Introduction Mathematics is a subject that studies concepts such as quantity, structure, change and spatial model. By using abstract and logical reasoning, the shape and motion of objects are counted, calculated, measured and observed. Mathematicians have extended these concepts in order to express new conjectures with formulas and establish strictly deduced truths from properly selected axioms and definitions. Mathematical attribute is the measurable attribute of anything, that is, mathematical attribute is the most basic attribute of things. The existence of measurable attributes has nothing to do with parameters, and the result depends on the selection of parameters. For example, time, whether measured in years, months, days or hours, minutes and seconds; Space, whether measured in meters, microns, inches or light years, always has their measurable properties, but the accuracy of the results is related to these reference coefficients. Mathematics is a science that studies quantitative relations and spatial forms in the real world. Simply put, it is the science of studying numbers and shapes. Due to the needs of life and labor, even the most primitive people know simple counting, and it has developed from counting with fingers or objects to counting with numbers. The knowledge and application of basic mathematics will always be an indispensable part of individual and group life. The refinement of its basic concepts can be found in ancient mathematical documents of ancient Egypt, Mesopotamia and ancient India. Since then, its development has made small progress until the Renaissance in16th century, and the mathematical innovation generated by the interaction with new scientific discoveries led to the acceleration of knowledge, until today. Today, mathematics is used in different fields of the world, including science, engineering, medicine and economics. The application of mathematics in these fields is usually called applied mathematics, and sometimes it will lead to new mathematical discoveries and the development of new disciplines. Mathematicians also study pure mathematics with no practical value, even though its application is often discovered later. French Bourbaki School, founded in 1930s, thinks that mathematics, at least pure mathematics, is a theory to study abstract structures. Structure is a deductive system based on initial concepts and axioms. Boone School believes that there are three basic abstract structures: algebraic structure (group, ring, domain …), ordered structure (partial order, total order …) and topological structure (neighborhood, limit, connectivity, dimension …). Etymological mathematics; ; Greek: μ α θ η μ α τ κ) comes from the western ancient Greek word μθξμα(máthēma), which has a narrow and technical meaning-"mathematical research", even in its etymology. Its adjective μ α θ η μ α τ κ (Math ? matikó s) indicates that it is related to study or hard work, and it can also be used to refer to mathematics. Its superficial plural form in English and its superficial plural form in French, les mathématiques, can be traced back to the neutral plural mathematica in Latin. It was translated by Cicero from the Greek plural τ α μ α θ α ι κ (Tamath \ Umatiká), and Aristotle used it to refer to the concept of "everything counts". (Latin: Mathemetica) means counting and counting technology. Ancient mathematics in China was called arithmetic, also called arithmetic, and later it was changed to mathematics. History chip, a counting tool used by the Inca Empire. Mathematics, which originated from the early production activities of human beings, is one of the six great arts in ancient China, and is also regarded as the starting point of philosophy by ancient Greek scholars. The Greek word for mathematics, μ α θ η μ α ι κ (Mathematikó s), means "the basis of learning" and comes from μθιμα(máthema) ("science, knowledge and learning"). The evolution of mathematics can be regarded as the continuous development of abstraction and the extension of subject matter. The first abstract concept is probably number, and its cognition that two apples and two oranges have something in common is a great breakthrough in human thought. In addition to knowing how to calculate the number of actual substances, prehistoric humans also learned how to calculate the number of abstract substances, such as time-date, season and year. Arithmetic (addition, subtraction, multiplication and division) will naturally occur. Ancient stone tablets also confirmed the knowledge of geometry at that time. In addition, writing or other systems that can record numbers are needed, such as Mu Fu or chips used by the Inca Empire to store data. There are many different counting systems in history. Since the historical era, the main principles in mathematics have been formed, which are used for tax and trade calculation, for understanding the relationship between numbers, for measuring land and predicting astronomical events. These needs can be simply summarized as the study of quantity, structure, space and time in mathematics. By16th century, elementary mathematics, such as arithmetic, elementary algebra and trigonometry, had been basically completed. The appearance of the concept of variables in the17th century made people begin to