Truth and Beauty in Mathematics

  From ancient times to the present, whether it is science and technology, mathematics or humanities, the content is more and more rich, and there are more and more branches. The reason for this is that, on the one hand, the ability to discover different phenomena is much greater than before due to more and more tools, and on the other hand, the population of the world has grown substantially, and people of different races, religions, and customs have communicated with each other. , the knowledge of different viewpoints has been integrated and sparked, resulting in new knowledge.
  The study of mathematics has its own uniqueness. It itself is a science that seeks the truth of nature, but mathematicians are also as imaginative as writers, creating works out of hobbies, so mathematics can be described as a bridge between the humanities and natural sciences.
  Mathematicians study all the materials provided by nature, find their common laws, and express them mathematically. Nature here speaks more widely than most people understand.
  We believe that numbers, geometric figures and all kinds of meaningful laws are part of nature. We hope to express the essence of these natural phenomena in concise mathematical language.
  Mathematics is an axiomatic science, and all propositions must be deduced by the logical method of three-part argumentation, but this is only the form of mathematics, not the essence of mathematics. Most mathematical works are boring, and some are breathtaking. What’s the difference?
  Roughly speaking, mathematicians decide the direction of their research with the depth of their feelings about nature. This feeling is both objective and subjective, the latter depending on the temperament of the individual. Temperament is related to cultural accomplishment. Whether it is choosing an unresolved problem or creating a new direction, cultural accomplishment plays a key role. Cultural accomplishment is based on the kung fu of mathematics, supplemented by natural science, but profound humanistic knowledge is also extremely important. Because humanistic knowledge is also devoted to describing the feelings of the mind towards nature, Sima Qian’s writing of historical records not only “understands the changes of the past and the present”, but also “studies the relationship between heaven and man”.
  The great mathematicians of the past dynasties, such as Archimedes and Newton, all took nature as their religion, and they saw the phenomenon of things and thought about the origin of mathematics, that is, the creation of calculus. Fermat and Euler’s seminal invention of variational methods also arose from the exploration of phenomena in nature.
  Gauss, the founder of modern geometry, believed that geometry and physics were inseparable. He said: “I am more and more convinced that the necessity of geometry cannot be verified, at least not now by or for humans, and that we may be able to grasp the nature of the unknowable space in the future. In the
  20th century, the development of geometry has repeatedly changed its course due to important breakthroughs in physics. When Paul Dirac applied the special theory of relativity to the quantized electron motion theory, he discovered the Dirac equation. Even Dirac himself was amazed at the subsequent development, thinking that his equation was more beautiful than his imagination. played a key role in the development of modern geometry.
  Our description of the vortex lacks intuitive geometric feeling, but it comes from nature, and the power of geometry endowed by nature can be said to be meticulous.
  General relativity presents the field equations, and its geometry becomes the coveted object of geometers because it gives space a harmonious and perfect structure. I have studied this kind of geometric structure for thirty years. Sometimes I am confused and sometimes excited. I feel that like the author of The Book of Songs and Chu Ci, or Tao Yuanming of the Jin Dynasty, I am one with nature and enjoy myself.
  Capturing the truth and beauty of nature is far better than all man-made creations. As “Wen Xin Diao Long” said: “The color of clouds and clouds is more wonderful than painting. Exterior, cover natural ears”.
  Whether there is a geometric structure in space that satisfies the gravitational field equation is an extremely important physical problem, and it has gradually become a great problem in geometry. Although other geometers do not believe it exists, I persevere and study it day and night, just as Qu Yuan said:
  ”I do what I do in my heart, even if I die, I still have no regrets.”
  I spent five After years of work, he finally found a supersymmetric gravitational field structure and created it into an important tool in mathematics.
  The state of mind at that time can be described by the following two sentences:
  ”The fallen flowers are independent, and the swift and the swallows fly.”
  Mathematical literary grace is manifested in simplicity. In just a few words, the laws of different phenomena can be expressed, and they can even play a role in nature. This is the beauty of mathematics.
  Chen’s class, created by my teacher, Mr. Chen Shishen, is very literary and admirable. It finds concise invariants in the distorted space, and becomes the main tool for quantization in the field of physics. It can be described as a poem describing the beauty of nature, as Tao Yuanming said “picking chrysanthemums under the eastern hedge, leisurely seeing the southern mountains” artistic conception.
  From the axiomatization of Euclidean geometry, to the analytic geometry created by Descartes, to the calculus of Newton and Leibniz, to the intrinsic geometry created by Gauss and Riemann, to the modern era that is in harmony with physics Geometry is based on simplicity and variety, and its literary style is by no means inferior to any literary creation. It is absolutely no coincidence that they were born in the same era as the rise of literature and art.
  Mathematicians can see elegant literary graces and new styles as they develop new mathematical ideas. For example, Euclid proved that there are infinitely many prime numbers, creating a precedent for proof by contradiction. Gauss studied the symmetry groups of the heptagon, making Galois groups the backbone of number theory. These studies have sprung up, and the judgment of Huamao is reminiscent of Su Lisang, the ancestor of five-character poetry, and Li Taibai’s “Remembering Qin’e”, the ancestor of poetry and ci.
