With the rapid development of society, the economic level continues to improve, and people's quality of life is getting better and better. But at the same time, people's desire for capital has expanded. People are paying more and more attention to practical interests and the development of industrial and heavy industry. Relatively speaking, some theoretical research is naturally regarded as a useless subject. . The first one to bear the brunt is mathematics. In China, almost everyone believes that studying pure mathematics in college will have no future. In fact, this is not the case in Western developed countries. In the eyes of philosophers, mathematics is so beautiful, so indescribably ingenious. Paul Erdos describes his view on mathematics: "Why are numbers beautiful? It's like asking why Beethoven's Ninth Symphony is beautiful. If you don't know why, no one else can tell you why. I know that numbers are beautiful, and if they were not beautiful, nothing in the world would be beautiful."
1. What is the beauty of mathematics?
The beauty of mathematics is the beauty of nature. objective reflection. In history, many scholars and celebrities have expressed their own opinions on the beauty of mathematics. The famous Chinese mathematician Hua Luogeng said: "As far as mathematics itself is concerned, it is magnificent, colorful, varied, and fascinating... People who think mathematics is boring just watch it. Mathematician Xu Lizhi said: "Mathematics as a scientific language has the unique beauty characteristics of general language and art, that is, mathematics is in its content structure. It also has its own certain beauty in terms of mathematics and methods, which is called mathematical beauty. The meaning of mathematical beauty is rich, such as the simplicity and unity of mathematical concepts, the coordination and symmetry of structural relationships, and the harmony of mathematical propositions and mathematical models. Generality, typicality and universality, as well as singularity in mathematics, etc. are all specific contents of mathematical beauty. "With the development of mathematics and the progress of human civilization, the concept of mathematical beauty will develop and the classification will be different. But its basic content is relatively stable, which is: the beauty of symmetry, simplicity, unity and singularity.
The symmetrical beauty of mathematics has been considered a basic content of mathematical beauty since ancient Greek times. The so-called symmetry refers to the equivalence of the two parts that make up a thing or object. This kind of symmetry can be seen everywhere in mathematics. The more vivid ones are some of the axially symmetrical figures we are accustomed to, especially circles, which can be said to be completely symmetrical at 360 degrees without dead ends. Pythagoras once said: "The most beautiful of all plane figures is the circle, and the most beautiful of all three-dimensional figures is the sphere." This is precisely based on the fact that these two shapes are symmetrical in all directions. For me, the most profound impression about symmetry was a math problem my teacher asked me to do when I was in fifth grade. At that time, the teacher saw this question in the newspaper and gave it to several teachers in the same office to do it. As a result, none of the teachers could do it, so the teacher called me to the office to do it on the spot to see how the children thought. Will it be more lively? The question is that the number obtained by multiplying a four-digit number by nine is equal to the reverse order of this number. When I saw this question, I thought that since it is symmetrical, then the first number must be 1, and then multiplied by nine, then the last number must be 9. Then I thought that the second number can be up to 1, but one generation can go in Obviously not, then it can only be 0. In this way, it is easy to guess that the third number is 8, so the answer is 1089*9=9801. I remember that I figured out the answer very quickly, teacher They were all surprised and praised him repeatedly. I was really happy at that time, and it was also the first time I had a basic concept of the symmetry of numbers. Now that I think about it, that question is actually very simple, but even such a simple math question contains the high degree of symmetry beauty of mathematics.
The simple beauty of mathematics is a reflection of the need for simplicity in human thought expression. Einstein once said: "Beauty is ultimately simplicity in nature." Mathematical language itself is the most concise text, and at the same time reflects extremely profound objective laws. Many complex objective phenomena, when summarized into certain laws, often appear very complex. Simple formula. The formula given by Euler: V-E+F=2, can be called a model of "simple beauty". No one can tell how many polyhedrons there are in the world. But their number of vertices V, number of edges E, and number of faces F must all obey the formula given by Euler. Such a simple formula summarizes the unique characteristics of countless types of polyhedrons, which is amazing. As the great Hilbert once said: "Every real advance in mathematics is closely connected with the discovery of more powerful tools and simpler methods." Such as the introduction of the Cartesian coordinate system.
