Comments:

10 pages, 2008, revised version 2009

Abstract:

The task of developing improved methodological tools for a Categorical Ontology of complex spacetime structures and the Ontological Theory of Levels is approached here from several perspectives including the quantum-molecular and complexity-relational-informational viewpoints. Our novel conceptual framework is thus primarily aimed at helping both mathematicians and philosophers or scientists to understand and `organize' the Ontology of Space-Time in Highly Complex Systems in terms of ontological levels and their sub-levels of existence. Another of our specific aims is to formalize in terms of categories, functors and higher order structures Roberto Poli's recent developments in the Ontological Theory of Levels. Highly Complex Systems, and indeed the corresponding meta-systems dynamics, are relevant to a wide variety of biosystems/organisms, as well as to societies, including human ones; furthermore, they appear to be indeed necessary for understanding the human brain and the mind, albeit in quite dierent form from the simpler organismic theories, such as Rashevsky's theory of Organismic Sets. This is indeed methodologically a monumental task, in and by, itself{to provide the `right kind of tools' for modern Ontology through exploring in some necessary detail the Philosophy of Highly Complex Systems. It would seem that we are now at such a stage of our development in both Philosophy and Science that the essential, formal tools are emerging mostly through non-reductionist developments in the `exact sciences' such as many-valued (non{standard) logics, mathematics, physics, molecular and population genetics, molecular biology, relational biology, and so on. This is in no small measure the result of recent trends towards unity in Logics, Mathematics and Physics. A paradigm shift towards non-commutative, or more generally, non-Abelian theories of highly complex dynamics is suggested to unfold now in physics, mathematics, life and cognitive sciences, thus leading to the realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. An outline of a Categorical Ontology of Space and Time can therefore be proposed now for super-complex dynamics in Emergent Biosystems, Species Evolution and Human Consciousness. The relational, dynamic patterns and the corresponding variable structures of organisms and the human mind are thus naturally represented in terms of novel variable (algebraic) topology concepts that are linked to special types of non-Abelian categories in Higher Dimensional Algebra. As an outstanding example, the ascent of man and other organisms through adaptation, evolution and especially through social co-evolution is represented here in relatively simple categorical terms such as variable biogroupoids that model the evolving organisms and species. Naturally, primordial organism structures are considered first in terms of the simplest Metabolic-Repair systems capable of self-replication, for example through autocatalytic reactions. The intrinsic dynamic `asymmetry' of such genetic networks in the organismic development and evolution is here investigated in terms of categories of many-valued, Lukasiewicz-Moisil logic algebras, and these are then compared with those obtained for other non-commutative, Quantum logics. The unifying theme of local-to-global approaches to organismic development, evolution and human consciousness leads to novel patterns of relations that emerge in super- and ultra- complex systems in terms of compositions of local procedures; the latter can be more precisely defined, for example, in terms of specific, locally compact structures such as locally Lie groupoids. Solutions to local-to-global problems in highly complex systems with `broken symmetry' may be found with the help of generalized van Kampen theorems in algebraic topology such as the Higher Homotopy van Kampen theorem. In this essay, the claim is defended that human consciousness is unique in the class of all organisms on Earth, and also that it should be viewed as an ultra-complex, global process of processes, at a meta-level which is not subsummed by-- but is compatible with--the underlying human brain dynamics. Therefore, the emergence of human consciousness and its existence seem to depend upon an extremely complex structural and functional unit with an asymmetric network topology and connectivities--the human brain. The latter super-structure is not just the result of biological evolution, but it seems to have emerged beyond animal brains through societal co-evolution and elaborate language/symbolic communication. Such meta-level dynamics in the human brain may involve `virtual', higher dimensional, non--commutative processes such as separate space and time perceptions that become however integrated at higher levels{those underlying consciousness. Philosophical theories of the mind are approached here from the theory of levels and ultra-complexity viewpoints which throw new light on both previous representational hypotheses, and proposed semantic models in cognitive science. Anticipatory systems and complex causality at the top levels of reality are then discussed in the context of Poli's Ontological Theory of Levels with its complex/entangled/intertwined ramifications in Psychology, Sociology and Ecology. The presence of strange attractors in the dynamics of modern society gives rise to very serious concerns for the future of mankind. A novel approach to QST construction in Algebraic/Axiomatic QFT involves the use of generalized fundamental theorems of algebraic topology from specialized, `globally well-behaved' topological spaces, to arbitrary ones (Baianu et al, 2007c). In this category, are the generalized, \emph{Higher Homotopy van Kampen theorems (HHvKT)} of Algebraic Topology with novel and unique non-Abelian applications. Such theorems greatly aid the calculation of higher homotopy of topological spaces. R. Brown and coworkers (1999, 2004a,b,c) generalized the van Kampen theorem, at first to fundamental groupoids on a set of base points (Brown,1967), and then, to higher dimensional algebras involving, for example, homotopy double groupoids and 2-categories (Brown, 2004a). The more sensitive \emph{algebraic invariant} of topological spaces seems to be, however, captured only by \emph{cohomology} theory through an algebraic \emph{ring} structure that is not accessible either in homology theory, or in the existing homotopy theory. Thus, two arbitrary topological spaces that have isomorphic homology groups may not have isomorphic cohomological ring structures, and may also not be homeomorphic, even if they are of the same homotopy type. Furthermore, several \emph {non-Abelian} results in algebraic topology could only be derived from the Generalized van Kampen Theorem (cf. Brown, 2004a), so that one may find links of such results to the expected \emph {`non-commutative} geometrical' structure of quantized space--time (Connes, 1994). In this context, the important algebraic--topological concept of a \emph{Fundamental Homotopy Groupoid (FHG) is applied to a Quantum Topological Space (QTS)} as a ``partial classifier" of the \emph{invariant} topological properties of quantum spaces of \emph{any} dimension; quantum topological spaces are then linked together in a \emph{crossed complex over a quantum groupoid} (Baianu, Brown and Glazebrook, 2006), thus suggesting the construction of global topological structures from local ones with well-defined quantum homotopy groupoids. The latter theme is then further pursued through defining locally topological groupoids that can be globally characterized by applying the Globalization Theorem, which involves the \emph{unique} construction of the Holonomy Groupoid. We are considering in a separate publication(Baianu et al 2007c) how such concepts might be applied in the context of Algebraic or Axiomatic Quantum Field Theory (AQFT) to provide a local-to-global construction of Quantum space-times which would still be valid in the presence of intense gravitational fields without generating singularities as in GR. The result of such a construction is a \emph{Quantum Holonomy Groupoid}, (QHG) which is unique up to an isomorphism.

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