Supercategorical Approach to Complex Systems, Meta-systems and Ontology Multi-Levels

Supercategorical Approach to Complex Systems, Meta-systems and Ontology Multi-Levels

SUPERCATEGORICAL ONTOLOGY OF COMPLEX SYSTEMS, META–SYSTEMS AND LEVELS:
The Emergence of Life, Human Consciousness and Society.

acomared19-TAO1proceeds3.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. AN-1. 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”.

AN-2.7:

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 useof generalized fundamental theorems of algebraic topology fromspecialized, ‘globally well-behaved’ topological spaces, toarbitrary ones (Baianu et al, 2007c). In this category, are the generalized, HigherHomotopy van Kampen theorems (HHvKT) of Algebraic Topology withnovel and unique non-Abelian applications. Such theorems greatly aidthe 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 algebraic invariant of topological spaces seems to be, however, captured only by cohomology theory through an algebraic ring structure that is not accessible either inhomology 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 alsonot be homeomorphic, even if they are of the same homotopy type.Furthermore, several non-Abelian results in algebraic topology could only be derived from the Generalized van Kampen Theorem (cf. Brown, 2004a), so thatone may find links of such results to the expected ‘non-commutative 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 propertiesof quantum spaces of any dimension; quantum topologicalspaces are then linked together in a crossed complex over aquantum groupoid (Baianu, Brown and Glazebrook, 2006), thussuggesting the construction of global topological structures fromlocal ones with well-defined quantum homotopy groupoids. The lattertheme is then further pursued through defining locally topologicalgroupoids that can be globally characterized by applying theGlobalization Theorem, which involves the unique constructionof 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 QuantumField 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 Quantum Holonomy Groupoid, (QHG) which is unique up to an isomorphism.

0.1.3. The Object-Based Approach vs Process-Based, Dynamic Ontology.

In classifications, such as those developed over time in Biologyfor organisms, or in Chemistry for chemical elements, theobjects are the basic items being classified even if the‘ultimate’ goal may be, for example, either evolutionary ormechanistic studies. Rutherford’s comment is pertinent in thiscontext: “There are two major types of science: physics or stamp collecting.” An ontology based strictly on object classification may havelittle to offer from the point of view of its cognitive content.It is interesting that many psychologists, especially behaviouralones, emphasize the object-based approach rather than theprocess-based approach to the ultra-complex process ofconsciousness occurring ‘in the mind’ –with the latter thought asan ‘object’. Nevertheless, as early as the work of William Jamesin 1850, consciousness was considered as a ‘continuousstream that never repeats itself’–a Heraclitian concept thatdoes also apply to super-complex systems and life, in general. Weshall see more examples of the object-based approach to psychologyin Section 8.

0.1.4. Procedures and Advantages of Poli’s Ontological Theory of Levels

According to Poli (2001), theontological procedures provide:

• coordination between categories (for instance, theinteractions and parallels between biological and ecologicalreproduction as in Poli, 2001);

• modes of dependence between levels (for instance, how theco-evolution/interaction of social and mental realms depend andimpinge upon the material);

• the categorical closure (or completeness) of levels.

Already we can underscore a significant component of this essaythat relates the ontology to geometry and topology; specifically, if alevel is defined via ‘iterates of local procedures’ (viz. ‘items initeration’, Poli, 2001), then we have some handle on describingits intrinsic governing dynamics (with feedback ) and, to quotePoli (2001), to ‘restrict the multi-dynamic frames to their linearfragments’. On each level of this ontological hierarchy there is a significantamount of connectivity through inter-dependence, interactions orgeneral relations often giving rise to complex patterns that arenot readily analyzed by partitioning or through stochastic methodsas they are neither simple, nor are they random connections. But weclaim that such complex patterns and processes have theirlogico-categorical representations quite apart from classical,Boolean mechanisms. This ontological situation gives rise to awide variety of networks, graphs, and/or mathematical categories,all with different connectivity rules, different types ofactivities, and also a hierarchy of super-networks of networks of sub-networks.Then, the important question arises what types ofbasic symmetry or patterns such super-networks of items can have,and also how do the effects of their sub-networks ‘percolate’ through thevarious levels. From the categorical viewpoint, these are of twobasic types: they are either commutative or non-commutative,where, at least at the quantum level, the latter takes precedence over the former.

