Modern mathematics is based on set theory. The significance of set theory can be seen from one side. With it, the abstract ability of mathematics extends to the depth 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 those objects that conform to the concept is called the extension of the concept. Extension is actually a collection. In this sense, sets can express concepts. In set theory,

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 defines that each set must be composed of clear elements. The subordination of elements to sets must be clear. You must not be ambiguous. For those concepts and things with unclear extension, classical set theory will not be reflected for the time being. It 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 used to avoid it. However, due to the increasingly complex system faced by modern science and technology, fuzziness is always accompanied by complexity.

The trend of mathematicization and quantification of various disciplines, especially humanities and social sciences, has pushed the mathematical treatment of fuzziness 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. The system is complex and fuzzy. From the cognitive point of view, fuzziness refers to the uncertainty of concept extension, which leads to the uncertainty of judgment.

In our daily life, we often encounter many vague things. There is no clear quantitative limit. We should describe them with some vague words. For example, the concepts of being relatively young, tall, fat, good, beautiful, kind, spicy and far away cannot be simply expressed by numbers. In people's work experience, there are often many vague things. For example, we should ensure that. In addition to accurate information such as smelting time, it is also necessary to refer to fuzzy information such as molten steel color and boiling situation. Therefore, in addition to early computational mathematics involving errors, fuzzy mathematics is also needed.

Generally speaking, the human brain has the ability to process fuzzy information and is good at judging and processing fuzzy phenomena, while the computer has a poor ability to identify fuzzy phenomena. 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. Mathematical tools to describe and deal with fuzzy information promote mathematicians' in-depth study of fuzzy mathematics. Therefore, with the development of science and technology and mathematics, the emergence of fuzzy mathematics is inevitable.

Research content of fuzzy mathematics

1965. Chad, an American cybernetic expert and mathematician, published a paper; It 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. Based on the set theory of precise mathematics, considering the modification and popularization of the concept of mathematical set, Chad proposed to use [fuzzy set] as a mathematical model to represent fuzzy things, gradually established the operation and transformation rules on [fuzzy set], and carried out related theoretical research. It is possible to construct a large number of fuzzy mathematical foundations for studying the real world.

In fuzzy sets, there are not necessarily only [yes] or [no] cases for elements in a given range, but real numbers between 0 and 1 are used to express the membership degree. There is also an intermediate transition state. For example, [old man] is a vague concept. A 70-year-old man must belong to the old man. A person with a membership degree of 1.40 is definitely not an old man. Its [old] degree is 0.5, which is [half old]. A 60-year-old [old] is 0.8. 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. 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 computer. This is the key to the application of 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 the standard sentence that conforms to grammar to 1, then the sentence that is slightly wrong in other grammars but can still express similar ideas can be characterized by the continuous number between 0 and 1. In this way, fuzzy language has been quantitatively described, and a set of operation and transformation rules have been formulated. At present, fuzzy language is still immature.

People's thinking activities often require the certainty and accuracy of concepts. Law of excluded middle's adoption of formal logic is neither true nor false. Then he makes judgment and reasoning, and draws a conclusion. The existing computers are all based on binary logic. It 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 rocchi 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 contribution lies in using the theory of fuzzy sets 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. The shortcomings of accurate mathematics and random mathematics description in the past can be made up. In fuzzy mathematics, there are fuzzy topology and fuzzy at present.

Application of fuzzy mathematics

Fuzzy mathematics is a new subject. It has been applied to fuzzy control, fuzzy identification, fuzzy cluster analysis, fuzzy decision-making, fuzzy evaluation, system theory, information retrieval, medicine, biology and so on. It has made concrete research achievements in meteorology, structural mechanics, control and psychology. However, the most important application field of fuzzy mathematics is computer function. Many people think that it 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. Yamagata Lielie of Japan successfully trial-produced the fuzzy inference machine for the first time. Its reasoning speed is100000 times per second. In 1988, several doctors under the guidance of Professor Wang Peizhuang in China have also successfully developed a fuzzy inference machine-a prototype of discrete components. The reasoning speed is as follows.

