Supercategorical Approach to Complex Systems, Metasystems and Ontology MultiLevels
Supercategorical Approach to Complex Systems, Metasystems and Ontology MultiLevels
SUPERCATEGORICAL ONTOLOGY OF COMPLEX SYSTEMS, META–SYSTEMS AND LEVELS:
The Emergence of Life, Human Consciousness and Society.
acomared19TAO1proceeds3.tex
August 19th, 2008
I. C. Baianu, R. Brown and J. F. Glazebrook
I. C. Baianu]ibaianu@uiuc.edu
R. Brown]r.brown@bangor.ac.uk
J. F. Glazebrook]jfglazebrook@eiu.edu
0.1. Mathematical and Metaphysics Notes.
0.1.1. AN1. On the Logical Foundations of Arithmetics
According to a website contributed entry (at $http://www.philosophypages.com/hy/6h.htm$): “The culmination of the new approach to logic lay in its capacity to illuminate the nature of the mathematical reasoning. While the idealists sought to reveal the internal coherence of absolute reality and the pragmatists offered to account for human inquiry as a loose pattern of investigation, the new logicians hoped to show that the most significant relations^{} among things could be understood as purely formal and external. Mathematicians like Richard Dedekind realized that on this basis it might be possible to establish mathematics firmly on logical grounds. Giuseppe Peano had demonstrated in 1889 that all of arithmetic^{} could be reduced to an axiomatic system with a carefully restricted set of preliminary postulates^{}. Frege promptly sought to express these postulates in the symbolic notation of his own invention. By 1913, Russell and Whitehead had completed the monumental “Principia Mathematica” (1913), taking three massive volumes to move from a few logical axioms through a definition of number to a proof that “ 1 + 1 = 2 .” Although the work of Gödel (less than two decades later) made clear the inherent limitations of this approach, its significance for our understanding of logic and mathematics remains”.
AN2.7:
0.1.2. Local–to–Global (LG) Construction Principles consistent with Quantum ‘Axiomatics’.
A novel approach to QST construction in Algebraic/Axiomatic QFT involves the use of generalized fundamental theorems of algebraic topology from specialized, ‘globally wellbehaved’ topological spaces^{}, to arbitrary ones (Baianu et al, 2007c). In this category^{}, are the generalized, Higher Homotopy van Kampen theorems (HHvKT) of Algebraic Topology with novel and unique nonAbelian^{} 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 2categories (Brown, 2004a). The more sensitive algebraic invariant of topological spaces seems to be, however, captured only by cohomology^{} theory through an algebraic 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 nonAbelian 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 ‘noncommutative geometrical’ structure of quantized space–time (Connes, 1994). In this context, the important algebraic–topological concept of a Fundamental Homotopy Groupoid (FHG) is applied to a Quantum Topological Space (QTS) as a “partial classifier” of the invariant^{} topological properties of quantum spaces of any dimension^{}; quantum topological spaces are then linked together in a crossed complex over a quantum groupoid^{} (Baianu, Brown and Glazebrook, 2006), thus suggesting the construction of global topological structures from local ones with welldefined 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 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 localtoglobal construction of Quantum spacetimes^{} 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 Quantum Holonomy Groupoid, (QHG) which is unique up to an isomorphism^{}.
0.1.3. The ObjectBased Approach vs ProcessBased, Dynamic Ontology.
In classifications, such as those developed over time in Biology for organisms, or in Chemistry for chemical elements, the objects are the basic items being classified even if the ‘ultimate’ goal may be, for example, either evolutionary or mechanistic studies. Rutherford’s comment is pertinent in this context: “There are two major types of science: physics or stamp collecting.” An ontology based strictly on object classification may have little to offer from the point of view of its cognitive content. It is interesting that many psychologists, especially behavioural ones, emphasize the objectbased approach rather than the processbased approach to the ultracomplex process of consciousness occurring ‘in the mind’ –with the latter thought as an ‘object’. Nevertheless, as early as the work of William James in 1850, consciousness was considered as a ‘continuous^{} stream that never repeats itself’–a Heraclitian concept that does also apply to supercomplex systems^{} and life, in general. We shall see more examples of the objectbased approach to psychology in Section^{} 8.
