Mr. Ding Shisun is a famous mathematician in China, and he has made important contributions to the development of mathematics in China. He has repeatedly spoken about the importance of mathematics and called on society to pay attention to mathematics education. Today, we share Mr. Ding Shisun's article "The Power of Mathematics" to remember and feel his life's work for mathematics education.
Written by |Ding Shisun.
The role of mathematics is not limited to a knowledge, but also a tool, and mathematics plays a very important role in the cultivation of talents in the whole educational process. There are many examples of a subject that can be developed by leaps and bounds once it is linked to a certain problem in mathematics. In the 80s of the 20th century, Hauptmann won the Nobel Prize in Chemistry for solving the problem of how to determine the crystal structure with X-rays. Hauptmann once said: My chemistry level is general chemistry after half a year of studying in college, and I don't know anything else. In fact, he relied mainly on mathematics to solve the problem of determining the crystal structure with X-rays. Mathematics is often able to work in different subjects, but it is not possible to predict in advance what subjects will work and in what way.
From the perspective of scientific development, mathematics and many disciplines have had a close relationship, and the development of mathematics and the development of many disciplines play a complementary role - either the development of mathematics promotes the development of other disciplines, or other disciplines put forward some specific problems to mathematics, which in turn promotes the development of mathematics.
It has been said that mathematics is the queen of science. Many mathematicians disagree with this statement. Mathematics is not isolated from other disciplines, but complements and promotes and develops together with other disciplines. It might be more appropriate to describe the relationship between mathematics and other disciplines as a partnership.
From the perspective of historical development, what role has mathematics played in promoting the development of science?
Let's take a look at the generation of computer design ideas. As you know, the world's first computer appeared in 1946. The earliest design ideas for computers can be traced back to the early 20th century. In 1900, the mathematician Hilbert proposed 23 questions at the Second World Congress of Mathematicians – a question that anyone who studied mathematics knows. On the one hand, these 23 questions summarize the development of mathematics in the 19th century, and at the same time point out the aspects in which mathematics should develop in the 20th century, which has had a great impact on the study of mathematics in the 20th century.
One of these 23 questions is whether there is a way to determine whether a multivariate polynomial group of integer coefficients has a rational number solution, or whether there is an integer solution. In the language of mathematics, it is whether there can be an algorithm. This question has caught the attention of some mathematicians, hoping for a clear answer. However, after more than 30 years, it was gradually discovered that this algorithm did not exist. In mathematics, it is easier to prove that there is to some extent, and to prove that there is, you have to give an algorithm that you can use to judge. However, if you want to say "no", you need to explain what an algorithm is. It can be said that algorithms are very dead methods, and of course this is not strictly mathematical language, but people can generally understand it. If you're going to prove that such a thing doesn't exist, that's not enough. So, it wasn't until around 1936 that mathematicians defined algorithms.
It is often the strange phenomenon in the development of science that a problem cannot be solved after many years, but at some point, several people at the same time solve the problem in different ways. The same is true for the problem of defining the algorithm. Around 1936, several people gave the definition of an algorithm. There is a definition of an algorithm, now called a turing machine. The turing machine is an ideal computer, and it is relatively close to the design ideas of today's computers. It can be said that the definition of turing machine is an important part of later computer design ideas. From here, it can be seen that at the beginning it was a purely mathematical problem, and it was not thought of designing a computer at all. But if you want to solve this problem, you have to define the algorithm. The root cause of the computers that can do so much today is a mathematical problem, and the study of this mathematical problem did not expect to have such a big impact in advance. This example illustrates that starting from a purely mathematical problem not only solves a mathematical problem, but also has an important impact on other disciplines.
Another example is "group theory". Now everyone who is involved in mathematics knows what the concept of group is. However, the definition of group appeared in the 50s of the 20th century, and it was first derived from solving equations. We all know that the solution to a quadratic polynomial is the so-called root number, and this problem has been known about 2000 years ago, and everyone has learned it in elementary mathematics. Here's an interesting process: to express the root through coefficients. When the quadratic equation is solved, it is easy to think about the triple, that is, whether there is a similar formula for the quadratic equation. Around the 15th century, the quadratic equation was solved, and the formula was very complicated. Soon the formula for solving the quadratic equation also came out. Mathematicians have a habit of always wanting to generalize, since there are quadratic formulas, cubic formulas, and quadruplicated formulas, how about five times?Everyone thought that there should be five times, but they did not expect to encounter a big problem in the problem of the fifth order equation, and after almost hundreds of years, it was not possible to solve this problem until the beginning of the 19th century. In the thirties of the 19th century, a young mathematician named Galois in France put forward a theory of Galois: he gave a method to determine how many times the root of an equation can be expressed by coefficients. The so-called expression is expressed in addition, subtraction, multiplication, division, and open square (not necessarily open square). In this way, the concept of swarm is proposed, and this problem is ultimately solved by the swarm method. At first, the results were sent to the French Academy of Sciences, and a very important scientist in the Academy of Sciences thought it was nonsense, so it was never ignored. It was not until the 50s of the 19th century that it was officially published, so that the concept of swarms was proposed. This result of Galois was not recognized until 20 years later.
