Indeed, like "a grain" and "a pile" above, "maximum" and "larger" are two different concepts. But their differences are gradual, not sudden, and there is no clear boundary between them. In other words, these concepts are vague to some extent. We say that a person is tall or fat, but how many centimeters is tall and how many kilograms is fat? Like here, height and fat are blurred.
Fuzzy mathematics Fuzzy mathematics is a mathematical tool to study many problems with unclear boundaries in reality, and one of its basic concepts is fuzzy set. Fuzzy mathematics and fuzzy logic can deal with all kinds of fuzzy problems well.
Pattern recognition is one of the important fields of computer application. The human brain can effectively deal with complex low-precision problems. If the computer uses fuzzy mathematics, it can greatly improve the ability of pattern recognition and simulate the activities of human nervous system. In the field of industrial control, the application of fuzzy mathematics can make the temperature control of air conditioner more reasonable, and the washing machine can save electricity and water and improve efficiency. In the large-scale system management of modern society, using fuzzy mathematics methods can form more effective decisions.
Fuzzy mathematics is a relatively new mathematical method and thinking method. Although it needs continuous improvement, its application prospect is very broad.
Fuzzy mathematics is a new branch of mathematics that studies and deals with fuzzy phenomena by mathematical methods. It is based on the theory of "fuzzy set". Fuzzy mathematics provides a new method to deal with uncertainty and inaccuracy, and it is a powerful tool to describe human brain thinking and deal with fuzzy information. It can be used in both "hard" science and "soft" science.
Fuzzy mathematics was founded by Professor L.A. Zadeh, an American cybernetic expert (192 1-). He published a paper entitled "Fuzzy Sets" in 1965, thus announcing the birth of fuzzy mathematics. Professor L.A. Zadeh has devoted himself to the study of the contradiction between "computer" and "large-scale system" for many years, focusing on why computers can't think and judge flexibly like human brains. Although the computer memory is superhuman and the calculation is fast, it is "at a loss" when facing the vague state with unclear extension. In the process of perception, recognition, reasoning, decision-making and abstraction, it is entirely possible for human brain thinking to accept, store and process fuzzy information. Why can't computers handle fuzzy information like human brain thinking? The reason is that traditional mathematics, such as Cantor's set, can't describe the phenomenon of "this and that". Set is a mathematical method to describe the recognition and classification of holistic and objective things by human brain thinking. Cantor's set theory requires that its classification must follow the law of excluded middle of formal logic. Any element in the universe (that is, all the objects considered) either belongs to set A or does not belong to set A, and both must be one and only one. In this way, Cantor set can only describe the "clear concept" with clear extension, and can only express "either one or the other", but can not reflect the "fuzzy concept" with unclear extension. This is an important reason why computers can't deal with fuzzy information as flexibly and agile as human brain thinking. In order to overcome this obstacle, Professor Zadeh put forward the "fuzzy set theory". On this basis, a fuzzy mathematics system is formed.
The so-called fuzzy phenomenon refers to the state that it is difficult to distinguish objective things with clear boundaries. It comes from people's identification and classification of objective things, and is reflected in the concept. Concepts with different extensions are called different concepts and reflect different phenomena. The concept of extensional fuzziness is called fuzzy concept, which reflects fuzzy phenomenon. Ambiguity is common. There are a lot of vague concepts in human common language and scientific language. For example, some opposing concepts, such as height and height, beauty and ugliness, cleanliness and pollution, minerals and non-minerals, and even humans and apes, vertebrates and invertebrates, living and non-living, have no absolute boundaries. Generally speaking, a clear concept is abstracted by sublating the fuzziness of the concept, and the concept is accurate and rigorous through absolute thinking. However, fuzzy sets do not simply sublate the fuzziness of concepts, but reflect the original meaning of people when using fuzzy concepts as truly as possible. This is the fundamental difference in methodology between fuzzy mathematics and general mathematics. Engels said: "Dialectics does not know what is an absolute and fixed boundary, nor does it know what is an unconditional and universally effective' either-or!'" "It makes fixed metaphysical differences transition to each other, except' this or that!' And make the opposite an intermediary; Dialectics is the only way of thinking, which is most suitable for this stage of the development of natural concepts.
