It is knowledge related to mathematics!
What is mathematics and what is knowledge are explained below.
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Mathematics:
Mathematics , its English is mathematics, which is a plural noun, "Mathematics used to be four disciplines: arithmetic, geometry, astronomy and music, in a higher status than the three disciplines of grammar, rhetoric and dialectics."
Since ancient times, most people have regarded mathematics as a knowledge system, a systematic sum of theoretical knowledge formed through rigorous logical reasoning. It not only reflects people's understanding of "the spatial form and quantitative relationships of the real world" (Engels)” also reflects people’s understanding of “possible quantitative relationships and forms.” Mathematics can come from direct abstraction from the real world, or it can come from the labor creation of the human mind.
From the perspective of the development history of human society, people’s understanding of the essential characteristics of mathematics is constantly changing and deepening. "The roots of mathematics lie in common common sense. The most obvious example is non-negative integers." Euclid's arithmetic originated from non-negative integers in common common sense, and until the middle of the 19th century, the scientific exploration of numbers still remained at the level of ordinary common sense. Common sense,” another example is the similarity in geometry, “geometry even precedes arithmetic in individual development,” and one of its “earliest symptoms is the knowledge of similarities,” the knowledge of which is discovered so early, “It’s like a big student. "Therefore, before the 19th century, people generally believed that mathematics was a natural science and an empirical science, because the connection between mathematics and reality at that time was very close. With the continuous deepening of mathematical research, from the middle of the 19th century, mathematics was The view of a deductive science gradually became dominant. This view was developed in the research of the Bourbaki school. They believed that mathematics is the science of studying structures. All mathematics is based on three types: algebraic structure, ordinal structure and topological structure. On top of the parent structure. Corresponding to this view, starting from Plato in ancient Greece, many people believe that mathematics is the study of patterns. The mathematician A. N. Whitehead (186-1947) wrote in "Mathematics and Goodness" "The essential characteristic of mathematics is the study of patterns in the process of abstraction from patterned individuals," said in 1931, "Mathematics is the most powerful technique for understanding patterns and analyzing relationships between patterns. The proof of G?del's (K, G0de1, 1978) incompleteness theorem announced the shortcomings in the axiomatic logical deduction system. In this way, people thought of the view that mathematics is an empirical science. The famous mathematician von Neumann It is believed that mathematics has the characteristics of both deductive science and empirical science.
We should analyze the above-mentioned views on the essential characteristics of mathematics from a historical perspective. In fact, the understanding of the essential characteristics of mathematics develops with the development of mathematics. Since mathematics originated from practices such as allocating goods, calculating time, measuring land and volume, the mathematical objects at this time (as the product of abstract thinking) are very close to objective reality, and people can easily find the realistic prototypes of mathematical concepts. In this way, 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, multivariate and high-dimensionality, people's attention Focusing on these abstract objects, the distance between mathematics and reality grew further and further, and mathematical proof (as a kind of deductive reasoning) occupied an important place in the study of mathematics, so the view that mathematics is the free creation of the human mind emerged Physics is the science of studying the relationship between quantities, the theory of abstract structures, the knowledge of patterns, and so on. These understandings not only reflect people's deepening understanding of mathematics, but are also the result of people's understanding of mathematics from different aspects. As someone said, "Engels's statement that mathematics is the study of quantitative relationships and spatial forms in the real world is not inconsistent with Bourbaki's structural views. The former reflects the origin of mathematics, and the latter reflects the development of modern mathematics. level, modern mathematics is an edifice built from a series of abstract structures. "The statement that mathematics is the study of patterns is an explanation of the essential characteristics of mathematics from the perspective of the abstract process and abstract level of mathematics. In addition, from the perspective of thought. From a fundamental point of view, the reason why people regard mathematics as a deductive science and a science that studies structures is based on human beings' innate belief in the inevitability and accuracy of mathematical reasoning, and is the root of human beings' own rational abilities. and power of confidence, it is therefore believed that the method of developing mathematical theories, namely deductive reasoning starting from self-evident axioms, is absolutely reliable, that is, if the axiom is true, then it can be deduced from it The conclusion must be true. By applying these seemingly clear, correct, and perfect logics, the conclusions drawn by mathematicians are obviously unquestionable and irrefutable.