0.1.4. Procedures and Advantages of Poli’s Ontological Theory of Levels
According to Poli (2001), the ontological procedures provide:
 •

•
modes of dependence between levels (for instance, how the coevolution/interaction of social and mental realms depend and impinge upon the material);

•
the categorical closure^{} (or completeness) of levels.
Already we can underscore a significant component^{} of this essay that relates the ontology to geometry and topology; specifically, if a level is defined via ‘iterates of local procedures’ (viz. ‘items in iteration’, Poli, 2001), then we have some handle on describing its intrinsic governing dynamics (with feedback ) and, to quote Poli (2001), to ‘restrict the multidynamic frames to their linear fragments’. On each level of this ontological hierarchy there is a significant amount of connectivity through interdependence, interactions or general relations often giving rise to complex patterns that are not readily analyzed by partitioning or through stochastic methods as they are neither simple, nor are they random connections. But we claim that such complex patterns and processes have their logicocategorical representations quite apart from classical, Boolean mechanisms. This ontological situation gives rise to a wide variety^{} of networks, graphs, and/or mathematical categories, all with different connectivity rules, different types of activities, and also a hierarchy of supernetworks of networks of subnetworks. Then, the important question arises what types of basic symmetry^{} or patterns such supernetworks of items can have, and also how do the effects of their subnetworks ‘percolate’ through the various levels. From the categorical viewpoint, these are of two basic types: they are either commutative^{} or noncommutative, where, at least at the quantum level, the latter takes precedence over the former.
It is often thought or taken for granted that the objectoriented approach can be readily converted into a processbased one. It would seem, however, that the answer to this question depends critically on the ontological level selected. For example, at the quantum level, object and process become intermingled. Either comparing or moving between
levels, requires ultimately a processbased approach, especially
in Categorical Ontology where relations and interprocess
connections are essential to developing any valid theory. At the
fundamental level of ‘elementary particle physics’ however the
answer to this question of processvs. object becomes quite
difficult as a result of the ‘blurring’ between the particle and
the wave concepts. Thus, it is wellknown that any ‘elementary
quantum object’ is considered by all accepted versions of quantum
theory^{} not just as a ‘particle’ or just a ‘wave’ but both: the
quantum ‘object’ is both wave and particle, at the
sametime, a proposition^{} accepted since the time when it was
proposed by de Broglie. At the quantum microscopic level, the
object and process are intermingled, they are no longer separate
items. Therefore, in the quantum view the ‘objectparticle’ and
the dynamic process‘wave’ are united into a single dynamic entity
or item, called the waveparticle quantum, which strangely enough
is neither discrete nor continuous, but both at the same time, thus
‘refusing’ intrinsically to be an item consistent^{} with Boolean
logic. Ontologically, the quantum level is a fundamentally important
starting point which needs to be taken into account by any theory
of levels that aims at completeness. Such completeness may not be
attainable, however, simply because an ‘extension^{}’ of Gödel’s theorem may
hold here also. The fundamental quantum level is generally accepted to be
dynamically, or intrinsically noncommutative, in the sense
of the noncommutative quantum logic and also in the sense of
noncommuting quantum operators for the essential quantum
observables such as position and momentum. Therefore, any comprehensive theory of levels, in the sense of incorporating the quantum level, is thus –mutatis
mutandis– nonAbelian. Furthermore, as the nonAbelian case is the more general one, from a strictly formal viewpoint, a nonAbelian Categorical Ontology is
the preferred choice. A paradigmshift towards a nonAbelian Categorical Ontology has already started (Brown et al, 2007: ‘NonAbelian Algebraic Topology’; Baianu, Brown and Glazebrook, 2006: NAQAT; Baianu et al 2007a,b,c).
0.1.5. Fundamental Concepts of Algebraic Topology with Potential Application to Ontology Levels Theory and SpaceTime Structures.