It is often thought or taken for granted that the object-oriented approach can be readily converted into a process-based 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 inter-mingled. Either comparing or moving betweenlevels, requires ultimately a process-based approach, especiallyin Categorical Ontology where relations and inter-processconnections are essential to developing any valid theory. At thefundamental level of ‘elementary particle physics’ however theanswer to this question of process-vs. object becomes quitedifficult as a result of the ‘blurring’ between the particle andthe wave concepts. Thus, it is well-known that any ‘elementaryquantum object’ is considered by all accepted versions of quantumtheory not just as a ‘particle’ or just a ‘wave’ but both: thequantum ‘object’ is both wave and particle, at thesame-time, a proposition accepted since the time when it wasproposed by de Broglie. At the quantum microscopic level, theobject and process are inter-mingled, they are no longer separateitems. Therefore, in the quantum view the ‘object-particle’ andthe dynamic process-‘wave’ are united into a single dynamic entityor item, called the wave-particle quantum, which strangely enoughis neither discrete nor continuous, but both at the same time, thus‘refusing’ intrinsically to be an item consistent with Booleanlogic. Ontologically, the quantum level is a fundamentally importantstarting point which needs to be taken into account by any theoryof levels that aims at completeness. Such completeness may not beattainable, however, simply because an ‘extension’ of Gödel’s theorem mayhold here also. The fundamental quantum level is generally accepted to bedynamically, or intrinsically non-commutative, in the senseof the non-commutative quantum logic and also in the sense ofnon-commuting quantum operators for the essential quantumobservables such as position and momentum. Therefore, any comprehensive theory of levels, in the sense of incorporating the quantum level, is thus –mutatismutandis– non-Abelian. Furthermore, as the non-Abelian case is the more general one, from a strictly formal viewpoint, a non-Abelian Categorical Ontology isthe preferred choice. A paradigm-shift towards a non-Abelian Categorical Ontology has already started (Brown et al, 2007: ‘Non-Abelian Algebraic Topology’; Baianu, Brown and Glazebrook, 2006: NA-QAT; Baianu et al 2007a,b,c).

0.1.5. Fundamental Concepts of Algebraic Topology with Potential Application to Ontology Levels Theory and Space-Time 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 space-times. 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 theoreticaland 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 notionof a path. How then is one to construct a homotopy theory?One possibility is to take the route signalled by Čech, and whichlater developed in the hands of Borsuk into ‘Shape Theory’ (see,Cordier and Porter, 1989). Thus a quite general space is studied bymeans of its approximation by open covers. Yet another possibleapproach is briefly pointed out in AN-2.6. A few fundamental concepts of Algebraic Topology and Category Theory are summarized in AN-2.6 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 ontologically-relevant 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, andindeed 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 toDarwin’s natural selection of species and the ‘survival of the fittest principle’, the emergentrelational structure thus defining adaptive organisms:

1. The Principle of Optimal Design,and2. 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 extremalcriterion, similar to that of Maupertuis; the optimally ‘designed’organism is that which acquires maximum functionality essential tosurvival of the successful species at the lowest ‘cost’ possible.The ‘costs’ are defined in the context of the environmental nichein terms of material, energy, genetic and organismic processesrequired to produce/entail the pre-requisite biological function(s) andtheir supporting anatomical structure(s) needed for competitive survivalin the selected niche. Further details were presented by RobertRosen in his short but significant book on optimality (1970). The ‘Principle ofBiological Epimorphism’ on the other hand states that the highly specializedbiological functions of higher organisms can be mapped (through an epimorphism) onto thoseof the simpler organisms, and ultimately onto those of a (hypothetical) primordial organism (which was assumed to be unique up to an isomorphism or selection-equivalence). The latter proposition, as formulated by Rashevsky, is more akin to a postulatethan a principle. However, it was then generalized and re-stated in the form of theexistence of a limit in the category of living organisms and theirfunctional genetic networks (${\text{𝐆𝐍}}^i$), as a directed family ofobjects, ${\text{𝐆𝐍}}^i(-t)$ projected backwards in time (Baianu and Marinescu, 1968), orsubsequently as a super-limit (Baianu, 1970 to 1987; Baianu, Brown, Georgescu and Glazebrook, 2006); then, it was re-phrased as the Postulate of Relational Invariance, represented by a colimit with the arrow of time pointing forward (Baianu, Brown, Georgescu andGlazebrook, 2006). Somewhat similarly, a dual principle and colimit construction was invokedfor the ontogenetic development of organisms (Baianu, 1970), and also forpopulations 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 topologyof 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 quiteselective/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 context-dependent, 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 semi-permeable membrane such as a cellophane, thin-walled ’closed’ tube–that allows water and small molecule fluxes to go through but blocks the transportof large molecules such as polymers through its pores– is selectiveand 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 membraneboundaries –albeit much more sophisticated and dynamic than the simple/fixed topology of the cellophane membrane; such variable structures are essential to maintainingtheir 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 space-time 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 fieldsor by quantum wave-particles 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. AN-4.1). Boundaries may be fixed, clear-cut, 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 multi-cellular 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 head-to-tail 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. (AN-4.2.)AN4.2

Brown and Higgins, 1981a, showed that certain multiple groupoidsequipped with an extra structure called connections wereequivalent to another structure called a crossed complexwhich had already occurred in homotopy theory. such asdouble, 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/oralgebraic, structures: groupoids, multiple groupoids, Heytingalgebras, $n$-valued logic algebras and $C^{*}$-convolution-algebras. One might incorporate a 3 or 4-valued logic to representgenetic dynamic networks in single-cell organisms such as bacteria. Anotherexample from the ultra-complex system of the human mind is synaesthesia–the case ofextreme communication processes between different types of‘logics’ or different levels of ‘thoughts’/thought processes. Thekey point here is communication. Hearing has to communicateto sight/vision in some way; this seems to happen in the humanbrain in the audiovisual (neocortex) and in the Wernicke (W)integrating area in the left-side hemisphere of the brain, thatalso communicates with the speech centers or the Broca area, alsoin the left hemisphere. Because of this dual-functional,quasi-symmetry, or more precisely asymmetry of the human brain, itmay be useful to represent all two-way communication/signallingpathways in the two brain hemispheres by a double groupoidas an over-simplified groupoid structure that may represent suchquasi-symmetry of the two sides of the human brain. In this case,the 300 millions or so of neuronal interconnections in thecorpus callosum that link up neural network pathwaysbetween the left and the right hemispheres of the brain would berepresented by the geometrical connection in the double groupoid.The brain’s overall asymmetric distribution of functionsand neural network structure between the two brain hemispheres maytherefore require a non-commutative, double–groupoid structurefor its relational representation. The potentially interestingquestion then arises how one would mathematically represent thesplit-brains that have been neurosurgically generated by cuttingjust the corpus callosum– some 300 millioninterconnections in the human brain (Sperry, 1992). It would seemthat either a crossed complex of two, or several, groupoids, orindeed a direct product of two groupoids $G_1$ and $G_2$, $G_1\times G_2$ might provide some of the simplest representations ofthe human split-brain. The latter, direct product construction hasa certain kind of built-in commutativity: $(a,b)(c,d)=(ac,bd)$,which is a form of the interchange law. In fact, from any twogroupoids $G_1$ and $G_2$ one can construct a double groupoid $G_1⨝G_2$ whose objects are $\mathrm{Ob}(G_1)\times \mathrm{Ob}(G_2)$. Theinternal groupoid ‘connection’ present in the double groupoidwould then represent the remaining basal/‘ancient’ brainconnections between the two hemispheres, below the corpum callosumthat has been removed by neurosurgery in the split-brain humanpatients.

