Http://www.qiji.cn/news/reading/2004/08/16/20040816135223.htm
Qiu Chengtong: The content, method and significance of mathematics
http://www.qiji.cn/news/ Miracle Report
[Science and Technology News] Today is to talk about the content, method and significance of mathematics. This is the title of a book written by the Soviet Union, and today's speech is borrowed as the name of the speech.
Today is the hundredth anniversary of the North, the May 4th exercise is the Northern University student launched. As the gland of the speech, let us briefly review the "May 4th" dispute between China and Western culture. After the Nineteenth Century, China's understanding of Xiwen Science and Technology is "ship vertical artillery". After repeated war, Zhang Zhidong proposed "the middle school as the body, the West", and the traditional Confucian. The spirit is main, join the Western technology. Before and after the May Movement, there was a ceremonial battle. The one-to-art civilization, defending Confucianism, to determine the natural and knowledge to conquer the natural, lack of life, and the Chinese culture is self-satisfied, and the Chinese culture is self-sufficient. It will be able to carry forward. At the end of the first World War, the Western philosopher Russell, etc. Another faction of Hu Shi is reversed, they think that in the field of science, life is intended to be scientific and historical methods, without criticism and logic research, it cannot be knowledgeable.
Could not finalize the battle, and there is no conclusion. The two factions have no deep research on modern times, nor collecting data, and theory cannot strictly derive, and finally become empty. In fact, this is a characteristic of Chinese traditional culture. On the one hand, it is extremely abstract, quality and no quantity, Confucianism is all in the clouds, and Zen is not a text, straightforward. On the other hand, it is extremely practical. Zhuangzi said "hovering in the sky." Ancient science is practical, everything is a service, four inventions, compass, paper, printing, gunpowder is not the case. It is basic science based on the foundation of Western technology in science, actual and abstract bridges, and basic science tools and languages are mathematics.
Many scientists in the past generation have high evaluation of mathematics. We will take some physicists' words as examples. R.Feynman said in the book of "Physical Law", says all of our laws, each of which is described by the pure mathematics in the esoteric mathematics. Why? I don't know at all. E.Wigner said that mathematics is uncommon with common science. F.Dyson said: One item in physical science history is therefore, it is a key contribution to mathematical imagination. Basic physics are represented by a deep mathematics. All phenomena of nature, etc., as long as you can get mature understanding, you can use mathematical description. The philosopher Thoreo written by "Lakeside Scatter" also said that the truth is the most clear and beautiful, and it will eventually be present in mathematics.
In fact, mathematicists do not only absorb nutrients from nature, and they are also implicated from social sciences and engineering. In the human mind, it is an insight into the phenomenon of enlightenment, as long as you can use rigorous logic to process the object of mathematicians. The difference between mathematics and other science is to allow abstraction. As long as it is beautiful, it is enough to dominate everything, mathematics and literature are different, and all propositions can be launched by recognized minority axioms. Mathematics officially became a systematic scientific scientific European European European, his "geometric original" is the immortal. Ming and Xu Ma si and Xu Guangqi translated it into Chinese, and pointed out "" Thirteen volumes in more than 500 questions, one pulse, volume and volume, questions and questions, first can't, after one afterwards, Incurrent accumulation, gradually accumulating, and finally it is the name of the Olympics. " Complex and deep theorem can be derived from a small number of concise axioms, and it is true to determine the meaning of beauty, and the watermast is blended, and it is difficult to separate. It is worth pointing out that European mid-style mathematical thinking directly affects Newton's physics's three laws, the Newton is interspersed with "Mathematics Principles" from "Natural Philosophy". From Einstein to the current physicists, they want to complete the unified field, and can use the same principle to explain all the power fields between the universe. The true and beauty of mathematics, the experience of mathematicians deep. Sylvester said, "They expose or clarify the concept world, which leads to the meditation to the beauty and order, the harmonious relationship between its parties, is the most solid foundation in human eyes." Mathematics History M.Kline said, "A wonderful certificate, the spirit of nearly a poem." When the mathematician absorbs the essence of natural science, use the beauty and logic to guide, and the imagination is exhausted, and the author is also amazing. Mathematics often have a grand idea, guided by beauty, such as Weil guess, has contributed to a huge plan for reorganization calculations, integrating topology and algebrane into integral equations. The Weil guess completed by A.Grothendieck and P.Deligne can be said to be the great victory of abstract methods. Review the history of mathematics, the results of several different important concepts are naturally integrated, and they have become milestones for mathematical development. Einstein integrates the concept of time and space, becoming the cornerstone of physics in the past 100 years; three years ago A.Wiles A.Wiles's research on self-defeating and FERMAT finalized is more exciting. Mathematics can reluctantly rely on the achievements of natural sciences, amazing, because the numbers and space itself are part of nature, and their structure is also part of the universe structure. However, we must keep in mind that the mystery of nature is unspeakable, not only in numbers and space, it is perfect, and mathematician can't resist this beauty.