We shall consider briefly the potential impact of novel Algebraic Topology concepts, methods and results on the problems of defining and classifying rigorously Quantum spacetimes. With the advent of Quantum Groupoids–generalizing Quantum Groups^{}, Quantum Algebra and Quantum Algebraic Topology, several fundamental concepts and new theorems of Algebraic Topology may also acquire an enhanced importance through their potential applications to current problems in theoretical and mathematical physics, such as those described in an available preprint (Baianu, Brown and Glazebrook, 2006), and also in several recent publications (Baianu et al 2007a,b; Brown et al 2007).
Now, if quantum mechanics is to reject the notion of a continuum^{}, then it must also reject the notion of the real line and the notion of a path. How then is one to construct a homotopy theory? One possibility is to take the route signalled by Čech, and which later developed in the hands of Borsuk into ‘Shape Theory’ (see, Cordier and Porter, 1989). Thus a quite general space is studied by means of its approximation by open covers. Yet another possible approach is briefly pointed out in AN2.6. A few fundamental concepts of Algebraic Topology and Category Theory^{} are summarized in AN2.6 that have an extremely wide range of applicability to the higher complexity levels of reality as well as to the fundamental, quantum level(s). We have omitted in this section the technical details in order to focus only on the ontologicallyrelevant aspects; full mathematical details are however also available in a recent paper by Brown et al (2007) that focuses on a mathematical/conceptual framework for a completely formal approach to categorical ontology and the theory of levels.
0.1.6. Towards Biological Postulates and Principles.
Often, Rashevsky considered in his Relational Biology papers, and indeed made comparisons, between established physical theories and principles. He was searching for new, more general relations in Biology and Sociology that were also compatible with the former. Furthermore, Rashevsky also proposed two biological principles that add to Darwin’s natural selection of species and the ‘survival of the fittest principle’, the emergent relational structure thus defining adaptive organisms:
1. The Principle of Optimal Design, and 2. The Principle of Relational Invariance (phrased by Rashevsky as “Biological Epimorphism^{}”).
In essence, the ‘Principle of Optimal Design’ defines the ‘fittest’ organism which survives in the natural selection process of competition between species, in terms of an extremal
criterion, similar^{} to that of Maupertuis; the optimally ‘designed’
organism is that which acquires maximum functionality essential to
survival of the successful species at the lowest ‘cost’ possible.
The ‘costs’ are defined in the context of the environmental niche
in terms of material, energy, genetic and organismic processes
required to produce/entail the prerequisite biological function(s) and
their supporting anatomical structure(s) needed for competitive survival
in the selected niche. Further details were presented by Robert
Rosen in his short but significant book on optimality (1970). The ‘Principle of
Biological Epimorphism’ on the other hand states that the highly specialized
biological functions of higher organisms can be mapped (through an epimorphism) onto those
of the simpler organisms, and ultimately onto those of a (hypothetical) primordial organism (which was assumed to be unique up to an isomorphism or selectionequivalence). The latter proposition, as formulated by Rashevsky, is more akin to a postulate
than a principle. However, it was then generalized and restated in the form of the
existence of a limit in the category of living organisms and their
functional^{} genetic networks (${\text{\mathbf{G}\mathbf{N}}}^{i}$), as a directed family of
objects, ${\text{\mathbf{G}\mathbf{N}}}^{i}(t)$ projected backwards in time (Baianu and Marinescu, 1968), or
subsequently as a superlimit (Baianu, 1970 to 1987; Baianu, Brown, Georgescu and Glazebrook, 2006); then, it was rephrased as the Postulate of Relational Invariance, represented by a colimit^{} with the arrow of time pointing forward (Baianu, Brown, Georgescu and
Glazebrook, 2006). Somewhat similarly, a dual principle and colimit construction was invoked
for the ontogenetic development of organisms (Baianu, 1970), and also for
populations evolving forward in time; this was subsequently applied to biological evolution although on a much longer time scale –that of evolution– also with the arrow of time pointing towards the future in a representation operating through Memory Evolutive Systems (MES) by A. Ehresmann and Vanbremeersch (2006).
