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1、1,數(shù)學(xué)與科技 丘成桐教授 北京二零零四年三月二十四日  由于促進(jìn)中國與其它國家的科技交流而得到中國政府的表揚，本人深感榮幸。 我生于汕頭,長于香港，接受的是英國式的殖民地教育?？尚蚁雀钢匾曋袊幕盐宜偷街形闹袑W(xué)就讀。,Mathematics and its role in Science and TechnologyProfessor Shing T

2、ung YauBeijingMarch 24, 2004  It is a great honor for me to be recognized by the Chinese government on my efforts to promote scientific interchange between China and other counties.  I was bo

3、rn in south China and grew up in Hong Kong. I was educated under the colonial system set up by the British. Fortunately, my father was proud of Chinese culture and sent me to a Chinese middle school.,2,I was also educa

4、ted in The Chinese University of Hong Kong. While my training on mathematics and science in Hong Kong could not be said to be great, I was fortunate to learn Chinese culture in a reasonable thorough manner. 

5、The elegance of Chinese literature had deep influence on my thinking. I am proud of the deep and continuous developing civilization of my mother country. While I devoted my whole life to study fundamental science, I al

6、so consider promotion of science, especially for my mother country, to be a responsibility of my life.,其后我肄業(yè)于香港中文大學(xué)。相對而言，在那里學(xué)到的數(shù)學(xué)和科學(xué)知識并不算很多，但卻對中國文化有了一個比較全面的了解。  中國文化博大精深，對我有很大的影響。我引以自傲的是，祖國源遠(yuǎn)流長，迄今猶自欣欣向榮的文明。我雖

7、然畢生研究基礎(chǔ)科學(xué)，但亦以推廣、普及科學(xué)為己任，對與祖國有關(guān)的工作，尤其珍惜。,3,While I left Hong Kong in 1969, I did not have passport of any country. At that time, it was not clear at all that the Chinese government would build a good relation with the Un

8、ited States. I did watch with great joy when Nixon visited China. In 1979, I was invited by Professor Hua to visit the Academic Sinica. I learnt a lot from Professor Hua’s book when I was in high school. I admired hi

9、m and was flattered to be invited.,1969年離開香港時，我并沒有拿英國護(hù)照。當(dāng)時中國政府是否會和美國修好，還是一個謎。尼克松訪華，我在電視上看到了，感到十分高興。1979年華羅庚教授邀請我訪問中國科學(xué)院。早在中學(xué)時，我已經(jīng)讀過不少華先生的著作，獲益良多，他是我敬佩的人物。他的來函，令我有受寵若驚之感。,4,When I stepped out from the airplane in the Beij

10、ing airport, I touched the soil of Beijing and felt great joy to return to my mother country. I am proud to say that when I was awarded the Field’s Medal in mathematics, I held no passport of any country and should cert

11、ainly be considered as Chinese. I did put in a lot of efforts to help Chinese mathematics. On the other hand, I regretted very much that I still could not settle down in China. Whatever I have done for Chinese

12、 mathematics can in no way be compared with those Chinese who have grown up and stayed here or those who returned to China on a permanent basis.,甫出機場，接觸到首都的泥土，回到了祖國母親的懷抱，我心潮澎湃，激動萬分?；叵氆@頒數(shù)學(xué)上的菲爾茲獎時，我并無持有任何國家的護(hù)照，因此我是以堂堂正正中國人

13、的身份去領(lǐng)獎的。  我為中國數(shù)學(xué)的發(fā)展出了不少力。不無遺憾的是，至今我尚未能回國定居。當(dāng)然，我對中國數(shù)學(xué)的貢獻(xiàn)，與在此間土生土長，或自海外歸來，長期工作的同行相比，是微不足道的。,5,It will be an exaggeration to attach too much importance for those who hold a permanent position in foreign soil t

14、o talk about development of this great country. I will therefore only make some humble comments based on my observations. As we know, China has developed education system since Confucius time. Knowl