The concept of a group is purely a mathematical problem, but the first thing that is used to solve the problem of how many crystals there are in a compound. In the late 19th and early 20th centuries, chemists used the concept of groups to solve the problem of how many kinds of crystal structures there are. The concept of a group is actually a good measure of symmetry, which can solve the problem of what symmetry is measured in, that is, what is the structure of its transformation group and how much. Again, this is a purely mathematical question, but one that goes far beyond mathematical examples. There are many such examples in mathematics.
Here's an example of how practical needs can contribute to the development of mathematics. During World War II, Germany's air force was strong, with a large number of aircraft and good quality. In order to solve the problem of how to defeat the Luftwaffe with an inferior air force, the United States sought out a group of mathematicians, and von Neumann was one of them. As a result, von Neumann discovered game theory by studying this problem. In recent decades, one of the most important uses of game theory has been the study of economic mathematics, which has developed into an indispensable foundation of economic mathematics.
What exactly is the object of mathematical research?It's not easy to articulate.
In the past, the definition of mathematics was put forward by Engels in "Dialectics of Nature", he said that mathematics is the study of the quantitative relations and spatial forms of the objective world. Engels' definition was put forward in the 19th century, and with the development of mathematics in the 20th century, many things cannot be summarized by this definition. When it comes to the relationship between numbers, it means that mathematics is the study of the operation of numbers, but with the development of mathematics, the object of number calculation goes far beyond numbers. For example, group theory calculates group elements. There are even others, so to speak, which have nothing to do with operations, so it is no longer enough to say that mathematics is the study of quantitative relations. There is also a form of space that was understood at the time as the objective world, which is commonly referred to as three-dimensional space. However, the study of geometry has gone far beyond the three-dimensional, involving four-dimensional, five-dimensional, multi-dimensional, and even innumerable dimensions. Therefore, it is not enough to summarize mathematics with 19th-century definitions.
How do you define mathematics?Until now, there has not been a definition satisfactory. This also shows that the definition of mathematics is difficult to come up with. For example, someone pointed out that mathematics is the study of quantity, and if the word "number" is removed, he said that there is "number", which is too dead, and number is an integer and a fraction. So what is quantity?Quantity is a philosophical concept. Now some people say that mathematics is the study of order, that is, the purpose of studying mathematics is to give order to the world, but this statement is not the language of mathematics. There is some truth to it, but it is not clear. It can be seen from this that because the object of study of mathematics is abstract, mathematics is different from other natural and social sciences, these disciplines have very specific objects, while mathematics does not. The reason why mathematics can be used in both the natural sciences and the social sciences and even the humanities is because it is abstract. The abstractness of the object of mathematical research first has one of the following, that is, it can train people to think a way - abstract thinking method. In mathematics, even if we start with natural numbers, they are already very abstract concepts, and we have to go through many layers of abstraction to arrive at the concept of numbers. Therefore, it has taken a long time in history for the majority and the singular to be distinguished. As long as you study the history of mathematics, you will find that the formation of the concept of number is not easy. Therefore, learning mathematics can train people's abstract thinking skills.
Why is abstraction so important?Because if people want to grasp the essence of things, they must remove many unimportant things, and if they want to discard many non-essential things, they must solve them through abstract thinking. Abstract methods of thought are important for the study of science and even for dealing with the problems that arise in everyday life. Without the ability to abstract, it is not easy to discern what the problem is to be solved. This is the outstanding feature of mathematics, namely its abstraction. The abstract nature of mathematics allows it to be applied in a wide range of ways, even in completely different ways.
The second feature is that because of the abstract nature of mathematics, the definition of mathematical objects must be very clear. Other disciplines have different requirements for definitions, and you can generally describe what it is, and the listener will be able to understand it. However, mathematics is not descriptive because of its abstract objects, and must be strictly defined. Definitions are very important in mathematics, and everyone knows that. In my teaching, I have found that teachers from other departments often encounter a great difficulty when they come to the mathematics department to give lectures. For example, when a teacher in the physics department gives a lecture and talks about "force", the student asks for a definition of "force", which is very difficult. It is difficult for the teacher to portray the "force" clearly in a few words. Unlike the "circle" in mathematics, which is a trajectory at an equal distance from a point, it is very clear.
Many things in chemistry can also be understood by description, and they are very clear and do not need to be defined. Why is mathematics so strict about definitions?Because of its abstract object, it cannot be discussed without defining it by definition. Therefore, mathematics requires that the description of concepts be very accurate. I often joke that a math student is very stupid, and he doesn't understand what he hears as long as the definition is not clear. In this sense, there are its advantages as well as its disadvantages. The disadvantage is that you have to ask for a definition of everything, and there will be problems, and not everything can be defined. So, the second characteristic of mathematics is that it requires a very accurate depiction of concepts.