0.1.7. Selective Boundaries and Homeostasis. Varying Boundaries vs Horizons.
Boundaries are especially relevant to closed systems. According to Poli (2008): “they serve to distinguish what is internal to the system from what is external to it”, thus defining the fixed, overall structural topology of a closed system. By virtue of possessing boundaries, a whole (entity) is something on the basis of which there is an interior and an exterior (viz. Baianu and Poli, 2007). One notes however that a boundary, or boundaries, may either change/vary or be quite selective/directional–in the sense of dynamic fluxes crossing such boundaries– if the system is open. In the case of an organism that grows and develops it will be therefore characterized by a variable topology^{} that may also depend on the environment, and is thus contextdependent, as well. Perhaps one of the simplest example of a system that changes from closed to open, and thus has a variable topology, is that of a pipe equipped with a functional valve that allows flow in only one direction. On the other hand, a semipermeable membrane such as a cellophane, thinwalled ’closed’ tube– that allows water and small molecule fluxes to go through but blocks the transport of large molecules such as polymers through its pores– is selective and may be considered as a primitive/’simple’ example of an open, selective system. Organisms, in general, are open systems with specific types or patterns of variable topology which incorporate both the valve and the selectively permeable membrane boundaries –albeit much more sophisticated and dynamic than the simple/fixed topology of the cellophane membrane; such variable^{} structures are essential to maintaining their stability and also to the control of their internal structural order, of low microscopic entropy. (The formal definition of this important concept of ‘variable topology’ was introduced in our recent paper (Baianu et al 2007a) in the context of the spacetime evolution of organisms, populations and species.)
As proposed by Baianu and Poli (2008), an essential feature of boundaries in open systems is that they can be crossed by matter; however, all boundaries may be crossed by either fields or by quantum waveparticles if the boundaries are sufficiently thin, even in ’closed’ systems. The boundaries of closed systems, however, cannot be crossed by molecules or larger particles. On the contrary, a horizon is something that one cannot reach. In other words, a horizon is not a boundary. This difference^{} between horizon and boundary appears to be useful in distinguishing between systems and their environment (v. AN4.1). Boundaries may be fixed, clearcut, or they may be vague/blurred, mobile, varying/variable in time, or again they may be intermediate between these any of these cases, according to how the differentiation is structured. At the beginning of an organism’s ontogenetic development, there may be only a slightly asymmetric distribution in perhaps just one direction, but usually still maintaining certain symmetries along other directions or planes. Interestingly, for many multicellular organisms, including man, the overall symmetry retained from the beginning of ontogenetic development is bilateral–just one plane of mirror symmetry– from Planaria to humans. The presence of the headtotail asymmetry introduces increasingly marked differences among the various areas of the head, middle, or tail regions as the organism develops. The formation of additional borderline phenomena occurs later as cells divide and differentiate thus causing the organism to grow and develop
v. (AN4.2.) AN4.2
Brown and Higgins, 1981a, showed that certain multiple groupoids^{} equipped with an extra structure called connections were equivalent^{} to another structure called a crossed complex which had already occurred in homotopy theory. such as double, or multiple groupoids (Brown, 2004; 2005). For example, the notion of an atlas of structures should, in principle, apply to a lot of interesting, topological and/or algebraic, structures: groupoids, multiple groupoids, Heyting algebras, $n$valued logic algebras and ${C}^{*}$convolution algebras^{}. One might incorporate a 3 or 4valued logic to represent genetic dynamic networks in singlecell organisms such as bacteria. Another example from the ultracomplex system of the human mind is synaesthesia–the case of extreme communication processes between different types of ‘logics’ or different levels of ‘thoughts’/thought processes. The key point here is communication. Hearing has to communicate to sight/vision in some way; this seems to happen in the human brain in the audiovisual (neocortex) and in the Wernicke (W) integrating area in the leftside hemisphere of the brain, that also communicates with the speech centers or the Broca area, also in the left hemisphere. Because of this dualfunctional, quasisymmetry, or more precisely asymmetry of the human brain, it may be useful to represent all twoway