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Wednesday, November 23, 2005

Motivation for a New Direction in Mathematical Modeling

(DNA Topology as a Dual Function of Homotopy Induced Solitonic and Discrete Resonance)

This quality of duality cannot be resolved merely by assuming a stochastic frame of reference for the unique problem inherent in phase space transformations. This is as true for the field quantum physics; and in general self organization theory that induces the formation of discrete attractor and emergent states from any “arbitrary” random preconditions. The toroidal dynamics (hence torsion, chirality, asymmetry, as well as orientation) of n dimensional Riemannian space as it undergoes homotopy to Costa Surface of n degree embedded in such space, is the equivalence of subsumption of transfinite ordinals extended outside the simple number theoretical domain of the “continuum.” The Vernadskyian definition of a modular continuum, delimited by irreducible boundary conditions into interoperating domains, i.e., the inert, biotic, and noosphere conforms to this conception. Further, the Hopf positively curved minimal surface’s connection with Costa in the sense of phase transformation allows for the orientability via Cartan Repere Mobile construction of Kiehn to evolve naturally, rather than stochastically. (It is of no small note that Vernadsky references Cartan’s geometry within his text explicating the Noosphere.) The model for the positively curved space wave function transformation to the hyperbolic horn (Mach cone) is Riemann’s paper on shock wave propagation. Toroidal action is a metastable configuration of Alfven’s plasma cosmogony in the macrocosmic scale and is invariant down to the infinitesimal. In this context, the metastable formation of solitons of determinate orderings is properly located. The connection these have with catalysis, DNA replication, and far from equilibrium systems in general is instructive. (See Davia) The fact that autowave dynamics are differentia specifica in the biotic domain as in contrast with the inert is not negligible and is suggestive of a further refinement of possible wave dynamics within the higher transfinite dimension of the noosphere. The application of Hausdorf fractional dimension and its fractal sequelae does not conform with my use of term transfinite dimension in the foregoing. However, it does properly situate the question of induced singularities from a continuous manifold.

Neurogenesis and conformality resonance

It is proven that an enriched environment is directly related to adult neurogenesis in certain cortical regions. As classical music is the semi-mediated expression of human creativity, the case for sonic resonance in this instance is not a trivial topological problem. Again, the connection at the cosmological scale of star formation and topological invariance is intriguing. This, if you will, is the true locus of “connectionism.” It is decidedly not the theory of “encoding” prevalent in academia today. (As a matter of fact the craze of trying to undo the relevance of Goedel’s proof among logicians and information theorists has led to a rather degenerate embracing of non-reality as substance. This has its approximate correlative in the attempt at ergodic modeling and distribution theory as a replacement for causality.) In this light, it is intriguing that current studies show that the orientation of the plane of cleavage in neurogenesis that regulates brain tissue growth. The Gbetagamma subunits of heterotrimeric G proteins are likely causal elements. Additionally, the vortical-like chiral nature of the enzyme ATP synthase is well understood. It is further evidence of a topological refinement for the biological equivalent of the model of toroidal torsion. It is also astounding that this enzyme exhibits nearly 100% mechanical chemical efficiency as an engine that makes possible the cellular consumption of free energy for the biosphere, including both aerobic and anaerobic types. (Also, separation in these two domains is defined as counter- or clockwise rotation on the mitochondria, exclusively (!)) All of which further reflects on the need for a model that can account for the formation of solitonic singularities within this transfinite hierarchical domain. Another qualifying refinement of in the arena of bioenergetics is that the quantum mechanical “laws” governing regimes of possible spin topologies are violated by biophysical macromolecular states.

Dynamics of Human Thought

Media of transmission for human thought are certainly of the most complex and excitable types imaginable. The first question to be addressed is that of the quality of hylozoic monism of this subject matter. We can view many different aspects or dimensions of human thinking and their impact upon various portions or manifolds of relationships of the continuum of reality. (This notion of continuum is not restricted to the idea of linear extension and types of number, as will be developed further on.) However, it is essential to comprehend that there exists a world line of development whereby ideas are characterized as ordered in the sense of Cantor’s transfinite. In this sense, the entirety of human history of advancement of thought and praxis is invariantly self similar. (By the way, in this regard the principle of fractal generation is simply a convenient mathematical device for producing discreteness out of such self similarity. It is not the generating principle but merely a relic of that generation, in reality the generation of singularities rarely proceeds by such a uniformly simple manner.)
For humanity, it is the ability of scientific discovery to overcome the relatively fixed limits or boundary conditions of any particular mode of functioning of society from the standpoint of thermodynamic disequilibrium, i.e. negative entropy. By this, it must be understood that due to the necessary nature of directedness or hylozoism, if you will, one mode must be superceded by another. This quality of the vectoring of society is not merely accidental, but is built into the very ordering of the inorganic and biological substrates. For example, the fact that there exists a birefringent axis or anisotropy for electromagnetic propagation in the universe, which at the most macroscopic regime exhibits precisely this quality. Similarly, the anti-racemic and optical chirality distinguishing the dynamics of living systems is a further higher ordered transfinite development of this same anisotropy that characterizes the inorganic. Electromagnetic propagation is now ordered once again according to a new lawfulness. Now, human thought or what Vernadsky labeled the Noosphere, is of a yet further transfinite such ordering distinguished by its own particular power over the lower two subsumed realms. It is only here, that an awareness for consciousness of these ordering principles has any meaning.

