1965, the American cybernetic scholar L.A. Zadeh published the paper Fuzzy Sets, which marked the birth of this new discipline. Modern mathematics is based on set theory. A set of objects determines a set of attributes, and people can explain concepts by specifying attributes or objects. The sum of the objects that conform to the concept is called the extension of the concept, and the extension is actually a set. All realistic theoretical systems may be included in the mathematical framework of set description. Classical set theory only limits its expressive force to those concepts and things with clear extension. It clearly stipulates that every set must be composed of certain elements, and the subordinate relationship of elements to the set must be clear. The mathematical treatment of fuzziness is based on the extension of classical set theory to fuzzy set theory, and the fuzzy subset in product space gives the fuzzy relationship between a pair of elements. On this basis, the fuzzy phenomenon is dealt with mathematically.
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 algebra, fuzzy analysis, fuzzy measure and integral, fuzzy group, fuzzy category, fuzzy graph theory, fuzzy probability statistics, fuzzy logic 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. For example, fuzzy clustering analysis, fuzzy pattern recognition, fuzzy comprehensive evaluation, fuzzy decision-making and fuzzy prediction, fuzzy control, fuzzy information processing 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 applied to expert system and knowledge engineering, played a very important role in various fields, and achieved great economic benefits.
The computing speed and storage capacity of modern computers are almost unparalleled. It can not only solve complex mathematical problems, but also participate in controlling the space shuttle. Since computers are so powerful, why are they sometimes inferior to human brains in judgment and reasoning? Professor Zade of the University of California has studied this problem carefully, so she is engaged in scientific research.
It is often in contradiction with "human brain thinking", "large system" and "computer". 1965, he published the concept of "membership function" in his paper "fuzzy set theory" to describe the intermediate transition in the phenomenon difference, thus breaking through the absolute relationship of belonging or not in classical set theory. Professor Zadeh's pioneering work marks 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 model, and then input instructions to the computer to establish an appropriate 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 with similar meanings and 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, adopt the law of excluded middle of formal logic, that is, truth and falsehood, and then make judgments and inferences to draw 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.