Heylighen F. (1990): "Relational Closure: a mathematical concept for distinction-making and complexity analysis", in: Cybernetics and Systems '90, R. Trappl (ed.), (World Science, Singapore), p. 335-342. RELATIONAL CLOSURE: a mathematical concept for distinction- making and complexity analysis Francis HEYLIGHEN * PESP, Free University of Brussels Pleinlaan 2, B-1050 Brussels, Belgium ABSTRACT. Complexity is defined as the combination of distinction and connection. Analysing a complex problem hence demands making the most adequate distinctions, taking into account connections existing between them. The concept of closure in mathematics and cybernetics is reviewed. A generalized formal concept is introduced by reformulating closure in a relational language based on connections. The resulting "relational closure" allows to reduce low level, internal distinctions and to highlight high level, external distinctions in a network of connections, thus diminishing the complexity of the description. 1. Complexity and Distinction-Making Many attempts to define complexity in a precise way have already been carried out (Serra, 1988), but none of them seems to cover all the aspects which we intuitively asso- ciate with the concept of “complex”. All these definitions try to characterize complexity in a quantitative way, as a kind of measure of difficulty. I think it is more important first to understand complexity in a qualitative (but precise) way. Let us go back to the original Latin word complexus, which signifies "entwined", "twisted together". This may be inter- preted in the following way: in order to have a complex you need: 1) two or more distinct parts; 2) these parts must in some way be connected, so that you cannot separate them without destroying the complex (Heylighen, 1988b, 1989a,d). Intuitively then a system would be more complex if more parts could be distinguished, and if more connections between them existed. This allow us to reduce the concept of complexity to two aspects: distinction and connection. Distinction corresponds to variety, to heterogeneity, to the fact that different parts of the complex behave differently. Connection corresponds to relational constraint, to redundancy, to the fact that different parts are not independent, but that the knowledge of one part allows to determine features of the other parts. Distinction leads in the limit to disorder and entropy, connection leads to order and negentropy. Complexity can only exist if both aspects are present: neither perfect disorder (which can be described statisti- cally through the law of large numbers), nor perfect order (which can be described by traditional deterministic methods) are really complex (Heylighen, 1988b). A provisional, quantitative definition of complexity C might express this as the product of variety V (or entropy, which corresponds roughly to the logarithm of variety) with redundancy R (corresponding to the difference between actual variety or entropy, and maximum poten- tial variety): C = V. R where R = V max - V C becomes zero in case of both maximum variety (V = V max ) and minimum variety * Senior Research Assistant NFWO (Belgian National Fund for Scientific Research)