The use of logarithmic notation and the introduction of complex units. The emergence of calculus reflects the simpler external form of mathematics and the deeper content. Most formulas in mathematics embody "simplicity of form and richness of content." The simple beauty of mathematics is also reflected in its form, that is, the external form of mathematical beauty, which is the beauty presented in the external structure of mathematical theorems and mathematical formulas (or expressions). The main characteristic of form beauty lies in its simplicity.
The unified beauty of mathematics is a certain homogeneity, correlation or consistency of aesthetic objects in form or content, which can give people an overall harmonious aesthetic feeling. All objective things are interconnected. Therefore, mathematical concepts, mathematical theorems, mathematical formulas, and mathematical rules that reflect objective things are also interconnected and can be in a unity under certain conditions. For example, from a structural analysis, specific methods such as analytical method, trigonometric method, complex number method, vector method, and diagramming can all be unified into the number-shape combination method. Euclid's "Elements of Geometry" simplified some spatial properties into several abstract concepts such as points, lines, planes, and solids, as well as five postulates and five axioms. This resulted in an elegant deductive theoretical system, showing a high degree of of unity. The "Elements of Mathematics" of the Bourgeois school uses structural ideas and language to reorganize various branches of mathematics, essentially revealing the internal connections of mathematics, making it an organic whole, and inspiring people with the high degree of unity of mathematics. .
2. How can the beauty of mathematics be achieved?
The beauty of mathematics is so beautiful in the eyes of prophets and philosophers, so how can mathematics be transformed from a few simple Arabic numerals and Latin How did letters develop into such a magnificent and legendary world of mathematics? It is obviously not enough to rely on personal strength alone. It has been accumulated by ancestors from generation to generation for thousands of years.
The beauty of mathematics is the crystallization of people’s wisdom in mathematics. People always encounter some small problems that need to be solved with mathematics in their daily lives. Then someone comes up with a small improved method to make calculations easier. Over time, this gradually makes the soil of mathematics more and more popular. It will become more and more fertile and cultivate more fragrant fruits of mathematics, making the world of mathematics richer and more beautiful. I am not an expert in mathematical archaeology, and I cannot investigate any specific people's small improvements in mathematics. But I can tell you my own example. Everyone around me knows that my speed calculation is very good. It's not that I'm very smart, but that I can make some transformations in my mind to calculate some difficult formulas and then calculate them again. This makes it much easier. It's just me. Personally speaking, although this improvement is very small, or it cannot be called an improvement, it is because of the accumulation of bit by bit by the people that mathematics becomes more and more beautiful.
The beauty of mathematics is the source of inspiration for wise men in mathematics. Chinese mathematician Chen Jingrun lives in a shabby house, but in order to break Goldbach's conjecture, a difficult mathematical problem in the world, he kept doing calculations and finally won the crown jewel of mathematics through hard work. Next, I will talk about an experiment in which Buffon used a needle to find the approximate value of pi. One day, Buffon invited many guests and friends to his home and conducted a strange experiment. He had drawn equidistant parallel lines on white paper in advance, spread the paper on the table, and took out some small needles with a uniform mass and a length half the distance between the parallel lines, and asked the guests to put the needles one by one. It was still on the paper, and Buffon counted on the side. The result was 2212 times, of which 704 times intersected any parallel lines. Buffon did another simple division, and then he announced that this was the approximate value of pi. , also said that the more times you cast, the more accurate it will be. This experiment is shocking. Pi is connected with a seemingly unrelated random needle experiment. However, this does have a theoretical basis. This method of calculating pi is novel, wonderful and amazing.
The beauty of mathematics is the development need of society for mathematics. We are facing an era of rapid development of science and technology. The digitization of information and the mathematical processing of information have become the common core technologies of almost all high-tech projects. From pre-design and plan formulation, to experimental exploration, continuous improvement, to command and control, and specific operations, mathematics and technology are relied on everywhere. Many countries realize that the development of high-definition television is one of the main battlefields in future economic and technological competition. It should be pointed out that TV screens are not only indispensable in modern people's daily lives, but may also become a work surface for information transmission and processing through the Internet. Almost all important jobs will be related to it. Mathematical techniques played a decisive role in the fierce competition for such an important project.
