There was a famous baldness paradox in ancient Greece, the key of which was that baldness and non-baldness could not be defined in precise language. Similarly, in real life, there are many things that cannot be precisely defined. For example, like "noble", "vulgar", "beautiful" and "ugly", it cannot be said that a person is either beautiful or ugly. This kind of behavior is to blur the uncertainty of things. Compared with the certainty of things in classical mathematics, it is more general, and you can't get a well-defined category when dividing things. In other words, it clearly reflects the either-or nature of things and categories, and fuzziness reflects the either-or nature of things and categories. It should be pointed out here that fuzziness and randomness are different. Randomness is relative to inevitability, which means that whether an event happens or not is uncertain, but the nature and characteristics of the event itself are certain. In the random experiment, there is no third possibility for the occurrence or non-occurrence of an event, so the random phenomenon obeys law of excluded middle, while the fuzziness does not obey law of excluded middle.
From the perspective of set, we can see the difference between fuzzy mathematics and classical mathematics more clearly. In classical mathematics, a set refers to a set of individuals with a certain property, and individuals without this property do not belong to this set. In order to express this either-or property, we introduce the characteristic function in mathematics:
In this way, the characteristic function value of each element is either 1 or 0, and the elements in the whole universe are divided into two categories: A and C (the complement of A). Things that are clearly described by ordinary sets can accurately describe the related quantitative relations.
However, in fuzzy mathematics, the uncertainty of the research object determines that it cannot be represented by a common set, and its characteristic function value is not limited to 0 and 1, but also includes other intermediate things, so we extend the characteristic value to any real number between [0, 1]. 0 means not belonging at all, 1 means completely belonging, and values between 0 and 1 indicate the degree of membership. The higher the numerical value, the higher the degree of membership. This generalized characteristic function is called membership function.
Zadeh defined fuzzy sets with membership functions.
Membership function describes the gradual change process of elements from belonging to a set to not belonging to a set, which makes the application of fuzzy mathematics possible. Zadeh once gave the membership function of the old people's collection, thus calculating the membership of people aged 50, 55, 60, 65 and 70.
From this, it is easier to see whether a person is old or not, not only yes or no, but also can reflect the essence of things and conform to the objective reality.
In the comparison between the human brain and the computer, people find the deficiency of the computer based on binary logic: it can't handle the ubiquitous fuzzy information. Therefore, Zadeh said, "Fundamentally speaking, in order to achieve high efficiency, it is necessary to apply a computer specially designed for processing fuzzy information." We have reason to believe that with the development of fuzzy mathematics, this highly intelligent computer will inevitably become the mainstream of computer development in the future.