Fuzzy mathematics is far from mature, and there are still different opinions and views on it, which need to be tested by practice.

Fuzzy mathematics is a new subject in mathematics, and its future is limitless.

1965.； The paper was published. The author is a famous cybernetics major.

Go home. L.A.Zadeh of California State University, USA. Professor Cantor's set theory has become the basis of modern mathematics. Now someone wants to modify the concept of set. Of course, it is unprecedented. Zadeh's concept of fuzzy sets laid the foundation of fuzzy theory. This theory is simple and powerful in dealing with complex systems, especially those with human intervention, which makes up for the shortcomings of classical mathematics and statistical mathematics to some extent. In recent 40 years, this field has received extensive attention. From theory to application, from soft technology to hard technology.

There is an ancient Greek paradox. Here's the thing:

【 A seed is definitely not called a heap. Two seeds are not. Three seeds is not-on the other hand, everyone agrees that 100 million seeds must be called a pile. So, where is the appropriate boundary? Can it be said that 123585 seeds are not called heaps, but 123586 seeds constitute heaps? "

Indeed, [a grain] and [a pile] are two different concepts. However, their differences are gradual, not sudden. There is no clear boundary between them. In other words, the concept of [a pile] is a bit vague. Similar concepts, such as [old], [high], [young] and [big].

In classical set theory, when determining whether an element belongs to a set, there are only two answers: [Yes] or [No]. We can use 0 or 1 to describe it. Elements belonging to a set are represented by 1. Elements that do not belong to the collection are represented by 0. However, the [old] mentioned above. [high]. [Young]. From the point of view of classical set theory, it is not, but it seems quite unreasonable. If a circle is used to represent the set A, there are points inside and around the circle, and points outside the circle indicate that the boundary that does not belong to A.A. is obviously a circle, which is a graph of the classical set. Now, imagine using a graph to represent a high set, and its boundary will be fuzzy and variable. Because an element (for example, the height is 65433) is not 100% high, but it is still relatively high. To some extent, it belongs to a set of tall people. At this point, whether an element belongs to a set can not only be represented by two numbers, 0 and 1, but can be any real number between 0 and 1. For example, height 1.75 meters, it can be said that 70% belong to a group of tall people.

Accuracy and fuzziness are contradictory. According to different situations, sometimes accuracy is required. Sometimes ambiguity is necessary. For example, in the war, the commander issued an order: [launch a general attack at dawn. "This is too messy. At this time, we must demand accuracy: [launch a general attack on XXX at 6 am] But what is "good"? Who can give a precise definition of "good"?

Some phenomena are fuzzy in nature. If we insist on making them accurate, it will naturally be difficult to conform to reality. For example, students are required to be over 60 to be qualified. But what's the difference between 69 and 60? Only the difference of 1 is used to distinguish between passing and failing. Insufficient basis.

There are not only sets with blurred boundaries, but also human thinking. Some phenomena are accurate. However, proper fuzziness may simplify the problem and greatly improve flexibility. For example, finding the biggest one in the wild is very troublesome. It's just pedantic. We must measure and compare all the corn in the corn field before we can be sure. Its workload is in direct proportion to the corn field area. Land area. The more difficult the job is, however, just change the formulation of the problem slightly: instead of looking for the biggest corn, look for a bigger one, that is, pick a big corn in the field as usual. At this time, the problem changed from precision to vagueness, but at the same time it changed from unnecessary complexity to unexpected simplicity. You can meet the requirements by choosing a few. The workload has nothing to do with land. So excessive accuracy has actually become pedantic. Proper fuzziness is flexible.