In fact, the above-mentioned understanding of the essential characteristics of mathematics is based on the source, way of existence, abstraction level, etc. of mathematics, and is mainly based on the results of mathematical research.
Obviously, the results (as a theoretical deduction system) do not reflect the whole picture of mathematics. Another very important aspect that makes up the whole of mathematics is the process of mathematical research, and generally speaking, mathematics is a dynamic process and a "The experimental process of thinking" is the process of abstract generalization of mathematical truths. Logical deduction systems are a natural outcome of this process. In the process of mathematical research, the rich, vivid and changing side of mathematical objects can be fully displayed. G. Poliva (1888-1985) believed that “mathematics has two sides. It is a rigorous science of Euclidean style, but it is also something else. The mathematical view proposed by the Euclidean method It seems to be a systematic deductive science, but mathematics in the creative process looks like an experimental inductive science,” Freidenthal said, “Mathematics is a very special activity. Opinion "is different from mathematics as something printed on books and inscribed in the mind. ” 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’ own organization (activities) of mathematics (activity) content. but for most people, they regard mathematics as a tool. They cannot live without mathematics because they need to apply mathematics. That is, for the public, they need to learn the content of mathematics through the form of mathematics. So as to learn the corresponding (applied mathematical) activities. This is probably what Freidenthal meant when he said that "mathematics is an activity of discovery and organization in the mutual influence of content and form." (Efraim Fischbein) said, “The fact that the ideal of a mathematician is to obtain rigorous, well-organized, and logically structured knowledge entities does not exclude the need to view mathematics as a creative process: mathematics is essentially a human activity. Mathematics was invented by humans. "Mathematical activities consist of the interaction between three basic components: formal, algorithmic and intuitive. Courani and Robbins also said, "Mathematics is the expression of human will. Reflecting active intention, thoughtful reasoning, and refined and complete desires, its basic elements are logic and intuition, analysis and construction, generality and particularity. Although different traditions may emphasize different aspects, it is only the interaction of these opposing forces, and the struggle for their synthesis, that constitutes the life, utility and high value of mathematical science. ”
In addition, there are some broader understandings of mathematics. For example, some people believe that “mathematics is a cultural system”, “mathematics is a language”, and mathematical activities are social. In the historical process of the development of human civilization, mathematics is the crystallization of a high degree of wisdom for human beings to understand nature, adapt to and transform nature, and improve themselves and society. Some people also believe that mathematics is a key influence on human thinking. An art, "I almost prefer to think of mathematics as an art than as a subject, because of the enduring creative activities of mathematicians under the guidance (although not under the control) of the rational world , has similarities with the activities of artists, such as painters, which are real and not imaginary. The mathematician's rigorous deductive reasoning can be likened here to a dedicated attention to technique. Just as one cannot be a painter without possessing a certain amount of skill, one cannot be a mathematician without possessing a certain level of precise reasoning. These qualities are fundamental and stand apart from others that are much more subtle. *The same qualities that make up a good artist or a good mathematician, the chief one in both cases is imagination. "Mathematics is the music of reasoning," and "music is the mathematics of images." This discusses the nature of mathematics from the perspective of the process of mathematical research and the qualities that mathematicians should possess. Some people regard mathematics as a way of dealing with things. Basic attitudes and methods, a spirit and concept, that is, the spirit of mathematics, mathematical concepts and attitudes, Mogens Niss and others believe in the article "Mathematics in Society" that mathematics is a discipline, "in an epistemological sense. It is a science whose goal is to establish, describe and understand objects, phenomena, relationships and mechanisms in certain fields. Mathematics plays the role of a pure science if the field is made up of what we normally think of as mathematical entities. In this case, mathematics aims at inner self-development and self-understanding, independent of the external world. On the other hand, mathematics plays the role of science if the field under consideration exists outside mathematics. These two aspects of mathematics The difference between the two aspects is not a problem of the mathematical content itself, but a different focus of people's attention. Whether pure or applied, mathematics as a science helps generate knowledge and insight. Mathematics is also a system of tools, products, and processes that help us make decisions and actions related to the mastery of practical areas other than mathematics. Mathematics is an area of ??aesthetics that can provide many people who are fascinated with it with a sense of beauty, pleasure. and exciting experiences. As a subject, the spread and development of mathematics require that it be mastered by a new generation of people. The learning of mathematics does not happen simultaneously and automatically, it needs to be taught by people. Therefore, mathematics is also a teaching subject in the education system of our society. ”
It can be seen from the above that people approach mathematics from within (and from several perspectives such as the content, expression, and research process of mathematics).
