Mathematical Topology Proposition Table _ Mathematical Topology Proposition Table Picture

Mathematics is a subject that studies concepts such as quantity, structure, change and spatial model. By using abstract and logical reasoning, the shape and motion of objects are counted, calculated, measured and observed. Mathematicians have extended these concepts in order to express new conjectures with formulas and establish strictly deduced truths from properly selected axioms and definitions.

Name source

Mathematics shù xué (Greek: μ α θ η μ α ι κ? ) the west originated from the Greek word ancient μ? θξμα(máthēma) has learning, learning and science, and it has another narrow and technical meaning-"mathematical research", even in its etymology. Its adjective meaning is related to study or hard work, and it can also be used to refer to mathematics. Its superficial plural form in English and les mathématiques in French can be traced back to the neutral plural mathematica in Latin, which is called math by Cishjt. In ancient China, mathematics was called arithmetic, also called arithmetic, and was finally changed to mathematics.

meaning

Mathematics, as an expression of human thinking, embodies people's aggressive will, meticulous logical reasoning and pursuit of perfection. Its basic elements are: logic and intuition, analysis and reasoning, generality and individuality. Although different traditional schools can emphasize different aspects, it is the interaction of these opposing forces and their comprehensive efforts that constitute the vitality, availability and lofty value of mathematical science.

History of mathematics

The knowledge and application of basic mathematics is an indispensable part of individual and group life. The refinement of its basic concepts can be found in ancient mathematical documents of ancient Egypt, Mesopotamia and ancient India. Since then, its development has made small progress until the Renaissance in16th century, and the mathematical innovation generated by the interaction with new scientific discoveries led to the acceleration of knowledge, until today. Today, mathematics is used in different fields of the world, including science, engineering, medicine and economics. The application of mathematics in these fields is usually called applied mathematics, and sometimes it will lead to new mathematical discoveries and the development of new disciplines. Mathematicians also study pure mathematics, that is, mathematics itself, without any practical application. Although many people started their research from pure mathematics, many applications will be found later. The French Bourbaki School, founded in 1930s, believes that mathematics, at least pure mathematics, is a theory to study abstract structures. Structure is a deductive system based on initial concepts and axioms. Boone School believes that there are three basic abstract structures: algebraic structure (group, ring, domain …), ordered structure (partial order, total order …) and topological structure (neighborhood, limit, connectivity, dimension …).

classify

Discrete mathematics fuzzy mathematics

Five branches of mathematics

1 classical mathematics 2. Modern mathematics iii. Computer mathematics. Random mathematics 5. Economic mathematics

Branch of mathematics

1 .arithmetic 2. Elementary algebra 3. Advanced algebra iv. Number theory 5. Euclidean geometry 6. Non-Euclidean geometry 7. Analytic geometry 8. Differential geometry 9. Algebraic geometry 10. Projective geometry 1 1. Geometric topology 12. Topology 65438. 38+05. Theory of real variable function 16. Probability statistics 17. Complex variable function theory 18. Functional analysis 19. Partial differential equation 20. Ordinary differential equation 2 1. Mathematical logic. Fuzzy mathematics. Operational research. Computational mathematics 25.

Mathematical classification

In modern symbols, simple expressions can describe complex concepts. This image is generated by a simple equation. Most of the mathematical symbols we use today were invented after16th century. Before that, mathematics was written in words, which was a hard procedure that would limit the development of mathematics. Today's symbols make mathematics easier to be controlled by experts, but beginners are often afraid of it. It is extremely compressed: several symbols contain a lot of information. Like music notation, today's mathematical symbols have clear grammar and information codes, so it is difficult to write them in other ways. Mathematical language is also difficult for beginners. How to make these words have more accurate meanings than everyday language? Novices are also troubled. Words such as openness and domain have special meanings in mathematics. Mathematical terms also include proper nouns such as embryo and integrability. But these special symbols and terms are used for a reason: mathematics needs accuracy more than everyday language. Mathematicians call this requirement for linguistic and logical accuracy "rigor". Stiffness is a very important and basic part of mathematical proof. Mathematicians hope that their reasoning and axioms of the definite reason system can be inferred. This is to avoid the wrong "theorem", relying on unreliable intuition, there have been many examples in history. The rigor expected in mathematics changes with time: the Greeks expected careful argumentation, but in Newton's time, the methods used were not so rigorous. Newton's definition of solving problems was not carefully analyzed and formally proved until the19th century. Today, mathematicians have been arguing about the rigor of computer-aided proof. When a large number of measurements are difficult to verify, it is hard to say that they are effective and rigorous.

phylogeny

Mathematics, the history of world mathematics development, originated from early human production activities. It is one of the six great arts in ancient China, and it is also regarded as the starting point of philosophy by ancient Greek scholars. Mathematical Greek μ α θ η μ α κ? Mathematickó s) means "the basis of learning" and comes from μαρθξμα(máthema) ("science, knowledge and learning"). The evolution of mathematics can be regarded as the continuous development of abstraction and the extension of subject matter. The first abstract concept is probably number, and its cognition that two apples and two oranges have something in common is a great breakthrough in human thought. In addition to knowing how to calculate the number of actual substances, prehistoric people also knew how to calculate the number of abstract substances, such as time-date, season and year. Arithmetic (addition, subtraction, multiplication and division) will naturally occur. Ancient stone tablets also confirmed the knowledge of geometry at that time. In addition, writing or other systems that can record numbers are needed, such as Mu Fu or chips used by the Inca Empire to store data. There are many different counting systems in history. Since the historical era, the main principles in mathematics have been formed to do many calculations related to taxation and trade, understand the relationship between numbers, measure land and predict astronomical events. These needs can be simply summarized as the study of quantity, structure, space and time in mathematics. By16th century, elementary mathematics, such as arithmetic, elementary algebra and trigonometry, had been basically completed. The appearance of the concept of variables in the17th century made people begin to study the relationship between variables and the mutual transformation between graphs. In the process of studying classical mechanics, the method of calculus was invented. With the further development of natural science and technology, set theory and mathematical logic, which are produced for studying the basis of mathematics, have also begun to develop slowly. Mathematics has been continuously extended since ancient times, and has rich interaction with science, and both of them have benefited a lot. There are many discoveries in mathematics in history, and they are still being discovered today. According to Mikhail B. Sevryuk's record in the Bulletin of the American Mathematical Society of June 5438+ 10, 2006: "Since 1940 (the first year of mathematical review), the number of papers and books in the database of mathematical reviews has exceeded1900,000, with an annual increase of more than 750,000. Most of this learning sea is a new mathematical theorem and its proof. "