Mathematics, whose English is Mathematics, is a plural noun. "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. "The root of mathematics lies in common sense, and the most obvious example is non-negative integers." 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. "One of its earliest signs is the understanding of similarity. "Similar knowledge is discovered so early", just 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.

In fact, the above-mentioned understanding of the essential characteristics of mathematics is carried out from the origin, existing mode and abstract level of mathematics, mainly from the achievements of mathematical research. Obviously, the result (as a theoretical deduction system) cannot reflect the whole picture of mathematics. Another very important aspect that constitutes the whole of mathematics is the process of mathematical research. On the whole, mathematics is a dynamic process, an "experimental process of thinking" and an abstract generalization process of mathematical truth. Logical deduction system is the natural result of this process. In the process of mathematical research, the rich, vivid and changeable side of mathematical objects can be fully displayed. Paulia (G. Poliva, 1888- 1985) believes that "mathematics has two sides, it is Euclid's strict science, but it is also something else. Mathematics proposed by Euclid's method seems to be a systematic deductive science, but mathematics in the process of creation seems to be an experimental inductive science. " Friedenthal said, "Mathematics is a very special activity, and this view" is different from what mathematics is printed in books and engraved in the mind. "He believes that mathematicians or mathematics textbooks like to describe mathematics as" a well-organized state ",that is," the form of mathematics "is formed by mathematicians through their own organizations (activities); But for most people, they regard mathematics as a tool. They can't live without mathematics because they need to apply mathematics. That is to say, for the public, it is necessary to learn the content of mathematics through mathematics, so as to learn the corresponding (applied mathematics) activities. This is probably what Friedenthal said: "Mathematics is an activity discovered and organized in the interaction of content and form". Efraim Fischbein said, "The ideal of mathematicians is to acquire rigorous, coherent and logical knowledge entities. This fact does not exclude that mathematics must be regarded as a creative process: mathematics is essentially a human activity, and mathematics is invented by human beings. "Mathematical activities consist of the interaction of three basic components: form, algorithm and intuition. Courand and Coulani Robbins also said, "Mathematics is the expression of human will, reflecting positive will, thoughtful reasoning and exquisite and perfect desire. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Although different traditions may emphasize different aspects, only the interaction of these opposing forces and the struggle for their integration constitute the life, utility and high value of mathematical science. "

In addition, there are some broader understandings of mathematics. For example, some people think that "mathematics is a cultural system" and "mathematics is a language", and mathematical activities are social. It is a highly intelligent crystallization of human beings' understanding, adaptation and transformation of nature, self-improvement and social improvement in the historical process of human civilization development. Mathematics has a key influence on the way of human thinking. Some people think that mathematics is an art. "Compared with mathematics as a discipline, I almost prefer to regard it as an art, because the persistent creative activities of mathematicians under the guidance of the rational world (although uncontrolled) are similar to those of artists, such as painters. This is real, not imaginary. Mathematicians' strict deductive reasoning can be compared to special attention skills here. Just as one can't be a painter without certain skills, one can't be a mathematician without a certain level of precise reasoning ability. These qualities are the most basic. Together with other more subtle qualities, they constitute the quality of an excellent artist or an excellent mathematician. In both cases, the most important thing is imagination. " "Mathematics is the music of reasoning" and "Music is the mathematics of images". This is to discuss the essence of mathematics from the process of mathematical research and the qualities that mathematicians should have. Some people regard mathematics as a basic attitude and method to treat things, a kind of spirit and concept, that is, mathematical spirit, mathematical concept and attitude. In the article "Mathematics in Society", Mogens Nice believes that mathematics is a discipline. "In the epistemological sense, it is a science, and its goal is to establish, describe and understand objects, phenomena, relationships and mechanisms in certain fields. If this field is composed of what we usually think of as mathematical entities, then mathematics plays the role of pure science. In this case, mathematics aims at internal self-development and self-understanding and is independent of the outside world. On the other hand, if the field under consideration exists outside mathematics, mathematics plays a role in the use of science. The difference between these two aspects of mathematics is not the problem of mathematics content itself, but the focus of people's attention. Whether it is pure theory or application, mathematics as a science is helpful to produce knowledge and insight. Mathematics is also a system of tools, products and processes, which helps us to make decisions and actions related to mastering practical fields other than mathematics. Mathematics is an aesthetic field, which can provide beauty, pleasure and excitement for many people who are addicted to it. As a discipline, mathematics needs a new generation of people to master its dissemination and development. Mathematics learning will not be carried out automatically at the same time, and it needs to be taught. Therefore, mathematics is also a teaching subject in our social education system. "

