Saturday, September 10, 2011

Dialogue on the Marxian Critique of Nonlinear Dynamical Systems Theory

Dialogue on the Marxian, Immanent Critique of Nonlinear Dynamical Systems Theory

Dear Readers,

Below is my "genericization" of a recent dialogue on Marxian & F.E.D. Dialectics vis-a-vis Nonlinear Dynamical Systems Theory in which I participated.

My theses are labelled "M.D." below, and the interlocutor's theses by "D".

Dialogue Excerpt --

M.D.: I strongly agree with the points regarding systems theory / complexity theory / cybernetics that you stated below, namely --

D.:  The idea of an 'information ecosystem' is an example of idealism / dualism since it conveniently detaches the societal mental realm from the underlying, productive activities of society and its politics.

M.D.:   Agreed, and I further note that such ideo-versus-physio detachment is a typical symptom of capitalist ideology in general.

D.:  Complexity theory is socially valuable and progressive in that it can enable frameworks and descriptions of natural and social systems that don't overlook those systems' intricate interconnections and vigorous dynamics.

M.D.: Agreed, that "complexity theory" is partially progressive [a view on the partially-progressive, partially-ideological character of "complexity theory", which, from some of your earlier theses, I understand that you also hold], and that, in particular, it has the potential to overcome that <<species>> of reductionism which wants to reduce the reality of each given sub-system of a super^w-system to the reality of just one of the super-system levels -- e.g., to the "highest"/latest-to-emerge level -- that contains them, retaining only the description of that super-system level, or that want to reduce, e.g., the reality of everything above the deepest sub-system level, including the "highest"/latest-to-emerge super-system level, to that of only the "deepest" sub-system level, retaining only the description of that deepest sub-system level as if complete and accurate to describe the whole reality, thus ignoring "complex multi-level interdependencies", to say the least!

An example would be an attempt to describe the total <<kosmos>>, at the stage in its self-development that includes the present moment of our locality, strictly in terms of pre-nuclear "particles", e.g., strictly in terms of quarks, electron, neutrinos, photons, etc., ignoring all description in terms of other/later-to-emerge levels of reality, e.g., the position that:  "Everything is just quarks, floating in the void".

D.: (Conventional linear equations cannot hope to capture the complex multi-level interdependencies that are recognized and accounted for with a complexity-systems approach.)

M.D.Agreed. Indeed, linear integro-differential equations tend to assert a kind of "equilibriumist anti-dynamics", or, at best, a kind "pseudo-dynamics", limited to single, global basins of a single, monolithic fixed-point attractor, repellor, or "neutral" solution-geometry in the state-space, that are most generally vastly unrealistic as models of the dynamics of the natural systems, including of the human-social systems, that we actually observe and, to some extent, measure.

Claims [shared with

. The ease with which general, "closed-form", "analytical" solution of systems of linear integro-differential equations have been obtained by modern mathematics, and the great difficulty encountered with attempts at "closed-form", "analytical" solution of systems of non-linear integro-differential equations in general, is a symptom of [unconscious] capitalist <<mentalite'>>, and resultant [again, unconscious] capitalist ideology, embedded in modern mathematics, the result of which is to block the development of a mathematics which would be at home with, and successful in solving, general nonlinear integro-differential systems.

. The requirement that differentiation [and/or integration] of the differential equation solution-function unknown, aptly modeling a given empirical phenomenon / phenomenology, produce a non-linear, "higher degree" [higher in absolute value of degree] expression, to achieve fitness to the empirical data/measurements on that phenomenon/phenomenology, is a sign-post of the "dialecticality" of that empirical phenomenon/phenomenology.   Nonlinearity in dynamical systems is, in general, an expression of their "dialectical-ness".

So bad is this unconscious enfetterment of the progress of mathematical science by [unconscious] capitalist <<mentailte'>> and ideology, that the highest expressions of the "laws" of nature so far attained -- typically taking the forms of systems of nonlinear partial differential equations -- HAVE NEVER, TO THIS DAY, BEEN SOLVED, so that the vast riches of potential information, and of potential technologically applicable knowledge, that these unsolved equations implicitly "contain", and which could be made "explicit" by their general solution, has yet to be made accessible, and usable, for the progress of the social forces of "quanto-qualitatively" expanded social reproduction of humankind, TO THIS VERY DAY, except in a few particular cases, of mostly "barely-nonlinear" equations[-systems], which have nonetheless proven richly applicable in the advance of the human productive forces.