Obviously, the size of corn depends on its length, volume and weight. Although size is a vague concept, length, volume and weight can be accurate in theory. However, when people actually judge the size of corn, they usually don't need to measure these accurate values. Similarly, the vague concept of "heap" is based on the accurate "grain", when people judge whether what is in front of them is called heap. A physical phenomenon. You can see things nearby very clearly. Things in the distance can't be seen clearly. Generally speaking, the farther you go, the more blurred you become. But there are exceptions: standing by the sea, the coastline is blurred and looking down from the sky. The coastline is clear, too high, fuzzy, accurate and fuzzy. There are essential differences, but there are internal relations. They are contradictory. They can also be transformed into each other. therefore ..

The discussion about fuzziness can be traced back to a long time ago. B.Russel, a great philosopher in the 20th century, wrote an article entitled Vagueness, which specifically discussed what we call "fuzziness" today (strictly speaking, there is a difference between the two) and clearly pointed out: [It is a big mistake to think that fuzzy knowledge must be unreliable. "Although Russell is famous, this article published in the Journal of Southern Hemisphere Philosophy did not arouse great interest in fuzziness or fuzziness in academic circles at that time. This is not because the question is unimportant, nor because the article is not profound, but [the time has not yet come]. Russell's incisive view is ahead of time. For a long time, people have always regarded vagueness as a derogatory term. Ge is full of respect. At the beginning of the 20th century, the development of society, especially the development of science and technology, no longer needs to study fuzziness. In fact, fuzzy theory is the product of the era of electronic computers. It is the invention and wide application of this very precise machine that makes people more deeply understand the limitations of precision and promotes the study of its opposite or its [other half]-fuzziness.

Zadeh was born in Baku, Soviet Union in February of 192 1. He graduated from the Department of Electrical Engineering of Tehran University in Iran with a bachelor's degree. 1944 obtained a master's degree in electrical engineering from Massachusetts Institute of Technology, and 1949 obtained a doctorate from Columbia University. Later, he studied in Princeton, Colombia and other places.

Zadeh engaged in the research of engineering cybernetics in 1950s, and made a series of important achievements in the design of nonlinear filters, which were regarded as classics and widely cited in this field. In the early 1960s, Zadeh turned to study multi-objective decision-making problems and put forward some important concepts such as non-inferior solutions. For a long time, from the success or failure of applying traditional mathematical methods and modern electronic computers to solve such problems, Zadeh gradually realized the limitations of traditional mathematical methods. He pointed out: [In the field of human knowledge, the only department where non-fuzzy concepts play a major role is classical mathematics. If we deeply study the cognitive process of human beings, we will find that human beings can regard vague concepts as great wealth, not a burden. This is the key to understand the profound difference between human intelligence and machine intelligence. "The concept of precision can be described by a common set. Fuzzy concepts should be used in stages. Zadeh grasped this point. First of all, he made a breakthrough in the quantitative description of fuzzy sets, which laid the foundation for fuzzy theory and its application.

Set is the foundation of modern mathematics. The presentation of fuzzy sets. The concept of [fuzziness] has also penetrated into many branches of mathematics. The development speed of fuzzy mathematics is also quite fast. Judging from the published papers, it is almost exponential growth. The research of fuzzy mathematics can be divided into three aspects: one is to study the theory of fuzzy mathematics and its relationship with precise mathematics and statistical mathematics, and the other is to study fuzzy language and fuzzy logic. The third is to study the application of fuzzy mathematics. In the study of fuzzy mathematics, there are fuzzy topology, fuzzy group theory, fuzzy convex theory, fuzzy probability and fuzzy ring theory. Although fuzzy mathematics is a new discipline, it has been initially applied to automatic control, pattern recognition, system theory, information retrieval, social science, psychology, medicine and biology, and fuzzy logic circuits, fuzzy hardware and fuzzy software may appear in the future.

Is there any objection? Of course. Some probability theorists believe that fuzzy mathematics is only an application of probability theory. Some people who do pure mathematics say that it is not mathematics. Those who engage in application say that the truth is very good, but the real practical effect is not. However, Professor A. Kaufman, an internationally renowned applied mathematician, said during his visit to China: [Their attack is unreasonable. Don't worry about what others say. Let's work hard. "