The earliest recorded ancient literature in China is 300 poems, including elegies, folk lyrics, court etiquette works and songs praising or satirizing rulers. By the time of Confucius, literature had served gentlemen and cultivated folk customs. During the Warring States Period, hundred schools of thought wrote books and made great contributions to literature, but scholars such as Han Fei looked down on literary scholars. Qu Yuan initiated ancient Ci Fu, but his lifelong ambition was to revitalize Chu State. Literature itself did not occupy an important position in ancient society. As for mathematics, China Confucianists put it at the end of the Six Arts as auxiliary knowledge. Politicians even regard it as a skill of carving insects. Compared with literature, they don't even have the ability to praise the court. The government's respect for mathematics has not been greatly improved until recent years.
This is not the case in the west. Greek philosophers regard mathematics as the basis of all learning. Plato took general geometry as the premise of introduction, so mathematicians gained a lofty position and flourished in the West for more than two thousand years.
Many people will find my topic a bit strange. China literature seems to have nothing to do with mathematics, but I'll discuss it. In fact, this is related to personal feelings and hobbies, and not necessarily other mathematicians have the same feelings. "If people drink water, they know what it is." Everyone's growth and style have a lot to do with his cultural background and family education. I was trained by the imperial court when I was young. What has the greatest influence on me is China literature, and my biggest interest is mathematics, so it is quite meaningful for me to compare them.
First, the basic meaning of mathematics
Mathematics, as a science, has its own uniqueness, which can be said to be a bridge between humanities and natural sciences.
Mathematicians study all the substances provided by nature, look for their similar laws and express them mathematically. The nature mentioned here is more extensive than most people know. We believe that numbers, geometric figures and various meaningful laws are all part of nature, and we hope to show the essence of these natural phenomena with concise mathematical language.
Mathematics is an axiomatic science, and all propositions must be deduced by the logical method of syllogism, but this is only the form of mathematics, not the essence of mathematics. Most mathematical works are boring, while some are amazing. What is the difference?
In a word, mathematicians decide their research direction based on their deep and superficial feelings about nature. This feeling is both objective and subjective, and the latter depends on personal temperament and is related to cultural literacy. Cultural literacy plays a key role in choosing unsolved problems or creating new directions. Cultural literacy is mainly based on mathematical kung fu, supplemented by natural science, but profound humanistic knowledge is also extremely important, because humanistic knowledge is also devoted to describing the soul's feelings about nature, so Sima Qian wrote Historical Records not only to "understand the changes of ancient and modern times" but also to "study the time between man and nature".
Liu Xie's Wen Xin Diao Long thinks that the value of the article lies in nature and literary talent. Great mathematicians of all ages, such as Archimedes and Newton, all took nature as their religion, and when they saw things, they thought about where mathematics came from, that is, they created calculus. Fermat and Euler's pioneering invention of variational method was also caused by exploring natural phenomena.
The general theory of relativity puts forward the field equation, and its geometric structure has become the dream object of geometricians because it can give space a harmonious and perfect structure. I have studied this geometric structure for 30 years, sometimes confused, sometimes excited, and consciously immersed myself in nature and enjoyed myself, like the author of The Book of Songs of the South or Tao Yuanming of the Jin Dynasty.
Whether there is a geometric structure satisfying the gravitational field equation in space is an extremely important physical problem, and it has gradually become a major problem in geometry. Although other geometricians don't believe it exists, I persevere and study day and night. "Although I died nine times, I still don't regret it."
It took me five years to finally find a supersymmetric gravitational field structure and create it as an important tool in mathematics. The mood at that time can be described by the following two sentences: "Falling flowers are independent, Swift Qi Fei."
Later, a large number of string theorists participated in the study of this structure and got many in-depth results. At first, all my friends stayed away from such problems and were unwilling to deal with physicists. However, I am convinced that nature will not fool people. Looking back more than ten years, I am quite satisfied with the research in this field. Now Calabi-Hill space theory has become a mainstream of mathematics.