  Let’s now look at another example to explain what mathematics and the humanities have in common: both writers and scientists want to construct a perfect picture, but each author has a different approach.
  In the Han Dynasty, Chinese mathematicians had begun to study how to solve equations, including calculating the cube root. By the Song Dynasty, it was possible to solve multiple equations, hundreds of years earlier than in the West, but the solution was numerical solution, and there was no understanding of the structure of the equation. In-depth understanding.
  One of the simplest problems is to solve the quadratic equation:
  x2 + 1 = 0.
  This equation has no real solution, in fact, no matter x is any real number, the left side of the equation is always greater than zero, so this equation has no real solution, so China Ancient mathematicians did not discuss this equation.
  About four hundred years ago, Western mathematicians began to pay attention to this equation, and Italian mathematicians after the Renaissance found that it was related to solving cubic and quartic equations. Knowing that the quadratic equation above has no real solution, they assume that it does, and call this imaginary solution an imaginary number.
  The discovery of imaginary numbers is amazing! It is comparable to the discovery of the wheel. With imaginary numbers, Western scholars found that all polynomial equations have solutions, and the number of solutions is exactly the degree of the polynomial. Therefore, with imaginary numbers, the theory of polynomials becomes a perfect theory. Perfect mathematical theories soon have endless applications.

  In fact, physicists and engineers later discovered that imaginary numbers are the best way to explain all wave phenomena, including music, fluids, and wave dynamics in quantum mechanics. An important part of the study of number theory is integers, but in order to study integers, we cannot avoid using the theory of complex numbers to help. At the beginning of the 19th century, Cauchy and Riemann started the study of complex variable functions, extending our vision from one-dimensional to two-dimensional, changing the development of modern mathematics.
  Riemann also introduced the Zeta function and found that the analytic properties of complex functions can give the basic properties of prime numbers in integers. On the other hand, he also developed the subject of high-dimensional topology.
  Because of the success of complex numbers, mathematicians attempted to generalize it to create new fields of numbers, and soon found that it was impossible unless some conditions were waived. However, Mr. Hamilton and Mr. Kelly introduced two new number fields, quaternion and octonion, after giving up some properties in the field of complex numbers.
  These new number fields influenced Dirac’s conception of quantum mechanics, creating the Dirac equation. Here we can see that mathematicians and physicists have achieved important results in pursuit of perfection.
  In fact, many great discoveries in physics are made by great scientists through some thought experiments and their in-depth insights.
  When Einstein created the general theory of relativity, the universe observed by humans was really small, but he received great help from mathematicians.
  After Einstein completed the general theory of relativity, Weyl and many scientists began to integrate gravitational field theory and electromagnetic field theory. Weyl first proposed the theory of gauge field. After ten years of struggle, Maxwell’s electromagnetic theory was regarded as similar to general relativity. The gauge field theory is a great breakthrough in physics.
  Interestingly, Weyl said: “If there is a conflict between the theory and the phenomenal world I see, and the theory is beautiful and concise, I would rather believe the theory.” This view is of great help to the development of gauge field theory!
  Here we see similarities between writers and scientists. After Dirac completed his equation, he said that his equation was deeper than his own because it beautifully described the properties of elementary particles and proved in the laboratory, some properties that Dirac had before he created the equation. There is no way to imagine. This is a wonderful phenomenon produced by scientific innovation. The tools we use to understand the truth often lead us to move forward and keep groping forward!
  Perfecting a problem or phenomenon, and then applying the perfected results to new mathematical theories to explain new phenomena, this is the usual method of mathematicians, which has many similarities with writers, but writers use This is the way to express their feelings.
  For example, in ancient China, there were many legends, many of which were based on imagination, some knowledge acquired, and some works were completed according to the needs of the author or the ruler at that time. Therefore, we see the apocryphal scriptures made by Liu Xiang and his son in the Eastern Han Dynasty, and we also see the mountains and seas. The writings of the classics exaggerately describe many unprovable events.
  There are also many examples of Chinese poetry. For example, Li Shangyin and Li Bai wrote two exaggerated verses: “Jinse has fifty strings for no reason” and “white hair three thousand feet”. In the legendary novels of the Ming and Qing dynasties, this way of writing is more popular. Many things described in Journey to the West are only a small part of the truth. In the Romance of the Three Kingdoms, Kong Ming borrowed the east wind. The author wrote it to exaggerate Zhuge Liang’s ability.
  Writers exaggerate and perfect in order to appreciate phenomena or soothe their feelings, but mathematicians construct perfect backgrounds in order to understand phenomena. We may not see the background of the virtual structure of mathematicians in the phenomenon world, but just like the process of creating imaginary numbers, these virtual backgrounds have the ability to explain the wonderful phenomena of nature. In the eyes of mathematicians, these virtual backgrounds are often Emerging in the phenomenal world, for many mathematicians, the idea of ​​imaginary numbers and spheres can be seen as part of nature.
  Now there is a successful theory in particle physics called quark theory, which is similar to imaginary number theory. People have never seen quarks, but we feel its existence.
