The cultural value of mathematics. Mathematics is an important foundation of philosophical thinking. Its position in scientific culture also makes it an important foundation of philosophical thinking. Many important debates in the field of philosophy in history often involve the understanding of some basic problems in mathematics. Thinking about these problems will help us to understand mathematics and the related debates in philosophy correctly. (1) Mathematics-rooted in the external expression of practical mathematics, more or less related to human intellectual activities. Therefore, in the relationship between mathematics and practice, mathematics has always been advocated as "the free creation of human spirit", denying that mathematics comes from practice. In fact, all the development of mathematics comes down to practical needs to varying degrees. We can see from the Oracle Bone Inscriptions of Yin Dynasty in China that our ancestors had used decimal counting method at that time. In order to meet the needs of agriculture, they matched "ten branches" and "twelve branches" into sixty jiazi. Thousands of years of history show that this calendar calculation method is effective. Similarly, due to the calculation of business and debt, the ancient Babylonians had multiplication tables and reciprocal tables, and accumulated a lot of materials belonging to the category of elementary algebra. In Egypt, due to the need to re-measure the land after the Nile flooded, they accumulated a lot of geometric knowledge to calculate the area. Later, with the development of social production, especially the astronomical survey to meet the needs of agricultural farming and navigation, elementary mathematics gradually formed, including most of the mathematical knowledge we learned in middle schools today. Later, the industrial revolution triggered by the invention of steam engine and other machinery required a more detailed study of motion, especially variable speed motion, and a large number of mechanical problems appeared, which prompted calculus to appear after a long period of brewing. Since the 20th century, with the rapid development of modern science and technology, mathematics has entered an unprecedented period of prosperity. During this period, many new branches of mathematics appeared: computational mathematics, information theory, cybernetics, fractal geometry and so on. In short, the need of practice is the most fundamental driving force for the development of mathematics. The abstraction of mathematics is often misunderstood. Some people think that the axioms, postulates and theorems of mathematics are only the products of mathematicians' thinking. Mathematicians work with a piece of paper and a pen, which has nothing to do with reality. In fact, even in the case of Euclidean geometry, which first appeared as an axiomatic system, the geometric intuition of practical things and the phenomena developed by people in practice, although not in line with various axiomatic systems of mathematicians, still contain the core of mathematical theory. When a mathematician aims at establishing an axiomatic system of geometry, geometric drawing and intuitive phenomena are bound to be linked with his mind. A person, even a talented mathematician, must be guided by practice consciously or unconsciously in the process of mathematical theory research in addition to strict mathematical thinking training if he wants to obtain scientific achievements in mathematical research. It can be said that without practice, mathematics will become passive water and rootless wood. In fact, even for Euclidean geometry, which was first introduced as an axiomatic system, the geometric intuition of actual things and phenomena found by people in practice, although not in line with the procedures of the axiomatic system of mathematicians, still contains the core of mathematical theory. When a mathematician takes the establishment of geometric axiom system as his goal, his mind is bound to be linked with geometric drawing and intuitive phenomena. A person, even a talented mathematician, can get scientific achievements in mathematical research, except that he has received strict mathematical thinking training. In the process of learning mathematical theory, he is bound to be consciously or unconsciously guided by practice in raising questions, choosing methods and prompting conclusions. It can be said that without practice, mathematics will become passive water and a tree without roots. However, due to the characteristics of mathematical rational thinking, it will not be satisfied with only studying the realistic quantitative relations and spatial forms, but also strive to explore all possible quantitative relations and spatial forms. In ancient Greece, mathematicians have gone beyond the method of measuring line segments within the limited calibration accuracy, and noticed the existence of unfair measurement line segments, that is, the existence of irrational numbers. This is actually one of the most difficult concepts in mathematics-continuity and infinity. It was not until two thousand years later that the same problem led to the in-depth study of limit theory and greatly promoted the development of mathematics. Imagine what kind of situation we would face without the concept of real numbers today. At this time, people can't measure the diagonal length of a square, nor can they solve a quadratic equation. As for limit theory and calculus, it is even more impossible to hold water. Even if people can apply calculus like Newton, they will feel at a loss when judging the truth of the conclusion. How far can technology go in this case? For example, when Euclidean geometry was produced, people doubted the independence of one of the postulates. In the first half of19th century, mathematicians changed this postulate and got another possible geometry-non-Euclidean geometry. The founder of this geometry showed great courage. Because this geometric conclusion is very absurd from common sense. For example, "the area of a triangle will not exceed a positive number." It seems that there is no such geometric position in the real world. But nearly a hundred years later, non-Euclidean geometry is the most suitable geometry in the theory of relativity discovered by physicist Einstein. For another