Second, the literary talent of mathematics
The literary talent of mathematics is simple, and a few words can tell the laws of different phenomena.
Chen Ban, founded by my teacher, is brilliant in literary talent. It looks for simple invariants in distorted space and becomes the main tool of quantization in physics. It can be said that it is a poem describing the beauty of nature, just like Tao Yuanming's artistic conception of "picking chrysanthemums under the east fence and seeing Nanshan leisurely"
From the axiomatization of Euclidean geometry to the analytic geometry founded by Descartes, to the calculus founded by Newton and Leibniz, to the intrinsic geometry founded by Gauss and Riemann, and to the modern geometry integrated with physics, all of them are based on simplicity and diversity, and their literary talent is by no means inferior to any literary creation. It is no coincidence that they were born at the same time as the rise of literature and art.
Mathematicians can see elegant literary talent and brand-new style when they create new mathematical ideas. For example, Euclid proved that there are infinitely many prime numbers, which initiated the reduction to absurdity. Gauss studied the symmetric group of heptagon, which made Galois group the pillar of number theory. These studies have sprung up all over the world, and judging Hua Mao reminds people of Su Li, the originator of five-character poetry, and Li Taibai, the originator of poetry talk.
China's poems pay attention to metaphor, and profound literary works must have implication, irony and metaphor. The same is true of mathematics. When we seek true knowledge, we can often only rely on our existing experience, follow the general direction of research, and move forward with our feelings about nature, which is quite subjective and depends on our personal cultural literacy.
In order to achieve the best description of artistic conception, writers may not faithfully describe the phenomenon world. In order to create a beautiful theory, mathematicians don't have to follow the laws of nature. As long as there is no problem with logical deduction, they can give full play to their imagination. However, after all, there are differences in the articles. Generally speaking, a good article "Bi Xing" is always rich in techniques.
The common comparison method in mathematics is the comparison of phenomena in low-dimensional space and high-dimensional space. Although we can't see things in high-dimensional space, we can see one-dimensional or two-dimensional phenomena and infer high-dimensional changes from them. When I was a graduate student, I tried to extend the univalent principle of two-dimensional space to high-dimensional space and got some beautiful conjectures. I think positive curvature or negative curvature can be used as the direction of complex structures. The influence of this view can be traced back to the study of the relationship between curvature and conformal mapping in19th century and early 20th century.
In fact, Einstein's general theory of relativity was successfully created by comparing different kinds of knowledge. It is the greatest thought in the history of science, and it can be said to be an earth-shattering work. It unifies the classical theory of gravity and special relativity. Based on the principle of equivalence, Einstein spent ten years comparing various methods to describe the gravitational field, skillfully expressing the gravitational field with geometric tensors, and completely innovating the concept of time and space.
Very similar to the literature, the development from local structure to large-scale structure is also the process of modern mathematics development, and it is often handled in a comparative way. Both geometry and number theory have this history. Algebraic geometricians study singularities by explosion, which is like focusing the whole world on one point. The singularity seen by differential geometry and general relativity is more complicated than algebraic manifold, but it is also hoped that the whole structure can be understood gradually from the local point of view. When studying the local structure, number theory experts use the modular method of prime numbers to turn arithmetic manifold into geometry over a finite field, and then compare it with a large range of arithmetic geometry, and get rich results.
Because writers have different feelings about things, the same thing or the same thing can produce different chants. After people have different feelings about things, they often refer to them in a comparative way. For example, "beauty" has multiple meanings. Besides beauty, it can also refer to monarch. Qu Yuan's "Nine Chapters" said: "With words and expressions, a beautiful woman's husband is right." It can also refer to people with good moral character. The Book of Songs: "Whoever wants to go into the clouds is Xi Shi." Su Shi's Red Cliff Fu: "Looking at the beauty of heaven."