  Sometimes, mathematicians use thousands of pages of theory to unify, describe, and explain some vaguely specific phenomena in extremely abstract ways. This is the ultimate pursuit of perfection by mathematicians. It is surprising that these abstract methods can solve some extremely important concrete problems. The most famous example is Grothendieck’s great work on Weil’s conjecture.
  The supersymmetry introduced by physicists in the 1970s also greatly promoted the concept of symmetry. Although we have not seen supersymmetry in the laboratory, it has already triggered a lot of important physical and mathematical thinking.
  Modern mathematicians have achieved great results in different branches of mathematics, very similar to the methods of writers. So I said that good mathematicians should have humanistic training, and get inspiration from the changing life and the natural world to perfect our science and mathematics, rather than restricting their own pace and vision and only following the works of predecessors , make small improvements, and think that you are a great scholar.
  Chinese mathematicians paid too much attention to application, and did not care about the strict derivation of mathematics, let alone the perfection of mathematics. By the Ming and Qing Dynasties, Chinese mathematicians could not compare with the mathematicians of the Renaissance.
  In the Qing Dynasty, mathematics was even worse, and there was no originality! Perhaps due to the influence of Qianjia’s research, most of the good mathematicians went to research Jiuzhang Arithmetic and the mathematical works of the Tang and Song Dynasties, and did not do original work. From the same era, the attitudes of Italian, British, German and French scholars after the Renaissance were completely different. The search for original mathematical ideas influenced Newtonian mechanics, which resulted in many industrial revolutions.
  To this day, China’s theoretical scientists are still not as advanced as the world in terms of originality. I think an important reason is that our scientists are not well educated in the humanities, and they are not rich enough in the truth and beauty of nature! This feeling is actually common to scientists and writers. Our Chinese nation is a nation full of emotion and depth. The works of the above-mentioned writers, poets, and novelists are comparable to those in the world!
  But our scientists don’t pay much attention to the cultivation of humanities, and our education officials have very strange education policies. They probably think that the education of Chinese and history is not important, and they use some general education that is superficial but not in-depth. To replace these important knowledge, they probably think that foreign countries pay attention to general education. But it’s a matter of scrutiny.
  Frankly speaking, I have yet to see a country and city of a high standard that does not teach its citizens the history of their own country or locality over and over again. My two children are studying in a small town in the United States. In elementary school and middle school, they read the 300-year history of the United States thoroughly! Because that’s the foundation of American culture.
  I dare to say: citizens who do not understand or are not familiar with history must think that they are a generation without roots. Generally speaking, their cultural roots are relatively shallow, and they are easily fooled and misled by others. This is because they cannot see clearly the cause and effect of what is happening now.
  History is a mirror, it not only points out the reasons for the success and failure of ancient great men, it also passes on to us the feelings left by our ancestors for thousands of years. Wrong way and sigh! China’s five thousand years of rich culture make us full of self-confidence! Why don’t we make good use of our ancestors’ legacy?
  Maybe someone said, I don’t want to be a big scientist, so I don’t have to take the path you said. Actually there is no contradiction. When a young man has a strong feeling for the subject he wants to learn, it will be easy to learn any subject! As for both mathematics and language, advanced countries (such as the United States, etc.) have always taken it for granted. The best universities in the United States all look at the SAT scores when they accept students. The most important part is Chinese and mathematics.
  In addition to examinations, good middle schools in the United States also encourage children to diversify, and try to get involved in subjects including humanities and mathematics. There are many high-quality popular science magazines in the United States, and the sales are often more than one million copies. There are not many good science popularization in China, and the sales volume is also pitiful. From this point, we can see the similarities and differences between Chinese and Western cultures. I hope we will gradually improve!
  I saw Rodin’s will in the Chinese museum, in which we see that sculptors and scientists have the same goal. The excerpt is as follows:
  Masters who were born before you, love them devoutly.
  But be careful not to imitate your predecessors. Respecting tradition and distinguishing what is always alive in tradition – love for “nature” and sincerity, these are the two strong desires of talented writers. They both worship nature and never lie. So tradition gives you the key, and by relying on this key you can escape the shackles of old-fashionedness. It is also tradition that admonishes you to constantly seek truth and prevents you from blindly following any one master.
  May “nature” be your only goddess.
  You must have absolute faith in nature. You must be sure that “nature” is never ugly, and be faithful to nature wholeheartedly.
  In the eyes of the artist, everything is beautiful, because his sharp eyes can penetrate any person or thing and discover its “character”, in other words, the inner truth revealed by its appearance; and this truth is beauty itself. Study religiously, and you will find beauty, for you will find truth. Work hard! be patient! Don’t expect inspiration. Inspiration does not exist. The good qualities of an artist are nothing more than wisdom, concentration, sincerity, and will. Do your job like an honest worker.
  In my opinion, what Rodin taught us is more than art. Every word of his can be used in scientific research innovation. We use sincere and simple feelings to find the beauty of nature and the truth of nature.
  We are all grateful to the previous masters, we groped forward on their shoulders, but we also know that their path is not the only one, let us go forward and build our own path to understand nature!