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1、1,数学与科技 丘成桐教授 北京 二零零四年三月二十四日 由于促进中国与其它国 家的科技交流而得到中国政 府的表扬，本人深感荣幸。 我生于汕头,长于香港， 接受的是英国式的殖民地教 育。可幸先父重视中国文化， 把我送到中文中学就读。,Mathematics and its role in Science and Technology Professor Shing Tung Yau Beijing March 24, 2004 It is a great honor for me to be recognized by the Chinese government on my efforts
2、to promote scientific interchange between China and other counties. I was born 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 educated in
3、 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. The elegance of Chinese literature had deep influence on my thinking. I am proud of the deep and co
4、ntinuous developing civilization of my mother country. While I devoted my whole life to study fundamental science, I also consider promotion of science, especially for my mother country, to be a responsibility of my life.,其后我肄业于香港中文大学。相对而言，在那里学到的数学和科学知识并不算很多，但却对中国文化有了一个比较全面的了解。 中国文化博大精深，对我有很大的影响。我引以
5、自傲的是，祖国源远流长，迄今犹自欣欣向荣的文明。我虽然毕生研究基础科学，但亦以推广、普及科学为己任，对与祖国有关的工作，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 United States. I did watch with great joy when Nixon visi
6、ted China. In 1979, I was invited by Professor Hua to visit the Academic Sinica. I learnt a lot from Professor Huas book when I was in high school. I admired him and was flattered to be invited.,1969年离开香港时，我并没有拿英国护照。当时中国政府是否会和美国修好，还是一个谜。尼克松访华，我在电视上看到了，感到十分高兴。1979年华罗庚教授邀请我访问中国科学院。早在中学时，我已经读过不少华先生的著作，
7、获益良多，他是我敬佩的人物。他的来函，令我有受宠若惊之感。,4,When I stepped out from the airplane in the Beijing 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 Fields Medal in mathematics, I held no passport of any country and should certai
8、nly 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 mathematics can in no way be compared with those Chinese who have grown up and stayed here o
9、r those who returned to China on a permanent basis.,甫出机场，接触到首都的泥土，回到了祖国母亲的怀抱，我心潮澎湃，激动万分。回想获颁数学上的菲尔兹奖时，我并无持有任何国家的护照，因此我是以堂堂正正中国人的身份去领奖的。 我为中国数学的发展出了不少力。不无遗憾的是，至今我尚未能回国定居。当然，我对中国数学的贡献，与在此间土生土长，或自海外归来，长期工作的同行相比，是微不足道的。,5,It will be an exaggeration to attach too much importance for those who hold a perm
10、anent position in foreign soil to 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. Knowledge is accessible even to non-nobleman. It was a break through.,海外学者对
11、国家发展种种意见，虽然每有精警之言，但也不必奉之为金科玉律，全盘接受。 故此，本人谨就亲眼所见，亲耳所闻，略抒管窥之见，如有一二中的，则于愿足矣。 我国自从孔子开始，便建立了完整的教育体系，这是大家都知道的。从此教育不再是贵族的专利，这可是件石破天惊的大事。,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
12、the central government. The system was reasonably 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 became system of examinations. The scope of
13、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 past four hundred years. It was thought that a thorough study of Confucius thinking is enough to run the government. Hence th
14、e test has been focused on a very narrow band of knowledge. The damage to the creative thinking in China is almost fatal.,及至汉代，地方举荐贤良文章之士于庙堂，于是乡党小子，亦有望大用于朝廷。这种颇为公允的做法，无远弗届，整个国家大一统的局面，或多或少亦由此而维系。值得一提的是，甚至外国人也曾在朝廷供任高职。 这种制度逐渐演变,最后便形成考试制度了。在这种制度的早期例如唐代考试的范围还是颇为广泛的,数学也包括在内。但在过去四百年间,考试的范围便大大地缩窄了。大家以为熟读四书
15、五经,便足以治国平天下。因此，考试的知识面变得异常狭窄，国人思想上的原创力在这种钳制下，遂变得奄奄一息了。,7,It is important to realize that Confucius considered knowledge to be virtue: 大學之道，在明明德， 在新民，在止於至善。 But Confucius did train students on practical matter. Some of his students became diplomats. Some of them became businessman. Some of them became
