Judging 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 obvious example is non-negative integers." Euclid's arithmetic comes from non-negative integers in common sense. Until the middle of19th century, the scientific exploration of numbers remained in common sense. "Another example is the similarity in geometry." Geometry even precedes arithmetic in individual development ",and its" one of the earliest signs is the knowledge of similarity "was discovered so early," just like being born. "Therefore, before the19th century, it was generally believed that mathematics was a natural science and an empirical science, because mathematics was closely related to reality at that time. With the deepening of mathematical research, mathematics has been a deductive science since the middle of19th century. This view has been developed in the research of Bourbaki school. He believes that mathematics is a science of studying structure, and all mathematics is based on three parent structures: algebraic structure, sequential structure and topological structure. Corresponding to this view, from Plato in ancient Greece, many people think that mathematics is a kind of knowledge of research mode, and the mathematician Whitehead (A.
Noun (short for noun)
Whiiehead, 186- 1947) said in Mathematics and Goodness, "The essential feature of mathematics is to study patterns in the process of abstracting from patterned individuals." Mathematics is the most powerful technology to understand the relationship between patterns and analyze patterns. "+09363.1000606661978) declares the defects in the axiomatic logical deduction system, from which people think that mathematics is an empirical science, and the famous mathematician von Neumann thinks that mathematics has the characteristics of both deductive science and empirical science.
For the above viewpoints about the essential characteristics of mathematics, we should analyze them from a historical perspective. In fact, the understanding of logarithmic essential characteristics develops with the development of mathematics. Because mathematics comes from the practice of distributing goods, calculating time and measuring land and volume, the mathematical object at this time (as the product of abstract thinking) is very close to the objective reality, and it is easy for people to find the realistic prototype of mathematical concepts, so people naturally think that mathematics is an empirical science. With the deepening of mathematical research, non-Euclidean geometry, abstract algebra and * * * theory have emerged, especially modern mathematics is developing towards abstraction, pluralism and high dimension. People's attention has been focused on these abstract objects, and the distance between mathematics and reality is getting farther and farther. Mathematical proof (as a deductive reasoning) occupies an important position in mathematical research. Therefore, mathematics has emerged as the free creation of human thinking and the science of studying the relationship between quantity and quantity. It is the theory of studying abstract structures, knowledge about patterns, and so on. These understandings not only reflect the deepening of people's understanding of mathematics, but also are the result of people's understanding of mathematics from different aspects. As someone said, "Engels' idea that mathematics is the study of the quantitative relations and spatial forms in the real world is not contradictory to bourbaki's structural viewpoint. The former reflects the origin of mathematics, while the latter reflects the level of modern mathematics. Modern mathematics is a building built by a series of abstract structures. "And mathematics is a kind of knowledge of research methods, which is an explanation of the essential characteristics of mathematics from the perspective of abstract process and level of mathematics. In addition, from the ideological source, people regard mathematics as a science of deduction and study of structure, which is based on human innate belief in the inevitability and accuracy of mathematical reasoning. It is a concentrated expression of confidence in human rational ability, root and strength. Therefore, people think that this method of developing mathematical theory, that is, deductive reasoning from axioms that are self-evident, is absolutely reliable, that is, if axioms are true, then the conclusions derived from them must be true. Applying these seemingly clear, correct and perfect logics, the conclusions drawn by mathematicians are obviously beyond doubt and irrefutable.
In fact, the above-mentioned understanding of the essential characteristics of mathematics is carried out from the aspects of the origin, existing mode and abstract level of mathematics, mainly from the achievements of mathematical research. Obviously, the result (as a theoretical deduction system) cannot reflect the whole picture of mathematics, and another very important aspect that constitutes the whole of mathematics is the process of mathematical research. On the whole, mathematics is a dynamic process. It is an "experimental process of thinking" and an abstract generalization process of mathematical truth. Logical deduction system is the natural result of this process. In the process of mathematical research, the rich, vivid and changeable side of mathematical objects can be fully displayed. Paulia (g.