The 1991 Gulf War was a modern high-tech war, and its core technology was actually mathematics. This fact caused quite a bit of surprise. Summarizing the experience of the Gulf War, the United States concluded: "The battlefield of the future is a digital war."
2. What is the use of knowing the beauty of mathematics?
Nowadays, more and more college students are unwilling to fill in the major of mathematics when filling in their university majors. The reason is that after graduation, Jobs are hard to find. I am the same. In fact, I personally love mathematics very much. I can do a math problem there without eating or drinking for a whole day and enjoy it. But in the end, due to pressure from family and society, I chose the electronics major, which is generally considered to be a better job in the future. Although I don't like it very much, I'll settle for it now. However, I still want to say here that learning mathematics is useful, and it is very useful. The future society must be a digital era.
Social application of the beauty of mathematics - revealing the laws of nature and guiding engineering design. In January 1995, after the great earthquake in Shenzhou, the United States used mathematical models to predict earthquakes, predicting that a major earthquake might occur in southern California at the end of this century; in March 1995, my country's Central People's Broadcasting and Television Station announced the use of digital broadcasting, pointing out that previous The analog broadcast method was ineffective, so a new broadcast method was used; in June 1995, the European Union held a meeting to discuss a unified standard for future digital communications; in February 1996, the Ministry of Electronics Industry of my country announced the development focus of the "Ninth Five-Year Plan": digitalization information Technology. The two key development projects ordered are: digital high-definition television receiver prototype and digital laser disc; in April 1996, my country's National Science and Technology Commission issued a bidding announcement and officially announced the digital high-definition television development project. With just a few examples, we can clearly see the important role that mathematics plays in contemporary people's production and life.
An outstanding expression of the beauty of mathematics - the golden ratio. The golden section, also known as the golden rule, refers to a certain mathematical proportional relationship between the parts of a thing, that is, the whole is divided into two parts. The ratio of the larger part to the smaller part is equal to the ratio of the whole to the larger part. The ratio is 1:0.618 Or 1.618:1, that is, the long segment is 0.618 of the entire segment. 0.618 is recognized as the most aesthetically significant ratio number. The use of this ratio can arouse people's sense of beauty and is widely used in real life. The ratio of certain line segments in buildings uses the golden section scientifically. The announcer on the stage does not stand in the center of the stage, but in the center of the stage. It is located on one side of the stage. It is most beautiful when standing at the golden section of the length of the stage and has the best sound transmission. Even in the plant world, there are places where the golden section is used. If you look down from the top of a twig, you will see that the leaves are arranged according to the rules of the golden section. In many scientific experiments, a 0.618 method is commonly used to select a plan, that is, the optimization method, which allows us to rationally arrange a smaller number of tests to find reasonable western and suitable process conditions. It is precisely because it has extensive and important applications in architecture, literature and art, industrial and agricultural production, and scientific experiments that people preciously call it the "golden section".
Bertrand Russell described his feelings about the beauty of mathematics in the following words: Mathematics, if viewed correctly, has... a supreme beauty - just like the beauty of sculpture, it is a kind of beauty. Cold and serious beauty, this kind of beauty does not cater to the weak aspects of our nature. This kind of beauty does not have the gorgeous decorations of painting or music. It can be pure to the point of sublimity, and can achieve strictness that only the greatest art can display. That perfect situation. A true spirit of joy, a spiritual excitement, a consciousness that feels higher than human beings - these are the standards of perfection and beauty, which can be found in poetry and mathematics.
References:
(1) (American) Sione Pappas. The Movement of Reason-Feeling the Beauty of Mathematics from Quotes. Translated by Wang Youjun. Shanghai: Shanghai Science and Technology Education Press, 2010.
(2) (English) Post. The Beauty of Mathematical Proof. Translated by He Junjie and Tie Hongling. Hunan: Hunan Science and Technology Press, 2012
（ 3) (US) Clifford A. Pikov. Translated by Ma Dongxi. Hunan: Hunan Science and Technology Press, 2010
(4) Wu Jun. Series of articles on the beauty of mathematics. 2006——2007.< /p>