好一陣子沒發文了，因為在當兵，幾乎沒時間發文， 之前在若水堂買了大連理工大學出版的 「 數學家思想文庫 04：數學的統一性」，這本書選編並翻譯了 Michael Atiyah 的文章，通過本書可以了解他的數學和哲學思想。 書中「如何進行研究」這一篇， 特別另人驚豔，文中 Atiyah 談到一些數學研究者可能遇到的問題， 並提供自己的看法，非常值得一讀， 可惜網路上找不到原文， 不過 Google 圖書到是有幾頁可以預覽「 Michael Atiyah: Collected Works 」， 本文中的原文就是從 Google 圖書節錄下來的。 原文是發表在
M. Atiyah, How research is carried out. Bulletin of the Institute of Mathematics and its Applications, 1974, 10, p.232–234.
我的筆記如下： 解決問題與建立理論之間的關係 [top]我提出的第一件事是解問題和建立理論之間的關係。當然，對這兩方面都可以提出些質疑：如果一個理論不能解決問題，其效用何在; 研究無窮多個毫無聯繫的問題，儘管每個也許都很有趣，又有什麼用處。我想，我們也許可以這樣來看問題：你從現存的問題出發，其中許多問題最初都有物理背景; 為了解問題，你必須要有一個聰明的想法以及某種訣竅，當這種訣竅足夠精巧又有足夠多的類型相當的問題，你就可以把訣竅發展成一種技術，若存在大量的這一類型的問題，你就可以發展起一套方法; 最後，假如你涉及的是一個非常廣闊的領域，你就能獲得某種理論。這就是從問題到理論的演化過程。The first thing I want to mention is the alternative between problem solving and theory. Now of course there is something to be said on both sides: what is the good of a theory if it does not solve problems and what is the good of an infinite collection of disjoint problems, however interesting each individual one may be. I think perhaps we can look at it like this: you start off by having problems presented, many of them arriving from a physical context originally, and in order to solve a problem you have got to have a clever idea, some kind of trick. If it is a sufficiently good trick and there are enough problems of a similar type you go on and develop this trick into a technique. If there is a large number of problems of this kind you then have a method and finally if you have a very wide range you have a theory; this is the process of evolution from problem to theory.
直覺式的證明還是嚴格的數學論證? [top]當然，理論之所以成為理論，並不僅僅在於它把你通過解各種問題而掌握的所有東西歸在一起。我們必須在心中牢牢記住，數學是人類的一項活動。解問題或者做數學的目的大概是為了把我們獲得的信息傳遞給後代。我們必須記住人的智力是有限的，肯定不能連續不斷地去領會和消化無窮多的問題並把它們全部記住。理論的真正目的在很大程度上著眼於把過去的經驗加以系統地組織，使得下一代人—我們的學生及學生的學生，等等，能夠盡可能順利地汲取事物的本質內容。只有如此，你才能不斷進行各種科學活動而不會走進死捐同。我們必須設法把我們的經驗濃縮成便於理解的形式，這即是理論之基本所為。也許我可以引用龐加萊在談論這個主題時不得不說的話：「科學由事實建造，正如房屋由石頭建成一樣; 但是事實的收集並非科學，恰如石頭的堆積並非屋宇。」
Now of course the point of theory is not just that of putting together all the facts that you have learnt by solving your problems. We must bear in mind that mathematics is a human activity, that the aim of solving problems or doing mathematics is presumably to pass on information you have gained for posterity. Now we have to remember that the human mind is finite and cannot really digest infinite numbers of problems one after the other and remember them all. The aim of theory really is, to a great extent, that of systematically organising past experience in such a way that the next generation, our students and their students and so on, will be able to absorb the essential aspects in as painless a way as possible, and this is the only way in which you can go on cumulatively building up any kind of scientific activity without eventually coming to a dead end. We have somehow to condense our past experience into an easily understandable form, and that is what theory is basically about. Perhaps here I could quote what Poincare had to say on this topic: "Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house."