The third characteristic of mathematics is the rigor of its logic. Because it is abstract, it can only be developed by logic, which is also very important training for people, which can be understood in terms of plane geometry. What exactly does plane geometry play when you learn it?When I was younger, after I went to college in mathematics, I declared that plane geometry was useless. In the 50s of the 20th century, I participated in the reform of the teaching of mathematics in middle schools, and I often said that plane geometry should be abolished. After a few years as a teacher, I found that there is a difference between students who have studied plane geometry and those who have not, that is, if you want to prove a problem, it is easy for students who have studied plane geometry to accept, and it is more difficult for students who have not studied plane geometry to accept. During the "Cultural Revolution", if the students said that the sum of the three angles of a triangle is equal to 180 degrees, many of them would ask it, do they still need to prove such a simple question?If you just take a protractor and measure it, it will make our instructors laugh. This shows that the ability of logical thinking needs to be cultivated through some specific things, and plane geometry is a good medium for cultivating people's logical thinking ability.
There are three characteristics of math classes. By learning mathematics, you can acquire good thinking habits and thinking methods, which will work invisibly on people. The relationship between mathematics and other disciplines is not only mutually reinforcing, but also important is that mathematics not only gives people knowledge, but also ways of thinking, and mathematics is actually part of culture. Mathematics is a typical example of rationality and thinking, and the above three characteristics of mathematics are all about rational thinking.
Mathematics is also part of the culture. In traditional Chinese culture, rational thinking is not very valued. For example, after the Renaissance, many Western philosophers liked to engage in philosophical systems, which was their habit. Whether this habit is good or not is another matter. China is very different, and the Analects, a very important Chinese classic, is in the form of quotations. Many of the words in it are aphorisms, pointing out the main points, and there is no discussion, and there is no need to discuss them, but you will understand it as soon as you hear it. This is a reflection of China's habits of thinking. One of the characteristics of traditional Chinese math books is that they are full of examples. For example, Sun Tzu's residual theorem, a very important theory in mathematics, in Chinese mathematics books, Sun Tzu's residual theorem is to tell you the remaining number of three or three, the remaining number of five or five, and the remaining number of seven or seven. It neither proves nor forms a system, which is a characteristic of Chinese mathematics and a characteristic of Chinese culture.
Mathematics is a part of culture, mathematics is reflected in a mode of thinking, and mathematics learning is also an important way to train people's logical thinking ability. In the past, we proposed in the teaching reform that logical thinking ability was directly acquired through logic classes, and formal logic courses were also opened in middle schools, but in the end it proved that the effect was very poor. Some of the rules of logical thinking have been taught for a long time, and students can't understand them, let alone use them. Later, it was recognized that people's logical thinking ability cannot be cultivated by taking logic classes. The biggest advantage of plane geometry is that its content is very intuitive, and through the carrier of plane geometry, it can effectively cultivate people's logical thinking ability. The logic of mathematical theory is very tight, and people cannot extract the logic from the specific content and talk about it alone, so no one can understand it and cannot learn it. Through the study of mathematics, the ability to think logically is gradually improved. Mathematics is a part of culture, and through the study of mathematics, people's abilities can be cultivated and their qualities can be improved.
Mathematical knowledge can be divided into two kinds, one is more basic and must be learned;There is also a kind of improvement, which is too late to learn when you want to use it. For example, more than ten years ago, everyone felt that the use of computers was promising, so they learned computer languages. I had to learn a little bit about computer languages, but my later experience was that it was useless to learn more language. There is a peculiarity of the language that you don't have to forget it soon after learning it. Another point is that computer technology is developing rapidly, and the relationship between computers and people is getting closer and closer, and it is easy to learn.
To sum up, mathematics is not only knowledge, but also cultivates people's abilities and improves their quality. Quality is a little bit more vain. Some comrades often say that mathematics is the enjoyment of beauty, but I don't understand this very well. You say that mathematics is beautiful, and sometimes you can say that it is very beautiful, but I don't really appreciate how much this beauty is useful to me. Mathematics is the enjoyment of beauty, it can be said, but it should not be exaggerated. In any case, mathematics is a very special science, and it can give people an invisible influence.
Remember a mathematician who said that the quality of mathematics education today determines the level of scientific talent of tomorrow.
About the Author. 
Ding Shisun (1927.)9.5-2019.10.12) Famous mathematician and educator, former president of Peking University, former director of the Department of Mathematics of Peking University.
In order to commemorate Mr. Ding Shisun, the School of Mathematical Sciences of Peking University has set up a memorial webpage for Mr. Ding Shisun
This article**: School of Mathematical Sciences, Peking University, editor: nhyilin.
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