Introduction

Non-commutative vs. commutative dynamic modeling diagrams

where $\delta$ is the real system dynamics and $\aleph$ is an algorithm implementing the numerical computation of the mathematical model ($\mathbf{M}$ ) on a digital computer. Firstly, one notes the ominous absence of the logical model, L, from Rosen's diagram published in 1987. Secondly, one also notes the obvious presence of logical arguments and indeed (non-Boolean) `schemes' related to the entailment of organismic models, such as MR-systems, in the more recent books that were published last by Robert Rosen (1994, 2001, 2004). Further mathematical details are provided in the paper by Brown, Glazebrook and Baianu (2007). Furthermore, Elsasser (1980) pointed out a fundamental, logical difference between physical systems and biosystems or organisms: whereas the former are readily represented by homogeneous logic classes, living organisms exhibit considerable variability and can only be represented by heterogeneous logic classes. One can readily represent homogeneous logic classes or endow them with `uniform' mathematical structures, but heterogeneous ones are far more elusive and may admit a multiplicity of mathematical representations or possess variable structure. This logical criterion may thus be useful for further distinguishing simple systems from highly complex systems.

The importance of logic algebras, and indeed of categories of logic algebras, is rarely discussed in modern Ontology even though categorical formulations of specific ontology domains such as biological Ontology and Neural Network ontology are being extensively developed. For a recent review of such categories of logic algebras the reader is referred to the concise presentation by Georgescu (2006); their relevance to network biodynamics was also recently assessed (Baianu, 2004, Baianu and Prisecaru, 2005; Baianu et al, 2006).

Super-complex systems, such as those supporting neurophysiological activities, are explained only in terms of non-linear, rather than linear causality. In some way then, these systems are not normally considered as part of either traditional physics or the complex, chaotic systems physics that are known to be fully deterministic. However, super-complex (biological) systems have the potential to manifest novel and counter-intuitive behavior such as in the manifestation of `emergence', development/morphogenesis and biological evolution. The precise meaning of supercomplex systems is formally defined here in the next section.

Simple and super-complex dynamics: Closed vs. open systems

In an early report (Baianu and Marinescu, 1968), the possibility of formulating a super-categorical unitary theory of systems (that is, of both simple and complex systems, etc.) was pointed out both in terms of organizational structure and dynamics. Furthermore, it was proposed that the formulation of any model or computer simulation of a complex system- such as living organism or a society-involves generating a first-stage logical model (not-necessarily Boolean!), followed by a mathematical one, complete with structure (Baianu, 1970). Then, it was pointed out that such a modeling process involves a diagram containing the complex system, (CS) and its dynamics, a corresponding, initial logical model, L, `encoding' the essential dynamic and/or structural properties of CS, and a detailed, structured mathematical model $\mathbf{M}$ ; this initial modeling diagram may or may not be commutative, and the modeling can be iterated through modifications of L, and/or $\mathbf{M}$ , until an acceptable agreement is achieved between the behaviour of the model and that of the natural, complex system (Baianu and Marinescu, 1968; Comoroshan and Baianu, 1969). Such an iterative modeling process may ultimately converge to appropriate models of the complex system, and perhaps a best possible model could be attained as the categorical colimit of the directed family of diagrams generated through such a modelling process. The possible models $\mathbf{L}$ , or especially $\mathbf{M}$ , were not considered to be necessarily either numerical or recursively computable (that is, with an algorithm or software program) by a digital computer (Baianu, 1971b, 1986-87). The mathematician John von Neumann regarded and defined complexity as a measurable property of natural systems below the threshold of which systems behave `simply', but above which they evolve, reproduce, self-organize, etc. It was claimed that any `natural' system fits this profile. But the classical assumption that natural systems are simple, or `mechanistic', is too restrictive since `simple' is applicable only to machines, closed physicochemical systems, computers, or any system that is recursively computable. Rosen (1987) proposed a major refinement of these ideas about complexity by a more exact classification between `simple' and `complex'. Simple systems can be characterized through representations which admit maximal models, and can be therefore re-assimilated via a hierarchy of informational levels. Besides, the duality between dynamical systems and states is also a characteristic of such simple dynamical systems. Complex systems do not admit any maximal model. On the other hand, an ultra-complex system- as applied to psychological-sociological structures- can be described in terms of variable categories or structures, and thus cannot be reasonably represented by a fixed state space for its entire lifespan. Simulations by limiting dynamical approximations lead to increasing system `errors'. Just as for simple systems, both super-complex and ultra-complex systems admit their own orders of causation, but the latter two types are different from the first-by inclusion rather than exclusion- of the mechanisms that control simple dynamical systems.