深度與廣度 [top]現在你可能會問。什麼是嚴格性？一些人把嚴格定義為 rigor mortis，相信伴隨純粹數學而來的，是對那些知道如何得到正確答案的人的活動的抑制。我想，我們必須再次記住數學是人類的一種活動。我們的目標不僅是要發現些什麼 而且要把信息傳下去。有些人，比如歐拉，他們知道如何寫出一個發散級數又得到正確的答案。他們對該做什麼和不該做什麼必定具有某種非凡的感覺。歐拉從大量的經驗中獲得了某種直覺，而直覺是很難傳達給別人的。下一代人不知道他的結果是怎麼得出來的。嚴格的數學論證的作用正在於使得本來是主觀的、極度依賴個人直覺的事物，變得具有客觀性並能夠加以傳遞。我完全不想拒絕這類直覺帶來的好處，只是強調為了能向其他人傳播，所獲得的發現最終應以如下方式表述：清晰明確，毫不含糊，能被並無開創者那種洞察力的人所理解。此外，只要你在鑽研某個範圍的問題，你的直覺自然也能把你引向正確的答案，儘管你可能尚不能肯定如何去証明它。但是，一旦你進入研究的下一階段，對已得到的結構開始提出更複雜、更精緻的問題時，對最初的基礎性工作的深入理解就會變得越來越重要了。所以，正是你所從事的研究本身，需要嚴格的論證。如果缺乏牢固的基礎，你修建的整座建築將岌岌可危。
Now you may well ask what is the point of rigour? Some of you may define rigour as "rigor mortis" and believe that pure mathematics comes along to stifle the activities of people who really know how to get the answers. Again, I think, we ought to bear in mind that mathematics is a human activity and our aim is not only to discover things but to pass this information on. Now somebody like Euler, who knows how to write down divergent series and get correct answers, must have a good feeling of what ought to be done and what ought not to be done. Euler had an intuition built up out of a great variety of experience, and this kind of intuition is very hard to convey. So the next generation will come along and will not know how it is done, and the point of having a rigorous mathematical statement is so that something which in the first place is subjective and depends very much on personal intuition, becomes objective and capable of transmission. I have no wish at all to deny the advantages of having this kind of intuition, but only to emphasis that in order for this to be conveyed to other people it must eventually be presented in such a way that it is unambiguous and capable of being understood by someone who does not necessarily have the same kind of insight as the originator. Beyond this, of course, as long as you deal with a certain range of problems then your intuition is quite capable of leading to the right answer although you may not be sure how to justify it. But when you go to the next stage of development and start to build a more elaborate problem on the structure you already have, it becomes more and more important that the initial groundwork should be fairly firmly understood. So the necessity for having rigorous arguments is again because you are going to be building, and if you do not build on solid foundations the whole structure will be in danger.
我的下一個論題是有關數學中的深度和廣度之間的曲別的。我的意思是，當你研究一個特殊的領域或問題時，可以搞得 非常精細鑽得越來越深，得到越來越具體的成果; 或者，你可以選擇另一種途徑，分身於數學的眾多領域之中，對相當大的範圍的課題都達到某種程度的理解，然後看看自己可以在哪方面發展作出努力。
My next point is the distinction between depth and breadth in mathematics. By this I mean that you can study a particular area or problem in very great detail delving deeper and deeper into it and getting harder and harder results, or you can choose to spread yourself much more thinly over a wide range of mathematics so you have some kind of understanding of a large area and then try and see what you can develop.
