Wednesday, September 04, 2013

Dialectics and Dynamical Systems Theory -- An Excerpt from the book DIALECTICAL IDEOGRAPHY, by Karl Seldon and Sophya St. Germain

Dear Readers,

Given the focus of the previous blog-entry, on "Complexity Theory", I thought it might be helpful for readers to post the passage from the book Dialectical Ideography:  A Contribution to the Immanent Critique of Arithmetic, by Karl Seldon and Sophya St. Germain, which addresses an immanent, dialectical critique / extension of Nonlinear Dynamical Systems Theory, the latter being the discipline that forms the mathematico-scientific heart and historical source of "Complexity Theory".

To date, only excerpts from Dialectical Ideography have been approved to be made public by the General Council of F.E.D., and by its Special Council of Psychohistorians, whose [elected] chairperson is Karl Seldon himself.

It is to be hoped, at least from the point-of-view of my -- still fledgling -- grasp of psychohistorical dialectics, that the optimal timing is near when the full text of Dialectical Ideography can rightly be published in book form.


Regards,

Miguel







Dialectical 'Meta-Systems' as via-Conversion Singularity Self-Bifurcating 'Meta-Systems'. Classical Dynamical Systems Theory uses the ideographic mathematical language of total differential equations to model the dynamics of natural systems. Its findings simulate and corroborate classical notions of dialectical process in many ways, especially in the case of the unsolved nonlinear dynamical systems, largely suppressed until recent decades. It also echoes much of classical Aristotelian 'essential-dynamics' or 'essence-dynamics'. It developed mathematical concepts which are highly homeomorphic to essentialist concepts of essence [ousia], dynamis [potentia], energeia, ergon, entelecheia, telos, etc. This sub-section introduces connexions of Dynamical Systems Theory to 'Dialectical Meta-Systems Theory' as 'Dialectical Meta-Dynamics', via the 'Self-Bifurcation' paradigm of dialectical process.   



Nonlinear Dynamical Systems Theory and Dialectics. The nonlinear integrodifferential equations that formulate the so-called "laws" of nature are primarily "partial" differential equations.  This means that they involve solution-functions S = s(x, y, z, t, . . .), whose values vary with physical-spatial position -- with the space-coordinates x, y, and z -- as well as with the time-coordinate, t, plus, in some cases, with other independent variables as well.  The equations thus involve "partial differentiation operators" ∂/∂x,  ∂/∂y,  ∂/∂z, and ∂/∂t, which measure the variation of S in terms of "infinitesimal" variations in x only, y only, z only, or t only, respectively.

The closed-form 'solution-operation' or solution-function for such an equation, here denoted by s, is an algorithm that "predicts", i.e., a 'recipe' that tells the user how to compute, the state of any point of space (x, y, z), in terms of the phenomena-measures that the equation models, for any value t, past or future, from the input values x, y, z plus from the initial 'state of [the] space' "occupied by" this system, that is, from the phenomena-measurements -- the states -- of the points-set {(xo, yo, zo)}, as measured "at" initial time to.

Dynamical Systems Theory traditionally models with "ordinary" or "total" differential equations, linear or nonlinear.  These involve solution-functions of the form X = x(t). There is but one ultimate independent variable to "differentiate with respect to" -- namely t, the time-variable. Time differentiation of X, using the 'non-partial' differentiation operator, d/dt, is thus "total" differentiation of X.  The state-"vector" x(t), for any value of t, is an ordered list of values of the various "state-variables" or 'system-attribute measurements', which are the model's [pre]dictions or predications of these key 'total' or 'holistic' aspect-metrics [vs. the partial-differential, spatially-distributed aspect-metrics] of the dynamical system modeled, if taken at that t value.

The closed-form 'solution-operation' or solution-function for such an equation, here denoted by x, is an algorithm that "[pre]dicts" or [pre]states, i.e., a 'recipe' that tells the user how to compute, the state of the system, the value of each of the "state-variables" or modeled 'attribute-measurements' of that system, for any value of t, past or future, from the input value t, and from the original 'state' of the system, that is, from the original values of all of the state-variables, their values as of the modeler-chosen 'initial' time denoted to.

