于说“因为我愿意这样做”,我就接受这一回答——那也是一个好回答。然而还有其它的回答,它们要实在些。

Why do mathematicians study? There are several answers to this question. My favorite answer is: Be curious - we need to know. This is almost equal to saying "Because I am willing to do this", I accept this answer - that is also a good answer. However, there are other answers, they are more realistic.

我们给未来的工程师,物理学家,生物学家,心理学家,经济学家,还有数学家教数学。如果我们只教会他们解课本中的习题,那不等他们毕业,他们受到的教育便过时了。即使从粗糙而世俗的工商业观点来看,我们的学生也得准备回答未来的问题,甚至在我们课堂上从未问过的问题。只教他们已为人们所知的一切东西是不够的——他们也必须知道如何去发现尚未被发现的东西。换句话说,他们必须接受独立解题的训练——去做研究工作。一个教师,如果他从不总是在考虑解题——解答他尚不知道答案的题目——从心理上来说,他就是不打算教他的学生们解题的本领。

We teach mathematics to future engineers, physicists, biologists, psychologists, economists, and mathematicians. If we only teach them to solve the exercises in the textbook, then the education they receive will be out of date before they graduate. Even from a crude and secular business perspective, our students are prepared to answer future questions, even questions that have never been asked in our class. It’s not enough to teach them everything they already know – they must also know how to discover things that have not yet been discovered. In other words, they must be trained in independent problem solving – to do research work. A teacher, if he is never always thinking about solving a problem - answering a question he still does not know the answer - psychologically, he is the skill of the students who do not intend to teach him to solve the problem.

做研究工作,有一点我不擅长因而也从不喜欢的是竞争。我不太善于抢在别人前面已获得荣誉。我争当第一的另一办法是离开研究主流方向去独自寻找属于我自己的一潭小而深的洄水。我讨厌为证明一个著名猜想而耗费大量的时间却得不到结果,所以我所干的事无非是分检出被别人漏掉的概念和阐明富有结果的问题。这样的事在你一生当中不可能常做,如果那概念和那些个问题真是“正确”的,它们便会被广泛接受,而你则很有可能在你自己的课题发展中,被更有能力和更有眼光的人们甩在后面。这很公平,我能受得了;这是合理的分工,当然我希望次正规不变子空间定理是我证明的,但至少我在引入概念和指出方法方面做过一点贡献。

Doing research work, one thing that I am not good at and therefore never like is competition. I am not very good at grabbing honors in front of others. Another way I strive to be the first is to leave the mainstream of research and find a small and deep drowning water that belongs to me. I hate spending a lot of time to prove a famous conjecture but not getting results, so what I do is to sort out the concepts that are missed by others and to clarify the results that are fruitful. Such things can't be done often in your life. If the concepts and those questions are really "correct," they will be widely accepted, and you are likely to be more capable in your own development. And more discerning people are behind. This is fair, I can stand it; this is a reasonable division of labor. Of course, I hope that the subnormal invariant subspace theorem is proved by me, but at least I have made some contributions in introducing concepts and pointing out methods.

不介入竞争的另一个方面就是我对强调抢时间争速度不以为然。我问我自己,落后于最近的精美的成果一两年又有什么关系呢?一点关系都没有,我这样对自己说,但即使对我自己来说,这样的回答有时也不管用,对那些心里构成和我相异的人们来说,这样的回答总是错的.当罗蒙诺索夫(Lomonosov)(关于交换紧算子的联立不变子空间)和斯科特.布朗(ScottBrown)的(关于次正规算子)消息传开时,我激动的就像我是第二位算子理论家似的,急切的想迅速的知道详情.然而这种破例的情形是少有的。所以我仍然可以在我一生大部分时间中心安理得地生活于时代之后。

Another aspect of not engaging in competition is that I don't agree with the speed of rushing to compete. I asked myself, what is the relationship between the recent beautiful results and one or two years? There is no relationship at all. I say this to myself, but even for myself, such an answer sometimes doesn't work. For those who have a heart that is different from me, such an answer is always wrong. When Lomonosov (the simultaneous invariant subspace of the exchange compact operator) and Scott Brown's (about the subnormal operator) spread, I was excited like I was The second operator is like a theorist, eager to know the details quickly. However, such an exception is rare. So I can still live in the center of my life most of my life.