As can be seen from the above, people speak from the inside of mathematics (and from the perspectives of mathematical content, expression and research process). The relationship between mathematics and society, mathematics and other disciplines, mathematics and human development is discussed. They all reflect the essential characteristics of mathematics from one side and provide a perspective for us to fully understand the essence of mathematics.

Based on the above understanding of the essential characteristics of mathematics, people have also discussed the specific characteristics of mathematics from different aspects. The general view is that mathematics has the characteristics of abstraction, accuracy and wide application, in which abstraction is the most essential feature. One, 20%. Alexander Love said, "Even superficial mathematical knowledge can easily perceive these characteristics of mathematics: first, it is abstract; Second, accuracy, or better, he said is the rigor of logic and the certainty of conclusions; Finally, it is the extreme universality of its application. " Wang Zikun said, "The characteristics of mathematics are: abstract content, extensive application, rigorous reasoning and certainty of conclusions." This view is mainly based on the content of mathematics. In addition, from the process of mathematical research and the relationship between mathematics and other disciplines, mathematics is also vivid, realistic and quasi-empirical. The characteristics of "falsifiability". The understanding of mathematical characteristics also has the characteristics of the times. For example, the rigor of mathematics has different standards in each historical development period of mathematics. From Euclid's geometry to Luo Baltscheffskij's geometry to Hilbert's axiomatic system, the evaluation criteria for rigor are quite different. Especially after Godel put forward and proved the "Incompleteness Theorem ……", people found that even axiomatization, a rigorous scientific method that was once highly respected, was flawed. Therefore, the rigor of mathematics is reflected in the history of mathematics development and has relativity. Regarding the paradox of mathematics, Paulia pointed out in his "Mathematics and Conjecture" that "mathematics is regarded as an argument science. However, this is only one aspect. The final form of stereotyped mathematics seems to be pure demonstration materials, only proof. However, the process of creating mathematics is the same as any other knowledge. Before proving a mathematical theorem, you should guess the content of this theorem. Before you make a detailed proof, you have to guess the idea of proof. You have to synthesize the observed results and make an analogy. You must do it again and again. The result of mathematicians' creative work is demonstration, that is, proof; But this proof was discovered through reasonable reasoning and conjecture. As long as the learning process of mathematics can slightly reflect the process of mathematical invention, then conjecture and reasonable reasoning should occupy an appropriate position. "It is from this perspective that we say that the certainty of mathematics is relative and conditional, and it is vivid, realistic and quasi-empirical for mathematics. The emphasis on the characteristics of "falsifiability" actually highlights the observation, experiment and analysis in mathematical research. The importance of thinking processes such as comparison, analogy, induction and association.

Humans have been dealing with natural numbers since they learned to count. Later, due to the need of practice, the concept of number was further expanded. Natural numbers are called positive integers, while their opposites are called negative integers, and neutral numbers between positive and negative integers are called 0. They add up to an integer.

For integers, four operations can be performed: addition, subtraction, multiplication and division, which are called four operations. Among them, addition, subtraction, multiplication and division can be carried out in an integer range without obstacles. That is to say, any two or more integers are added, subtracted and multiplied, and their sum, difference and product are still an integer. However, the division between integers may not be carried out smoothly within the integer range.

In the application and research of integer operation, people are gradually familiar with the characteristics of integers. For example, integers can be divided into two categories-odd and even (usually called odd and even) and so on. Using some basic properties of integers, we can further explore many interesting and complex mathematical laws. It is the charm of these characteristics that has attracted many mathematicians to study and explore continuously throughout the ages.