Methodological issues of the fuzzy set theory (generalizing article)

The theory of fuzziness is an important area of modern theoretical and applied mathematics. The methodology of the theory of fuzziness is a doctrine of organizing activities in the field of development and application of the scientific results of this theory. We discuss some methodological issues of the theory of fuzziness, i.e., individual components of the methodology in the area under consideration. The theory of fuzziness is a science of pragmatic (fuzzy) numbers and sets. The ancient Greek philosopher Eubulides showed that the concepts «Heap» and «Bald» cannot be described using natural numbers. E. Borel proposed to define a fuzzy set using a membership function. A fundamentally important step was taken by L. A. Zadeh in 1965. He gave the basic definitions of the algebra of fuzzy sets and introduced the operations of intersection, product, union, sum, negation of fuzzy sets. The main thing he did was demonstration of the possibilities of expanding («doubling») mathematics: by replacing the numbers and sets used in mathematics with their fuzzy counterparts, we obtain new mathematical formulations. In the statistics of non-numerical data, methods of statistical analysis of fuzzy sets have been developed. Interval and triangular fuzzy numbers are often used specific types of membership functions. The theory of fuzzy sets in a certain sense is reduced to the theory of random sets. We think fuzzy and that is the only reason we understand each other. The paradox of the fuzzy theory is that it is impossible to consistently implement the thesis «Everything in the world is fuzzy». For ordinary fuzzy sets, the argument and values of the membership function are crisp. If they are replaced by fuzzy analogues, then their description will require their own clear arguments and membership functions, and so on ad infinitum. System fuzzy interval mathematics proceeds from the need to take into account the fuzziness of the initial data and the prerequisites of the mathematical model. One of the options for its practical implementation is an automated system-cognitive analysis and the intellectual system «Eidos».