communication/signalling pathways in the two brain hemispheres by a double groupoid^{} as an oversimplified groupoid structure that may represent such quasisymmetry of the two sides of the human brain. In this case, the 300 millions or so of neuronal interconnections in the corpus callosum that link up neural network pathways between the left and the right hemispheres of the brain would be represented by the geometrical connection in the double groupoid. The brain’s overall asymmetric distribution of functions and neural network structure between the two brain hemispheres may therefore require a noncommutative, double–groupoid structure for its relational representation. The potentially interesting question then arises how one would mathematically represent the splitbrains that have been neurosurgically generated by cutting just the corpus callosum– some 300 million interconnections in the human brain (Sperry, 1992). It would seem that either a crossed complex of two, or several, groupoids, or indeed a direct product^{} of two groupoids ${G}_{1}$ and ${G}_{2}$, ${G}_{1}\times {G}_{2}$ might provide some of the simplest representations of the human splitbrain. The latter, direct product construction has a certain kind of builtin commutativity: $(a,b)(c,d)=(ac,bd)$, which is a form of the interchange law. In fact, from any two groupoids ${G}_{1}$ and ${G}_{2}$ one can construct a double groupoid ${G}_{1}\u2a1d{G}_{2}$ whose objects are $\mathrm{Ob}({G}_{1})\times \mathrm{Ob}({G}_{2})$. The internal groupoid ‘connection’ present in the double groupoid would then represent the remaining basal/‘ancient’ brain connections between the two hemispheres, below the corpum callosum that has been removed by neurosurgery in the splitbrain human patients.
The remarkable variability observed in such human subjects both between different subjects and also at different times after the splitbrain (bridgelocalized) surgery may very well be accounted for by the different possible groupoid representations^{}. It may also be explained by the existence of other, older neural pathways that remain untouched by the neurosurgeon in the splitbrain, and which relearn gradually, in time, to at least partially reconnect the two sides of the human splitbrain. The more common health problem –caused by the senescence of the brain– could be approached as a localtoglobal, supercomplex ageing process represented for example by the patching of a topological double groupoid atlas connecting up many local faulty dynamics in ‘small’ un–repairable regions of the brain neural network, caused for example by tangles, locally blocked arterioles and/or capillaries, and also low local oxygen or nutrient concentrations. The result, as correctly surmised by Rosen (1987), is a global, rather than local, senescence, supercomplex dynamic process.
Social Autopoiesis Within a social system the autopoiesis of the various components is a necessary and sufficient condition for realization of the system itself. In this respect, the structure of a society as a particular instance of a social system is determined by the structural framework of the (autopoietic components) and the sum total of collective interactive relations. Consequently, the societal framework is based upon a selection of its component structures in providing a medium in which these components realize their ontogeny. It is just through participation alone that an autopoietic system determines a social system by realizing the relations that are characteristic of that system. The descriptive and causal notions are essentially as follows (Maturana and Varela, 1980, Chapter III):

(1)
Relations of constitution that determine the components produced constitute the topology in which the autopoiesis is realized.

(2)
Relations of specificity that determine that the components produced are the specific ones defined by their participation in autopoiesis.

(3)
Relations of order that determine that the concatenation^{} of the components in the relations of specification, constitution and order are the ones specified by the autopoiesis.
0.2. Propagation and Persistence of Organisms through Space and Time. Autopoiesis, Survival and Extinction of Species.
The autopoietic model of Maturana (1987) claims to explain the persistence of living systems in time as the consequence of their structural coupling or adaptation as structure determined systems, and also because of their existence as molecular autopoietic systems with a ‘closed’ network structure. As part of the autopoietic explanation is the ‘structural drift’, presumably facilitating evolutionary changes and speciation. One notes that autopoietic systems may be therefore considered as dynamic realizations of Rosen’s simple MR s. Similar arguments seem to be echoed more recently by Dawkins (2003) who claims to explain the remarkable persistence of biological organisms over geological timescales as the result of their intrinsic, (super) complex adaptive capabilities.
