From simple to highly-complex systems: A paradigm shift towards non-Abelian emergent system dynamics and meta-levels

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 "uni-versal", or generalised Topos–a logico-mathematical construction that covers both the commutative and non-commutative structures based on logic classi-fiers 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 dis-tinguish 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 appli-cations to logical programming involving categorical logic in computer science [58] , as well as the categorical foundations of mathematics [59]-[60]. 131 Acta Universitatis Apulensis, ISSN 1582-5329, 2011, Special Issue on Understanding Complex Systems, Eds. Barna Iantovics, Ladislav Hluch and Roumen Kountchev Each phase is characterized by reference to distinct concepts of the general system and operational logic levels–including meta-levels, that were meant in many–but not all cases–to include classes of specific, actual, or concrete sys-tems. The exceptions to such concrete cases are several, perhaps one of the most notable example being that of abstract metabolic-replication, or (M,R)-systems introduced by Robert Rosen in 1958 in terms of categories of finite sets [81]-[82]. Such categories were later shown to be part of the category of ab-stract automata or sequential machines [8],[100]. Rosen's abstract, relational approach to biological systems was then complemented by the (M,R)-system representations as dynamical systems with added mathematical structure [88]-[89] ; it was still clearly unable to include all classes of actual, concrete systems because of the paradigm shifts needed to do so: from simple to complex sys-tems [91], and then again to extremely complex [46], or super-complex [19] and ultra-complex systems [22],[25] classes of systems. Furthermore, each sub-sequent phase generalised the previous one, thus addressing previously ne-glected, major problems and aspects, as well as involving new paradigms and also including the higher ontological levels. The second part deals with the deeper problems of providing a sufficiently flexible mathematical framework, and yet well-defined, that might be suitable for various classes of systems ranging from simple to super-complex, and beyond. As we shall see, this is something still in wait as mathematics itself is undergoing development from symmetric (commutative, or natural) categories and commutative homology to the non-commutative, higher dimensional structures of Non-Abelian Alge-braic Topology [33], that are capable of representing dynamic asymmetry, and thus enabling the development of theories that are more general and much less restrictive than any static modelling. Among such novel mathematical constructs are those addressing problems associated with multiple geometric structures that are of great interest not only in mathematics (for example, in the case of "foliation bundles"), but also in applications to a variety of classes of non-linear dynamic systems that exhibit symmetry-breaking, ranging from super-conductors to percolating neural networks. Examples of the need for such non-Abelian theories are abundant in Complex Systems Biology, Popu-lation Genetics, Morphogenesis, Evolutionary Biology, Ecology, Neurophysiol-ogy and other biomedical sciences, Cognitive Sciences, Psychology, Algebraic Quantum Field Theory and Categorical Ontology. Its potential importance for the future of a human society without Mumford's megamachines, or the human use of human beings, is also pointed out.