Fuzzy Sets: Math, Applied Math, Heuristics? Problems and Interpretations

Number of Disciplines and Theories changed their status from status of Natural Science discipline to Mathematics. The Theory of Probability is the classical example of that kind. The main privilege of the new Math status is the conception of Math truth, which distinguishes Math from other theories. Some disciplines used in Applications pretended to be Math not really being it. It is entirely true of Fuzzy Subsets Theory with its pretension to be Math and to be exclusive tools in uncertainty handling. Fundamental pretensions of classical Fuzzy subset theory including pretension to be math as well as some gaps in the theory are discussed in the article. Statistical interpretation of membership functions is proposed. It is proved that such interpretation takes place for practically all supporters with minimal constraints on it. Namely, a supporter must be a space with a measure. Proposed interpretation explains modification of classical fuzzy objects to fill the gaps. It is then possible to talk about observations of fuzzy subset within the conception of modification and to extend likelihood method to the new area. Fuzzy likelihood equation is adduced as an example of new possibilities within the proposed approach. One more interpretation for the Fuzzy subset theory is proposed for a discussion: multiset theory.