The point is being often made that it is not the component atoms that are preserved in organisms (and indeed in ‘living fosils’ for geological periods of time), but the structurefunction relational pattern, or indeed the associated organismic categories, higher order categories or supercategories^{}. This is a very important point: only the functional organismic structure is ‘immortal’ as it is being conserved and transmitted from one generation to the next. Hence the relevance here, and indeed the great importance of the science of abstract structures and relations, i.e., Mathematics.
This was the feature that appeared paradoxical or puzzling to Erwin Schrödinger from a quantum theoretical point of view when he wrote his book “What is Life?” As individual molecules often interact through multiple quantum interactions, which are most of the time causing irreversible, molecular or energetic changes to occur, how can one then explain the hereditary stability over hundreds of years (or occasionally, a great deal longer, NAs) within the same genealogy of a family of men? The answer is that the ‘actors change but the play does not!’. The atoms and molecules turnover, and not infrequently, but the structurefunction patterns/organismic categories remain unchanged/are conserved over long periods of time through repeated repairs and replacements of the molecular parts that need repairing, as long as the organism lives. Such stable patterns of relations are, at least in principle, amenable to logical and mathematical representation without tearing apart the living system. In fact, looking at this remarkable persistence of certain gene subnetworks in time and space from the categorical ontology and Darwinian viewpoints, the existence of live ‘fossils’ (e.g., a coelacanth found alive in 1923 to have remained unchanged at great depths in the ocean as a species for 300 million years!) it is not so difficult to explain; one can attribute the rare examples of ‘live fossils’ to the lack of ‘selection pressure in a very stable niche’. Thus, one sees in such exceptions the lack of any adaptation apart from those which have already occurred some 300 million years ago. This is by no means the only long lived species: several species of marine, giant unicellular green algae with complex morphology from a family called the Dasycladales may have persisted as long as 600 million years (Goodwin, 1994), and so on. However, the situation of many other species that emerged through supercomplex adaptations–such as the species of Homo sapiens–is quite the opposite, in the sense of marked, supercomplex adaptive changes over much shorter time–scales than that of the exceptionally ‘lucky’ coelacanths. Clearly, some species, that were less adaptable, such as the Neanderthals or Homo erectus, became extinct even though many of their functional genes may be still conserved in Homo sapiens, as for example, through comparison with the more distant chimpanzee relative. When comparing the Homo erectus fossils with skeletal remains of modern men one is struck how much closer the former are to modern man than to either the Australopithecus or the chimpanzee (the last two species appear to have quite similar skeletons and skulls, and also their ‘reconstructed’ vocal chords/apparatus would not allow them to speak). Therefore, if the functional genomes of man and chimpanzee overlap by about $98\%$, then the overlap of modern man functional genome would have to be greater than $99\%$ with that of Homo erectus of 1 million years ago, if it somehow could be actually found and measured (but it cannot be, at least not at this point in time). Thus, one would also wonder if another more recent hominin than H. erectus, such as Homo floresiensis– which is estimated to have existed between 74,000 and 18,000 years ago on the now Indonesian island of Flores– may have been capable of human speech. One may thus consider another indicator of intelligence such as the size of region 10 of the dorsomedial prefrontal cortex, which is thought to be associated with the existence of selfawareness; this region 10 is about the same size in H. floresiensis as in modern humans, despite the much smaller overall size of the brain in the former (Falk, D. et al., 2005).
