The formalization of mathematics includes three levels: symbolicization, logic, and axiomatization

The question is not precise enough.

The General High School Mathematics Curriculum Standards state that "formalization is one of the basic characteristics of mathematics. In mathematics teaching, learning formal expression is a basic requirement, but it should not be limited to formal expression, and the understanding of the essence of mathematics should be emphasized, otherwise the vivid and lively mathematical thinking activities will be submerged in the ocean of formalization.

Modern developments in mathematics have also shown that total formalization is impossible. Therefore, the high school mathematics curriculum should return to the basics and strive to reveal the essence of the development process of mathematical concepts, laws, and conclusions. ”

The so-called "mathematical form" refers to the use of specific mathematical language, including mathematical symbolic language, image language and written language, to express the spatial structure and quantitative relationship of natural and social phenomena, that is, mathematical concepts, laws and conclusions with a relatively fixed style, which have the following characteristics:

1) Stability. Once mathematical concepts, laws, conclusions, and other contents become "forms", they have relatively stable characteristics and will never change due to changes in the environment and conditions.

2) Generalization. The mathematical form is the result of the abstract generalization of countless concrete things, and should be a reaction to the study of quantitative relations or the essential properties of figures.

3) Brevity. The simplest is often the most profound, and the more concise things are, the more vital and valuable they are. The mathematical form is known for the simplicity of its expression.

4) Extensiveness. The generalization of the form of mathematics determines that it is extensive, and it can truly achieve what Professor Hua Luogeng said: "Mathematics is a principle, countless contents, and a method, which is useful everywhere." ”

5) Operability. Stylized operations in the relevant mathematical form can be referred to as behavioral patterns. There are two kinds of human behavior patterns, one is the behavior mode that requires intellectual investment and thinking participation; One is a pattern of behavior that requires less intellectual input and mental involvement.

In all the activities of mathematics learning and solving mathematical problems, the content of creative thinking accounts for only a small part, and the application is more of a stylized operation. This kind of operation is skilled, accurate, fast and efficient. Most of the students' problem solving is modeled according to the established rules.

Even if the more difficult ones require a certain amount of creative thinking, the "roots" of creation are still rooted in the solid soil of basic mathematical forms. The basic mathematical form is the source and archetype of creation. Of course, even if the operation is simple, mechanical, and procedural, it is necessary to strive to increase the content of intelligence and thinking.

To be precise, symbolic logic is mathematical rather than mathematical.

Mathematical logic is also known as symbolic logic and theoretical logic. It is both a branch of mathematics and a branch of logic. It is a discipline that studies logic or formal logic with mathematical methods.

The object of study is the formal system after the symbolization of the two intuitive concepts of proof and computation. Mathematical logic is an indispensable part of the foundations of mathematics. Although the word logic is included in the name, the juxtaposition does not belong to the category of pure logic.

mathematical

Logic) is a branch of mathematics closely related to the foundations of mathematics, theoretical computational science, and philosophical logic. His research interests include the study of logical mathematics and the application of formal logic to other areas of mathematics. The scope of mathematical logic is the part of logic that can be mathematically modeled.

Mathematical logic can be roughly divided into four parts:1Set theory; 2.

simulation theory; 3.Rent conjecture and 4Theory of Evidence and Constructive Mathematics.

I think predicate logic is to treat the predicate Zaodou as a function f(x) and then treat the subject as an individual, and Lu Somo assigns these individuals to this function to get the truth value of the sentence.

Propositional omission logic is concerned with the relationship between propositions and propositions, and propositions and propositions are connected with or without the truth of these logical symbols, so predicate logic can explain the truth conditions of atomic propositions and the truth conditions of combinatorial propositions, while propositional logic only explains the truth conditions of propositions after atomic propositions are combined by various relations.

For example, p (Xiao Ming is a human) is a true proposition, q (the puppy is a human) is a false proposition, and p&q is a false proposition according to propositional logic, and it is impossible to explain why p or q is a true proposition or a false proposition.

Predicate logic can explain why Xiao Ming is a human being and is not a human being, because the function of being a human is true to Xiao Ming and false to the puppy.

The first stage in the development of the axiomatic method was from Aristotle's complete syllogism to the advent of Euclid's Geometry Around the 3rd century BC, the Greek philosopher and logician Aristotle summarized the rich materials of geometry and logic, systematically studied the syllogism, took mathematics and other deductive disciplines as examples, took the complete syllogism as an axiom, and derived all other syllogisms, so that the entire syllogism system became an axiomatic system. Aristotle proposed the first codified axiomatic system in history

Aristotle's thinking and method deeply influenced the Greek mathematician Euclid applied the axiomatic deductive method of formal logic to geometry, thus completing the important work in the history of mathematics "Geometric Originals" He used abstract analysis methods to extract a series of basic concepts and axioms from the ancient geometry and the original intuition about geometric shapes He summarized and summarized 14 basic propositions, including 5 axioms and 9 axioms, and then started from this, Using deductive methods to deduce all the geometric knowledge known at that time and organize them into a deductive system The book "Geometric Originals" applies the axiomatic methods initially summarized by Aristotle to mathematics, collating, summarizing and developing a large amount of mathematical knowledge in the classical period of Greece, and sets up an immortal monument in the history of mathematical development

The objects, properties, and relations studied in axioms are called "domains", and these objects, properties, and relations are represented by initial concepts For example, in Euclidean's Geometry Primitives, only "points", "lines", and "planes" need to be taken; "In ......Above", "on ......The three types of objects represented by the first three concepts and the three relations represented by the last three concepts are the fields of this kind of geometry The axioms established according to the view that "an axiom system has only one field" are called substantive axioms This axioms are the systematic arrangement of empirical knowledge, and the axioms are generally self-evident Therefore, Euclidean's "Geometry Originals" is a model of substantive axioms

Development of axiomatic methods.

The development of axiomatic methods has gone through three stages: substantive (or substantial) axiomatic stage, formal axiomatic stage and pure formal axiomatic stage, and the theoretical system models constructed with them are respectively the Geometric Original, the Geometric Foundation, and the ZFC axiomatic system.

Although Geometry Primitives has created a precedent for mathematical axiomatic methods, its axiomatic system still has many imperfections, which are mainly manifested in the following aspects: (1) some definitions use some concepts whose meanings have not yet been determined; (2) some definitions are redundant; (3) The process of proving some theorems often depends on the intuition of graphs; (4) Whether some axioms (i.e., parallel axioms) can be proved or replaced by other axioms These problems have become the subject of research by many mathematicians in the future, and through the study of these problems, the axiomatic methods have been continuously improved.