First, introduce the essence of mathematics in detail.
1, understand the basic concepts of mathematics. The teaching of basic concepts is very important, and students' understanding of concepts will be different with different learning processes.
2. Master mathematical thinking methods. The basic concepts of mathematics often contain important mathematical thinking methods, which are extremely rich.
3. Understanding of the unique way of thinking in mathematics. Every subject has its own unique way of thinking and perspective of understanding the world, and mathematics is no exception.
4. Appreciate the beauty of mathematics. Being able to grasp the essence of mathematical beauty is also helpful to cultivate students' attitude towards mathematics and mathematics learning, and then affect the process of mathematics learning and academic performance.
5. Pursuing the spirit of mathematics, rationality and inquiry. The rational spirit of mathematics and the inquiry spirit of mathematics are the driving forces that support mathematicians to study mathematics and then the world.
Second, the introduction of mathematics
Mathematics is a subject that studies concepts such as quantity, structure, change, space and information. Mathematics is a universal means for human beings to strictly describe and deduce the abstract structure and mode of things, and can be applied to any problem in the real world. All mathematical objects are artificially defined in essence. In this sense, mathematics belongs to formal science, not natural science.
Three main types of mathematical definitions:
I. Logic
The early definition of mathematical logic is Benjamin Peirce's Science of Drawing Inevitable Conclusions (1870). Experts such as Principia and Mathematica put forward a philosophical program called logicism, trying to prove that all mathematical concepts, statements and principles can be defined and proved by symbolic logic. The logical definition of mathematics is Russell's "All mathematics is symbolic logic".
Second, the definition of intuitionism
The definition of intuitionism comes from mathematician L.E.J.Brouwer, who equates mathematics with some psychological phenomena. An example of the definition of intuitionism is that "mathematics is a psychological activity constructed one after another". Intuitionism is characterized by rejecting some mathematical ideas that are considered effective according to other definitions.
Third, the definition of formalism.
Formalism defines mathematics through mathematical symbols and operational rules. Haskell Curry simply defined mathematics as "formal system science". A formal system is a set of symbols, or marks, and there are some rules that tell how the marks are combined into formulas.
In the formal system, the word axiom has a special meaning, which is different from the ordinary meaning of "self-evident truth" in the formal system. Axiom is a combination of symbols contained in a given formal system, without using the rules of the system to deduce it.