The impact of fuzzy set theory on contemporary mathematics [survey]

In this paper we first outline the shortcomings of classical binary logic and Cantor's set theory in order to handle imprecise and uncertain information. Next we briefly introduce the basic notions of Zadeh's fuzzy set theory among them: definition of a fuzzy set, operations on fuzzy sets, the concept of a linguistic variable, the concept of a fuzzy number and a fuzzy relation. The major part consists of a sketch of the evolution of the mathematics of fuzziness, mostly illustrated with examples from my research group during the past 35 years. In this evolution I see three overlapping stages. In the first stage taking place during the seventies only straightforward fuzzifications of classical domains such as general topology, theory of groups, relational calculus, ... have been introduced and investigated w.r.t. the main deviations from their binary originals. The second stage is characterized by an explosion of the possible fuzzifications of the classical structures which has lead to a deep study of the alternatives as well as to the enrichment of the structures due to the non-equivalence of the different fuzzifications. Finally some of the current topics of research in the mathematics of fuzziness are highlighted. Nowadays fuzzy research concerns standardization, axiomatization, extensions to lattice-valued fuzzy sets, critical comparison of the different so-called soft computing models that have been launched during the past three decennia for the representation and processing of incomplete information.