單打獨鬥還是合作研究? [top]一開始就涉足廣闊的前沿領域有個優點，即年輕的學生學習新東西相對容易些。如果開始就在一個合理的範圍內盡你所能，學習較多的前沿課題，你會從中獲得較豐厚的儲備以圖後進。當數學時尚或人們關注的問題發生變化時，你便能隨之而變。反對如上看法的人可能會說，數學最重要的是解問題，追求"廣度"充其量只是一種佯攻，你們應該去搞硬問題。關於廣度的討論涉及數學的本質。在很大程度上，數學是一門將完全不同的和毫無聯繫的事物組織成一個整體的藝術。畢竟，數學在所有科學領域中達到了抽象的頂峰，它應該適用於廣闊的現象領域。也許，我又可以引用龐加萊的一段話，我覺得它跟我的幾個論題有關。他說：「什麼是值得研究的數學事物呢，通過跟其他事物的類比，它們應能引導我們獲得有關的數學定律的知識，正如實驗事實引導我們獲得物理定律的知識一樣。數學中的結論將向我們揭示其他事實間意想不到的親緣關係。」將眾多來自經驗科學或數學本身的不同事物結合在一起，乃是數學的本質特征之一。我們之中必定要有這樣的人，他們努力把數學中的不同部分聯接起來; 也要有另一種人，他們把自己約束在一個領域內，在此方向儘可能獲得更多的成果。
The advantages of starting on a wider front are that as a young student it is relatively easy to learn new things, and if you start off by learning on as wide a front as you reasonably can, you have a larger store from which to draw in the future. As mathematical fashions and problems change you can then attempt to change with them. Against this others will say that the important thing in mathematics is to solve problems and therefore all this "breadth" is a diversion and you should get down to hard work. The argument for breadth is that the essence of mathematics is to a great extent the art of putting together very disparate things. After all mathematics is the ultimate of abstraction in scientific fields and claims to apply to a wide range of phenomena. Again perhaps I can quote a little paragraph from Poincare, which I suppose is relevant to several of these headings of mine. He says, "The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law, just as experimental facts lead us to the knowledge of a physical law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another." Putting together widely differing facts drawn either from the experimental sciences or from within mathematics itself is one of the essential ingredients of mathematics. We must have people who try to connect up different parts of mathematics as well as those who restrict themselves to one area and try to get as far as possible in that direction.
另一對問題跟數學的具體內容關係不大，而涉及數學家的工作方式：是個人奮鬥還是合作研究。處理這類問題顯然隨人而異，差別很大。有些人不喜歡或無法跟別的數學家合作。他們最善於個人思考，自己寫文章，這是他們的工作方式。另一些人喜歡跟同事聯手，許多研究都是合作進行的。我認為有相當多的證據說明，後一種做法有優越性， 將來也還會有更多的證據出現。首先，如果你和其他數學家合作，實際上大大增加了你所擁有的技巧，也開闊了自己對數學的看法，它必然會影響到研究工作。假定數學的多樣性在增加，對任何單個人而言，想要熟悉所有的領域確實太困難了。正如我說過的，很多有趣的問題來自數學不同部分間的相互影響。數學家越來越需要在一起工作，集中他們的智謀，在某個特定的範圍進行攻堅。當然，你不要想把觀點完全不同的數學家拉到一起。你需要基本志趣非常相近的人，他們以多少有點相似的方式思考，有相似的情感，但在創造個別具體事物時有足夠的差異。合作研究另有一個好處。當你直接攻一個數學問題時，常會走進死胡同，你所做的似乎一概不起作用，你會期盼能巧遇什麼轉機，那樣問題也許就容易解決了。可是不會有人來幫忙，因為別人通常也在那兒等待轉機。僅僅一處障礙就會耽擱你多少年，這在數學研究中是屢見不鮮的。可能因為出現簡單的智力方面的阻滯，某個愚蠢的念頭使你看不到下一步該怎麼走，而你的同伴可能容易直指要害之處，這種現象十分普遍。這恰是合作的用武之地。另一方面，合作也有利於聽到批評意見; 我們大家都容易犯錯誤，容易帶著不完全的論證匆匆向前冒進。有個人在你身邊就大有好處，他會以批判的眼光檢查你給出的論證並挑出其中的漏洞。顯然，挑別人的錯比挑自己的容易。
The next sort of dichotomy is concerned not so much with the content of mathematics as with the way in which mathematicians work, namely whether they choose to work on their own or in collaboration with others. Again this differs a great deal from person to person. Some people just do not like or are incapable of collaborating with other mathematicians. They think best by themselves, they write by themselves and that is the way they work. Others prefer to conduct a lot of their work in conjunction with their colleagues and I think there are a lot of arguments I favour of this and there will be many more in the future. In the first place if you collaborate with other mathematicians you greatly increase the actual techniques and range of mathematical points of view that brought to bear on any problem. Given the increasing diversity of mathematics it is very hard for anyone to be at home in all fields and since, as I have said, many of the interesting problems arise from an interaction between different parts of mathematics, it becomes more and more necessary for mathematicians to get together and pool their resources in a common attack on a given area. Of course you do not want to bring together mathematicians with entirely different points of view. You need people who have basically very much in common so that they think in a somewhat similar manner and have similar sympathies, but with enough variations to create something individual. There is another great advantage in collaboration. If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else alongside you, because he can usually peer round the corner. Very often in mathematics one single obstacle can hold you up for years; it may simply be a mental block, some stupid reason why you have not seen the next step and the person alongside you can easily point it out. That is a very common and helpful aspect of collaboration. On the other hand there is also the critical aspect; we are all prone to make mistakes, and to rush ahead with an incomplete argument. It is very helpful to have some one alongside who can critically examine the arguments put forward and pick holes in them. Of course it is much easier to pick holes in other people's arguments than in your own.
