Shulou(Shulou.com)11/24 Report--

As the saying goes: if you lose an inch, you will lose a thousand miles.

A man-made satellite cannot be sent to its intended orbit over the earth without precise mathematical calculation. It is inconceivable that a hundred-storey skyscraper can rise from the ground without accurate mathematical calculation.

Mathematics has always been known for its rigour and precision. However, in the 1960s, a new branch of mathematics called "fuzzy mathematics" emerged as a new force.

Is it possible that mathematical calculations do not need to be precise and accurate, but need to be "vague"? Of course not. The disciplines of natural science are mature only when they can be described in mathematical language. In Engels' time, the application of mathematics in biology was almost zero. However, today's biology is completely inseparable from mathematics. Even many social sciences are constantly pursuing quantification and mathematization. So, why kill a "fuzzy mathematics" halfway at this moment? We have to start with two different concepts.

In daily life, we encounter no more than two kinds of concepts. One is a clear concept, and it is clear whether the object belongs to this concept or not. For example, people, natural numbers, squares, etc. Either a person or not a person; either a natural number or a non-natural number; either a square or a non-square. It must be either this or that.

The other kind of concept object subordinate boundary is vague, depending on the judgment of people's thinking. For example, whether it is beautiful, early, cheap and so on. Xi Shi is recognized as a beautiful woman in ancient China, but there is a saying that "beauty is in the eye of the beholder", that is to say, people who may not be so beautiful in the eyes of some people are so beautiful in the eyes of others that they can be compared with Xi Shi. It can be seen that there is no precise boundary between "beauty" and "unbeauty".

Besides, "early" and "not early", 5 o'clock in the morning may be too late for cleaners who "dress up" for the city, but for most people, it is very early. As for whether it is cheap or not, it varies with people's feelings.

In the objective world, there are much more vague concepts than clear concepts. For this kind of fuzzy phenomenon, the existing mathematical models are difficult to apply, so it is necessary to form new theories and methods, that is, to build a bridge between mathematics and fuzzy phenomena. This is what we want to talk about "fuzzy mathematics".

What accelerates the construction of this bridge is the rapid development of computer science. As we all know, the human brain has an extraordinary ability to distinguish and deal with vague things. Take a child who identifies her mother as an example. Even if the mother changes her new clothes and hairstyle, her child will still quickly make accurate judgments from height, fat and thin, voice, appearance, posture, and so on.

If the computer were to do this, it would be necessary to calculate the mother's height, weight, walking speed and shape curve to a dozen places after the decimal point before starting to judge. Such "precision" is really contrary to one's wishes, and goes to the opposite of things.

Perhaps it was because the mother had a small furuncle on her face that the average height of this part was more than 0 mm higher than the original, which made the computer make the judgment of "rejection". No wonder the founder of fuzzy mathematics, professor of the University of California, automatic control expert L.A. Zade (L.A. Zadehjue 1921-2017) said: "the more complex the system you are faced with, the less people will be able to make it meaningfully accurate."

He gave a vivid example of the parking problem, saying that it was not difficult for experienced drivers to park their cars in the open space between two cars in a crowded parking lot. However, even a large electronic computer is not easy to solve with an accurate method.

So, where is the way out for computers to imitate the human brain and identify and judge complex systems?

Professor Zade advocates a step back in terms of precision in the face of extreme complexity. He proposes to mathematize fuzzy concepts with membership functions. For example, "baldness" is obviously a vague concept.

There are five types of hair in the picture above.

(a) the head does not have any hair, and the membership degree of its own standard "baldness" is 1.

(d) the head is typically bald, so the membership of "bald" can be set at 0.8.

(C) his head is covered with jet-black hair, which has nothing to do with "baldness" at all, so the membership degree of "baldness" is 0.

(B) the "baldness" of (e) is less than that of (a) and (d), but more than that of (c), and the degree of membership can be set at 0.5 and 0.3 respectively.

In this way, the fuzzy concept of "baldness" can be quantitatively defined by the following methods: [baldness] = 1 / a / b / 0.5 / b / cm 0.8 / d = 0.3 / e

The "+" and "/" here are not the usual addition and division, but just a sign. "1 / a" indicates that the subordinate degree of state an is "1", and "+" indicates the juxtaposition of various situations.

Next, let's look at the two vague concepts of "young" and "old".

Professor Zade himself fitted the membership function images of these two concepts according to the statistical data. In the picture, the Abscissa indicates the age and the ordinate indicates the degree of membership.

For example, it can be seen from the coordinate diagram that people under the age of 50 do not belong to "old age", and when they are over 50 years old, with the increase of age, the degree of membership of "old age" also increases.

"Life is rare since ancient times," and the subordinate degree of "old age" of 70-year-olds has reached 94%. Similarly, in the coordinate chart, we can see that people under the age of 25, the degree of membership of "young" is 100%, over 25 years old, the degree of "young" is getting smaller and smaller. The age of 40 is already "middle age", and the subordinate degree of "young" is only 10%. Suppose someone asks you, "is your math teacher young?" Your answer is: "he is' young 'with a membership of 25%." Of course, there is nothing wrong with such an answer, but it is obviously awkward.

In order to give people a definite impression, we can fix a percentage, such as 40%. Those whose membership is greater than or equal to 40% are called "young", otherwise they are not called "young".

Under this premise, your answer to your friend is yes, and you can tell your friend clearly that your math teacher is not young. Because at this time, the word "young" has changed from a vague concept to a clear concept.

Of course, the fixed percentage as the dividing line of membership should be selected through scientific analysis or through the statistics of opinion polls.

Taking the staging of ancient Chinese history as an example, "slave society" is a vague concept.

[slave Society] = 1 / Xia + 1 / Shang + 0.9 / Western Zhou + 0.7 / Spring and Autumn + 0.5 / warring States + 0.4 / Qin + 0.3 / Western Han + 0.1 / Eastern Han Dynasty

Take 0.5 degree of membership as the dividing line of the slave society, then those belonging to the slave society should be the Xia, Shang, Western Zhou, Spring and Autumn period and the warring States period. Qin and Han dynasties did not belong to the slave society.

In exact mathematics, the words "very", "very" and "no" are difficult to express in quantity. But in fuzzy mathematics, they can be quantified. For example, "very" means the square of the degree of membership, and "no" means using 1 to subtract the original degree of membership, and so on. If the membership degree of "young" at the age of 30 is 0.5, then the membership degree of "very young" is only (0.5) 20.25, while that of "not very young" is 1-(0.5) 2 = 0.75.

As we can see above, when we quantitatively depict the fuzziness of things, we also need to use the means of probability and statistics and accurate mathematical methods. Thus it can be seen that "fuzzy mathematics" is actually not fuzzy.

The birth of fuzzy mathematics extends the application field of mathematics from clear phenomenon to fuzzy phenomenon, which makes mathematics break into many "forbidden areas" which are difficult to reach in the past. Using the model of fuzzy mathematics to program and let the computer simulate the thinking activity of human brain has been successful in character recognition, disease diagnosis, meteorological prediction, rocket launch and so on.

Although fuzzy mathematics has been studied in our country for only a few decades, this new discipline has developed rapidly and has shown strong vitality. At present, the discipline has begun to show its edge in the application of industrial, agricultural and national defense technology.

Source: "Mathematics Story Book for Children" author: Zhang Yuannan Zhang Chang part of the picture is from the network copyright belongs to the original author. This article comes from Wechat official account: ID:tupydread, author: Zhang Yuannan Zhang Chang, Editor: Zhang Runxin

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