From the point of view of pure mathematics, the expansion of the concept of set has added new contents to many branches of mathematics. Such as fuzzy topology, fuzzy linear space, fuzzy measure and integral, fuzzy group, fuzzy category, fuzzy graph theory and so on. Some of these areas have been thoroughly studied.
The mainstream of fuzzy mathematics development lies in its application. Because the concept of fuzziness finds the description of fuzzy sets, the process of people's judgment, evaluation, reasoning, decision-making and control by using concepts can also be described by fuzzy mathematics. Such as fuzzy cluster analysis, fuzzy comprehensive evaluation, fuzzy decision-making, fuzzy control and so on. These methods constitute the embryonic form of fuzzy system theory and speculative mathematics, and have made concrete research achievements in the fields of medicine, meteorology, psychology, economic management, petroleum, geology, environment, biology, agriculture, forestry, chemical engineering, language, control, remote sensing, education and sports. The most important application field of fuzzy mathematics should be computer intelligence. It has been used in expert system and knowledge engineering. [Edit this paragraph] The emergence of fuzzy mathematics Modern mathematics is based on set theory. The significance of set theory can be seen from one side, which extends the abstract ability of mathematics to the depths of human cognition. A set of objects determines a set of attributes. People can explain the concept (connotation) by explaining the attribute, and can also explain it by specifying the object. The sum of the objects that conform to the concept is called the extension of the concept, and the extension is actually a set. In this sense, sets can represent concepts, while relations and operations in set theory can represent judgments and reasoning. All realistic theoretical systems may be included in the mathematical framework of set description.
However, the development of mathematics also has stages. Classical set theory can only limit its expressive force to those concepts and things with clear extension. It clearly stipulates that each set must be composed of clear elements, and the subordinate relationship of elements to the set must be clear and unambiguous. For those concepts and things with unclear extension, the classical set theory has not been reflected for the time being and belongs to the category to be developed.
For a long time, precise mathematics and stochastic mathematics have made remarkable achievements in describing the laws of motion of many things in nature. However, there are still a lot of vague phenomena in the objective world. People are used to avoiding, but because the system faced by modern science and technology is more and more complex, fuzziness is always accompanied by complexity.
The mathematization and quantification of various disciplines, especially "soft sciences" such as humanities and social sciences, have pushed the problem of fuzzy mathematics to the central position. More importantly, with the rapid development of electronic computers, cybernetics and system science, it is necessary to study and deal with fuzziness in order to make computers have the ability to identify complex things like human brains.
We study the behavior of human systems, or deal with complex systems that can be compared with the behavior of human systems, such as aerospace systems, human brain systems, social systems and so on. There are many parameters and variables, and various factors are intertwined, so the system is complex and fuzzy. In terms of cognition, fuzziness refers to the uncertainty of concept extension, which leads to the uncertainty of judgment.
In daily life, we often encounter many vague things, and there is no clear quantitative boundary. We should use some vague words to describe them. For example, young, tall, fat, good, beautiful, kind, hot and far away. These concepts cannot be simply expressed by yes, no or numbers. In people's work experience, there are often many vague things. For example, to determine whether a heat of molten steel has been smelted, it is necessary to know not only the precise information such as the temperature, composition ratio and smelting time of molten steel, but also the fuzzy information such as the color and boiling situation of molten steel. Therefore, besides computational mathematics involving errors, fuzzy mathematics is also needed.
Compared with computers, generally speaking, the human brain has the ability to deal with fuzzy information and is good at judging and dealing with fuzzy phenomena. However, the ability of computer to identify fuzzy phenomena is very poor. In order to improve the computer's ability to identify fuzzy phenomena, it is necessary to design the commonly used fuzzy language into instructions and programs acceptable to the machine, so that the machine can make corresponding judgments simply and flexibly like the human brain, thus improving the efficiency of automatically identifying and controlling fuzzy phenomena. In this way, we need to find a mathematical tool to describe and deal with fuzzy information, which promotes mathematicians' in-depth study of fuzzy mathematics. Therefore, the emergence of fuzzy mathematics has its inevitability in the development of science, technology and mathematics. [Edit this paragraph] The research content of fuzzy mathematics 1965, Chad, an American cybernetic expert and mathematician, published his paper "Fuzzy Sets", which marked the birth of fuzzy mathematics.