Among these still-unsolved nonlinear integro-differential equations-systems are the Newtonian, gravitational equations for a system of 3 or more mutually-gravitating bodies [the very founding instance of the mathematical "laws of nature"]; the Navier-Stokes Equations for electrically neutral "rheid" -- gas and liquid -- "hydrodynamics", whose general solution might enable greater in-advance-time predictability of, and reveal the "Achilles heels" of, hurricanes, tornadoes, and even tsunamis, so that, with a disproportionately [nonlinearly] small, affordable human input of energy, these storms could be stopped in their tracks [or, in capitalist hands, would reveal how to generate hurricanes, tornadoes, and tsunamis as weapons of class war/other contracted social reproduction]; the equations of rock visco-plasticity and brittle-fracture mechanics, whose general solution might enable greater in-advance-time predictability of, and even reveal the "Achilles heels" of, earthquakes and volcanic irruptions
[or, in capitalist hands, would reveal how to generate earthquakes as weapons of class war/other contracted social reproduction]; the non-constant-throttle rocket equations; the Klimontovich Equation and the Maxwell-Boltzmann-Vlasov Equation, for electrically-charged "rheid" "hydrodynamics", e.g., for the plasma phase of matter-energy, whose general solution might immediately reveal successful designs for Boron-Helium, radioactivity-free, electron-releasing-only fusion reactors, making abundant energy affordable for all of humanity, and; the 10, simultaneous nonlinear partial-differential equations [or the single tensor equation] of the Einstein General-Relativity Equations for the Universal Gravity Field.

All that is standardly available toward solution of these "laws of nature" equations, today, are "numerical simulations" of the unsolved equations, which generate only a single state-space/physical-space "track", for a single starting-state, in a potentially infinite domain of unsolved other such "starting states", and their still-uncharted "special solutions", consuming vast amounts, even of massively-parallel super-computer time, and memory space, in this process of deriving limited, approximate special solutions, that are typically also insensitive to essential characteristics of the general solutions, such as singularities.

The "natural law" equations that are linear -- and, that, therefore, have long-since been solved -- are telling in themselves: like the linear Schroedinger Equation of "Quantum Mechanics", which can only "predict" spatio-temporal probability distributions, they are "missing something" -- something that the capitalist <<
mentalite'>>, with its atomistic/reductionistic bias, chronically misses!

Nonlinear versions, e.g., cubically-nonlinear versions, of the Schroedinger Equation, along with a vast expanse of other, usually "barely-nonlinear" partial differential equations, describing 1-dimensional "nonlinear waves", many of which have been solved analytically, in "closed form", e.g., using a "transcendental function" known as the the "hyperbolic secant function", have revealed mathematical models of a phenomenon that had been observed, at the "meso-scale" of our ordinary experience, but that is now known to exist also at "micro-scales" accessible to our experience only as mediated via advanced scientific instrumentation: that of "nonlinear waves" now called "solitary waves", or "particle"-like "solitons" -- typically "one-hump" waves of "infinite period", i.e., which consist of a single wave-"mound" of heightened amplitude, without even any second "mound", "until" t "=" +oo, a "time" which never arrives.

The air-masses of Earth's atmosphere -- "low-[surface-]pressure areas/air masses" ["cyclones"] and "high-[surface-]pressure areas/air masses" [inverted cyclones, or "anti-cyclones"] -- "toroidal vortices" all, may be physical correlates of not-so-barely-nonlinear, 3-dimensional "solitons", or "3-D nonlinear waves" -- solitary, vortex/anti-vortex paired, or otherwise -- that a general solution function for the highly-nonlinear Navier-Stokes Equations would model.

