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The origin of mathematical knowledge Mathematics, like other branches of science, is a kind of intellectual accumulation developed through human social practice and production activities under certain social conditions. Its main content reflects the quantitative relationship and spatial form of the real world, as well as the relationship and structure between them. This can be confirmed from the origin of mathematics The Nile River in ancient Africa, the Tigris River and Euphrates River in Western Asia, the Indus River and Ganges River in central and South Asia, and the Yellow River and Yangtze River in East Asia were the birthplaces of mathematics. Because of the need of agricultural production, the ancestors of these areas used to control floods and irrigation, measure the area of fields, calculate the volume of warehouses, calculate calendars suitable for agricultural production, and calculate related wealth We have accumulated rich experience in long-term practice, such as calculation, product exchange and so on, and gradually formed corresponding technical knowledge and related mathematical knowledge
Mathematics, whose English name is mathematics, is a plural noun. "Mathematics used to be four subjects: arithmetic, geometry, astronomy and music. It was in a higher position than grammar, rhetoric and dialectics."
Since ancient times, most people regard mathematics as a kind of knowledge system, which is the sum of systematic theoretical knowledge formed by rigorous logical reasoning. It not only reflects people's understanding of "the spatial form and quantitative relationship of the real world" (Engels), but also reflects people's understanding of "the spatial form and quantitative relationship of the real world" (Engels), Mathematics can not only come from the direct abstraction of the real world, but also from the labor creation of human thinking
From the development history of human society, people's understanding of the essential characteristics of mathematics is constantly changing and deepening. "The root of mathematics lies in common sense, and the most prominent example is non negative integers." Euclidean arithmetic comes from non negative integers in common sense, and until the middle of the 19th century, the scientific exploration of numbers remained in common sense, ”Another example is the similarity in geometry. "In the individual development, geometry even precedes arithmetic." one of the earliest signs is the knowledge of similarity. "The knowledge of similarity was discovered so early that" it's like a big life. "Therefore, before the 19th century, people generally believed that mathematics was a natural science and empirical science, At that time, there was a close relationship between mathematics and reality. With the deepening of mathematical research, the view that mathematics was a deductive science gradually occupied a dominant position after the middle of the 19th century. This view was developed in the research of burbaki school. They believed that mathematics was a science of structure, All mathematics is based on the three parent structures of algebraic structure, ordered structure and topological structure. Correspondingly, from Plato in ancient Greece, many people think that mathematics is the study of models. Mathematician A. N. whiiehead (186-1947) said in mathematics and the good, "The essential characteristic of mathematics is to study patterns in the process of abstracting from patterned individuals." mathematics is the most powerful technology for understanding the relationship between patterns and analyzing patterns. "In 1931, the proof of Goethe's Incompleteness Theorem declared the defects in axiomatic logical deductive system, The famous mathematician von Neumann thinks that mathematics has both deductive science and empirical science
In fact, the understanding of the essence of logarithm develops with the development of mathematics. Because mathematics originates from the practice of distributing goods, calculating time, measuring land and volume, the mathematical object (as the product of abstract thinking) at that time is very close to the objective reality, People can easily find the real prototype of mathematical concepts, so people naturally think that mathematics is an empirical science. With the deepening of mathematical research, the emergence of non Euclidean geometry, abstract algebra and set theory, especially the development of modern mathematics towards abstraction, pluralism and high dimension, people's attention is focused on these abstract objects, and the distance between mathematics and reality is getting farther and farther, Moreover, mathematical proof (as a kind of deductive reasoning) occupies an important position in mathematical research. Therefore, there are some viewpoints that mathematics is the free creation of human thinking, the science of studying the relationship between quantity, the theory of studying abstract structure, the knowledge of model, and so on, It is also the result of people's understanding of mathematics from different aspects. As someone said, "Engels's proposition that mathematics is the study of quantitative relations and spatial forms in the real world is not contradictory to burbaki's structural view. The former reflects the source of mathematics, while the latter reflects the level of modern mathematics, Modern mathematics is a building built by a series of abstract structures. "The statement that mathematics is the knowledge of research mode is the explanation of the essential characteristics of mathematics from the perspective of the abstract process and level of mathematics. In addition, from the ideological root, people regard mathematics as deductive science and the science of research structure, It is based on human's innate belief in the inevitability and accuracy of mathematical reasoning, and it is the concentrated embodiment of human's own rational ability, root and strength. Therefore, people believe that the method of developing mathematical theory, that is, deductive reasoning from self-evident axioms, is absolutely reliable, that is, if the axioms are true, Then the conclusion deduced from it must be true. By applying these clear, correct and perfect logics, the conclusion drawn by mathematicians is undoubtedly irrefutable