Poliva, 1888- 1985) thinks, "Mathematics has two sides. It is Euclid's strict science, but it is also something else. The mathematics proposed by Euclid's method looks like a systematic deductive science, but it is like an experimental inductive science in the process of creation. " Friedenthal said, "Mathematics is quite. This view "is different from mathematics, which is printed in books and engraved in the heart." "He believes that mathematicians or mathematics textbooks like to describe mathematics as" a well-organized state ",that is," the form of mathematics "is formed by mathematicians through their own organizations (activities); But for most people, they regard mathematics as a tool. They can't do without mathematics because they need to apply it. That is to say, for the public, it is necessary to learn the content of mathematics in the form of mathematics, so as to learn the corresponding (applied mathematics) activities. This is probably what Friedenthal said: "Mathematics is an activity discovered and organized in the interaction of content and form". Fitzbain
Fischbein) said, "The ideal of mathematicians is to acquire a rigorous, coherent and logical knowledge entity. This fact does not exclude that mathematics must be regarded as a creative process: mathematics is essentially a human activity, and mathematics is invented by human beings. "Mathematical activities are composed of three basic components, namely, the interaction between form, algorithm and intuition. Courand and Robinson (Courani
Robbins) also said, "Mathematics is the expression of human will, which embodies 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. "
In addition, there are some broader understandings of mathematics. For example, some people think that "mathematics is a cultural system" and "mathematics is a language", and mathematical activities are social. It is the crystallization of human's high wisdom in understanding nature, adapting to and transforming nature, and improving self and society in the historical process of human civilization development. Mathematics has a key influence on the way of thinking of human beings. Others think that mathematics is an art. "Compared with mathematics as a discipline, I almost prefer to regard it as an art, because the persistent creative activities of mathematicians under the guidance of the rational world (although uncontrolled) are similar to those of artists, such as painters. This is real, not imaginary. Mathematicians' strict deductive reasoning can be compared to paying special attention to skills here. Just as a person can't be a painter without certain skills, you can't be a mathematician without a certain level of precise reasoning ability. These qualities are the most basic. Together with other more subtle qualities, it constitutes the quality of an excellent artist or an excellent mathematician. The most important thing in both cases is imagination. " "Mathematics is the music of reasoning" and "Music is the mathematics of images". This paper discusses the essence of mathematics from the process of mathematical research and the qualities that mathematicians should have.
Niss) and others believe that mathematics is a discipline, "in the epistemological sense, it is a science, and its goal is to establish, describe and understand objects, phenomena, relationships and mechanisms in some fields. If this field is composed of what we usually think of as mathematical entities, then mathematics plays the role of pure science. In this case, mathematics is independent with self-development and self-understanding as the goal. On the other hand, if the field of consideration exists outside mathematics, then mathematics plays a role in using science. The difference between these two aspects of mathematics is not the problem of mathematics content itself, but the focus of people's attention. Whether it is pure theory or application, mathematics as a science is helpful to produce knowledge and insight. Mathematics is also a system composed of tools, products and processes. It helps us to make decisions and actions related to mastering practical fields other than mathematics. Mathematics is an aesthetic field, which can provide beauty, pleasure and excitement for many people who are addicted to it. As a discipline, the spread and development of mathematics requires it to be mastered by a new generation. Mathematics learning will not be carried out automatically at the same time, and it needs to be taught. Therefore, mathematics is also a teaching subject in our social education system. "