0.3. NeuroGroupoids and CatNeurons
Categorical representations on nerve cells in the terminology of Ehresmann and Vanbremeersch (1987,2006) are called ‘categorical neurons’ (or cat–neurons for short). ‘Consciousness loops’ (Edelmann 1989, 1992) and the neuronal workspace of Baars (1988) (see also Baars and Franklin, 2003) are among an assortment of such local models that may be consistent with such local categorical representations. Among other notions, there were proposed several criteria for studying the overall integration of neuronal assemblies into an archetypal core: the cat–neuron resonates as an echo that propagates to target concepts through series of thalamocortical loops in response to the thalamus response to stimuli. Mimicking the neuron signalling through synaptic networks, cat–neurons would interact according tp certain linking procedures (Baianu, 1972), that could be then studied in the context of categorical logic, which in its turn may be applied either to semantic modelling of neural networks (Healy and Caudell, 2006) or possibly the schemata of adaptive resonance theory of Grossberg (1999). For such interactive network systems one would expect global actions and groupoid atlases to play more instrumental roles as possible realizations of various types of multi–agent systems (Bak et al, 2006). Let one be aware however, that such models tend to be reductionist in character^{}, falling somewhere between simple and chaotic (‘complex’) systems. Although useful for the industry of higher level automata and robotics, they are unlikely to address at all the ontological problems of the human mind.
As regards to the role played by quantum events in mental processes the situation is different. Although there can be no reasonable doubt that quantum events do occur in the brain as elsewhere in the material world, there is no substantial, experimental evidence that quantum events are in any way efficacious or relevant for those aspects of brain activity that are correlates of mental activity. Bohm (1990), and Hiley and Pylkkännen (2005) have suggested theories of active information enabling ‘self’ to control brain functions without violating energy conservation laws. Such ideas are relevant to how quantum tunneling is instrumental in controlling the engagement of synaptic exocytosis (Beck and Eccles, 1992) and how the notion of a ‘(dendron) mind field’ (Eccles, 1986) could alter quantum transition probabilities as in the case of synaptic vesicular emission (nevertheless, there are criticisms to this approach as in Wilson, 1999).
Attempting to define consciousness runs into somewhat similar problems to those encountered in attempting to define Life, but in many ways far less ‘tangible’; one can make a long list of important attributes of human consciousness from which one must decide which ones are the essential or primary properties, and which ones are to be derived from such primary attributes in a rational manner.
Kant considered that the internal structure of reasoning, or the ‘pure reason’, was essential to human nature for knowledge of the world but the inexactness of empirical science amounted to limitations on the overall comprehension. At the same time,
Kozma et al. (2004) used network percolation models to analyze phase transitions of dynamic neural systems such as those embedded within segments of neuropil. This idea of neuropercolation so provides a means of passage via transition states within a neurophysiological hierarchy (viz. levels). But the actual substance of the hierarchy cannot by itself explain the quality of intention. The constitution of the latter may be in part consciousness, but actual neural manifestations, such as for example pain, are clearly not products^{} of a finite state Turing machine^{} (Searle, 1983).
AN52 section 5 Point (5a) claims that a system should occupy either a macroscopic or a microscopic spacetime region, but a system that comes into birth and dies off extremely rapidly may be considered either a shortlived process, or rather, a ‘resonance’ –an instability rather than a system, although it may have significant effects as in the case of ‘virtual particles’, ‘virtual photons’, etc., as in quantum electrodynamics and chromodynamics. Note also that there are many other, different mathematical definitions of systems, ranging from (systems of) coupled differential equations to operator formulations, semigroups^{}, monoids, topological groupoid dynamic systems and dynamic categories. Clearly, the more useful system definitions include algebraic and/or topological structures rather than simple, discrete structure sets, classes or their categories (cf. Baianu, 1970, and Baianu et al., 2006).
It can be shown that such organizational order must either result in a stable attractor or else it should occupy a stable spacetime domain, which is generally expressed in closed systems by the concept of equilibrium. On the other hand,
Quantum theories (QTs) were developed that are just as elegant mathematically as GR, and they were also physically ‘validated’ through numerous, extremely sensitive and carefully designed experiments. However, to date, quantum theories have not yet been extended, or generalized, to a form capable of recovering the results of Einstein’s GR as a quantum field theory over a GRspacetime altered by gravity is not yet available
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Title  Supercategorical Approach to Complex Systems, Metasystems and Ontology MultiLevels 

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