15、edge is accessible even to non-nobleman. It was a break through.,海外學(xué)者對國家發(fā)展種種意見，雖然每有精警之言，但也不必奉之為金科玉律，全盤接受。 故此，本人謹(jǐn)就親眼所見，親耳所聞，略抒管窺之見，如有一二中的，則于愿足矣。 我國自從孔子開始，便建立了完整的教育體系，這是大家都知道的。從此教育不再是貴族的專利，這可是件石破天驚的大事

16、。,6,By Han dynasty, the government started to build a system to select educated man and man with virtue to serve the country. Even kids in villages have potential to be in the central government. The system was reasona

17、bly fair and China is united partially based on such outreached system. It may be interesting to note that even foreigners were able to hold high position in the court at that time. The system was polished and

18、became system of examinations. The scope of examinations was rather large during early dynasties, like Tang dynasty. Even mathematics was included in the test. However, the examination has been much more focused in the p

19、ast four hundred years. It was thought that a thorough study of Confucius thinking is enough to run the government. Hence the test has been focused on a very narrow band of knowledge. The damage to the creative thinking

20、in China is almost fatal.,及至漢代，地方舉薦賢良文章之士于廟堂，于是鄉(xiāng)黨小子，亦有望大用于朝廷。這種頗為公允的做法，無遠(yuǎn)弗屆，整個國家大一統(tǒng)的局面，或多或少亦由此而維系。值得一提的是，甚至外國人也曾在朝廷供任高職。  這種制度逐漸演變,最后便形成考試制度了。在這種制度的早期──例如唐代──考試的范圍還是頗為廣泛的,數(shù)學(xué)也包括在內(nèi)。但在過去四百年間,考試的范圍便大大地縮窄了。大家以為熟讀四書

21、五經(jīng),便足以治國平天下。因此，考試的知識面變得異常狹窄，國人思想上的原創(chuàng)力在這種鉗制下，遂變得奄奄一息了。,7,It is important to realize that Confucius considered knowledge to be virtue:大學(xué)之道，在明明德，在新民，在止於至善。 But Confucius did train students on practical matter.

22、Some of his students became diplomats. Some of them became businessman. Some of them became general. The Greek philosophers Socrates also consider knowledge to be virtue. But pursuing of truth and beauty is the essen

23、ce of Greek education. In this daunting new world, knowledge is the key to human well-being. Any great country will have to substain long term investment in learning, without which social advancement is an empty dream.

24、 Knowledge has to be based on the following: 1. Virtue, Ethics 2. Knowledge of humanity 3. Basic science 4. Applied Science,重要的是，孔子以為知識是一種美德︰ 大學(xué)之道，在明

25、明德， 在新民，在止于至善。  雖然如此，孔子也傳授實際的學(xué)問。他的門人當(dāng)中，有的當(dāng)上外交官，有的作生意，有的做了將軍。希臘哲人蘇格拉底也以知識為善。追求真善美乃是希臘教育的宗旨。在無畏的新時代 (daunting new world) 里，知識乃是人類通往幸福的鑰匙。任何大國都必須長期投資于教育，不這樣做，社會的進(jìn)步只能是空談。知識必須建基于 1. 道德倫理 2. 人文知識

26、 3. 基礎(chǔ)科學(xué) 4. 應(yīng)用科學(xué),8,Since the defeat of Opium war in the early nineteenth century, China realized its own weakness in modern technology and tried several times to modernize. The major development was on bui

27、lding ships, railways, mining and building weapons. After two centuries of trial and error, we finally see a true opportunity to develop China into a great country. While economic growth is unpreceding strong in modern

28、Chinese history, we should remember that the true engine of modernization is based on acquisition and application of knowledge. However, for a developing country, knowledge is usually associated to applied scie

29、nce only. One tends to overemphasize short-term payoffs and ignores long-term strength. One should note that Fundamental science is the base of all modern technology. China’s modernization should realize the importance