16、 general. The Greek philosophers Socrates also consider knowledge to be virtue. But pursuing of truth and beauty is the essence 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
17、which social advancement is an empty dream. 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，孔子以为知识是一种美德 大学之道，在明明德， 在新民，在止于至善。 虽然如此，孔子也传授实际的学问。他的门人当中，有的当上外交官，有的作生意，有的做了将军。希腊哲人苏格拉底也以知识为善。追求真善美乃是希腊教育的宗旨。在无畏的新时代 (daunting ne
18、w world) 里，知识乃是人类通往幸福的钥匙。任何大国都必须长期投资于教育，不这样做，社会的进步只能是空谈。知识必须建基于 1. 道德伦理 2. 人文知识 3. 基础科学 4. 应用科学,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 building ships, rai
19、lways, 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 Chinese history, we should remember that the true engine of modernization is based on acquisition
20、 and application of knowledge. However, for a developing country, knowledge is usually associated to applied science 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. Chinas modernization
21、 should realize the importance of basic science.,自十九世纪中鸦片战争失败后，中国便深深感受到技术落后的弱点，尝试改革、现代化也不只一次了。当时主要的做法是造船、筑铁路、开矿、生产武器等等。经过了差不多两个世纪的努力与失误，到了今日，我们终于看到了中华民族复兴的契机。当前我国经济迅速发展，是近代史上空前的。但是，我们必须牢牢记住，汲取知识和应用知识，才是现代化的真正动力。 然而,发展中国家往往以为知识只指应用科学而言。人们追求立竿见影的效果,忽视长期的利益。我们必须认识到,只有基础科学,才是现代科技之母。中国的现代化,必须要意识到基础科学的重要性
22、。,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 world 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
23、 in fundamental research. Also the US is the most advanced 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.,阿提亚教授 (Michael Atiyah) 担任英国皇家学会会长时，曾对我说了这番话: 中国既望跻身经济大国之列，就必须
24、雄心万丈，志不在小。日本维新之初，一意仿效西洋，但旋即改变方向，致力发展基础研究。 美国虽是当今经济最强体，但它依然大力注资于科研。我想中国要与日本,美国分庭抗礼,就必须在各方面与它们并驾齐驱。,10,There are several important branches of sciences and technology that will play fundamental role in this century: Information technology Life science Energy science Material science Environment scienc
25、e 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 how things actually work. Interdependence of scientific disciplines recurs frequently throughout history. A succes
26、sful merge of ideas of two apparently independent fields always brings unexpected glory to both subjects.,在这个世纪，有几门科技会发挥根本的作用，它们包括 信息技术 生命科学 能源科学 材料科学 环境科学 经济与金融 社会科学 这几门学科互相渗透，它们同样依赖于基础科学的发展，因为后者指出了事物的根本原理。回顾历史，科技领域互相依赖，屡见不鲜。两门看似无关的领域，其中的概念一旦能成功地融合，肯定对大家都大有好处。,11,In the nineteenth century, we saw t
27、he great success of unification of electricity and magnetism. In the twentieth century, we saw the great success of application of quantum 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
28、 and technology. Nowadays, we are witnessing the 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 economic growth for those countries who initiated
29、 and enhanced such development.,在十九世纪，人们看到了电学与磁学的结合；在二十世纪，人们看到了量子力学在化学上的应用，同时也看到了数学和物理如何应用于现代计算机，使之成为所有科学技术中不可或缺的工具。当前人们正在见证物理科学应用于生命科学。凡此种种，都是人类文明的伟大成就。 学科之间的融合，始于其基础部分。当融合完成之时，往往导致技术上的突破。对于带动或支持这些发展的国家，其在经济上的利益，是不可低估的。,12,In the past two centuries, we saw the accumulation in wealth of European cou
30、ntries based on scientific and technological advances. The first and second world wars brought 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
31、 the investment 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.,在过去的两个世纪,欧洲各国因科学及技术而累积了大量的财富。二次世界大战导致大量科学家及工程人员移民美国。当今之世,美国的影响力可说是无远弗届。它空前的繁荣，实归功于技术工艺的进步，而后者多少源自其在基础科学的投资。
32、美国公司和院校所拥有的大量专利权，都拜基础科学研究之赐。,13,The strength as a world power relies on the ability to educate the population in basic science and to recruit the best 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 sho