In fact, the above understanding of the essence of mathematics is from the source of mathematics, the way of existence, the level of abstraction, and mainly from the results of mathematical research. Obviously, the results (as a theoretical deductive system) can not reflect the whole picture of mathematics. Another very important aspect of the whole of mathematics is the process of mathematical research, On the whole, mathematics is a dynamic process, an "experimental process of thinking" and an abstract generalization process of mathematical truth. Logical deduction system is a natural result of this process. In the process of mathematical research, the rich, vivid and changeable side of mathematical objects can be fully displayed. G. poliva (1888-1985) holds that the logical deduction system is the natural result of this process, "Mathematics has two sides. It's Euclidean rigorous science, but it's something else. Mathematics proposed by Euclidean method seems to be a systematic deductive science, but mathematics in the process of creation seems to be an experimental inductive science." friedenthal said, "mathematics is a very special activity, He believes that mathematicians or mathematics textbooks like to express mathematics as "a well-organized state," that is, "the form of mathematics" is formed by mathematicians organizing the content of Mathematics (activities) through their own organization (activities); however, for most people, mathematics (activities) is not the same as mathematics, They regard mathematics as a tool. They can't do without mathematics because they need to apply mathematics. That is, for the public, they need to learn the content of mathematics in the form of mathematics, So as to learn the corresponding (Applied Mathematics) activities. This is probably the meaning of friedenthal's saying that "mathematics is an activity of discovery and organization in the interaction of content and form". Efriam Fischbein said that "the ideal of mathematicians is to obtain a rigorous, well-organized and logical knowledge entity, This fact does not rule out that mathematics must be regarded as a creative process: mathematics is essentially a human activity, and mathematics is invented by human beings. "Mathematical activity is composed of the interaction of three basic components: formal, algorithmic and intuitive. Courani Robbins also said," mathematics is the expression of human will, It reflects positive will, thoughtful reasoning and exquisite and perfect desire. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Although different traditions may emphasize different aspects, only the interaction of these opposing forces and the struggle for their integration constitute the life, utility and high value of mathematical science. "
For example, some people think that "mathematics is a cultural system", "mathematics is a language", and mathematical activities are social. It is in the historical process of human civilization development, Mathematics has a key influence on the way of thinking of human beings. Some people think that mathematics is an art. "Compared with mathematics as a discipline, I almost like to see it as an art, The reason is that the long-lasting creative activities of mathematicians under the guidance of the rational world (though not under control) are similar to those of artists, such as painters. This is true rather than fictitious. The strict deductive reasoning of mathematicians can be compared to special skills here, just as a person can not become a painter without a certain amount of skills, If you don't have a certain level of precise reasoning ability, you can't become a mathematician. These qualities are the most basic. Together with other qualities that are much more subtle, they constitute the qualities of an excellent artist or mathematician. The most important one is imagination in both cases, ”And "music is the mathematics of image". This is to discuss the essence of mathematics from the process of mathematical research and the quality that mathematicians should have. Some people regard mathematics as a basic attitude and method to treat things, a spirit and concept, that is, mathematical spirit, mathematical concept and attitude. In the article "Mathematics in society", Mogens NISS and others believe that mathematics is a discipline, "In the sense of epistemology, it is a science, the goal is to establish, describe and understand the objects, phenomena, relations and mechanisms in some fields. If this field is composed of mathematical entities that we usually think of, mathematics plays the role of pure science. In this case, mathematics's internal self-development and self understanding are the goals, independent of the external world, On the other hand, if the field under consideration exists outside of mathematics, mathematics will play a role of practical science. The difference between these two aspects of mathematics is not the problem of mathematical content itself, but the focus of people's attention, Mathematics, as a science, helps to produce knowledge and insight. Mathematics is also a system of tools, products and processes, which helps us make decisions and actions related to mastering practical fields other than mathematics. Mathematics is a field of aesthetics, which can provide aesthetic, pleasure and exciting experience for many people who are intoxicated with it, The spread and development of mathematics require that it can be mastered by a new generation of people. Mathematics learning will not be carried out automatically at the same time. It needs to be taught by people. Therefore, mathematics is also a teaching subject in our social education system. "