30、 of basic science.,自十九世紀(jì)中鴉片戰(zhàn)爭失敗后，中國便深深感受到技術(shù)落后的弱點，嘗試改革、現(xiàn)代化也不只一次了。當(dāng)時主要的做法是造船、筑鐵路、開礦、生產(chǎn)武器等等。經(jīng)過了差不多兩個世紀(jì)的努力與失誤，到了今日，我們終于看到了中華民族復(fù)興的契機。當(dāng)前我國經(jīng)濟迅速發(fā)展，是近代史上空前的。但是，我們必須牢牢記住，汲取知識和應(yīng)用知識，才是現(xiàn)代化的真正動力。  然而,發(fā)展中國家往往以為知識只指應(yīng)用科學(xué)而言。人們

31、追求立竿見影的效果,忽視長期的利益。我們必須認(rèn)識到,只有基礎(chǔ)科學(xué),才是現(xiàn)代科技之母。中國的現(xiàn)代化,必須要意識到基礎(chǔ)科學(xué)的重要性。,9,Michael Atiyah who is president of Royal Society of England, told me the following: For a country like China which has aspirations to become a w

32、orld economic power, it cannot be lack of ambition. Certainly the Japanese who started off with the plan of just copying the west soon changed to invest in fundamental research.  Also the US is the most advanced

33、economically and is supporting fundamental research in a big way. I would think that China world be receptive to the challenge of competing in all respects with Japan and the USA.,阿提亞教授 (Michael Atiyah) 擔(dān)任英國皇家學(xué)會會長時，曾對我說

34、了這番話:  中國既望躋身經(jīng)濟大國之列，就必須雄心萬丈，志不在小。日本維新之初，一意仿效西洋，但旋即改變方向，致力發(fā)展基礎(chǔ)研究。  美國雖是當(dāng)今經(jīng)濟最強體，但它依然大力注資于科研。我想中國要與日本,美國分庭抗禮,就必須在各方面與它們并駕齊驅(qū)。,10,There are several important branches of sciences and technology th

35、at will play fundamental role in this century: – Information technology – Life science – Energy science – Material science – Environment science

36、 – Economics and finance – Social science  While these branches do interact with each other, they depend heavily on the development of fundamental science who provide the principle: why and ho

37、w things actually work. Interdependence of scientific disciplines recurs frequently throughout history. A successful merge of ideas of two apparently independent fields always brings unexpected glory to both subjects.,

38、在這個世紀(jì)，有幾門科技會發(fā)揮根本的作用，它們包括︰ – 信息技術(shù) – 生命科學(xué) – 能源科學(xué) – 材料科學(xué) – 環(huán)境科學(xué) – 經(jīng)濟與金融 – 社會科學(xué) 這幾門學(xué)科互相滲透，它們同樣依賴于基礎(chǔ)科學(xué)的發(fā)展，因為后者指出了事物的根本原理。回顧歷史，科

39、技領(lǐng)域互相依賴，屢見不鮮。兩門看似無關(guān)的領(lǐng)域，其中的概念一旦能成功地融合，肯定對大家都大有好處。,11,In the nineteenth century, we saw the great success of unification of electricity and magnetism. In the twentieth century, we saw the great success of application of q

40、uantum mechanics to chemistry. We saw application of mathematics and physics to the building of modern computer which in turn become the fundamental tool of all science and technology. Nowadays, we are witnessing the

41、glory of application of physical science to life science. Each unification started from basic science. By the time when it is successful, great breakthrough in technology was rewarded. It brought great economi

42、c growth for those countries who initiated and enhanced such development.,在十九世紀(jì)，人們看到了電學(xué)與磁學(xué)的結(jié)合；在二十世紀(jì)，人們看到了量子力學(xué)在化學(xué)上的應(yīng)用，同時也看到了數(shù)學(xué)和物理如何應(yīng)用于現(xiàn)代計算機，使之成為所有科學(xué)技術(shù)中不可或缺的工具。當(dāng)前人們正在見證物理科學(xué)應(yīng)用于生命科學(xué)。凡此種種，都是人類文明的偉大成就。 學(xué)科之間的融合，始于其基礎(chǔ)部分。當(dāng)