As can be seen from the above, people discuss the essence of mathematics from several aspects, such as the interior of mathematics (from its contents, forms of expression, research process, etc.), 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 and provide a perspective for us to fully understand the essence 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. The general view is that mathematics has the characteristics of abstraction, accuracy and wide application, and the most essential feature is abstraction. A. Aleksandrov said, "Even with very superficial mathematical knowledge, you can easily perceive these characteristics of mathematics: first, its abstraction, and second, its accuracy. Or better, what he said is the rigor of logic and the certainty of conclusions, and finally the extreme universality of application. " Wang Zikun said, "The characteristics of mathematics are: abstract content, extensive application, rigorous reasoning and certainty of conclusions." This view is mainly from the content, expression and function of mathematics to understand the characteristics of mathematics, which is one aspect of its characteristics. Judging from the process of mathematical research and the relationship between mathematics and other disciplines, mathematics also has the characteristics of visualization, realism and quasi-experience. The understanding of mathematical characteristics also has the characteristics of the times. For example, there are different standards for the rigor of mathematics in different historical development periods. From Euclid geometry to Robard Chevsky geometry to Hilbert axiom system, the evaluation standards for rigor are quite different. Especially after Godel put forward and proved the "Incompleteness Theorem ……", people found that even axiomatization, a rigorous scientific method that was once praised, was flawed. Therefore, the rigor of mathematics is shown in the history of mathematics development, which is relative. Regarding the rationality of mathematics, Paulia pointed out in his "Mathematics and Conjecture" that "mathematics is regarded as an argument science. However, this is only one aspect. The final mathematics seems to be pure demonstration material, only proof. However, the creation process of mathematics is the same as that of any other knowledge. Before proving a mathematical theorem, you should guess the content of this theorem. Before you make a detailed proof, you have to guess the idea of proof. You have to synthesize the observed results and make an analogy. You must try again and again. Mathematicians' creative work is demonstration. However, this proof was found through rational reasoning and speculation. As long as the learning process of mathematics can slightly reflect the process of mathematical invention, then guessing and rational reasoning should occupy an appropriate position. " It is from this perspective that we say that the certainty of mathematics is relative and conditional, and we emphasize the characteristics of mathematics, such as visualization, authenticity and quasi-experience. In fact, it highlights the importance of thinking processes such as observation, experiment, analysis, comparison, analogy, induction and association in mathematical research. Mathematics, whose English is Mathematics, is a plural noun. "Mathematics used to be four disciplines: arithmetic, geometry, astronomy and music, which had a higher status than grammar, rhetoric and dialectics."
Since ancient times, most people regard mathematics as a kind of knowledge system, which is the systematic summation of theoretical knowledge formed through strict logical reasoning. It not only reflects people's understanding of "Engels' spatial form and quantitative relationship in the real world", but also reflects people's understanding of "possible quantitative relationship and form" Mathematics can come from the direct abstraction of the real world or from the labor creation of human thinking.
Judging 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 obvious example is non-negative integers." Euclid's arithmetic comes from non-negative integers in common sense. Until the middle of19th century, the scientific exploration of numbers remained in common sense. "Another example is the similarity in geometry." Geometry even precedes arithmetic in individual development ",and its" one of the earliest signs is the knowledge of similarity "was discovered so early," just like being born. "Therefore, before the19th century, it was generally believed that mathematics was a natural science and an empirical science, because mathematics was closely related to reality at that time. With the deepening of mathematical research, mathematics has been a deductive science since the middle of19th century. This view has been developed in the research of Bourbaki school. He believes that mathematics is a science of studying structure, and all mathematics is based on three parent structures: algebraic structure, sequential structure and topological structure. Corresponding to this view, from Plato in ancient Greece, many people think that mathematics is a kind of knowledge of research mode, and the mathematician Whitehead (A.
Noun (short for noun)
Whiiehead, 186- 1947) said in Mathematics and Goodness, "The essential feature of mathematics is to study patterns in the process of abstracting from patterned individuals." Mathematics is the most powerful technology to understand the relationship between patterns and analyze patterns. "+09363.1000606661978) declares the defects in the axiomatic logical deduction system, from which people think that mathematics is an empirical science, and the famous mathematician von Neumann thinks that mathematics has the characteristics of both deductive science and empirical science.