It can be seen from the above that people discuss the nature of mathematics from the following aspects: the relationship between mathematics and society, the relationship between mathematics and other disciplines, and the relationship between mathematics and human development. They all reflect the essential characteristics of mathematics from one side, It provides a perspective for us to fully understand the nature of mathematics
Based on the above understanding of the essential characteristics of mathematics, people have also discussed the specific characteristics of mathematics from different aspects. A more common view is that mathematics has the characteristics of abstraction, accuracy and wide application, among which the most essential characteristic is abstraction, "Even if you only have superficial knowledge of mathematics, you can easily perceive these characteristics of Mathematics: first, its abstractness, second, its accuracy, or better still, the strictness of logic and the certainty of its conclusion, and finally, its extremely extensive application," said Wang zikun
RELATED INFORMATIONS
- 1. The origin of mathematical knowledge is less than 50 words
- 2. The discovery and origin of mathematical knowledge, and make a brief introduction and elaboration
- 3. Let | Z | denote the distance from the point corresponding to complex Z in the plane to the origin. Given | Z | = 2 + z-4i, find complex Z
- 4. The distance from the corresponding point of complex number 1 + 1I in the complex plane to the origin is () A. 12B. 22C. 1D. 2
- 5. In the complex plane, the distance from the point corresponding to the complex number 1 + 2I to the origin
- 6. Let the complex Z satisfy ZZ - + (2-I) Z + (2 + I) Z - + 4 = 0, and prove the distance from the point corresponding to Z on the complex plane to the point corresponding to complex - 2-I on the complex plane constant Z - is the conjugate complex number of Z, which is convenient to type
- 7. Fish plural
- 8. Fish plural
- 9. On the complex number of fish I remember n years ago when fish was said to be an uncountable noun, isn't that right? It should be a singular plural word, right? But what I doubt is that there is a problem .There __ fish in the lake . The answer is is, I am very confused, and there is another question in the book. What's your opinion? I hope the answer is wrong
- 10. Is fish plural? Besides fish on the table, is fish plural? Can you say: "there are many fishes in my home"? People on the third floor, but sheet also has a plural. The plural of sheet is itself
- 11. The discovery and origin of mathematical knowledge To be brief, no more than 50 words 2. What kind of diseases are easily caused by insufficient oxygen and dirty air A. Respiratory system B, blood circulation system C, digestive system D, nervous system 3. Brazil and Australia are in the forefront of the world in terms of mineral resources, reserves and production
- 12. Change these sentences into simple and plural ones 1.These are my parents. ______ _______ _______ my____ 2.That boy is my sister‘s friend _______________ _________ are my sister's friend
- 13. Give me five plural sentences
- 14. Change the following sentences to plural 1.He is a hard-working employee.she is a hard-working employee,too 2.I am a nurse.My friend is a nurse,too 3.she is a taxi driver.he is a taxi driver,too 4.i an a smart student,my friend is a smart student,too
- 15. Make the following sentences plural This is a watermlon. it is a yellow shirt. that's my watch.
- 16. He's a policeman
- 17. Complete the following sentences as required, with one word left blank 1. We of talk about news when we have breakfast We often talk about news_____ _____ . 2. Don't phone me when you are driving a car Don’t_____ _____ _____ _____ me when you are driving a car 3. They all rnjoy watching TV plays What ______ they all _______ ________ ? 4、He returns to his hometown as soon as he has no money? _____ he _____ to his hometown as soon as he has no money? 5. He is gone and you can go on with your story He is gone and you can ______ _______ your story According to the Chinese prompt, complete the sentence, each empty word. I've never failed the same exam. My deskmate never ______ ______ ______
- 18. Complete every word in the sentence Mary was so nervous that she didn't and t do well the exam Mary was ____ ____ ____ ____ well in the exam. I went to (the computer club) yesterday after noon _____ ____ _____ you ______ yesterday afternoon?
- 19. Complete each empty word in the sentence Every July, our city gathers thousands of tourists from all over the world There are _____ of _____ from all over the world in our city every year. It's helpful to bring a shopping list when you go shopping It's very _____ to take a _____ list with you when you buy something. You are welcome —Thank you for your help.—_____ _____ . If meng0902 and AEE_ The last question is two blanks
- 20. First grade English complete sentences, each empty word Xiao Ming is going to review his lessons for the exam Xiaoming is going to ______ ______ his test. How about a picnic this weekend? ______ ______ having a picnic this weekend? What are you going to do tonight? _____ are you ______ to do tonight? I plan to make a plan for my study on Monday morning I am going to _____ ______ ______ for my study on Monday morning.