From Simple to Highly-Complex Systems: A Paradigm Shift Towards Non-Abelian Emergent System Dynamics and Meta-Levels (original) (raw)

The evolution of non-linear dynamical system theory and super-complex systems-that are defined by classes of variable topologies and their associated transformations-is presented from a categorial and generalised, or extended topos viewpoint. A generalisation of dynamical systems, general systems theory is then considered for the meta-level dynamical systems with variable topology and variable phase space, within the framework of an "universal", or generalised Topos-a logico-mathematical construction that covers both the commutative and non-commutative structures based on logic classifiers that are multi-valued (MV) logic algebras. The extended topos concept was previously developed in conjunction with dynamic networks that were shown to be relevant to Complex Systems Biology. In so doing, we shall distinguish three major phases in the development of systems theory (two already completed and one currently unfolding). The three phases will be respectively called The Age of Equilibrium, The Age of Complexity and The Age of Super-Complexity. The first two may be taken as lasting from approximately 1850 to 1960, and the third which is now rapidly developing in applications to various types of systems after it began in the 1970s after the works of Rosen, Maturana and others. The mathematical theory of categories-which began in the 1940s [44],[45] with a seminal paper by Eilenberg and Mac Lane in 1945 [45]-is an unifying trend in modern mathematics [40], and has proved especially suitable for modeling the novelties raised by the third phase of systems' theory, which became associated with applications to system super-complexity problems in the late 1950s and 70s [84]-[85],[2],[6],[8], [88]-[89]; it was continued by applications to logical programming involving categorical logic in computer science [58] , as well as the categorical foundations of mathematics [59]-[60].