Mathematics is a science that studies quantitative relations and spatial forms, with strict symbol system, unique formula structure and vivid image language. It has three remarkable characteristics: high abstraction, strict logic and wide application.

1. Highly abstract.

The abstraction of mathematics is different from that of other disciplines in object and degree. Mathematics is established and developed in an abstract way.

The abstraction of mathematics abandons the concrete content of the object, and only retains the quantitative relationship and spatial form. In the eyes of mathematicians, there is no difference between five stones, five mountains, five golden flowers and five poisonous snakes. Mathematicians only care about "five"

Another example is the concepts of points, lines and surfaces in geometry, and the concepts of sets, equations and functions in algebra are all products of abstract thinking. Think of "point" as something without size, and the grain length has no width and height; A "line" is considered as an infinite extension without width and height, and a "face" is considered as an infinite extension without height. In fact, theoretical "points", "lines" and "surfaces" do not exist in reality. Only by giving full play to our spatial imagination can we really understand them.

2. The logic is strict.

Mathematics has strict logic, and any mathematical conclusion must be strictly proved by logical reasoning before it can be recognized. Strict logic is not unique to mathematics. Any science must use logical tools, which has its rigorous side. However, the requirements of mathematics for logic are different from other sciences, because the research object of mathematics is highly abstract quantitative relations and spatial forms, and it is a formal ideological material. Many mathematical results are difficult to find realistic prototypes with intuitive significance, and they are often studied under ideal conditions. For example, the derivation of the root formula of a quadratic equation in one variable, the determination of the positional relationship between two straight lines, the derivation of an infinitesimal quantity and so on. The correctness of mathematical operation, mathematical reasoning, mathematical proof, mathematical theory, etc. It can't be tested by repeatable experiments like natural science, but can only be realized by strict logical methods.

3. Wide applicability. As a tool or means, mathematics is applied to almost any science and technology and all social fields. Mathematicization of various sciences is a major trend in the development of modern science. Professor Hua, a famous mathematician in China, once pointed out: "The universe is big, the particles are tiny, the speed of rockets, the cleverness of chemical engineering, the change of the earth, the mystery of biology, the daily use, and mathematics is everywhere." This is an incisive summary of the wide application of mathematics.

There are countless examples of mathematical application. The discovery of Neptune, one of the nine planets in the solar system, and the discovery of electromagnetic waves are brilliant examples of mathematical applications in history.

These three remarkable characteristics of mathematics are interrelated. The high abstraction of mathematics determines the rigor of its logic and ensures its wide application. These characteristics also profoundly reflect that practice is the source of mathematics, and the need of practical application is the purpose of learning mathematics.