The two most important discovery of physics in this century: relativity and quantum mechanics cause great impact on mathematics. Generalized relative to the differential geometry "There is an object of words", Liman Geometry is no longer talking on abstract paper. Quantum fields are confused from the beginning, it is like magic in mathematics. For example, the Dirac equation is in geometric applications that make people feel hard, but it is so strong and strongly affects the development of geometries. Super-tone is the idea of the 1990s of physicists, whether they are querial or theoretical, but by the help of super string theory, mathematician can solve the unresolved problem for more than 100 years. UHCA Theory is unbearable in mathematical, unless it makes people, it is physically a seat.
At the end of the last century, the mathematics axictification movement made mathematics, and mathematicians thought that the tool was already prepared, and the work will not be unfavorable. Hilbert, this century, I thought that any mathematics can use a complete axiom to derive all the propositions. However, there is no common, and Godel issued a famous papers in 1991, "Mathematical Principles" in the form of unconnected propositions and related system I. "Prove that there is a non-flexible transctor of a system that contains usual logic and number theory. Established. This means that Hilbert's ideas are not comprehensive, but also science can't be universal. However, the problems generated by nature, we still believe that Hilbert's ideas are basically correct. Mathematician due to their own products, rough It can be divided into three of the following:
(1) Creating theory of mathematicians. These mathematicians work mode, and they can be crude.
● Peek from the phenomenon in the phenomenon. Thus, a set of theory can systematically explain a lot of similar problems. An obvious example is that the last century LIE has created continuous conversion group theory of differential equations after observing a large number of symmetrical symmetrics in mathematics and physics. Li Qun has become the basic concept of modern mathematics.
● Promote existing theoretical or transplanted to other structures. For example, the subporal score is extended from finite-dimensional space to unlimited dimension, and the solution is used to get the surface to obtain a contact theory. When Ricci, Christofel is studying a contact with the coordinate theory on the surface, they are hard to imagine its importance in yang-mills.
● Use a comparison method to seek the joint part of different disciplines to develop new results. For example: Weil compares integer equation and algebraic geometric development calculation geometry: thirty years ago, Langlands combined group expressed theory and self-adapt form and proposed "LangLands Program", which will be able to exchange domain theory to uncharged fields.
● Develop theory to interpret new mathematics phenomena. For example, GAUSS discovered that the curvature of the surface was an intrinsic (that is, the Riemann has created the geometric development of his name and has achieved the geometric development of his name; H.Whitney After discovering the invariance of the sexual class appeared on the fiber, Pontryagin and Chen province will be promoted to more general, and the charter class has become the most basic invariant in the topology and algebra geometry today.
● Develop theory to solve important problems. For example, J.NASH is a hidden function that is developing in the general Limann's branch isometric, in the future self-sufficient discipline, which is large in the differential equation. S.SMale has solved five-dimensional or more Poincare guess with H-coordination theory, this theory has become the most important tool for differential topology.