43、融合完成之時，往往導(dǎo)致技術(shù)上的突破。對于帶動或支持這些發(fā)展的國家，其在經(jīng)濟上的利益，是不可低估的。,12,In the past two centuries, we saw the accumulation in wealth of European countries based on scientific and technological advances. The first and second world wars bro

44、ught a large group of international scientists and engineers to the United States. We saw the American dominance of the world. The unparalleled prosperity of United States is driven by technology growing out of the inv

45、estment in basic scientific research over the past fifty years. Large portions of the patent rights held by American companies and universities came from the strength of basic science.,在過去的兩個世紀(jì),歐洲各國因科學(xué)及技術(shù)而累積了大量的財富。二次世界大

46、戰(zhàn)導(dǎo)致大量科學(xué)家及工程人員移民美國。當(dāng)今之世,美國的影響力可說是無遠(yuǎn)弗屆。它空前的繁榮，實歸功于技術(shù)工藝的進(jìn)步，而后者多少源自其在基礎(chǔ)科學(xué)的投資。美國公司和院校所擁有的大量專利權(quán)，都拜基礎(chǔ)科學(xué)研究之賜。,13,The strength as a world power relies on the ability to educate the population in basic science and to recruit the b

47、est mend in the world to work for them. In the case of United States, a lot of foreigner talents may not even speak English fluently. I believe China should recruit many non-Chinese talents even if they may not underst

48、and about China at the beginning. The reason is that science, especially pure science, has no border. We can be benefited much more by foreigners who are not biased. In the twenty first century, mathematics wi

49、ll be the most basic subject to be studied. Mathematics gives the underpinning structure for all science. It is not only the language for all science. It has a life by itself.,一個國家的國力是否強盛，表現(xiàn)于其國民的科學(xué)知識水平，以及其吸引外來精英的能力。就以

50、美國為例，很多在美國工作的海外人材，連英語都說不好。我認(rèn)為中國應(yīng)吸引非華裔人材來華工作，不管他們是否認(rèn)識中國，畢竟科學(xué)是沒有疆界的。只有不分中外，兼收并蓄，我們才能取得成功。 在二十一世紀(jì)，數(shù)學(xué)會成為最基本的學(xué)科。數(shù)學(xué)會成為所有科學(xué)的框架，它不但是科學(xué)的語言，還有其本身的價值。,14,I. Mathematics as a basic language: Geometry is the language

51、 of spacetime. Calculus is the language of astronomy. Operator theory is the language of quantum mechanics. Fourier analysis is the language for wave motion. All these are subjects that are considered exceedingly bea

52、utiful by mathematicians. They were developed in its own right. The common feature of these subjects that were understood by mathematicians. They then give rise to the most fundamental reason why we have interdiscipla

53、ry interactions between seemingly unrelated subjects.,一.數(shù)學(xué)是基本語言 時空的語言是幾何，天文學(xué)的語言是微積分，量子力學(xué)要透過算子理論來描述，而波動理論則靠傅立葉分析來說明。數(shù)學(xué)家研究這些科目，最先都由于其本身之美所感召，但最后卻發(fā)現(xiàn)這些科目背后，竟有些共通的特性。這個事實說明了看起來并不相關(guān)的科目，它們之間有甚多交纏互倚的地方。,15,Let us look at

54、common language. While language may be a symbol, it does convey our emotion and our feeling of the subject that we describe. Chinese poems are not the same as Western poems because we do emphasize the use of each Chine

55、se character which carry different meanings in general. Even among Chinese poems, the number of characters that are allowed in composing the poem do change the feeling that we want to convey. Ancient Poems were more fl

56、exible. Han poems used five characters per sentence. Tang poems use seven characters. Sung developed Qi which had different number of characters for each sentences. Each style of poems reflects the feeling of schlors