Mathematicians will also put forward many different proofs for some important theorems. For example, there are more than ten different proofs of Pythagorean theorem, five or six proofs of isoperimetric inequality, and six different viewpoints of dual law of number theory are given by Gauss. Different proofs let us understand the same fact from different angles, which often leads to different development of mathematics.
I remember thirty years ago, after I proved that a complete but not compact positive curvature space has infinite volume, the geometer gromov didn't believe this proof at first. Later, he discovered the geometric intuitive meaning of my proof method and developed his geometric theory. These two different concepts have their own importance.
For a surface in space, differential geometricians will ask about its curvature. Some analysts hope to push it along the curvature direction and see what happens. Algebraic geometricians can consider whether it can be expressed by polynomials, and number theorists will ask if there are integer lattice points on it. These subjective feelings are dominated by our cultivation.
Third, the evaluation and evolution of mathematics
There are talented people in the country, and what can lead us into a new realm is good mathematics.
A good job should be that the text is exhausted, and the intention is more than enough. Most math articles are of poor quality and popular, but they have only been heard for two or three years. But creative articles are not necessarily better than the times, and it often takes more than ten years to see the results.
I used to study harmonic function in a brand-new way, and later I improved this method with a few friends and became an important tool for thermal equations. It was not appreciated by others at first, and it was not until the last five years that everyone realized its potential. But we still insist on research and feel that we are not finished yet.
Gorgeous works of mathematics can be found in the extensive knowledge of functional analysis. Although they are beautiful and important, they are always out of touch with nature. For example, an important concept abstracted from function space is called Banach space, which plays a very important role in differential equations. However, many mathematicians continue to popularize this kind of space in the future, such as whether bounded operators have invariant spaces. It's really beautiful, but it doesn't stir up any waves in the mathematical flow.
There are few jobs that can stand the test of time, and government auditors should take this as their first choice. Over the years, guided by the number of articles and citations, the level of domestic mathematics workers is far less than that of people, not only isolated from nature, but also difficult to see gorgeous articles.
The evolution of mathematics and literature has very similar changes. From plane geometry to solid geometry, to differential geometry and so on. On the one hand, it is the improvement of tools, on the other hand, we have a better understanding of nature, and we need to enter a new realm after exerting the beauty of mathematical structure we have known before. There are talented people in the country, and what can lead us into a new realm is good mathematics. The high-dimensional topology mentioned above is exhausted. If we can combine differential geometry, mathematical physics and arithmetic geometry, we can also shake it.
When a big problem is not solved, we often think that mathematics is more difficult than this. After the problem is solved, the future will suddenly open up, and you will feel different when you see a brighter spark than before. Scientists' understanding of nature is gradual, and they naturally have different feelings in different time and space. Some students don't know the difficulty of creation, and even disdain Mr. Chen Shengshen's masterpieces. They think they are more knowledgeable, so they can't see themselves. People can make progress only if they have self-knowledge. That is, as Zhuangzi said: "If you look at the sea from the cliff today, you will know its ugliness and you will be able to talk with Dali."
I once visited the University of G? ttingen in Germany and saw the manuscripts of1great scientists in the 9th and 20th centuries. Their handed down works are only a part of their works, and many masterpieces have not yet been published, which makes me deeply ashamed and admires their hearts. Modern people, on the other hand, imitate a lot, even slightly change the famous works, take them for themselves and publish them as soon as possible. Or apply for an academician, or dazzle yourself as an academic master, what is it to the ancients?
Fourth, the artistic conception and feelings of mathematics
How to find the soul of mathematics depends on our cultural accomplishment.
Wang Guowei said in "Ci on earth": "Ci takes the realm as the top. If there is a realm, it will be a family. " So he distinguished between "creating environment" and "writing environment", "having my environment" and "having no environment".
Of course, mathematical research also has the concept of realm, and to some extent, it can also be said that there is a realm of self.