This opens the possibility -- not yet actualized, to my knowledge -- of a new, Nonlinear Quantum "Mechanics", which models quantized "particles" as [solitary, nonlinear] waves in a universal, unitary field, and which replaces the "acausal" linear functions of a random [probability-distribution-spanning] variable of Linear "Quantum Mechanics", with causal but nonlinear functions of a determinate variable, whose physical-time-variation, and physical-space-variation, exhibits an "entangle-able", "chaotic" attractor, or "strange" attractor, subject to bifurcation "quantum leaps", from one such "attractor" to another -- with an asymptotically aperiodic -- pattern with respect to each such attractor, which asymptotically mimic the patterns of the probability distributions of linear quantum mechanics after long-duration "inter-nonlinear-wavicle" interaction-entanglement.

Claim: A key task of Marxians is to continue/extend Marx's dialectical, immanent critique of the capitalist ideological science of "political economy" / "economics", and its positive scientific advances/fruitions, not only within itself, and not only into the spheres of the other capitalist-ideological social sciences, but also into the spheres of the capitalist-ideological natural sciences, and of capitalist-ideological mathematics itself, as well.

The dialectical logic exemplified, but not explicitly defined, in the writings of Marx and Engels, coupled with the capability to understand the causes of, and to see beyond, capitalist <<mentalite'>> and ideology, coupled with the ability to not assume for all eternity, but to see beyond, the present social formation, that of global/descendant capitalism -- all of which is beyond the capability of, or is "taboo" to, unconsciously capitalist-ideology-permeated scientists, or consciously pro-capitalist scientists, or consciously capitalist-apologist scientists, affords Marxians a unique potential to perform "universal labor" [Marx] which contributes advances to the social, natural, and mathematical sciences -- advances that, in themselves, already constitute contributions to capital-relation-dissolving, capitalism-dissolving further growth of the human-social forces of human-societal [expanded self-re-]production, in the form of scientific / technological advances: "the most solid form of social wealth" [Marx, <<Grundrisse>>].

D.: - . . . (The 'balance of nature' model is obviously flawed and ideological, as noted, since it imposes a concept of a set equilibrium that does not actually exist in nature.)

M.D.: I strongly agree.

"Attractor" solution-geometries for linear differential equations consist of single "fixed points" in state-space, approached asymptotically by the state-space trajectories of the systems modeled by that state-space with ever-diminishing, exponentially-diminishing "state-space velocities" ["temporal deceleration, to zero "at"  

t "=" +oo].

Those "fixed point" attractors represent absolute "equilibria" -- points which, e.g., if ever reached in finite time, would represent states of the absolute death of all change for these systems; single states from which these systems can never again escape; single states that these systems must "stay" in "for all eternity": unreaslistic, state-space "black holes".

Such linear differential equation models are hopelessly unfit to actuality.

Even "black holes" do not represent the absolute abolition of all change which linear "dynamics" demands.

Even "black holes" are predicted to "self-evaporate" over a sufficiently long duration.

D.: The 'selfish gene' theory, or biological determinism generally, lends itself to ideology as well since it hyper-balkanizes humanity's common interests and props up the institution of private property as handed down through biological lineages.

M.D.: I strongly agree.

"Biological determinisms", "socio-biologies", and other such ideologies involve a capitalist-ideology-characteristic move of ontological reductionism, or "back-to-[prior-]nature-ism" in terms of "scientific" explanation.

They represent a denial that the new ontology that emerged with the dialectical self-transformation of the earlier-emerged ontology of asocial meta-zoa -- of pre-social, pre-language multicellular animals -- is actual, and they represent a reduction of those new qualities, new properties, new behaviors, and new "laws" that emerged with social animals and with their proto-language-based animal societies, and with human[oid](s)-led "meta-societies" that came next; a reduction back to those of the qualities, properties, behaviors, and "laws" characteristic of those asocial meta-zoa, attempting to conceptually collapse the very actuality of those emergences.

This is particularly relevant to this dialogue, I think, because, as I perceive them, some of our other interlocutors' formulations exhibit these sub-Marxian characteristics, lacking historical specificity, and abiding in the historically-generic realm of explaining the cognitive problems of humanity in strictly biological-genomic terms, ignoring entirely the historical succession of qualitatively-different human-social ideologies, founded in the historically self-developing
human phenome, and associated with the qualitatively different social relations of production predominating in each successive human-social formation, culminating in the capital-centered global formation of today, and in the ideologies historically-specific thereto.



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