57、 of these dynasties.,我們先看看通用的語言。語言是一種符號，用以傳情達(dá)意。中國詩與西洋詩不同之處，在于前者著重每個單字的用法，因為每個單字都具有不同的意義。然而，就算在中國詩內(nèi)，字體的多寡也左右了要表達(dá)的感情。古詩較隨意，漢詩以五言為主，唐代則重七言，到了宋代，流行的便是長短句─詞了。不同的體裁，微妙地反映了不同朝代文人的感受。,16,Hence mathematics research does change th

58、e scientific development that we are working with. For example, a deep understanding of Fourier analysis or wavelet can give much different understanding of wave motion or technique of imaging. Conversely, the practica

59、l world did influence the development of mathematics. The beauty of wave motions and their spectrum was the most basic driving force to develop the subject. Their important application to modern technology and theoreti

60、cal science cannot be exaggerated.,因之，數(shù)學(xué)的研究改變了科學(xué)發(fā)展的航道。舉例而言，對付立葉分析的理解越深入，我們就更能理解波的運動及圖像的技巧。反之，現(xiàn)實世界也左右了數(shù)學(xué)的發(fā)展。波運動及其譜所顯示的美，乃是這些科目發(fā)展的原動力。這些學(xué)科對現(xiàn)代技術(shù)及理論科學(xué)的影響極其深遠(yuǎn)。,17,It is difficult to imagine that Newton can develop classic

61、al mechanics without the great language of calculus whose development can be traced back to Archimedes. There is no doubt that Farady understood electricity and magnetism. But the final completion of the theory

62、 was due to the Maxwell equations. Its implications for understanding light, radio wave and modern science is tremendous. II. Mathematics as a science of order Besides being a language, and a subj

63、ect of great beauty on its own right, mathematics is a science of order. Let us read a citation of a Harvard professor, Andrew Gleason:,沒有微積分這種起源于亞基米德的偉大語言，很難想象牛頓能發(fā)展古典力學(xué)。 毫無疑問，法拉第精通電學(xué)和磁學(xué)。但電磁學(xué)的完整理論要歸功于麥克斯維方程。電磁學(xué)對光、無線

64、電波和現(xiàn)代科學(xué)的研究是極為重要的。 二.數(shù)學(xué)是秩序的科學(xué) 除了作為一種語言，以及一門純美的學(xué)科外，數(shù)學(xué)還是秩序的科學(xué) (a science of order)。我們引一段美國數(shù)學(xué)學(xué)會前會長、哈佛教授格臣 (Andrew Gleason) 的說話:,18,“Mathematics is the science of order: its object is to find, describe and und

65、erstand the order that underlies apparently complex situations. The principal tools of mathematics are concepts which enable us to describe this order. Precisely because mathematicians have been searching for centuries f

66、or the most efficient concepts for describing obscure instances of order. Their tools are applicable to the outside world; for the real world is the very epitome of a complex situation in which there is a great deal of o

67、rder.” Therefore it is not surprising to see that mathematics has strong applications in economics where several Nobel Laurent’s were awarded based on their works related to mathematics. We expect its applicati

68、on to Social science and history through Data mining and statistic.,數(shù)學(xué)乃是秩序的科學(xué)，它的目的是發(fā)現(xiàn)、刻劃、了解外觀復(fù)雜情況的秩序。數(shù)學(xué)中的概念，恰好能夠描述這些秩序。數(shù)學(xué)家花了幾百年來尋找最有效地描述這些秩序的精微曲折處。這種工具可用于外在世界，畢竟現(xiàn)實世界是種種復(fù)雜情況的縮影，其中包含大量的秩序。  由是觀之，數(shù)學(xué)能大用于經(jīng)濟學(xué)，是毫不奇怪的

69、。好幾個諾貝爾經(jīng)濟學(xué)得獎?wù)撸涔ぷ鹘耘c數(shù)學(xué)有關(guān)。,19,III. Mathematics as a tool A lot of important mathematics were developed with sole motivation to study problems in engineer. For example, N. Wiener and his students pioneer the stud

70、y of information science. However, the theory of stochastic differential equations, the theory of Wiener measure, and the theory of entropy have gone far beyond the original purpose of their study. The theory of Bucy-K