Basic Introduction Title: Mathematics and Its Understanding ISBN :7040 1035 1+0 Pricing: 18.0 Publishing House: Higher Education Press, Book Information, Introduction, Catalogue, Book Information Book Number: 106 1279 Author. Understand the logical category of mathematics; Recognition of real numbers; Periodic mathematics and its understanding; The basis of "marriage" between modern mathematics and social science. Introduction of the first chapter of the preface to the catalogue: epistemology and mathematical thinking of mathematics I. On the thinking characteristics of orientals II. On the deepening process of philosophy and its epistemology III. Mathematical thinking and rational reasoning II. On learning psychology I. Understanding cognitive process II. Re-understand learning. On the age characteristics of learning Chapter II A glimpse of mathematics 1 A glimpse of the history of mathematics. Basic Mathematics Graphics II. History of mathematical center migration iii. Several important stages in the history of mathematics ii. Understanding of the whole space of applied mathematics I. Schematic diagram of the whole space of applied mathematics II. Modeling-Mathematical Approximation III. The internal process of mathematics. Mathematics returning to the objective world. Show up for a glimpse. Two. The emergence of classical axiomatic thought II. The emergence of modern axiomatic system III. Axiomatization. Axiomatic system and formalization. The unified relationship of formal system 4. Axiomatic thought (generalized axiomatization) 5. Axiomatic praise (generation summary) Chapter III Several basic features in mathematics 1 Basic object of mathematics: set 1. Introduction to set cognition II. The concept of set and its extension. Abstraction and mathematics of set elements II. Basic mathematical relations 1. Ordering relationship 2. Operational relationship 3. Mapping relation 3. The basic structure of mathematics 1. Ordering structure 2. Three. A brief history of infinite cognition iv. A brief introduction to the cognitive state of infinity in modern mathematics Chapter IV Duality, Duality and Complete Space 1 Duality Space Cognition I. The concept of dual space with inner product characteristics and its generalization II. Duality principle and its application. Discrimination of system variables, arguments and influencing factors II. Principle of Duality 1. Duality characteristics and its mechanism in micro-world cognition 2. The dual structure of the macro world. The ubiquity of binary structure 3. Law of unity of opposites and theory of complete space. Introduction of the concept of complete space II. Characteristic cognition of duality in complete space III. Description of the mutual expansion relationship of duality in complete space (several model examples) 4. Understanding the theory of "soft science era" Chapter 5 Understanding the category of mathematical logic 1 logical concept 2 formal logic and symbolic logic 1. Basic concepts and characteristics. Basic content 3. Research on thinking methods. The development of formal logic. Symbolic logic 3. Simple understanding of mathematical logic 1. A brief review II. Basic feature 3. Main branch 4. Basic content 4. The essence of formal logic recognition 1. Understanding of thinking form II. Understanding of the law of thinking (four laws) 3. A basic law of formal logic is the Law of Causality. The basic feature of the material world is movement. The material world is a dynamic system. The law of causality coincides with the dynamic system characteristics of the physical universe X 7. Guess: the background space of logical thinking = complete universe = logical thinking object set 8. Inference 1: complete universe. X*) = is the largest dynamic system. Inference 2: The complete universe is a high-dimensional space. Note 5 Understanding of dialectical logic. The objective basis of dialectical logic. On dialectical logic. Law of dialectical logic. The essence and characteristics of dialectical logic. Mathematical logic and its understanding. Introduction to the problem. Mathematics has the basic characteristics of logic. The unique characteristics of mathematics. Concept definition of mathematical logic. Several main logics. Understanding of the relationship between categories 7 Understanding of multivalued logic 1. Concept introduction. Existence mechanism 3. Classification of uncertain problems. Fuzzy logic and its characteristics. Recognition of real numbers 1 Review of several important achievements of real number recognition 1. Arithmetic-operational properties of real numbers II. Number Theory-Combinatorial Characteristics of Integers III. Extension of number system 4. Study on the order, density and completeness of real numbers 5. Understanding of set theory 2 Macro appreciation of real number sets 1. Four stages of real number set ii. Re-understanding of real number set 3. Real number set and the history of mathematics 4. Square inches are embedded in Universe 5. The principle of centering. Appreciation of a point on the real axis 1. Divergence and the view of material structure II. Understanding of interval endpoint 3. Understanding the real number closest to any real number: dense (granular) view and continuous (stream) 3. Human measurement activities are almost inaccurate. 5 Can the real axis be understood thoroughly? Chapter 7: Additive Mathematics: Algebra Understanding the Basic Concept of Algebra 1 and its Notes I and II. The ring. Ideals and combinatorial rings III. Field. Body. Finite field. Extended fields and boolean algebras. Grid four. Linear space. Modulus sum algebra v. tensor algebra VI Notes and Notes 2 Text Algebra i. Text Algebra II. Symbolic algebra III. Equation theory four. Equation pure mathematics 2. Abstract algebra course: rings and hypercomplex systems III. The fourth lesson of algebra. The essence of complex number and its position in algebra V. Re-understanding the characteristics of algebra 4. Extension of the concepts of addition and multiplication I. Point multiplication and inner product, cross product and outer product II. ∪ .