还有写作。我在我的书桌前坐下,提起一杆黑色的圆珠笔,开始在一张81/2x11见方的标准用纸上写作.我在右上角上写上个“1”,然后开始:“这些笔记的目的是研究秩为1的摄动在…的格上的影响。”在这一自然段写完后,我在稿纸边上标上个黑体“A”字,然后开始写B段,页数字和段落字构成了参考系统,常常可以一连写上好一百页:87C意味着87页上C段。我将这些页手稿放入三环笔记夹中,在夹脊上贴上标签:逼近论,格,积分算子等等。如果一个研究项目获得成功,这笔记本便成为一篇论文,但不管成功与否,这笔记本是很难扔掉的。我常在我的书桌旁的书架上放上几十本,我仍然希望那些未完成的笔记将继续得到新的补充,希望那些已成为文章发表的笔记以后会被发现隐含着某种被忽视了的新思路的宝贵萌芽,而这种新思路恰恰是为解决某一悬而未决的大问题所需要的。

There is also writing. I sat down at my desk, lifted a black ballpoint pen and started writing on a standard paper of 81/2x11 square. I wrote a "1" in the upper right corner and started: "These notes The purpose is to study the influence of the perturbation of rank 1 on the lattice of .... After writing this natural paragraph, I marked a black body "A" on the edge of the manuscript paper, and then began to write the B paragraph, page number and Paragraph words form the reference system, and can often be written in a hundred pages: 87C means 87 on page 87. I put these manuscripts into a three-ring note and put labels on the ridges: approximation theory, lattice, integral operator, and so on. If a research project is successful, the notebook becomes a paper, but whether it is successful or not, this notebook is difficult to throw away. I often put dozens of books on the bookshelf next to my desk. I still hope that those unfinished notes will continue to receive new supplements. I hope that those notes that have become published will be found to imply some neglect. The precious sprout of the new ideas, and this new idea is precisely what is needed to solve a large and unresolved problem.

我继续尽可能长时间地坐在我的书桌前——这可以理解为,我只要有精力,或者只要有时间,我就这样坐在书桌前,我努力整理笔记到一个弱拍出现为止,如一个引理的确定,或者,在最坏的情况下,一个未经过仔细研究但明显不是没希望解答的问题被提出。那样,我的潜意识可以投入工作了,并且在最好的时候,在我走向办公室时,或者给一个班上课时,甚至在夜间睡眠中,我取得意外的进展。那捉摸不透的问题解答有时让我无法入睡,但我似乎养成了一种愚弄我自己的办法了.在我翻来覆去一会后,时间并不长——通常仅为几分钟——我“解决”了那问题;那问题的证明或反例在闪念中出现了,我心满意足了,翻了个身便睡着了。那闪念几乎总被证明是假的;那证明有个巨大的漏洞,或者那反例根本就不反对任何东西。可不管怎么说,我对那个“解”相信的时间,长的足够使我睡个好觉。奇怪的事情是,在夜间,在床上,在黑暗中,我从未记得我怀疑过那“思路”,我百分之百地相信它可是件大好事,对一些情形它甚至被证明是正确的。

I continue to sit at my desk for as long as possible - this can be understood as, as long as I have energy, or as long as I have time, I sit at my desk like this, I try to organize my notes until a weak beat appears, such as A lemma is determined, or, in the worst case, a problem that has not been carefully studied but is clearly not hopeless. In that way, my subconscious mind can be put into work, and at the best of times, when I go to the office, or when I attend a class, or even sleep at night, I make unexpected progress. The unpredictable questions sometimes made me unable to sleep, but I seemed to have developed a way to fool myself. After I turned over and over for a while, the time was not long—usually only a few minutes—I Solved the problem; the proof or counterexample of the problem appeared in the flash, I am satisfied, turned over and fell asleep. That flash is almost always proved to be false; it proves that there is a huge loophole, or that counterexample does not object at all. But in any case, the time I believe in that "solution" is long enough to make me sleep well. The strange thing is that at night, in bed, in the dark, I never remembered that I suspected the "thinking". I believe 100% that it is a good thing, and in some cases it has even proved to be correct.

