A category-theoretic approach to systems in a fuzzy world

The last 30 years have seen the growth of a new branch of mathematics called CATEGORY THEORY which provides a general perspective on many different branches of mathematics. Many workers (see Lawvere, 1972) have argued that it is category theory, rather than SET THEORY, that provides the proper setting for the study of the FOUNDATIONS OF MATHEMATICS. The aim of this paper is to show that problems in APPLIED MATHEMATICS, too, may find their proper foundation in the language of category theory. We do this by introducing a number of concepts of SYSTEM THEORY which we unify in our theory of MACHINES IN A CATEGORY. We write as system theorists, not as philosophers. Our hope is to stimulate a dialogue with philosophers of science as to the proper role for category theory in a systematic analysis of a fuzzy world. We do not discuss applications to biology or psychology the framework presented here is at a very high level of generality, and does not address the particularities which give these disciplines their distinctive flavor. This paper is divided into two Sections. In Section I, we sketch how the subjects of control theory, computers and formal language have grown out of the urdisciplines of MECHANICS and LOGIC; and then present the formal concepts of sequential machine, linear machine, and tree automaton. We sho how our notion of MACHINE IN A CATEGORY provides an uncluttered generalization of these three concepts. In Section II, we introduce the 'fuzzy world'. Although the study of quantum mechanics provides the best known framework, we stay within system theory, showing how PROBABILITY, MECHANICS and LOGIC gave rise to the study of markov chains, structural stability and multivalued logics. We then present the formal concepts of nondeterministic sequential machine, stochastic automaton and fuzzy-set automaton. Our notion of FUZZY MACHINE will generalize all three. Of particular interest will be the demonstration that, although fuzzy machines generalize