2 - From Classical Mathematics to Fuzzy Mathematics: Emergence of a New Paradigm for Theoretical Science

Abstract

This chapter reviews a major paradigmatic change that concerns the role of uncertainty in science. This change is manifested by a transition from the traditional attitude toward uncertainty in science, according to which uncertainty is undesirable and the ideal is to eliminate it, to an alternative attitude, according to which uncertainty is fundamental and its avoidance is counterproductive. The chapter focuses on the mathematics pertaining to fuzzy set theory and its role in science. The classical mathematical theories by which certain types of uncertainty can be expressed are the classical set theory and the probability theory. In terms of the set theory, uncertainty is expressed by any given set of possible alternatives in situations where only one of the alternatives may actually happen. The probability theory expresses uncertainty in terms of a classical measure on the subsets of a given universal set of alternatives. The measure is a function that, according to the situation, assigns a number in the unit interval [0,1] to each subset of the universal set. Due to the additivity of classical measures, the probability of each subset is uniquely determined from the probabilities assigned to the smallest non-empty subsets of the universal set, each consisting of exactly one alternative. The fuzzy set theory is a generalization of the classical set theory, and the fuzzy measure theory is a generalization of the classical measure theory, and hence it is also a generalization of the probability theory.