example, in the 1930s, Godel obtained that mathematical conclusions are uncertain results, some of which have found applications in the analysis of algorithmic languages in recent decades. In fact, many applications of mathematics in some fields or some problems, once practice promotes mathematics, mathematics itself will inevitably gain a kind of power, making it possible to go beyond the boundaries of direct application. This development of mathematics will eventually return to practice. In a word, we should vigorously advocate the study of mathematical topics directly related to the current practical application. Especially the mathematical problems in the real economic construction. However, we should also establish an organic connection between pure science and applied science, and establish a balance between abstract commonness and colorful individuality, so as to promote the coordinated development of the whole science. (2) Mathematics-full of dialectics. Because of the strict characteristics of mathematics, few people doubt the correctness of mathematical conclusions. On the contrary, mathematical conclusions often become the model of truth. For example, there is no doubt that people often use "as sure as one plus one equals two" to express their conclusions. In our primary and secondary school teaching, mathematics is only allowed to imitate, practice and recite. Is mathematics really an eternal absolute truth? In fact, the truth of mathematical conclusions is relative. Even a simple formula like 1+ 1=2 has its shortcomings. For example, in Boolean algebra, 1+ 1=0! Boolean algebra is widely used in electronic circuits. Euclidean geometry is always correct in our daily life, while non-Euclidean geometry is suitable for studying some problems of celestial bodies or the motion of fast particles. Mathematics is actually very diverse, and its research scope is constantly expanding with the emergence of new problems. Like all sciences, if mathematicians stick to the ideas, methods and conclusions of their predecessors, mathematical science will not progress. It is wrong to regard the rigor and axiomatic system of mathematics as a kind of "dogma", nor can it be said to Confucius by scholars in feudal times that "truth" has been included in what saints said, and future generations can only interpret it. The history of mathematics development can prove that it is the innovative spirit of mathematicians, especially young mathematicians, who dare to challenge the old-fashioned ideas that makes the face of mathematics constantly updated. Mathematics has grown into such a vibrant subject today. The axiomatic system of mathematics has never been an unquestionable and unchangeable "absolute truth". Euclid's geometric system is the earliest mathematical axiom system, but from the beginning, some people suspected that the fifth postulate was not independent, that is, it could be deduced from other parts of the axiom system. For more than two thousand years, people have been looking for answers. Finally, in the19th century, non-Euclidean geometry was discovered. Although people have been bound by Euclidean geometry for a long time, they finally accepted different geometric axioms. If some mathematicians in history were more innovative to challenge the old system, non-Euclidean geometry might have had a mathematical axiom system reflecting the requirements of internal logic rigor hundreds of years ago. In a subject field, when the relevant knowledge is accumulated to a certain extent, the theory will need a bunch of seemingly scattered achievements to be expressed in the form of a certain system. This requires re-understanding, re-examining and rethinking the existing facts, creating new concepts and methods, and making the theory contain the most common and newly discovered laws as much as possible. This is really a hard process of theoretical innovation. The same is true of mathematical axiomatization, which means that mathematical theory has developed to a mature stage. But this is not the end of understanding once and for all. Existing knowledge may be replaced by deeper knowledge in the future, and existing axioms may also be replaced by a more general axiom system containing more facts in the future. Mathematics is developing in the process of constant renewal. There is a view that the application of mathematics is to put familiar mathematical conclusions into practical problems. I think that teaching in primary and secondary schools is to teach students these eternal dogmas. In fact, the application of mathematics is extremely challenging. On the one hand, we need to deeply understand the actual problem itself, on the other hand, we need to master the true meaning of relevant mathematical knowledge, and more importantly, we need to creatively combine the two. As far as the content of mathematics is concerned, mathematics is full of dialectics. In the development period of elementary mathematics, metaphysics is dominant. In the eyes of mathematicians or other scientists in this period, the world is made up of rigid things. Accordingly, the object of mathematical research at that time was constant, that is, constant quantity. Descartes' variables are the turning point of mathematics. He combined geometry and algebra, two completely different fields in elementary mathematics, and established the framework of analytic geometry, which has the characteristics of expressing movement and change. Dialectics therefore entered mathematics. Calculus, which came into being shortly thereafter, abandoned the idea of taking the conclusion of elementary mathematics as eternal truth, often made opposite judgments and put forward some propositions that elementary mathematics representatives could not understand at all. Mathematics has come to such a field that even simple relations have taken a completely dialectical form, forcing mathematicians to become dialectical mathematicians unconsciously. The objects of mathematical research are full of contradictory opposites: curve and straight line, infinite and finite, differential and integral, accidental and inevitable, infinite and infinitesimal, polynomial and infinite series. Because of this, classical Marxist writers often mention mathematics in their discourses on dialectics. Learning some math will help us.