State-variables should be 'holistic', 'overall' metrics of facets of the system being modeled. I.e., they should characterize the entire physical body of the system 'all at once', not differing in their values substantially -- within the utility of the model -- from spatial/synchronic point to point on or within that body. Otherwise, they belong in a "partial differential" model. Take your body, for instance. To model its physiological dynamics, you might use "systemic" state-variables like temperature, T(t), blood pressure, P(t), and heart-rate, H(t), which can be approximated as uniform throughout the soma, to partially characterize your body's changing physiological state at various moments, t. Hair density, which varies widely over the body's surface, and vanishes for much of its interior, would not make a good "total differential" state-metric. Your "total-differential", 'solved' lifetime body-model, a "state vector valued" solution-function, would then be of the form  X = x(t) = ( T(t), P(t), H(t) ).

The first-order "total" or "ordinary" integrodifferential equation-model states the 'slope-invariant' or, more generally, the 'change-invariant' of the function-values, x(t), of the unknown function or operation x; the invariant "law" of its function-values' variations, the pattern of variation of the "state" of the system, x(t), as the time t varies.  Such equations are termed "nonlinear" if their expression of that change-"law" contains terms of degree > 1 in x(t), and/or in its differentials, and/or in its integrals, and/or in any products of itself, its differentials, or its integrals with any such forms of itself or of other function-unknowns, if any.  

Said differently, if the equation stating the change-rule of the values of the unknown operation, x, which is to be discovered from that equation, contains any 'self-reflexions' of those values, terms containing x(t)n, n > 1, or any terms containing 'flexions' with function-values of other operator-unknowns, with or without any order of integral or differential operators as 'coefficients', then the term is said to be "nonlinear". The equation containing such (a) term(s) is also said to be a "nonlinear" differential, integral, or integrodifferential equation. 

The equation may be termed just "differential" if it contains no integration operations, just "integral" if no differentiation operations, or "integrodifferential" if it contains either or both.  

If any equational occurrence(s) of the 'unknown function-values variable' or "dependent variable", x(t), is of the form x(t)n, n = 1, i.e., 'simple presences' of those function-values, without self-action, and without interaction with any other function-unknown(s)/dependent variable(s), then the integrodifferential equation is said to be "linear".



State-Space Trajectories, Control-Space Paths, and Bifurcations. The 'dynamical algebra' of "total" [or "ordinary"] integrodifferential equations involves new operations, "differentiation" and "integration", involving "limits" of conceptually infinitary processes, which, as such, are foreign to classical algebra. It also entails expressions involving "functions of time", or 'operations on time', like x(t), not encountered in that 'statical' algebra.  But this 'dynamical algebra' does have, like 'statical algebra', an "analytical geometry"; not the 'statical' analytic geometry of Descartes, but a special, dynamical analytical geometry called "Phase Space" or "State-Space".

Our hypothetical 'dynamical-algebraic' model, x(t) = ( T(t), P(t), H(t) ), corresponds to a 3-dimensional 'dynamical-geometric' model, formed by 'crossing' 3 mutually perpendicular numberlines, scales, or axes, one assigned to T(t), one to P(t), and one to H(t), at their origins or 0-points. Any value of t, representing a moment of time, an "exact date", corresponds to 3 coordinates, computed by applying the state-functions or operations T, P, and H to that value of t. These three values together define a single point in this conceptually-constructed, non-physical, imaginary 3-dimensional space. That point is identified with "the state of the System S at time t".  Obviously, if, as the time-value, t, changes, the values of one or more of the "state-variables", T(t), P(t), and H(t), also change, the position of this state-point will change as t changes. "Connecting the dots" of the different state-points computed for different t values forms a track in this space, called the "State-Space Trajectory" of system S. The totality of points representing possible combinations of T(t), P(t), and H(t), whether the state-point of a given instance of S ever gets to them or not, is called the "State-Space" of S.  If the integrodifferential equation solved by x(t) = ( T(t), P(t), H(t) ) is linear, the State-Space Trajectory will be rather simple.  The solution-geometry of x(t) must be dominated by a single "fixed point", or "equilibrium" point, essentially [0, 0, 0], the origin, surrounded by a field of "transient" trajectories that leave it, and/or approach it, or neutrally orbit it.  Any t = 0 starting point, or 'birth state', in the State-Space will be for all time attracted to and/or repelled by the origin, or will neutrally orbit it, without attraction or repulsion.  If attracting, the solution-point is called an "attractor"; if repelling, a "repellor", if of mixed effect, a "saddle", if neutral, a "center".  The "dynamics" of linear systems with attractor solutions is more aptly described as an 'anti-dynamics' -- a monotonic taxis toward a point of equilibrium, that is, a point of no further change, of eternal non-change.  Closed form solutions have long been known for general linear total differential equations.