The nature of mathematics is discussed from several aspects, including the relationship between mathematics and society, the relationship between mathematics and other subjects, 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 comprehensively 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. The more common view is that mathematics has the characteristics of abstraction, accuracy and wide application, among which the most essential characteristic is abstraction. A,. Alexanderlov said, “Even with a very superficial knowledge of mathematics one can easily perceive these characteristics of mathematics: first, its abstraction, second, its precision, or better to say, its logical rigor and The certainty of its conclusions, and finally the extreme breadth of its applications," Wang Zikun said, "The characteristics of mathematics are: the abstraction of content, the breadth of applications, the rigor of reasoning and the clarity of conclusions." This view is mainly Understanding the characteristics of mathematics from the aspects of its content, expression, and functions is one aspect of the characteristics of mathematics. In addition, from the perspective of the process of mathematical research and the relationship between mathematics and other disciplines, mathematics is also figurative, realistic, and quasi-empirical. The characteristic of “falsifiability”. The understanding of the characteristics of mathematics also has the characteristics of the times. For example, regarding the rigor of mathematics, there are different standards in various historical development periods of mathematics, from Euclidean geometry to Lobachevsky geometry to Hilbert's axiom system. , there are great differences in the evaluation standards of rigor, especially after G?del proposed and proved the "incompleteness theorem..." After that, people found that even axiomatic, a rigorous scientific method that was once highly respected, was flawed. Therefore, the rigor of mathematics is shown in the history of mathematics development and is relativistic. Regarding the plausibility of mathematics, Polya pointed out in his "Mathematics and Conjecture" that "mathematics is regarded as an argument. science. However, this is only one aspect of it. The stereotyped mathematics in its final form seems to be purely argumentative material containing only proofs. However, the creation process of mathematics is the same as the creation process of any other knowledge. Before you prove a mathematical theorem, you first have to guess the content of the theorem. Before you make a detailed proof, you first have to guess the idea of ??the proof. You first have to synthesize the observed results and then make analogies. You have to try again and again. The product of mathematicians' creative work is demonstrative reasoning, that is, proof; but this proof is discovered through reasonable reasoning and conjecture. As long as the process of learning mathematics reflects at all the process of mathematical invention, guesswork and reasoning should have their proper place. "It is from this perspective that we say that the certainty of mathematics is relative and conditional. The emphasis on the imageability, truth-likeness, and quasi-empirical nature of mathematics. The "falsifiability" feature actually highlights the The importance of observation, experimentation, analysis, comparison, analogy, induction, association and other thinking processes in research.
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Knowledge:
What exactly knowledge is is still controversial. The definition of knowledge in our country is generally made from a philosophical perspective. For example, the "knowledge" entry in the "Chinese Encyclopedia·Education" states: "The so-called knowledge, in terms of the content it reflects, is a reflection of the attributes and connections of objective things. It is the reflection of the objective world in the human brain. subjective image. In terms of its form of reflection activity, it sometimes appears as the subject's perceptual perception or representation of things, which belongs to perceptual knowledge; sometimes it appears as concepts or laws about things, which belongs to rational knowledge. "From this definition, we can see that knowledge is the product of the unity of subject and object. It comes from the external world, so knowledge is objective; but knowledge itself is not objective reality, but the characteristics and connections of things in the human brain. Reflection in is a subjective representation of objective things. Knowledge is produced through the reflection activities of the human brain based on the interaction between subject and object.
