Horatius mathematics is a plural noun in English. "Mathematics used to be arithmetic, geometry, astronomy and music, which was higher than grammar, rhetoric and dialectics." Since ancient times, most people regard mathematics as a kind of knowledge system, which is the systematic summation of theoretical knowledge formed through strict logical reasoning. It not only reflects people's understanding of "Engels' spatial form and quantitative relationship in the real world", but also reflects people's understanding of "possible quantitative relationship and form" Mathematics can not only come from the direct abstraction of the real world, but also from the labor creation of human thinking. Judging from the development history of human society, people's understanding of the essential characteristics of mathematics is constantly changing and deepening. "Mathematics is based on common sense, and the most obvious example is non-negative integer. "Euclid's arithmetic comes from non-negative integers in common sense. Until the middle of19th century, the scientific exploration of numbers remained in common sense." Another example is the similarity in geometry. "In individual development, geometry even precedes arithmetic", and its "one of the earliest signs is the understanding of similarity." Finding similar knowledge so early is "like being born." Therefore, before the19th century, it was generally believed that mathematics was a natural science and an empirical science, because mathematics was closely related to reality at that time. With the deepening of mathematical research, the view that mathematics is a deductive science gradually occupied a dominant position after the middle of19th century, which was developed in the research of Bourbaki school. They think that mathematics is a science of studying structure, and all mathematics is based on algebraic structure. Corresponding to this view, from Plato in ancient Greece, many people think that mathematics is the knowledge of research mode. Mathematician A.N. Whitehead (186- 1947) said in Mathematics and Goodness, "The essential feature of mathematics is to study patterns in the process of abstracting from patterned individuals. 193 1 year, the proof of Godel's incompleteness theorem (k, G0de 1, 1978) declared the deficiency in the deductive system of axiomatic logic, so people thought that mathematics is an empirical science, and the famous mathematician von Neumann thought that mathematics has both deductive science and empirical science. For the above viewpoints about the essential characteristics of mathematics, we should analyze them from a historical perspective. In fact, the understanding of logarithmic essential characteristics develops with the development of mathematics. Because mathematics comes from the practice of distributing goods, calculating time and measuring land and volume, the mathematical object at this time (as the product of abstract thinking) is very close to objective reality, and it is easy for people to find the realistic prototype of mathematical concepts, so people naturally think that mathematics is an empirical science; With the deepening of mathematical research, non-Euclidean geometry, abstract algebra and set theory have emerged, especially modern mathematics is developing in the direction of abstraction, pluralism and high dimension. People's attention is focused on these abstract objects, and the distance between mathematics and reality is getting farther and farther. Mathematical proof (as a deductive reasoning) plays an important role in mathematical research. Therefore, mathematics, as a free creation of human thinking, is a science that studies the relationship between quantity and abstract structure. These understandings not only reflect the deepening of people's understanding of mathematics, but also are the result of people's understanding of mathematics from different aspects. As someone said, "Engels thought that mathematics is the study of quantitative relations and spatial forms in the real world, which is not contradictory to bourbaki's structural view. The former reflects the origin of mathematics, while the latter reflects the level of modern mathematics. It is a building built by a series of abstract structures. "Mathematics is the knowledge of research methods, which explains the essential characteristics of mathematics from the perspective of abstract process and level of mathematics. In addition, from the ideological source, people regard mathematics as a science of deduction and research structure, which is based on human innate belief in the inevitability and accuracy of mathematical reasoning and a concentrated expression of self-confidence in their own rational ability, roots and strength. Therefore, people think that this method of developing mathematical theory, that is, deductive reasoning from axioms that are self-evident, is absolutely reliable, that is, if axioms are true, then the conclusions deduced from axioms must also be true. Applying these seemingly clear, correct and perfect logics, the conclusions drawn by mathematicians are obviously beyond doubt and irrefutable. The ancients used to tie knots with ropes to let them know how many livestock there were. One knot represents one ~ two knots represent two ~ so slowly mathematics comes out-