If the integrodifferential equation solved by x(t) = ( T(t), P(t), H(t) ) is nonlinear, the repertoire of possible State-Trajectories is vastly richer.  The ultimate or "asymptotic", "t = +oo" solution-geometry can involve (1) two or more fixed points, (2) various combinations of fixed points with attracting, repelling, mixed, or neutral asymptotically periodic orbits of vast shape-variety, and/or various multiplicities of so-called "chaotic", asymptotically aperiodic, "strange attractor" orbits of even vaster shape-variety. The latter represent fractal, never-repeating but ever self-similar, not "random" but deterministic patterns of state-flow, surrounded by complex flow-fields. 'Non-pointal', that is, 'orbital' attractor solution-geometries describe various kinds of sustained self-oscillations, regular or irregular, of the state-variables or measured aspects of the modeled nonlinear systems. Especially the irregular "self-oscillator" orbits analogize to business "cycles", climate "cycles", and myriad other "imperfect" or "never exactly repeating", 'fluctuatory' processes in nature. Orbital attractors, orbital repellors, and orbital saddles cannot arise in linear dynamical systems. Neutral orbits can arise in linear differential systems, but only in cases of systems with pure-"imaginary" eigenvalues, l = ar + bi, a = 0.

Closed form solutions have been discovered only for special cases, usually "barely" nonlinear total differential equations. However, those solved special cases have yielded great treasure, both theoretically and practically.

The states of a dynamical system will also be affected by "external conditions" and "accidents", not determined by its “internal” dynamics. The state of our hypothetical system, S, for example -- the temperature, blood pressure, and heart-rate of your body -- will be partly determined by current air temperature, oxygen concentration, and acoustical noise level, etc. in the physical space that surrounds it. 

Measurements of these conditions may appear in the integrodifferential equation of the system as "constant parameters" -- constant "coefficients" of terms involving the state-variable-function-unknowns; constant terms, etc. -- incorporated into the state-variable functions T(t), P(t), and H(t), or as time-varying "forcing functions" or "drivers"  All such parameters are mapped to mutually perpendicular numberlines or axes in what is usually conceived as a separate, second system-space, called the "Control Parameter-Space" of the system. In engineered environments, such parameters can be "shifted" or adjusted by agents operating external controls, such as thermostats. The "Parameter-Space" of a dynamical system is thus often also termed its "Control-Space".  Parameter "shifts" can, if they cross through certain "critical values" in the Control-Space, cause sudden, qualitative changes in the solution-geometry exhibited by the first space, that is, metamorphoses in the system's State-Space Trajectory and attractor(s), its Trajectory "flow" or "vector-field". An Attractor Trajectory, for example, may suddenly become a Repellor, Saddle, or Neutral Trajectory. Such deep breaks in behavior-pattern are traditionally termed "Bifurcations". They often involve the branching of one solution-geometry into two new solution-geometries, starting from the critical point of the "bifurcation diagram" of the system-behavior, hence the term "bifurcations".




State-Space, Control-Space [Parameter-Space], and State/Control-Metaspace. In classical Dynamical Systems Theory, a system's state-point and even its control-point may change location, but the state-space and the control-space do not change.  They are statical, not dynamical. Their structure does not vary with time, state, or parameter. Their dimensionality is fixed. They form a static backdrop against which state-change and control adjustment occur.  Even if the system develops partial 'self-control', so that the state-point begins to control the control-point; so that the control-point begins to move in correlation with the movements of the state-point, both spaces  remain both separate and 'unmoved' per the classical conception and convention. This convention restricts the scope and coherence of evolution-models. They tend to be limited to a single epoch or stage of 'meta-evolution'. The models tend to end with misleading, counter-empirical predictions, e.g.,  of asymptotic -- that is, infinitely-delayed -- approaches to final attractors, or with "singularities", apparently infinite values of state-metrics, attained "at" finite values of the time parameter. The actual dissipative systems soon abandon and bifurcate away from these, due to their 'essence-ial' dissipative depletion of the resources fueling their old dynamic, and the emergence of new dynamics, defining new resources.  Static state-space models tend, for example, to encompass but one phase of stellar burning, or even one generation of stars, but not repeated 'phase transitions'; not repeated generational transitions, not the cumulative enrichment of the interstellar medium that the latter entail, and its consequences for later-generation dynamics, e.g., 'planetogenesis'. They typically omit the ineluctable system self-subversion in the single epoch they cover.  Next epoch and preceding epoch models disconnect. Models must be reconfigured at every epochal transition. Successive models have trouble "passing the baton" across epochal, self-bifurcation boundaries, let alone merging into single, unitary models of natural history, covering entire successions of such transitions.  Rightly-formulated dynamical equations, and their solution-functions, should not be 'one-epoch models'.  They should describe both sides of the dual self-consequential process of the meta-evolutionary self-accumulation within each natural formation: both the self-growth, and the eventual self-bifurcation which that self-growth entails.
  