The above definition provides us with the connotation of knowledge. However, the understanding of macroscopic philosophical reflection theory still needs to be embodied from the perspective of individual cognition, so that it can be effectively used to guide specific teaching in schools.
Different from philosophy, cognitive psychology is. Knowledge is studied from the perspective of the source of knowledge, the production process and representation form of individual knowledge. For example, Piaget believed that experience (that is, knowledge) comes from the interaction between individuals and the environment, and this experience can be divided into two categories. : One type is physical experience, which comes from the external world, and is the understanding of objective things and their connections obtained by individuals acting on objects; the other type is logical-mathematical experience, which comes from the actions of the subject and is the individual's understanding of actions and actions. The result of mutual coordination. For example, children gain experience about the conservation of quantity through playing with objects, and students gain knowledge about mathematical principles through mathematical reasoning. Piaget's definition of knowledge is expressed from the production process of individual knowledge. In "Taxonomy of Educational Objectives", Lum believes that knowledge is "the recollection of specific things and general principles, the recollection of methods and processes, or the recollection of a pattern, structure, or framework", which is derived from what knowledge contains. From the perspective of content, it is a description of a phenomenon.
We believe that when understanding the meaning of knowledge, it is necessary to distinguish knowledge as the common wealth of human society from knowledge as the knowledge in the individual mind. Knowledge in human society exists objectively, but the knowledge in an individual's mind is not the objective reality itself, but a subjective representation of the individual, that is, the knowledge structure in the human brain, which includes both feelings, perceptions, representations, etc. Concepts, propositions, and schemas respectively mark the different breadth and depth of an individual's response to objective things, which are formed through individual cognitive activities. Generally speaking, individual knowledge is stored in the brain in the form of a hierarchical network structure (cognitive structure) from concrete to abstract. Philosophy mainly studies the nature of collective knowledge in human society, while psychology mainly studies the nature of individual knowledge.
Quotes about knowledge
Gorky: Take care of books, they are the source of knowledge.
Northcote: A learned man is a reservoir of knowledge, not a source.
If you don’t absorb the light of knowledge, your mind will be shrouded in darkness.
Flex: A university is an institution that consciously devotes itself to the pursuit of knowledge, strives to solve problems, critically evaluates people's achievements, and educates people at a truly high level. .
Chesterfield: When we enter old age, knowledge will be our comfortable and necessary retreat; if we do not plant the tree of knowledge when we are young, we will have no place to enjoy the shade when we are old. .
Zhu Xi of the Song Dynasty: What is urgent is that it is difficult to know without seeking; practice what you know and do not be afraid of doing what is difficult.
Chesterfield: Knowledge can be gained by reading; but more useful knowledge and understanding of the world can only be obtained by studying various people.
Cy Johnson: The thirst for knowledge is a natural tendency of human beings, and any sane person will do whatever it takes to obtain knowledge.
Engels: Complex labor involves the application of skills and knowledge that require more or less effort, time and money to acquire.
Custer: The manager does not bear the task of creating knowledge, his task is to use knowledge effectively.
·Riggs: A manager's management ability is a function of his quality, knowledge and experience. These three factors interact to form a special management style.
Deng Xiaoping: Modernization cannot be achieved by relying on empty talk. We must have knowledge and talents. Without knowledge and talent, how can you get ahead?
Kolmogorov: Science is the unique wealth of mankind, and the task of a real scientist is to enrich this treasure trove of knowledge that can benefit all mankind.
Herbert Spencer: Science is systematized knowledge.
Joseph Rue: Science is for those who are diligent and studious, poetry is for those who are knowledgeable.
O. Holmes: Science is the anatomy of ignorance.
Schopenhauer: Extensive thoughts and knowledge without deep experience are like a textbook with only two lines of text per page but forty lines of notes.
Argument: If a person has knowledge, he is powerful.
Practice is the mother of knowledge, and knowledge is the beacon of life.
Einstein: To learn knowledge, you must be good at thinking, thinking, and thinking again.