∩ Operation three. V. Action 4. Operation 5 Generalized Algebra i. Algebraic Structure II of Mathematics. Natural algebraic structure. Summary of Generalized Algebra VI Chapter 8 Periodic Mathematics and Its Understanding 1 Periodic Principle I. Periodic Structure of Nature II. Period: the basic way to express infinity with finiteness III. Period: the basic form of the fourth movement. Discussion on periodic function and related concepts 1. Definition and discussion of periodic function II. Understanding of taking the periodicity 3 of complex number and complex variable function as the periodic function of solution 1 Periodic solution 2. Periodic orbit level 3. Generalization of the concept of harmonic class 4- period. The development of periodic function theory 1. Generalization of the concept of periodicity II. Harmonic analysis 5. Basic understanding of wavelet analysis 1. Brief history and essence of Fourier series II. The triangular expression of f(x) and its conditions. Orthogonal basis and f(x) fourier series iv. Fourier transform and Fourier integral and their basic properties V. Basic development process of 6-period mechanics of wavelet transform I. Understanding of vibration theory II. Understanding of Wave Theory Chapter 9 Mathematical Understanding 1 Deterministic Mathematics I. Numerical Mathematics II. Analytical Mathematics I: Development of Mathematical Analysis III. Analytical Mathematics II: Development of Mathematical Reasoning Method 2- Time Mathematics I: T Variable Mathematics and Dynamic System. Unity. Order: time variables and time functions. Continuous time mathematics. Discrete Time Mathematics (Dynamical System and Chaos) 3 Time Mathematics II: Random Mathematics I: Random Mathematics II: Timeliness in Probability Concept III: Timeliness in Random Process IV: Timeliness in Statistics V: Time Series Analysis 4 Fuzzy Mathematics and Complexity Mathematics I: About Uncertainty Mathematics II: Fuzzy Mathematics III: Complexity Mathematics 5 Optimization Mathematics I: Value Mathematics and Optimization Mathematics II: Operational Research II: Basic Principles of Optimization Mathematics III: Finding the Optimal Scheme. Optimal mathematics for finding the optimal orbit: cybernetics, etc. Chapter 10 The development of mathematics according to its spatial form 1. 1 point mathematics 1. 16 19 years ago: line mathematics 2. 1665438 point mathematics 2 neighborhood mathematics 1. Function theory and analysis triggered by Cartesian coordinate concept II. Extension of Neighborhood Mathematics of Coordinate Concept I: Introduction to Point Set Topology and Topology V: Typical Neighborhood Mathematics II: Mathematics on Manifold VI. Application of neighborhood mathematics: social core topological model III. Spatial mathematics i. Spatial technology in mathematical research II. Euclid space mathematics III. Non-euclidean space and geometry iv. Curved Space: Manifold Understanding V. Mathematics of Function Space: Talking about Function VI. XI: The development of mathematics in spatial form. In-depth: infinitesimal theory 1 mathematical understanding of infinitesimal 1 review. The first challenge of infinitesimal to epistemology and methodology II. The second challenge of infinitesimal and the progress of cognition III. Axiomatic set theory iv. Nonstandard analysis: the second active challenge of mankind to infinitesimal II. Comment on limit theory 1. Comments and statements ii. Advantage 3. Essence 4. > 0 is just a dense set. 5. A new definition of infinitesimal. 6. Defects of limit theory under the concept of infinitesimal. 3. Summary of nonstandard analysis. 1. Introduction of background and ideas. 3. Overview of nonstandard analysis. 4. The preliminary properties of infinitesimal. 5. Lack of non-standard analysis. 4. Enlightenment from the micro-world. 2. The general characteristics of the micro-world. 3. The root of the micro-world. Features: Non-Newtonian Space 4. The relationship between infinitesimal world and non-newtonian space. Superstrings: the support of elementary particle theory 6. In the micro field, it is necessary to further "marry" 5. Understanding of the "dynamic" mechanism of the objective world 1. From the understanding of energy II. The essence of energy and the "dynamic" mechanism. Dynamic mechanism and dynamic neighborhood 4. Infinite understanding of dynamic neighborhood and high-dimensional space 6 and explanation of Zeno paradox 1. Some qualitative understandings of 0 ~ 5 2. Complex list: model description of infinitesimal 3. The applied cognition of infinitesimal and the explanation of Zeno's paradox Chapter 12 "Marriage" The basis of modern mathematics and social science 1 the characteristics of modern science and modern mathematics 1. Talking about the concept of "modernity" II. Characteristics of modern science. Features of modern mathematics I: Functionality 4. Feature 2 of modern mathematics: large-scale analysis 5. Modern. Non-linearity, high-dimensional space, uncertainty and abstract wind 2. Characteristics of social science and its similarity with modern mathematics 1. Social concept, attribute space and social cluster II. Characteristics of social science. The prospect of "marriage" between social science and modern mathematics III. Logical orientation of the prospect of marriage between social science and modern mathematics 1. The fact that modern mathematics and modern physics are married. The deepening of any subject needs mathematics and philosophy 3. Fractal thinking on marriage prospect. 4 An application example: competition mechanism in market economy 1. Market structure of society. Competition model x 3 in the market. Systematic discussion (4.2)' 4. Discussion of system parameters (4.2) or (4.3) 5. Competition between state-owned enterprises and private enterprises. Reorganization theory. Competitive potential and its transmission effect.