The remarkable variability observed in such human subjects bothbetween different subjects and also at different times after thesplit-brain (bridge-localized) surgery may very well be accountedfor by the different possible groupoid representations. It mayalso be explained by the existence of other, older neural pathwaysthat remain untouched by the neurosurgeon in the split-brain, andwhich re-learn gradually, in time, to at least partiallyre-connect the two sides of the human split-brain. The more commonhealth problem –caused by the senescence of the brain– could beapproached as a local-to-global, super-complex ageingprocess represented for example by the patching of atopological double groupoid atlas connecting up many localfaulty dynamics in ‘small’ un–repairable regions of the brainneural network, caused for example by tangles, locally blockedarterioles and/or capillaries, and also low local oxygen ornutrient concentrations. The result, as correctly surmised byRosen (1987), is a global, rather than local, senescence,super-complex dynamic process.

Social AutopoiesisWithin a social system the autopoiesis of the various components is a necessaryand sufficient condition for realization of the system itself. Inthis respect, the structure of a society as a particular instanceof a social system is determined by the structural framework ofthe (autopoietic components) and the sum total of collectiveinteractive relations. Consequently, the societal framework isbased upon a selection of its component structures in providing amedium in which these components realize their ontogeny. It isjust through participation alone that an autopoietic systemdetermines a social system by realizing the relations that arecharacteristic of that system. The descriptive and causal notionsare essentially as follows (Maturana and Varela, 1980, ChapterIII):

• Relations of constitution that determine the componentsproduced constitute the topology in which the autopoiesis is realized.

• Relations of specificity that determine that the components producedare the specific ones defined by their participation in autopoiesis.

• Relations of order that determine that the concatenation of thecomponents in the relations of specification, constitution and orderare 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 thepersistence of living systems in time as the consequence of theirstructural coupling or adaptation as structure determinedsystems, and also because of their existence as molecularautopoietic systems with a ‘closed’ network structure. As part ofthe autopoietic explanation is the ‘structural drift’, presumablyfacilitating evolutionary changes and speciation. One notes thatautopoietic systems may be therefore considered as dynamicrealizations of Rosen’s simple MR s. Similar arguments seem to beechoed more recently by Dawkins (2003) who claims to explain theremarkable persistence of biological organisms over geologicaltimescales as the result of their intrinsic, (super-) complexadaptive capabilities.

The point is being often made that it is not the component atomsthat are preserved in organisms (and indeed in ‘living fosils’ forgeological periods of time), but the structure-functionrelational pattern, or indeed the associated organismiccategories, higher order categories or supercategories. This is a very important point: only the functional organismic structure is ‘immortal’ as it isbeing conserved and transmitted from one generation to the next.Hence the relevance here, and indeed the great importance of thescience of abstract structures and relations, i.e., Mathematics.

This was the feature that appeared paradoxical or puzzling toErwin Schrödinger from a quantum theoretical point of view whenhe wrote his book “What is Life?” As individual molecules ofteninteract through multiple quantum interactions, which are most ofthe time causing irreversible, molecular or energeticchanges to occur, how can one then explain the hereditarystability over hundreds of years (or occasionally, a greatdeal longer, NAs) within the same genealogy of a family of men?The answer is that the ‘actors change but the play does not!’. Theatoms and molecules turn-over, and not infrequently, but thestructure-function patterns/organismic categories remainunchanged/are conserved over long periods of time throughrepeated repairs and replacements of the molecular parts that needrepairing, as long as the organism lives. Such stable patterns ofrelations are, at least in principle, amenable to logical andmathematical representation without tearing apart the livingsystem. In fact, looking at this remarkable persistence of certaingene subnetworks in time and space from the categorical ontologyand Darwinian viewpoints, the existence of live ‘fossils’(e.g., a coelacanth found alive in 1923 to have remained unchangedat great depths in the ocean as a species for 300 million years!)it is not so difficult to explain; one can attribute the rareexamples of ‘live fossils’ to the lack of ‘selection pressure in avery stable niche’. Thus, one sees in such exceptions the lack ofany adaptation apart from those which have already occurred some300 million years ago. This is by no means the only long livedspecies: several species of marine, giant unicellular green algaewith complex morphology from a family called theDasycladales may have persisted as long as 600 millionyears (Goodwin, 1994), and so on. However, the situation of manyother species that emerged through super-complexadaptations–such as the species of Homo sapiens–is quitethe opposite, in the sense of marked, super-complex adaptivechanges over much shorter time–scales than that of theexceptionally ‘lucky’ coelacanths. Clearly, some species, thatwere less adaptable, such as the Neanderthals or Homoerectus, became extinct even though many of their functionalgenes may be still conserved in Homo sapiens, as forexample, through comparison with the more distant chimpanzeerelative. When comparing the Homo erectus fossils withskeletal remains of modern men one is struck how much closer theformer are to modern man than to either theAustralopithecus or the chimpanzee (the last two speciesappear to have quite similar skeletons and skulls, and also their‘reconstructed’ vocal chords/apparatus would not allow them tospeak). Therefore, if the functional genomes of man and chimpanzeeoverlap by about $98\%$, then the overlap of modern manfunctional genome would have to be greater than $99\%$ with thatof Homo erectus of 1 million years ago, if it somehow couldbe actually found and measured (but it cannot be, at least not atthis point in time). Thus, one would also wonder if another morerecent hominin than H. erectus, such as Homofloresiensis– which is estimated to have existed between 74,000and 18,000 years ago on the now Indonesian island of Flores– mayhave been capable of human speech. One may thus consider anotherindicator of intelligence such as the size of region 10 of thedorsomedial prefrontal cortex, which is thought to be associatedwith the existence of self-awareness; this region 10 isabout 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. Neuro-Groupoids and Cat-Neurons