有效與優雅 [top]最後，我們不能忽略遭單獨監禁乃是人生最痛苦的經歷。數學研究非常艱辛，我想從人的角度考慮，合作的好處也是值得重視的，它可以使數學思維過程變得更有樂趣。儘管我承認喜歡合作，但在關鍵時刻誰也代替不了你自己的冥思苦索。
Finally, and this ought not to be ignored, the fact is that solitary confinement is a very painful way of spending your life. Mathematical research is really a very arduous activity and I think the advantages of collaboration from a human point of view are considerable. It helps to make the process of mathematical thought far more enjoyable. Having said this in favour of collaboration I admit, when it comes to the crunch, there is no substitute for really hard thinking on your own.
最後，我想對比一下數學論証是否「有效」和「優美」的問題。我們所有的人都不會漠視這方面的問題。一個有效的証明不一定是優美的，它可能是一副強蠻的面孔，完全靠力氣， 靠推土機式的技巧獲得成功。你寫下一頁又一頁的公式向前跋涉，看起來很不舒服，事實也確是如此！不過最終還是達到了目的。 至於優美的工作，你似乎並不費力，只要寫上幾頁紙，嗨，你瞧！耀眼的結果出現了，令四座皆驚。
Finally, I would like to contrast "power" and "elegance" as applied to mathematical arguments. All of us have some idea of what the distinction is. A powerful argument need not be elegant, it can be a brute force fact which succeeds by sheer strength, a bulldozer technique, you just plough ahead with pages full of formulae, it looks ugly, it is ugly, but it gets there. With the elegant approach you seem to be doing no work, you write along for a few pages and, lo and behold, a brilliant result appears at the end much to everybody's surprise.
這回，我們還是需要兩類數學家，這是毫無疑問的。許多結果首先是完全靠蠻力証明的。一些人堅韌不拔地一直往下算，不在乎它是否優美，最後得到答案。接下去，對此結論感興趣的人會繼續考慮，試圖理解它，最後把它打扮得很漂亮，富於感染力。當然，這並非簡單的粉飾門面。因為優美是一種評價標準，若想讓數學繼續保持旺盛的活力，堅持這一標準是非常重要的。如果你想讓其他人理解某個論証的實質，原則上它必須是簡單和優美的，這顯示了質量：表達最明朗，最容易被人類的心智在數學框架內所理解。事實上，龐加萊將簡明性視為數學理論的定向力，使我們選擇某個方向而不是另一個方向前進。所以，優美與否是非常重要的，不僅對基本結構如此，而且對次一層的結構亦然。
Again, we need both kind of mathematician. There is no doubt about it, many results are, in the first place, proved by sheer brute force. Somebody with a great deal of perseverance computes away, not worrying too much about how elegant it looks, and comes up with the answer. Subsequently other people, impressed with the result, look at it, try to understand it and finally dress it up in a manner which makes it look appealing, makes it look elegant. Now that is not simply window-dressing because elegance is the kind of criterion which is very important if you want mathematics to continue as a living activity. If you want other people to understand the essential ingredient of an argument, it ought in principle to be simple and elegant. These are the qualities which appeal most easily and are best understood by the human mind in the mathematical framework. In fact Poincare regarded simplicity as the guiding force in mathematical theory, which makes us choose one direction rather than another. So elegance is very important often coming not in the primary form but in the secondary form.
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