33、uld recruit many non-Chinese talents even if they may not understand 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 will be the most basic subje
34、ct 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.,一个国家的国力是否强盛，表现于其国民的科学知识水平，以及其吸引外来精英的能力。就以美国为例，很多在美国工作的海外人材，连英语都说不好。我认为中国应吸引非华裔人材来华工作，不管他们是否认识中国，毕竟科学是没有疆界的。只有不分中外，兼收并蓄，我们才能取得成功。 在二十一世纪，数学会成为最基本的学科。数学
35、会成为所有科学的框架，它不但是科学的语言，还有其本身的价值。,14,I. Mathematics as a basic language: Geometry is the language 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 excee
36、dingly beautiful 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 interdisciplary interactions between seemingly unrelated subjects.,一.数学是基本语言 时空的语言是几何，
37、天文学的语言是微积分，量子力学要透过算子理论来描述，而波动理论则靠傅立叶分析来说明。数学家研究这些科目，最先都由于其本身之美所感召，但最后却发现这些科目背后，竟有些共通的特性。这个事实说明了看起来并不相关的科目，它们之间有甚多交缠互倚的地方。,15,Let us look at 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 W
38、estern poems because we do emphasize the use of each Chinese 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 flexible. Han poems used f
39、ive 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 of these dynasties.,我们先看看通用的语言。语言是一种符号，用以传情达意。中国诗与西洋诗不同之处，在于前者着重每个单字的用法，因为每个单字都具有不同的意义。然而，就算在中国诗内，字体的多寡也左右了
40、要表达的感情。古诗较随意，汉诗以五言为主，唐代则重七言，到了宋代，流行的便是长短句词了。不同的体裁，微妙地反映了不同朝代文人的感受。,16,Hence mathematics research does change the 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 imag
41、ing. Conversely, the practical 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 theoretical science cannot be exaggerated.,因之，数学的研究改变了科学发展的航道
42、。举例而言，对付立叶分析的理解越深入，我们就更能理解波的运动及图像的技巧。反之，现实世界也左右了数学的发展。波运动及其谱所显示的美，乃是这些科目发展的原动力。这些学科对现代技术及理论科学的影响极其深远。,17,It is difficult to imagine that Newton can develop classical mechanics without the great language of calculus whose development can be traced back to Archimedes. There is no doubt that Farady und
43、erstood electricity and magnetism. But the final completion of the theory 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 subject of great beauty on its own
44、right, mathematics is a science of order. Let us read a citation of a Harvard professor, Andrew Gleason:,没有微积分这种起源于亚基米德的伟大语言，很难想象牛顿能发展古典力学。 毫无疑问，法拉第精通电学和磁学。但电磁学的完整理论要归功于麦克斯维方程。电磁学对光、无线电波和现代科学的研究是极为重要的。 二.数学是秩序的科学 除了作为一种语言，以及一门纯美的学科外，数学还是秩序的科学 (a science of order)。我们引一段美国数学学会前会长、哈佛教授格臣 (Andrew Gleaso
45、n) 的说话:,18,“Mathematics is the science of order: its object is to find, describe and understand 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 cen
46、turies for 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 order.” Therefore it is not surprising to see that mathematics has strong
47、 applications in economics where several Nobel Laurents were awarded based on their works related to mathematics. We expect its application to Social science and history through Data mining and statistic.,数学乃是秩序的科学，它的目的是发现、刻划、了解外观复杂情况的秩序。数学中的概念，恰好能够描述这些秩序。数学家花了几百年来寻找最有效地描述这些秩序的精微曲折处。这种工具可用于外在世界，毕竟现实
48、世界是种种复杂情况的缩影，其中包含大量的秩序。 由是观之，数学能大用于经济学，是毫不奇怪的。好几个诺贝尔经济学得奖者，其工作皆与数学有关。,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 study of information science. However, the theory
49、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-Kalman filter is absolutely fundamental for modern Control theory. The theory of Shock wave is important for design of airplanes.,三.作为工具的数学 大量重要的数学，原意是为解决工程上的问题。比如，维纳 (N. Wiener) 及其弟子，是讯息科学的先驱。他们发展出来的如随机微分方程、维纳测度沦、熵论等，最终都远远超出它们原来的动机。Bucy-Kalman 滤子理论在现在控制论中举足轻重，而冲击波则在飞机设计起着关键的作用。,20,IV. Mathematics as a subject of beauty The purest branch of mathematics is perhaps number theory. It goe