● The new theorem has been proven to establish a more in-depth theory. Such as the ATIYAH-Singer index theorem, Donaldson theory, etc., there are many different proofs. These certificates have aroused important work.
● Give new structures on the research object. Kahler introduced the scale of his name when studying a rehabilitation; in recent years, Thurston also introduced the concept of "geometry" when studying three-dimensional flow. In general, the introduction of new structures makes extensive concepts. Sometimes the structure is also added to restrictions, such as Kahler's streamline we have to concentrate on Kahler- Einstein scale, which is the result.
(2) Mathematics who find regularities from phenomena. These mathematicians or data experiments, or discovering the problems worth studying in nature and social phenomena, with the experience of extracting the essence, the meaningful guess. If GAUSS has a large number of rigid numbers, the prototypes distributed in an integer are proposed; Pascal and Fermat have a letter about gambling's odds, which focuses on the cornerstone for modern probability. In the fifty-years, the futures market just raised, Black and Scholes proposed an option pricing equation, and then widely used in transactions. Scholes also won Nobel's Economic Award last year. There are still many examples of such examples, and it is not an exclusive.
If you come back, you have to make meaningful guess is not easy, you must have a full understanding of the phenomenon facing. Taking the Red House Dream as an example, as long as you read six or seven hundred times before, you can guess how much it is. But if we don't know much about the poems, you can't understand its true meaning. There is no meaningful guess. (3) Mathematics who solve the problem. All mathematical theories must lead to some important problems, otherwise this theory is empty. The importance of theory must be proportional to the importance of it can solve problems. The importance of a mathematical problem is whether it is rich in the theory it takes out. Single is a beautiful certificate is not the true meaning of mathematics. For example, the four-color problem is a famous problem, but we have to benefit much after it is resolved, and some difficult problems are like a medium-sized column, you must break it, then you can go to the room. For example, I can't solve the Poincare guess. One day can't say we understand the three-dimensional space! I solved the Calabi guess, and what I encountered was similar.
After the mathematicians must be in the premium, the problem is "undressed", further development theory, and find new problems is "post". There is no new problem mathematics will die, so this "Release" is the common mission of our mathematician. The ultimate goal is based on mathematics, and integrates the entire natural science, social science and engineering.
Since A.Wiles solves the Fermat agencies in 1994, many people ask this. Everyone feels that the proven of Fermat is an epoch. It not only solves a problem of up to 350 years, but also makes us understand the elliptical curve on the rational domain; it is the mainstream of the two numbers - self-defeating and elliptical curves - and burst out Sparks. It is worth mentioning that the elliptical curve has developed rapidly in the coding theory in the past decade, and the coding theory will be used in computer trade, and its potential is not estimated.
Finally, we talked about the differences between physicists and mathematicians. In general, there is no eternal truth in physics, physicists continue to explore, hoping to identify the basic laws of the final big unity, thus achieving the purpose of conquering nature. In the kingdom of mathematics, each theorem can be strictly derived from the axiom system, so it is a true truth that is struggling. Mathematics use beauty as the main selection criteria, the good theorem makes us feel the true and beauty of nature from the mind, reaching the leisurely realm of "the heavens and the earth and me, all things and I am a one" leisurely, with physicist to conquer the big Natural is completely different.
In order to capture truths, physicists are often jumping on thinking, although they are not strict and easy to make mistakes, they want to see natural phenomena more and more, this is our very admirable. After all, mathematicians should be careful, step by step, take time to remove all possible mistakes, so that these two practices are mutual, and it is not possible.
In traditional culture, we speak Lid, but it will never discuss how to seek truth, don't seek real, why? We also say "gentle Towns, poetry", but only vague, mathematics, and true beauty, is the basic science of the Chinese nation.