For the above viewpoints about the essential characteristics of mathematics, we should analyze them from a historical perspective. In fact, the understanding of logarithmic essential characteristics develops with the development of mathematics. Because mathematics comes from the practice of distributing goods, calculating time and measuring land and volume, the mathematical object at this time (as the product of abstract thinking) is very close to the objective reality, and it is easy for people to find the realistic prototype of mathematical concepts, so people naturally think that mathematics is an empirical science. With the deepening of mathematical research, non-Euclidean geometry, abstract algebra and * * * theory have emerged, especially modern mathematics is developing towards abstraction, pluralism and high dimension. People's attention has been focused on these abstract objects, and the distance between mathematics and reality is getting farther and farther. Mathematical proof (as a deductive reasoning) occupies an important position in mathematical research. Therefore, mathematics has emerged as the free creation of human thinking and the science of studying the relationship between quantity and quantity. It is the theory of studying abstract structures, knowledge about patterns, and so on. These understandings not only reflect the deepening of people's understanding of mathematics, but also are the result of people's understanding of mathematics from different aspects. As someone said, "Engels' idea that mathematics is the study of the quantitative relations and spatial forms in the real world is not contradictory to bourbaki's structural viewpoint. The former reflects the origin of mathematics, while the latter reflects the level of modern mathematics. Modern mathematics is a building built by a series of abstract structures. "And mathematics is a kind of knowledge of research methods, which is an explanation of the essential characteristics of mathematics from the perspective of abstract process and level of mathematics. In addition, from the ideological source, people regard mathematics as a science of deduction and study of structure, which is based on human innate belief in the inevitability and accuracy of mathematical reasoning. It is a concentrated expression of confidence in human rational ability, root and strength. Therefore, people think that this method of developing mathematical theory, that is, deductive reasoning from axioms that are self-evident, is absolutely reliable, that is, if axioms are true, then the conclusions derived from them must be true. Applying these seemingly clear, correct and perfect logics, the conclusions drawn by mathematicians are obviously beyond doubt and irrefutable.
In fact, the above-mentioned understanding of the essential characteristics of mathematics is carried out from the aspects of the origin, existing mode and abstract level of mathematics, mainly from the achievements of mathematical research. Obviously, the result (as a theoretical deduction system) cannot reflect the whole picture of mathematics, and another very important aspect that constitutes the whole of mathematics is the process of mathematical research. On the whole, mathematics is a dynamic process. It is an "experimental process of thinking" and an abstract generalization process of mathematical truth. Logical deduction system is the natural result of this process. In the process of mathematical research, the rich, vivid and changeable side of mathematical objects can be fully displayed. Paulia (g.
Poliva, 1888- 1985) thinks, "Mathematics has two sides. It is Euclid's strict science, but it is also something else. The mathematics proposed by Euclid's method looks like a systematic deductive science, but it is like an experimental inductive science in the process of creation. " Friedenthal said, "Mathematics is quite. This view "is different from mathematics, which is printed in books and engraved in the heart." "He believes that mathematicians or mathematics textbooks like to describe mathematics as" a well-organized state ",that is," the form of mathematics "is formed by mathematicians through their own organizations (activities); But for most people, they regard mathematics as a tool. They can't live without mathematics because they need to apply mathematics. That is to say, for the public, it is necessary to learn the content of mathematics in the form of mathematics, so as to learn the corresponding (applied mathematics) activities. This is probably what Friedenthal said: "Mathematics is an activity discovered and organized in the interaction of content and form". Fitzpatrick
Fischbein) said, "The ideal of mathematicians is to acquire a rigorous, coherent and logical knowledge entity. This fact does not exclude that mathematics must be regarded as a creative process: mathematics is essentially a human activity, and mathematics is invented by human beings. "Mathematical activities are composed of three basic components, namely, the interaction between form, algorithm and intuition. Courand and Robinson (Courani
Robbins) also said, "Mathematics is the expression of human will, which embodies 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. "
In addition, there are some broader understandings of mathematics. For example, some people think that "mathematics is a cultural system" and "mathematics is a language", and mathematical activities are social. It is the crystallization of human's high wisdom in understanding nature, adapting to and transforming nature, and improving self and society in the historical process of human civilization development. Mathematics has a key influence on the way of thinking of human beings. Others think that mathematics is an art. "Compared with mathematics as a discipline, I almost prefer to regard it as an art, because the persistent creative activities of mathematicians under the guidance of the rational world (although uncontrolled) are similar to those of artists, such as painters. This is real, not imaginary. Mathematicians' strict deductive reasoning can be compared to paying special attention to skills here. Just as a person can't be a painter without certain skills, you can't be a mathematician without a certain level of precise reasoning ability. These qualities are the most basic. Together with other more subtle qualities, it constitutes the quality of an excellent artist or an excellent mathematician. The most important thing in both cases is imagination. " "Mathematics is the music of reasoning" and "Music is the mathematics of images". This paper discusses the essence of mathematics from the process of mathematical research and the qualities that mathematicians should have.