Categorical representations on nerve cells in the terminology of Ehresmannand Vanbremeersch (1987,2006) are called ‘categorical neurons’ (orcat–neurons for short). ‘Consciousness loops’ (Edelmann1989, 1992) and the neuronal workspace of Baars (1988) (see alsoBaars and Franklin, 2003) are among an assortment of such local models thatmay be consistent with such local categorical representations. Among other notions, therewere proposed several criteria for studying the overall integration of neuronal assembliesinto an archetypal core: the cat–neuron resonates as anecho that propagates to target concepts through series ofthalamocortical loops in response to the thalamus response to stimuli.Mimicking the neuron signalling through synaptic networks, cat–neurons would interactaccording tp certain linking procedures (Baianu, 1972), that could be then studied in the context ofcategorical logic, which in its turn may be applied either to semanticmodelling 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 expectglobal 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 beaware however, that such models tend to be reductionist in character,falling somewhere between simple and chaotic (‘complex’) systems. Althoughuseful 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 inthe brain as elsewhere in the material world, there is no substantial,experimental evidence that quantum events are in any way efficacious orrelevant for those aspects of brain activity that are correlatesof mental activity. Bohm (1990), and Hiley and Pylkkännen(2005) have suggested theories of active informationenabling ‘self’ to control brain functions without violatingenergy conservation laws. Such ideas are relevant to how quantumtunneling is instrumental in controlling the engagement ofsynaptic exocytosis (Beck and Eccles, 1992) and how the notion ofa ‘(dendron) mind field’ (Eccles, 1986) could alter quantumtransition probabilities as in the case of synaptic vesicularemission (nevertheless, there are criticisms to this approach asin 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 attributesin a rational manner.

Kant considered that the internal structure of reasoning, or the ‘pure reason’, wasessential to human nature for knowledge of the world but theinexactness of empirical science amounted to limitations on theoverall comprehension. At the same time,

Kozma et al. (2004) usednetwork percolation models to analyze phase transitions ofdynamic neural systems such as those embedded within segments ofneuropil. This idea of neuro-percolation so provides ameans of passage via transition states within a neurophysiologicalhierarchy (viz. levels). But the actual substance of the hierarchycannot by itself explain the quality of intention. Theconstitution of the latter may be in part consciousness, butactual neural manifestations, such as for example pain, areclearly not products of a finite state Turing machine (Searle, 1983).

AN5-2section 5Point (5a) claims that a system should occupy either a macroscopic or a microscopic space-time region, but a system that comes into birth and dies off extremely rapidly may be considered either a short-lived 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 space-time 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 andcarefully designed experiments. However, to date, quantum theorieshave 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 GR-space-time altered by gravity is not yet available

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