Some Fruitions of F.E.D.'s Dialectical, Immanent Critique of [Nonlinear] Dynamical Systems Theory
Here is the rest of the recent dialogue, from elsewhere, the earlier part of which was recorded in the immediately previous blog entry to this one, on another aspect of that earlier entry to this blog, on the "simple side" of F.E.D. Dialectics --
Further Response to "Simple Side" Blog Entry: "... The processive steps -- of this self-elaboration, this self-succession -- may go slowly or rapidly, as one sees in rapid evolution (which could be also called revolution)."
Reply: Your point about the different rapidities, or slownesses, with which [psycho]historical-dialectical self-successions proceed, in terms of "evolutions", for the slower periods of such successions, and in terms of "revolutions", for the faster phases, is also salient, in my view.
To my reading, F.E.D. defines as "EVOLUTIONS" phases of change wherein a system's "vital signs" or "key metrics" -- as represented by their "state variables and "control parameters" -- change ONLY QUANTITATIVELY, within a "state/control space" of FIXED dimensionality, with NO QUALITATIVE CHANGES in the sense of no changes to the number or content of the mathematical-space dimensions representing the system's "vital signs" or "key metrics" -- its state-variable axes [dimensions] -- and no changes to the number or content of the control parameter axes [dimensions] which typically represent the system-external forces co-determining the system's fate [though control parameters can also represent a dynamical system's "SELF-control", and hence its "SELF-bifurcation meta-dynamics"], generating a "state-space trajectory", and a "control-space path", which, however much, and however complexly, they wind, remain confined within that fixed mathematical space.
In such phases, the system so "trajecting" & so "pathing" is called a DYNAMICAL System, and the "law", i.e., the [typically NONlinear system of] differential equation(s), that describe(s) / predict(s) / model(s) that system's state-space "trajectory", and its control-space "path", is (are) called that system's "DYNAMIC", or "law(s) of motion".
Such "trajectories"/"paths" of the "time-driven", "time-varying" moving points that are used to represent -- "dynamic-geometrically" -- the system's "state" as a function of time, and its "control regime" as a function of time, may speed up some, and/or slow down some, at various times within such a phase of "EVOLUTION".
However, the really fast changes manifest, per F.E.D., with the "META-DYNAMICS" of "META-EVOLUTION" that arise in and as the REVOLUTIONARY transition from one DYNAMICAL system to its successor DYNAMICAL system, i.e., from one SYSTEM-DYNAMIC [dynamical "law of motion"], to its successor SYSTEM-DYNAMIC [dynamical "law of motion"].
Such REVOLUTIONARY transitions involve CHANGES OF SPACE, not just changes of coordinate-values within a fixed state/control space.
That is, the predecessor system's state and control spaces suddenly become "UNfixed" in the process by which the predecessor system gives rise to its successor system., e.g. [the following are some state-space-CHANGE manifestations, & some control-parameter-space-CHANGE manifestations, of QUALITATIVE, ONTOLOGICAL changes in the dynamical systems that they model] --
1. The state-space may lose old axes/dimensions;
2. The control parameter space may lose old axes/dimensions;
3. Some old-system control parameter [constants or] variables may be transformed/transferred into new-system state variables -- old dimensions or axes of the old system control-space moving to become new dimensions or axes of the successor system state-space;
4. Some old-system state-variables/metrics/dimensions/axes may be transformed/transferred into new-system control parameter [constants or] variables -- old dimensions or axes of the old state-space moving to become new dimensions or axes of the successor system control-space;
5. The new, successor system may irrupt new dimensions, new axes, adding themselves to those of the old system control-space, and representing unprecedented new key metrics, in the form of new system control-space control parameter variables [or constants];
6. The new system may irrupt new dimensions, new axes, adding themselves to the old system state-space, and representing unprecedented new key metrics, in the form of new system state-space state-variables.
Thus, slower predecessor-system EVOLUTION leads to relatively much faster system "META-EVOLUTION", or REVOLUTION, wherein and whereby the predecessor system gives rise to, or [partially] turns itself into, its successor system.
Thus the slower DYNAMICS of predecessor-system EVOLUTION engender, once they accumulate to a "critical point", the rapid "META-DYNAMICS" of the self-transition of the predecessor DYNAMIC/system/"law of motion" [in]to its successor DYNAMIC/system/"law of motion".
F.E.D. calls a diachronic [i.e., a historical, chronological] self-progression of systems/DYNAMICS, exhibiting EVOLUTION within each DYNAMIC/system, and exhibiting the "META-DYNAMICS" of the "META-EVOLUTIONARY" transition -- i.e., of the REVOLUTIONARY transition -- bridging between each predecessor system and its successor system -- bridging/mediating FROM each predecessor system TO its successor system -- a "DIACHRONIC META-SYSTEM".
F.E.D. also recognizes two different kinds of "meta-FINITE singularities" associated with such REVOLUTIONARY INTER-SYSTEM TRANSITIONS:
A. "Depletion Singularities"
B. "Resonance Singularities"
More about such qualitative, ontological "singularities" -- as reflected, "reductionistically", in [typically nonlinear] integro-differential equations that are "purely quantitative", or "unqualified", as "finite time zero divisions" [i.e., as finite time "purely-quantitative INFINITIES"] in purely-quantitative [typically nonlinear] integro-differential equations -- in future posts.
F.E.D. provides explicit procedures for the "RE-QUALIFICATION" of traditional, "UNQUALIFIED", or "purely-quantitative", differential equation-system models, using the "ontological-qualifier", "metrical-qualifier", and "system-qualifier" "meta-numbers", derived by F.E.D. via their "dialectical progression of the systems of dialectical arithmetic", which procedures result in the "semantification" of otherwise "meaningless" singularities [purely-quantitative "infinities", or "undefined values", where the purely-quantitative differential equations-models "breakdown", "fail", or become "infinitely wrong"].
Example: If F.E.D.'s predictions turn out to be correct, there is, upcoming in our near futures, a revolutionary transition from the human socio-political-economic system founded upon the "Capital- relation", i.e., upon the "Wage-Labor-relation", that is, upon the "alienated-[sold-]labor-relation", as predominating social relation of production -- i.e., upon CAPITAL EQUITY -- to a successor human socio-political-economic system, <<aufheben>>-conserving the Capital-Equity relation-of-production <<species>> of Social/Privatized equity in a "subsumed", socially/democratically constrained form, but adding also four new <<species>> of universal, all-citizens, Social Equity -- Citizen Externality Equity, Citizen Birthright Equity, Citizen Stewardship Equity, and Citizen Allocational Equity.
Thus, if this F.E.D. prediction proves out, AT LEAST four new state-variable metrics/axes/dimensions will have to be recognized as irrupting from inside the state-space of any good, "RE-QUALIFIED", nonlinear differential equations-system model of the human-social, historical/diachronic "meta-system" of the human-social relations of production, for the relatively brief period of extraordinarily rapid change in human history when the "meta-finite singularity" of exclusive, capital-relation, capital-equity domination of global human society self-explodes.
F.E.D. expositions of the concepts that I have attempted to summarize above are found in Supplements A & B to the F.E.D. "Introductory Letter" --
[Pages A-6 through A-14].
[Pages B-24 through B-37; especially Page B-33].
F.E.D. also asserts the existence of a phenomenon of "TEMPORAL ACCELERATION" that characterizes -- per F.E.D. -- historical-dialectical self-progressions overall and in general, in the form of a "diachronic meta-fractal" self-similarity durational scaling of the time intervals separating the successive self-unfolding contents of concrete time: