Key Words: Fuzzy Sets, Fuzzy Systems, System Theory, History of Science

Abstract

In this contribution, the reader will find original research work on the history of the theory of fuzzy sets and systems. It is a reconstruction of Lotfi Zadehs scientific work on his way to establish a new scientific discipline and many details on the historical background of the development of the theory of fuzzy sets are given.

The author’s original research work bases on studies of scientific articles, newspapers, letters, and interviews with the very first protagonists of the theory of fuzzy sets. The presented interpretation is a new view on the theory of fuzzy sets and on the history of “Fuzzy Sets”, the first paper in the journal Information and Control in 1965. The author of this article was Lotfi Zadeh, a professor of electrical engineering at the University of California, Berkeley since 1958. Years before he wrote his Ph.D thesis on Frequency Analysis of Variable Networks (1949) at the Columbia University in New York. Then, he was influenced by Shannons and Wieners information theory and he was interested in the theory of ideal and optimal filtering. Many of the mathematical techniques used in this theory are commonly employed in quantum mechanics. Ideal filters are defined as filters which achieve a perfect separation of signal and noise whereas optimal filters give the “best approximation” of a signal. Zadeh noticed that “best approximations” depend on reasonable criterions and he formulated these criterions in statistical terms.

In the 1950s Zadeh was one of the founders of system theory, a rising scientific discipline “to the study of systems per se, regardless of their physical structure“. Engineers in that time were, in general, inadequately trained to think in abstract terms, but nevertheless, Zadeh believed that it was only a matter of time before system theory attains acceptance.

He took notice that there would not be any chance to compute all system equations of any real system. He noticed that no new digital computers of the 40s and 50s could help to get exact knowledge of what happens in real world systems. To compute, to describe, or to control processes in complex systems in particular in large scaled systems there are too many state equations and as a consequence: too many systems of differential equations. Computers cannot solve that many differential equations in a limited time.

Zadeh became aware of the fact that the de­scription languages based on classical mathematics are not suffi­ciently expressive to serve as a means of char­acterization of input-output relations in an environment of imprecision, uncertainty and incompleteness of in­formation. In 1964, he found a new way to describe real world systems: fuzzy sets and systems. In the summer of this year, he gave a talk on pattern recognition in Dayton, Ohio. Think­ing about problems in pattern classification and separation of sets of points he got the idea to use grades of membership in this field.

Some weeks later he discussed these ideas with Richard Bellman, his friend and colleague at the Rand Corporation in Santa Monica. Then he wrote the first paper on fuzzy sets, the Rand memoran­dum RM-4307-PR (October 1964) that appeared two years later as the article Abstraction and Pattern Classification in the Jour­nal of Mathematical Analysis and Applications. Here and in his famous article Fuzzy Sets Zadeh introduced the “concept of a “fuzzy sets”, that is a class in which there may be a continuous infinity of grades of membership, with the grade of mem­bership of an object x in a fuzzy set A represented by a number m_A(x) in the interval [0,1].”

The long version of this paper will present a detailed view on the development of the theory of fuzzy sets from electrical engineering, system theory and pattern separation.