Niss) and others believe that mathematics is a discipline, "in the epistemological sense, it is a science, and its goal is to establish, describe and understand objects, phenomena, relationships and mechanisms in some fields. If this field is composed of what we usually think of as mathematical entities, then mathematics plays the role of pure science. In this case, mathematics is independent with self-development and self-understanding as the goal. On the other hand, if the field of consideration exists outside mathematics, then mathematics plays a role in using science. The difference between these two aspects of mathematics is not the problem of mathematics content itself, but the focus of people's attention. Whether it is pure theory or application, mathematics as a science is helpful to produce knowledge and insight. Mathematics is also a system composed of tools, products and processes. It helps us to make decisions and actions related to mastering practical fields other than mathematics. Mathematics is an aesthetic field, which can provide beauty, pleasure and excitement for many people who are addicted to it. As a discipline, the spread and development of mathematics requires it to be mastered by a new generation. Mathematics learning will not be carried out automatically at the same time, and it needs to be taught. Therefore, mathematics is also a teaching subject in our social education system. "
As can be seen from the above, people discuss the essence of mathematics from several aspects, such as the interior of mathematics (from its contents, forms of expression, research process, etc.), 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 and provide a perspective for us to fully understand the essence 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. The general view is that mathematics has the characteristics of abstraction, accuracy and wide application, and the most essential feature is abstraction. A. Aleksandrov said, "Even with very superficial mathematical knowledge, you can easily perceive these characteristics of mathematics: first, its abstraction, and second, its accuracy. Or better, what he said is the rigor of logic and the certainty of conclusions, and finally the extreme universality of application. " Wang Zikun said, "The characteristics of mathematics are: abstract content, extensive application, rigorous reasoning and certainty of conclusions." This view is mainly from the content, expression and function of mathematics to understand the characteristics of mathematics, which is one aspect of its characteristics. Judging from the process of mathematical research and the relationship between mathematics and other disciplines, mathematics also has the characteristics of visualization, realism and quasi-experience. The understanding of mathematical characteristics also has the characteristics of the times. For example, there are different standards for the rigor of mathematics in different historical development periods. From Euclid geometry to Robard Chevsky geometry to Hilbert axiom system, the evaluation standards for rigor are quite different. Especially after Godel put forward and proved the "Incompleteness Theorem ……", people found that even axiomatization, a rigorous scientific method that was once praised, was flawed. Therefore, the rigor of mathematics is shown in the history of mathematics development, which is relative. Regarding the rationality of mathematics, Paulia pointed out in his "Mathematics and Conjecture" that "mathematics is regarded as an argument science. However, this is only one aspect. The final mathematics seems to be pure demonstration material, only proof. However, the creation process of mathematics is the same as that of any other knowledge. Before proving a mathematical theorem, you should guess the content of this theorem. Before you make a detailed proof, you have to guess the idea of proof. You have to synthesize the observed results and make an analogy. You must try again and again. Mathematicians' creative work is demonstration. However, this proof was found through rational reasoning and speculation. As long as the learning process of mathematics can reflect the process of mathematical invention, then guessing and rational reasoning should occupy an appropriate position. " It is from this perspective that we say that the certainty of mathematics is relative and conditional, and emphasize the image, truth and quasi-experience of mathematics. In fact, the emphasis on the characteristics of "falsifiability" highlights the observation, experiment and quasi-experience in mathematical research.