study the relationship between variables and the mutual transformation between graphs. In the process of studying classical mechanics, the method of calculus was invented. With the further development of natural science and technology, set theory and mathematical logic, which are produced for studying the basis of mathematics, have also begun to develop slowly. Mathematics has been continuously extended since ancient times, and has rich interaction with science, and both of them have benefited a lot. There are many discoveries in mathematics in history, and they are still being discovered today. According to Mikhail B. Sevryuk's record in the Bulletin of the American Mathematical Society of June 5438+ 10, 2006: "Since 1940 (the first year of mathematical review), the number of papers and books in the database of mathematical reviews has exceeded1900,000, with an annual increase of more than 750,000. Most of this learning sea is a new mathematical theorem and its proof. " Mathematics branch 1. Arithmetic 2. Elementary algebra 3. Advanced algebra iv. Number theory 5. Euclidean geometry 6. Non-Euclidean geometry 7. Analytic geometry 8. Differential geometry 9. Algebraic geometry 10. Projective geometry 1. Topological geometry 12. Topology 65438+. Calculus 15. Theory of real variable function 16. Probability and quantity statistics 17. Complex variable function theory 18. Functional analysis 19. Partial differential equation 20. Ordinary differential equation 2 1. Mathematical logic. Fuzzy mathematics. Operational research. At the end of primitive commune, after the germination of ancient mathematics in China, private ownership and commodity exchange appeared, the concepts of number and shape were further developed. The pottery unearthed during Yangshao culture period has been engraved with the symbol representing 1234. By the end of the primitive commune, written symbols had begun to replace knotted notes. Pottery unearthed in Xi 'an Banpo has an equilateral triangle composed of 1 ~ 8 dots, and a pattern of 100 small squares divided into squares. The houses in Banpo site are all round and square. In order to draw circles and determine straightness, people have also created drawing and measuring tools such as rulers, moments, rulers and ropes. According to Records of Historical Records Xia Benji, Yu Xia used these tools in water conservancy. In the middle of Shang Dynasty, a set of decimal numbers and notation had been produced in Oracle Bone Inscriptions, the largest of which was 30 thousand; At the same time, the Yin people recorded the date of 60 days with 60 names, including Jiazi, Yechou, Bingyin and Dingmao, which were composed of ten heavenly stems and twelve earthly branches. In the Zhou Dynasty, eight kinds of things were previously represented by eight diagrams composed of yin and yang symbols, which developed into sixty-four hexagrams, representing sixty-four kinds of things. The book Parallel Computation in 1 century BC mentioned the methods of using moments of high, deep, wide and distance in the early Western Zhou Dynasty, and listed some examples, such as hook three, strand four, chord five and ring moments can be circles. It is mentioned in the Book of Rites that the aristocratic children of the Western Zhou Dynasty have to learn numbers and counting methods since they were nine years old, and they have to be trained in rites and music, shooting, controlling, writing and counting. As one of the "six arts", number has begun to become a special course. During the Spring and Autumn Period and the Warring States Period, calculation has been widely used and decimal notation has been used, which is of epoch-making significance to the development of mathematics in the world. During this period, econometrics was widely used in production, and mathematics was improved accordingly. The contention of a hundred schools of thought in the Warring States period also promoted the development of mathematics, especially the dispute of rectifying the name and some propositions were directly related to mathematics. Famous experts believe that the abstract concepts of nouns are different from their original entities. They put forward that "if the moment is not square, the rules cannot be round", and defined "freshman" (infinity) as "nothing beyond the maximum" and "junior" (infinitesimal) as "nothing within the minimum". He also put forward the idea that "one foot is worth half a day, which is inexhaustible". Mohism believes that names come from things, and names can reflect things from different sides and depths. Mohist school gave some mathematical definitions. Such as circle, square, flat, straight, sub (tangent), end (point) and so on. Mohism disagreed with the proposition of "one foot" and put forward the proposition of "non-half" to refute: if a line segment is divided into two halves indefinitely, there will be a non-half, which is a point. The famous scholar's proposition discusses that a finite length can be divided into an infinite sequence, while the Mohist proposition points out the changes and results of this infinite division. The discussion on the definition and proposition of mathematics by famous scholars and Mohists is of great significance to the development of China's ancient mathematical theory.