Fuzzy mathematics seems incredible at first glance. Because mathematics is characterized by accuracy, how can it be linked to "fuzziness"? In fact, fuzzy mathematics is not "fuzzy mathematics", but its real meaning is to study and deal with fuzzy things by mathematical methods. This is a new discipline born in 1965, which has developed rapidly for more than ten years.
Starting from Aesop's fables
There is such a story in Aesop's Fables. On one occasion, Aesop's master was drunk and raving, and made a bet with others, vowing to drink the sea dry and gamble on all his property and slaves. He woke up the next day and regretted it. But the news caused a sensation in the whole city, and people were waiting for him at the seaside. So the master had to ask the clever Aesop for help, and Aesop gave him an idea after talking about the terms. When the master heard this, he was like a treasure. He rushed to the seaside and shouted at the crowd there, "Now, I want to say it again, I can drink the whole sea dry. But now thousands of rivers flow into the sea, and many rivers are mixed in the sea. If anyone can separate the river from the sea, I can drink the real sea dry! "
Aesop simply applied fuzzy linguistics to help his master tide over the difficulties. Because "sea water" is a vague concept, although we often use this word, the definition of it is often full of loopholes. Similarly, there is no clear boundary between "fruit" and "vegetable" and "spring, summer, autumn and winter". There are many vague expressions in our life, such as light and shade, depth, cold and warm, width, speed, shade, height and so on.
Fuzzy logic is produced when fuzzy things are reflected in people's thinking. In fuzzy logic, when judging whether a proposition is true or false, you can not only answer it with "Yes" (1) or "No" (0), but also answer it with decimals between 0 and 1. Therefore, it is a continuous value logic.
Ambiguity is not a crime.
It is generally believed that "fuzzy" is a derogatory term, and its reputation has indeed been "bad". In the primitive society with very low productivity, people can barely survive. At that time, there was no need for mathematical calculation, and it was a chaotic and fuzzy world. However, with the continuous improvement of productivity, surplus products and commodities have been exchanged, so people began to count with fingers and pebbles, and gradually formed the concept of natural numbers. Starting from natural numbers, mathematics began its glorious course and finally won the reputation of "the crown of science". Obviously, as the opposite of precision, fuzziness once represented backward productive forces and had a disgraceful history. However, with the development of electronic computers, in order to further improve the degree of automation and the activity of computers, people began to study the similarities and differences between human brains and computers. The human brain is good at distinguishing and dealing with imprecise and unmeasurable fuzzy things, and draws certain accurate conclusions from them. It is precisely because the human brain has this ability that we can recognize scrawled handwriting, understand incomplete or unconventional sentences, and make correct decisions even in uncertain and changeable situations. Von Neumann, known as the "father of electronic computers", said that the human brain is such a computer. Although its accuracy is extremely low, only equivalent to two or three decimal places, its work is very reliable and efficient. For example, to judge who is coming in front, we only need to compare the height, thinness and walking posture. Information stored in the brain.
By comparing the samples, we can draw the correct conclusion. But to make the computer do this, it is necessary to make a big move, not only to measure a lot of data such as people's height, weight, arm swing angle, frequency, speed and acceleration, but also to be accurate to dozens of decimal places before giving up. Such triviality makes accuracy go to the opposite side. It is full of living dialectics: precise and vague; Vague and accurate. Only the human brain can unify the two well, just right. This skill is beyond the reach of electronic computers. Von Neumann believes that there is no computer like the human brain in the world. Therefore, vagueness is by no means a sin. On the contrary, it is a gift from nature. Fuzzy methods always operate quietly in the human brain, which promotes the development of human society.
Thus, fuzzy mathematics was born. Fuzzy mathematics focuses on solving two problems: one is to provide new mathematical tools for complex systems, especially those forbidden areas of classical mathematics, such as humanities; Second, computers can imitate the human brain to identify and judge complex systems and improve the level of automation.