我不在乎坐在钟边工作,当因为到了上课的时间或者到了除去吃饭的时间,而我必须停止思考时,我总是高兴地将我的笔记收起来。我也许会在下楼去教室的路上,或者在发动我的汽车,关闭我车库门时仔细思考我的问题;但我并不因为这种打扰而生气(不像我的一些朋友们说的那样,他们讨厌被打断思绪)。这些都是生活的组成部分,一想到几小时后我俩——我的工作和我——又要相聚时,我就感到很舒坦。

I don't care about sitting at the clock. When I have to stop thinking because of the time of class or when I have to stop thinking, I am always happy to take my notes away. I might be thinking about my problems when I go downstairs to the classroom, or when I start my car and close my garage door; but I am not angry because of this interruption (not like some of my friends said, They hate being interrupted.) These are all part of life, and I feel very comfortable when we think about it in a few hours, when I am working together with me.

好的问题,好的研究问题,打哪儿来呢?它们也许来自一个隐蔽的洞穴,同在那个洞穴里,作家发现了他们的小说情节,作曲家则发现了他们的曲调——谁也不知道它在何方,甚至在偶然之中闯进一两次后,也记不清它的位置。有一点是肯定的:好的问题不是来自于做推广的模糊欲念。几乎正相反的说法倒是真的:所有大数学问题的根源都是特例,是具体的例子。在数学中常见到的一个似乎具有很大普遍性的概念实质上与一个小的具体的特例是一样的。通常,正是这个特例首次揭示了普遍性。阐述“在实质上是一样”的一个精确明晰的方法就如同一个定理表述。关于线性泛函的黎兹(Riesz)定理就很典型。固定一个在内积中的向量就定义了一个有界线性泛函;一个有界线性泛函的抽象概念表面上看来具有很大的概括性;事实上,每个抽象概念都是以具体特定的方式产生出来的,那定理也是。

Good questions, good research questions, where do you come from? They may come from a hidden cave, in the same cave, the writer discovers their novel plot, and the composer discovers their tune—no one knows where it is, or even breaks into it once or twice. After that, I can't remember its location. One thing is certain: a good question does not come from the ambiguous desire to do promotion. The opposite is true: the root of all major mathematical problems is a special case, a concrete example. A concept that seems to be very common in mathematics is essentially the same as a small concrete exception. Usually, it is this special case that reveals universality for the first time. An accurate and clear method of expounding "substantially the same" is like a theorem. The Riesz theorem about linear functionals is typical. The vector fixed in an inner product defines a bounded linear functional; the abstract concept of a bounded linear functional appears to be very general on the surface; in fact, each abstract concept is specific The way that is produced, the theorem is also.

这是我和狄多涅(Dieudonne)似乎各执己见的许多论题中的一个。在马里兰,我曾做过一次学术报告,那正好也是狄多涅访问那里的许多次中的一次。那次报告的主题是正逼近,我那次选定的问题是:已知一希尔伯特(Hilbert)空间上的任意算子A,求一个正(非负半定的)算子P极小化||A-P||。我很幸运:结果发现有一个小的具体的特例,它包含了一切概念,一切困难,一切为理解和克服它们所需要的步骤.我使我的报告紧紧围绕那个特例,由矩阵 /0100/定义的C^2上的算子,我当时感到很自豪:我认为我成功地讲清了一个很好的问题及其令人满意的解,却没有因此而陷入与此无关的分析的术语陈式之中去。狄多涅当时表现得礼貌且友好,但事后显然表现出不屑一顾的态度;我记不清他的原话了,但大意上,他祝贺我的滑稽表演,他对我的报告的印象似乎是“娱乐数学”,这在他的词汇中是个讥笑的字眼;他认为我的报告趣味有余,但是做作且轻浮,我认为(现在还继续认为)问题远不只如此。我俩评价的相异是我们观点上的差别造成的。我认为对于狄多涅来说,重要的是那个强大的一般性定理,从这一定理你很容易推出所有你需要的特例来;而对于我来说,最伟大的前进步骤是,很能说明问题的中心例子,从这一例子中我们很容易搞清楚围在该例子周围的所有带普遍性的东西。