The research content of fuzzy mathematics mainly includes the following three aspects:
Firstly, the theory of fuzzy mathematics and its relationship with precise mathematics and stochastic mathematics are studied.
Chad is based on the set theory of precise mathematics, taking into account the modification and popularization of the concept of mathematical set. He proposed using "fuzzy set" as a mathematical model to express fuzzy things. And gradually establish the operation and transformation rules on the "fuzzy set" and carry out relevant theoretical research, it is possible to build a mathematical basis for studying a large number of fuzziness in the real world and a mathematical method for quantitatively describing and dealing with seemingly complicated fuzzy systems.
In a fuzzy set, the membership relationship of elements in a given range is not necessarily only "yes" or "no", but the membership degree is expressed by real numbers between 0 and 1, and there is an intermediate transition state. For example, "old man" is a vague concept. A 70-year-old must be an old man, and his membership degree is 1. A 40-year-old is definitely not an old man, and his membership degree is 0. According to the formula given by Chad, the "old" degree is 0.5 at the age of 55, that is, "semi-old", and it is 0.8 at the age of 60. Chad thinks that indicating the subordinate set of each element is equivalent to specifying a set. When it belongs to a value between 0 and 1, it is a fuzzy set.
Second, study fuzzy linguistics and fuzzy logic.
The natural language of human beings is vague, and people often accept vague language and vague information, and can make correct identification and judgment.
In order to realize the direct dialogue between natural language and computer, it is necessary to refine human language and thinking process into mathematical models, and then input instructions to the computer to establish a harmonious fuzzy mathematical model, which is the key to using mathematical methods. Chad uses fuzzy set theory to establish a mathematical model of fuzzy language, which makes human language quantitative and formal.
If we set the subordinate function value of a grammatical standard sentence to 1, then other sentences that are slightly grammatically wrong but can still express similar ideas can be represented by continuous numbers between 0 and 1. In this way, fuzzy language is described quantitatively, and a set of operation and transformation rules are set up. At present, fuzzy language is not mature, and linguists are studying it deeply.
People's thinking activities often require the certainty and accuracy of concepts. Law of excluded middle, using formal logic, is not true or false, and then makes judgments and inferences and draws conclusions. The existing computers are all based on binary logic, which plays a great role in dealing with the certainty of objective things, but it does not have the ability to deal with the uncertainty or fuzziness of things and concepts.
In order to make the computer simulate the characteristics of advanced intelligence of human brain, it is necessary to turn the computer into multi-valued logic and study fuzzy logic. At present, fuzzy logic is still immature and needs further study.
Thirdly, the application of fuzzy mathematics is studied.
Fuzzy mathematics takes uncertain things as the research object. The appearance of fuzzy sets is the need for mathematics to adapt to the description of complex things. Chad's merit lies in the use of fuzzy set theory to find and solve fuzzy objects and make them accurate, so that the mathematics of deterministic objects can communicate with the mathematics of uncertain objects, making up for the shortcomings of accurate mathematics and random mathematics description in the past. In fuzzy mathematics, there are many branches such as fuzzy topology, fuzzy group theory, fuzzy graph theory, fuzzy probability, fuzzy linguistics and fuzzy logic. [Edit this paragraph] Application of fuzzy mathematics Fuzzy mathematics is a new discipline, which has been initially applied to fuzzy control, fuzzy identification, fuzzy cluster analysis, fuzzy decision-making, fuzzy evaluation, system theory, information retrieval, medicine, biology and so on. There are concrete research results in meteorology, structural mechanics, control and psychology. But the most important application field of fuzzy mathematics is computer function, which many people think is closely related to the development of a new generation of computers.
At present, the developed countries in the world are actively researching and trial-producing intelligent fuzzy computers. 1986, Dr. Ryder Yamakawa of Japan successfully trial-produced the fuzzy inference machine for the first time, and its inference speed was100000 times per second. 1988, under the guidance of Professor Wang Peizhuang, several Chinese doctors also successfully developed a fuzzy inference machine-a prototype of discrete components, and its inference speed is150,000 times per second. This shows that China has taken an important step in breaking through the difficulties of fuzzy information processing.
Fuzzy mathematics is far from mature, and there are still different opinions and views on it, which need to be tested by practice.