1. Mathematics is a universal means for human beings to strictly describe and deduce abstract structures and patterns of things, and can be applied to any problems in the real world. All mathematical objects are artificially defined in essence. In this sense, mathematics belongs to formal science, not natural science. Different mathematicians and philosophers have a series of views on the exact scope and definition of mathematics.
2. Basic mathematics: study the basic concepts, properties, axioms and theorems of mathematics, and explore the basic structure, properties and laws of mathematics. Applied mathematics: applying mathematical methods and theories to various practical problems, including natural science, social science, engineering and other fields. Computational mathematics: Study the theory and methods of mathematical calculation, including the design, implementation and application of computers.
3. Probability statistics: the mathematical theory and method of studying random phenomena, as well as the collection, collation, analysis and interpretation of data. Mathematical analysis: study the nature, limit, continuity, differential calculus and integral calculus of functions. Algebra: the theory and method of studying algebra, including groups, rings, fields, linear algebra, etc.
4. Geometry: Mathematical theories and methods to study the relationship between shape, size and position, including plane geometry and solid geometry. Discrete mathematics: the study of mathematical theories and methods of discrete structures, including graph theory and combinatorial mathematics.
Mathematics is widely used in various fields. For example, in natural science, mathematics is used to describe physical phenomena, study material properties, and deduce natural laws. In social science, mathematics is used to study human behavior, social phenomena and economic problems. In engineering, mathematics is used in design, manufacturing, optimization and so on.
6. The development of mathematics has gone through different stages, and each stage has its unique contribution and characteristics. For example, ancient Greek mathematics paid attention to logical reasoning and proof, medieval European mathematics paid attention to calculation and application, and modern mathematics paid attention to abstraction and generalization.
7. Mathematical research methods include deductive reasoning, inductive reasoning and analogical reasoning. Deductive reasoning is one of the most commonly used methods, which deduces new conclusions and theorems through known facts and axioms. Inductive reasoning is to summarize the general laws and trends through the observation and analysis of a large number of data.
8. Analogical reasoning is to draw a conclusion that two or more things may be similar in other aspects by comparing their similarities. The study of mathematics is not only of great significance in theory, but also widely used in practice. For example, in computer science, mathematics is used in algorithm design, data structure, cryptography and other fields.