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 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 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.
Application of fuzzy mathematics
Fuzzy mathematics is a new discipline, which has been applied to fuzzy control, fuzzy identification, fuzzy cluster analysis, fuzzy decision-making, fuzzy evaluation, system theory, information retrieval, medicine, biology and other fields. 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.
Fuzzy mathematics is a new subject in mathematics, and its future is limitless.
1965 published the paper "Fuzzy Sets". The author is a famous cybernetics major.
Home, Professor L.A. Zadeh of California State University. Cantor's set theory has become the basis of modern mathematics. It must be the first time for someone to modify the concept of set now. Zadeh's concept of fuzzy sets laid the foundation of fuzzy theory. Because this theory is simple and powerful in dealing with complex systems, especially those with human intervention, it makes up for the shortcomings of classical mathematics and statistical mathematics to a certain extent, and has quickly attracted widespread attention. Over the past 40 years, this field has achieved fruitful results from theory to application, from soft technology to hard technology, and has exerted more and more significant influence on the development of related fields and technologies, especially some high and new technologies.
There is an old Greek paradox that goes like this:
"A seed is never called a pile, nor two, nor three ... On the other hand, everyone agrees that 100 million seeds must be called a pile. So, where is the appropriate boundary? Can we say that 123585 seeds are not called heaps, but 123586 seeds form heaps? "
Indeed, "a grain" and "a pile" are two different concepts. However, the difference between them is gradual, not abrupt, and there is no clear boundary between them. In other words, the concept of "a pile" is somewhat vague. Similar concepts, such as "old", "high", "young", "big", "smart", "beautiful" and "cheap" are endless.
In classical set theory, when determining whether an element belongs to a set, there are only two answers: "Yes" or "No". We can describe it with two values 0 or 1. Elements belonging to the set are represented by 1, and elements not belonging to the set are represented by 0. The situation of "old", "tall", "young", "big", "smart", "beautiful" and "cheap" mentioned above is much more complicated. If it is stipulated that the height1.8m belongs to the height range, does the height1.79m count? According to the classical set theory: not counting. But this seems unreasonable. If a circle is used, the set A is represented by points inside the circle and on the circumference, and points outside the circle indicate that it does not belong to a .. The boundary of a is obviously a circle. This is a classic series of charts. Now, imagine that a set of tall people is represented by a graph, and its boundary will be fuzzy and changeable. Because an element (such as a person with a height of 1.75 meters) is relatively high, although it is not 100% high, and it belongs to a collection of tall people to some extent. At this time, whether an element belongs to a set can not be expressed by the numbers 0 and 1, but can be any real number between 0 and 1. For example, the height of 1.75 meters can be said that 70% of the height belongs to the high subset. It seems verbose, but it is more practical.
Accuracy and fuzziness are contradictions. According to different situations, sometimes it is required to be accurate and sometimes it is required to be vague. For example, in a war, the commander issued an order: "Launch a general attack at dawn." This is really a mess. At this time, it must be accurate: "The general attack is being launched at 6 am on XX." In some old movies, we can also see that commanders in various positions are looking at their watches before accepting orders, for fear of an error of half a minute and ten seconds. However, extremes meet. If everything is accurate, people can't communicate ideas smoothly-two people meet and ask, "How are you?" However, what is "good" and who can give a precise definition of "good"?
Some phenomena are fuzzy in nature, and it is naturally difficult to conform to reality if we insist on making them precise. For example, when evaluating students' grades, it is stipulated that more than 60 points are qualified. However, the difference between 59 points and 60 points, only based on the difference of 1 point to distinguish between passing and failing, is not sufficient.
There is not only a set of fuzzy boundaries, but also human thinking and fuzzy characteristics. Some phenomena are accurate, but proper fuzzification may simplify the problem and greatly improve the flexibility. For example, it is troublesome and almost pedantic to find the biggest corn in the field. We must measure and compare all the corn in the corn field before we can be sure. Its workload is proportional to the area of corn field. The bigger the land area, the more difficult the work is. But as long as the formulation of the problem is changed slightly, it is not required to find the biggest corn, but to find a bigger one, that is, according to the usual saying, pick a big corn in the field. At this time, the problem has changed from precision to vagueness, but at the same time, it has changed from unnecessary complexity to unexpected simplicity, and only a few can meet the requirements. The workload is not even related to the land. Therefore, excessive precision has actually become pedantic, while appropriate vagueness 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, it is usually not necessary to determine these precise values. Similarly, the vague concept of "heap" is based on accurate "particles", and people never need to count "particles" when judging whether something in front of them is called heap. Sometimes, people think that fuzziness is a physical phenomenon. You can see clearly what is near, but you can't see clearly what is far away. Generally speaking, the farther away, the more blurred. However, there are exceptions: standing by the sea, the coastline is blurred; Looking down from the air, the coastline is very clear. It's too high and fuzzy. There are essential differences between precision and fuzziness, but there are internal relations. They are contradictory, interdependent and can be transformed into each other. So the other half of accuracy is ambiguity.
The discussion of fuzziness can be traced back to a long time ago. In a paper entitled Vagueness in 1923, B.Russel, a great philosopher in the 20th century, specifically discussed what we call "fuzziness" today (strictly speaking, there is a difference between the two), and clearly pointed out: "I think 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 vagueness or vagueness in academic circles at that time. This is not because the question is unimportant, nor because the article is not profound, but because it is not yet time. Russell's incisive view is ahead of time. For a long time, people have always regarded vagueness as a derogatory term, and only paid tribute to accuracy and strictness. At the beginning of the 20th century, the development of society, especially the development of science and technology, did not need the study of 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 aware of the limitations of precision and promotes people's research on its opposite or its "other half"-fuzziness.
Zadeh 192 1 was born in Baku, Soviet Union in February. Graduated from the Department of Electrical Engineering, Tehran University, Iran, with a bachelor's degree 1942. 1944 received a master's degree in electrical engineering from Massachusetts Institute of Technology (MIT), and 1949 received a doctorate from Columbia University. Later, he worked in famous universities such as Columbia and Princeton. Since 1959, he has been a professor in the Department of Electrical Engineering and Computer Science at the University of California, Berkeley.
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 began to study multi-objective decision-making problems and put forward some important concepts such as non-inferior solutions. For a long time, based on the research on a series of related important issues such as decision-making and control, Zadeh gradually realized the limitations of traditional mathematical methods from the success or failure of applying traditional mathematical methods and modern electronic computers to solve such problems. 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 the use of vague concepts is a great wealth rather than a burden for human beings. This is the key to understanding the profound difference between human intelligence and machine intelligence. " The concept of precision can be described by the usual set. Fuzzy concepts should be described by corresponding fuzzy sets. Zadeh grasped this point, first made a breakthrough in the quantitative description of fuzzy sets, and laid the foundation of fuzzy theory and its application.
Set is the foundation of modern mathematics. Once fuzzy sets were put forward, the concept of "fuzzy" 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; The second is to learn 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. In the future, fuzzy logic circuits, fuzzy hardware, fuzzy software and fuzzy firmware may appear, and a new type of computer will appear, which can talk to people in natural language and be closer to human intelligence. Therefore, fuzzy mathematics will show its strong vitality more and more.
Is there any objection? Of course there is. Some probability theorists believe that fuzzy mathematics is only an application of probability theory. Some people who practice pure mathematics say that this is not mathematics. Those who engage in applications 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. No matter what others say, we just try our best."