As to the question of interaction of wave functions and the excitable media: firstly the approach of Riemann in his groundbreaking paper on the principle of the dynamics of propagation of sound and pressure in a physical medium. In this, he established the principle that the medium itself possesses a boundary of rate of transmissibility. The resulting shockwave formation induces a change in the quality of curvature of wave propagation. (For astrobiology compare galactic ram pressure.)The so-called Mach cone, taken as an accelerated series as the speed of sound (pressure) continuously surpassed, represents in its entirety a negatively curved hyperbolic horn. That is, what was a propagation of a radiative spherical kind now has gone over through a shockwave formation into a different species of curvature. In the instance of the propagation of electromagnetic at the speed of light, the medium is not able to rectify itself as for sound and instead, a transformation of a different sort occurs via Einstein’s equation. This being a different transfinite species of boundary.

This same principle certainly applies to the two remaining substrates, i.e. the biosphere and the noosphere. Mutagenic transformations, I contend, are nothing but the same shockwave principle applied to the biological realm. When an organization of life develops an ability to reduce energy through a new quality hitherto unknown, this is nothing but the surpassing of the previously fixed rate of transmissibility of the medium. In the same way that the speed of light represents a higher kind of boundary condition for the inorganic, the boundary separating humanity from all other biology is a sort of speed limit, if you will, for mutagenic transformations. This again, of course, is of a higher ordered transfinite than Einstein’s principle was for the inorganic.

Now, you are probably asking yourself, how might this same shockwave principle apply to human thought? In any mode of functioning of society there exists an active “media/manifold” of a set of relationships created by the successive discovery of principles of underlying organization of the three substrates, i.e. inert, biota, and human cognition and interrelationships among these (see for example the role of Deinococcus radiodurans). The human individual’s cognitive development occurs, of course in an historically concrete/discrete context. The previous associated development of society has made the medium of cognition transparent for the mental development of this individual. For example as a thought experiment, consider the following: (along the same line of dialectic of mnemonics from Plato’s Meno) how is it possible that a gifted young child in the modern era is able to seemingly intuitively grasp the principles of projective geometry and readily apply them in drawing? View the historical artifacts of the lengthy and arduous development of these principles; compare Massacio’s nearly completed perspectival artwork with that of DaVinci. So, I posit that the substrate of the mind is also a developing organic “compact” type of n + m dimensional manifold, a multi-sheeted hyper-surface, connected through topological singularities, (unfortunately sometimes referred to analytically as defects) and exhibiting negative and positive regions of curvature. Now, crucial scientific discoveries necessarily create for human practice free energy that allows society to overcome the bounding conditions of this thermodynamically relatively fixed reactive medium. This signifies the movement from n dimensions to n + m (topological cobordism.) Further, the geometrical transformation of such a succession, I posit, also acts upon the principle of least actional, minimal surface, topological constraints. For example, the latest in a centuries’ long mathematical development in this arena is the remarkable discovery of the Brazilian geometer Celsoe Costa of a principle of generation of such surface by way of homotopic deformation of toroidal types. Here, from this perspective, are properly situated considerations of self similar, relatively “force free” species of action, found in the subsumed substrates we have been considering, i.e. solitons and cellular automata or autowaves. Such species of hypertori and their deformations are intimately connected with what has come before as the geometrical analysis situs of instantiations of “compact” manifolds as a substrate capable of resolving and emerging whole the principle of shockwave transformations. Topological singularities that are either discovered in nature, e.g. Falaco, morning glories, hurricanes, spiral nebula, Davidov, aorta autowaves etc. or produced in the laboratory under extreme circumstances, e.g. plasma kink instabilities, light bullets, BEC breathers, sonoluminescence etc. are representative of anomalous conditions, which when their principles of organization are made explicit through scientific investigation act to transform the domain of human practice and create new thermodynamic boundary conditions.

Such qualities of human thinking themselves may be cognized as having a wavelike nature in terms of propagation through the medium of human practice. Consider for example, how the historical personage of Plato was directly and mediately responsible for sparking off a methodological renaissance centuries after his death. His opus may be viewed as propagating itself historically in two senses from the standpoint of curved space. In his own epoch, his attempts to win over the Syracusan tyrant to his system, create an academy, etc. represent a radiative propagation of his method analogical to the previously instanced propagation of plane waves below the limits that the medium is capable of supporting. Why? Primarily because his efforts failed to transform his own society. That is to say, even though the ideas were effectively transmitted in the medium of language and otherwise of his day, the power (potential energy) inhering in them failed to catalyze the desired transformation of society. However, centuries later Plato’s class of ideas were able to transform the practice of society very much in principle as a shockwave. So then, human thought, although it is insubstantial from the perspective of spatial extension, except as it adheres in some way to the nervous system, cortical manifold is able to act in a particular sense to transform the geometry of human practice. In the same oriented/vectored characteristic of action that birefringence has for electromagnetic inert propagation, chiral or helical enantiomorphisms for the biotic, thought has but on a transfinitely higher ordering for human practice.

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