The direct motive force of fuzzy mathematics is closely related to the development of system science. In multivariable, nonlinear and time-varying large-scale systems, complexity and accuracy have formed a sharp contradiction. Professor L.A. Zadeh summed up the principle of reciprocity from practice: "As the complexity of the system increases, our ability to make an accurate but meaningful description of the system characteristics will decrease accordingly until we reach such a threshold. Once we surpass it, accuracy and meaning will become two almost mutually exclusive features. " In other words, the higher the complexity, the lower the meaningful accuracy. Complexity refers to many factors, some of which are difficult for people to accurately grasp, and it is often impossible for people to accurately examine all the factors and processes, but only to grasp the main part and ignore the so-called secondary part. In this way, it actually brings fuzziness to the description of the system. "The application of conventional mathematical methods to the analysis of fuzzy systems is essentially uncoordinated and will cause a huge gap between theory and practice." Therefore, we must find a set of mathematical methods to study and deal with fuzziness. This is the historical inevitability of fuzzy mathematics. Fuzzy mathematics uses precise mathematical language to describe fuzzy phenomena. It represents an idea that is different from the traditional method of dealing with uncertainty and inaccuracy based on probability theory, … different from the traditional new methodology. It can better reflect the fuzzy phenomenon that exists objectively. Therefore, it provides a powerful tool for describing fuzzy systems.
Professor Zadeh's long series of papers "The Concept of Language Variables and Its Application in Approximate Reasoning" was published in 1975 ("The Concept of Language Variables &; Its application in approximate reasoning), the concept of language variable is put forward and its meaning is discussed. The concept of fuzzy language is one of the most important developments of fuzzy set theory, and the concept of language variables is an important aspect of fuzzy language theory. Language probability and its calculation, fuzzy logic and approximate reasoning can all be regarded as the application of language variables. The ability of human language to express subjective and objective fuzziness is particularly amazing. Perhaps starting from the study of fuzzy language, we can master subjective fuzziness and objective fuzziness, and find out how to deal with these fuzziness. It is expected that this theory and method will make an important contribution to control theory and artificial intelligence.
Fuzzy mathematics was born only 22 years ago, but it developed rapidly and was widely used. It involves pure mathematics, applied mathematics, natural science, humanities and management science. It has been widely used in image recognition, artificial intelligence, automatic control, information processing, economics, psychology, sociology, ecology, linguistics, management, medical diagnosis, philosophical research and other fields. Fuzzy mathematics theory is applied to decision-making research, and fuzzy decision-making technology is formed. As long as we study it carefully, we will find that in most cases, the decision-making objectives and constraints are fuzzy, especially for the decision-making process of complex large-scale systems. In this case, the application of fuzzy decision technology will be more natural and get better results.
Chinese scholars began to study fuzzy mathematics in the mid-1970s, but it developed rapidly. A strong research team was established, the China Society of Fuzzy Sets and Systems was established, and the magazine Fuzzy Mathematics was published. He has published many valuable works such as Fuzzy Sets and Random Sets edited by Professor Wang Peizhuang, Fuzzy Set Theory and Its Application, and Fuzzy Mathematics Basis edited by Professor Zhang Wenxiu. Chinese scholars apply fuzzy mathematics theory to meteorological forecast, which improves the forecast quality. At the international meteorological symposium held in 1980, the paper submitted by China was well received by the conference. In the aspect of medical diagnosis of traditional Chinese medicine, a computer diagnosis program for Professor Guan Youbo's treatment of liver diseases was also made. Practice shows that the medical effect of computer is good, which has made contributions to inheriting and carrying forward traditional Chinese medicine. This experience has also been applied to the treatment of acute abdomen. Chinese scholars have applied fuzzy mathematics theory to geological prospecting, ecological environment, enterprise management, biology, psychology and other fields, and achieved good results.