This is one of many topics that I and Diudonne seem to have their own opinions. In Maryland, I did an academic report, which happened to be one of the many times that Didione visited there. The subject of that report was approaching. The question I chose was: I knew an arbitrary operator A on the Hilbert space, and found a positive (non-negative and semi-determined) operator P. ||AP||. I am very fortunate: I found out that there is a small specific exception that contains all the concepts, all the difficulties, everything is necessary to understand and overcome the steps they need. I have made my report close to that exception, by matrix 0100/ I was very proud of the operator on C^2 defined: I thought that I successfully clarified a good question and a satisfactory solution, but did not fall into the term of the analysis that was not related to this. Go in. Didogne was polite and friendly at the time, but apparently showed disdain afterwards; I couldn’t remember his original words, but to the extent he congratulated me on his burlesque performance, his impression of my report seemed to It is "entertainment mathematics", which is a ridiculous word in his vocabulary; he thinks my report is more than fun, but it is contrived and frivolous, I think (and continue to think) the problem is far more than that. The difference between our evaluations is due to the difference in our views. I think that for Didogne, what is important is the powerful general theorem. From this theorem, it is easy to introduce all the special cases you need. For me, the greatest advancement step is that it is very illustrative. A central example of the problem, from this example we can easily figure out all the things that are universal around the example.

作为数学家,我最强的能力便是能看到两个事物在什么时候是“相同的“。例如,当我对大卫·伯格(David Berg)定理(正规等于对角加上紧致)苦苦思索时,我注意到它的困境很像那个证明:每个紧统(Compactam)是康托(Cantor)集的一个连续象,从那时起用不着很大的灵感就可使用经典的表述而不用它的证明了,结果是能取得伯格结果的一种意思明白的新方法。这样的例子我还可以举出很多,一些最突出的例子发生在对偶理论中,例如:紧阿贝尔群的研究与傅里叶(Fourier)级数的研究是一样的,正如布尔代数的研究与不连通的紧豪斯道夫(Hausdorff)空间的研究是一样的,其它的例子,不是对偶那一类的有:逐次逼近的经典方法与巴拿赫不动点定理是一样的,概率论与测度论也是一样的。

As a mathematician, my strongest ability is to see when two things are "identical." For example, when I thought about David Berg's theorem (regular equal diagonality plus firmness), I noticed that its dilemma is very similar to that proof: every compact (Compactam) is Kang A continuous image of the Cantor collection, from then on, can be used without a great inspiration to use the classic expression without its proof, and the result is a new way of understanding the results of the Berg. I can cite a lot of such examples. Some of the most prominent examples occur in the duality theory. For example, the study of the tight Abelian group is the same as the Fourier series, just as the study of Boolean algebra The study of the disconnected Hausdorff space is the same. Other examples are not the same ones: the classical method of successive approximation is the same as the Banach fixed point theorem, probability theory and measure The same is true.

这样一联系起来看问题,数学便清楚了;这样看问题去掉了表象,揭示了实质,他推进了数学的发展了吗?难道那些伟大的新思想仅仅是看清了两个东西是一样的而已吗?我常常这样想——但我并不是总有把握的。

When this is linked to see the problem, the mathematics is clear; thus, the problem is removed, the appearance is revealed, and the essence is revealed. Has he advanced the development of mathematics? Are those great new ideas just to see that two things are the same? I often think like this - but I am not always sure.

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