Mathematical Description of Fuzzy Things —— Fuzzy Sets
Modern mathematics is based on set theory, and fuzzy mathematics is also based on set theory. The first paper on fuzzy mathematics published by American scholar Chad 1965 is called Fuzzy Sets.
The sum of things combined according to any characteristics and laws is called a set. For example, "things on the table", "planets in the solar system" and "passengers in cars" can all form a set. The individuals that make up a set are called elements. In general set theory, a thing (element) either belongs to a set or does not belong to a set, and the two must be one of them. In other words, the boundaries of this set can be clearly defined, such as "male students" and "students who failed in mathematics", which are all ordinary sets. However, the sets of "fat man", "young man" and "tall man" are completely different in nature, and it is impossible to give a clear answer whether a person belongs to this set. This kind of ambiguous set is a fuzzy set, which is called a "soft set". Accordingly, a common set with clear boundaries is called a hard set.
Because in a fuzzy set, whether an element belongs to a set can't be absolutely answered, so we should use a number to express the "degree" of its belonging to a set. We previously recorded "Yes" as 1 and "No" as 0, so the membership degree can be continuously taken from 0 to 1. Membership is the most basic and active factor in fuzzy mathematics. The so-called fuzzy set operation is not a general numerical operation, but a special fuzzy operation for the membership degree of values between 0 and 1.
For example, a small number of people, such as A, B, C and D, belong to the fuzzy set of "fat people", and their membership degrees are 0. 1, 0.5 and 0.7 respectively.
1, which means that B is "half fat" and only D is really fat. This fuzzy set can be expressed as: {fatty} = 0.1/a+0.5/b+0.7/c+1/d.
In the formula, a plus sign indicates juxtaposition, which means no addition; The numerator of each fraction represents the degree of membership, and the denominator represents the name of the element. Compared with the common set in the same range: {boy} = 0/a+1/b+1/c+1/d; {girl} = 1/A+0/B+0/C+0/D。
It can be seen that only one of the four people is a girl. It is not difficult to see that ordinary sets are only special cases of fuzzy sets (membership is equal to 0 or1); Fuzzy set is a natural extension of ordinary set, and fuzzy set is a more advanced and general set.
In fuzzy mathematics, determining the degree of membership is an art. It can be given according to experience or statistical laws, and can also be determined by authoritative organizations, so it is subjective and relative. For example, the membership degree of fuzzy set "old man" given by Chad is:
(omitted here, see the original page for details)
Where y stands for age. When y≤50 (years old), its membership degree is 0, so none of them belong to the "old people" set; When y = 55 (years old), you can get 0.5 by substituting the above formula, that is, people who are 55 years old are "semi-old"; When y = 60 (years old), its membership degree is 0.8, that is, a person who is 60 years old is "0.8 years old", and so on.
Fuzzy mathematics is promising.
Because of the highly differentiated combination of contemporary science and technology, the huge scientific system has become a multi-level and multi-sequence three-dimensional structure. The study of science of science shows that modern science has developed from the study of things to the study of systems, from the study of single value to the study of multiple values, from static research to dynamic research, from vertical research to horizontal research. The characteristics of fuzzy mathematics determine that it will become a powerful tool to study complex systems. It builds a bridge between classical mathematics and the real world full of fuzziness. At present, the application of fuzzy mathematics has involved cluster analysis, image recognition, factory control, mechanical fault diagnosis, system evaluation, data structure, information retrieval, robotics, artificial intelligence, logic and many other aspects. For example, in environmental protection, environmental units are divided according to pollution degree, and parent crops are divided in improved seed cultivation. It's like:
"A ray of sunshine through the clouds, the more see Jiao Yan.
The truth of Vientiane in the world is becoming more and more blurred;
-I accidentally saw a little light in the blur and felt more beautiful. "
The philosophy expounded in Premier Zhou's poem "Youth Chasing Light" is not the best annotation to fuzzy mathematics!