The proposed 'meta-dynamics' merges state-space and control-space in a unified 'state/control metaspace'.   

State-shifts driving parameter-shifts is par for the course. State-Space Trajectory and Control-Space Path merge into a unified Course Of Development.  The resulting unified metaspace is also itself a dynamical object.  Its axial content changes.  Its dimensionality changes -- usually grows -- "as a function of time".  Each system-self-induced bifurcation builds new axes, new dimensions, new state-variables, into the "state-space" 'side' of metaspace, converting former control axes into state axes, and sprouting new control-axes out of the origin. This self-expanding metaspace is an integral part of a meta-dynamical model. 'Change of [meta-]space', as well as mere change of place of the state/control point inside a fixed 'metastate' of that [meta-]space, mirrors predicted quanto-qualitative, epochal changes in this unitary, multi-epochal, meta-dynamical model. Change of place models fulfillment of "laws". 'Change of space' models change of "laws". Dynamics change.  Dynamics change themselves, by self-bifurcation. Change-of-space, change-of-"laws", change-of metrics also imply ontology-change.

This 'Meta-Dynamics' is a dynamics of dynamics, 'dynamics squared', the nonlinear, second degree of dynamics.  

We claim that this 'Meta-Dynamics' is also Dialectics.



Self-Bifurcation.  A dialectical meta-system itself, its 'essence', its "law" of change, is expressed by its entire state/control meta-space, the total "flow" of its possible courses of development within that space, that is, the actions and defining mode of action of the entire family of meta-systems of which a given individual meta-system is an instance. This meta-system-action is also mediated through the control-path that the meta-system itself induces for itself in its parameter-space or control-space, by which it acts back upon its own state-space trajectory. The meta-system quanto-qualitatively changes itself, mediately, when the control parameter variables that its own state-motion drives cross their critical values. One visualization of this "change of [state/control meta-]space" is as a kind of "jump" from one meta-space to an other, separate meta-space, somehow located "elsewhere".  This is a 'convolute' paradigm of change at the level of the meta-space as a totality. Here we will visualize this change differently. 'Evolutely'. Cumulatively. The meta-space changes by expanding [occasionally, old axes will, in effect, wither away as well, so meta-spaces can change by at least partially contracting also]. A new axis, or several new axes, sprout from 0, the origin of the meta-space, each perpendicular to any other newcomer-axes as well as to all previously-sprouted axes. The new axes correspond to the new state-variables and new control-parameters, new measurements or metrics/metrical ontos needed to describe the meta-states of the mediately self-transformed meta-system going-forward, in the meta-system’s post-transformation epoch. The new axes or dimensions cover qualitative change(s) -- increment(s) of new qualities, meta-system ontology-expansions -- gained in that self-transformation.
  
Thus, typically, all or most of the metrics or state/control-variables of the preceding meta-system meta-state and of its old meta-space remain. The expansion of the meta-space is a qualitative as well as a quantitative expansion, because the new axes of the added state/control-variables measure newly-emerged qualities or attributes, tied to new metrical ontos, of the self-bifurcated meta-system. The meta-space expansion is thus a quanto-qualitative one.  It is also an ‘evolute’ one. The meta-space grows cumulatively, accumulating ever more new axes, metrics [qualities, attributes, predicates, metrical ontos], or dimensions, as the self-bifurcations sequence continues.  But some of the old metrics or state/control-variables may "vanish", collapse back into the origin, to intermittent or even steady 0 values, signifying the extinction or obsolescence of the system-qualities or metrical ontos they measured. Traditional approaches also visualize the control-space, as located "elsewhere", separate from the state-space, though as if exerting an 'action-at(from)-a-distance' upon it.

The proposed Meta-Dynamics visualizes the control-space as embedding -- engulfing, surrounding, and permeating -- the state-space. This view visualizes control-space as another set of orthogonal axes sharing the same origin as the state-space's state-variable axes. This approach views the control-space as also a dynamical entity; as changing.  When the action of a dialectical meta-system, as recorded in its state-space by its state-space trajectory, drives that system's parameter-space path to a critical, self-bifurcation threshold value, and beyond, that old control-parameter axis ceases to exist as such.  Instead, it transfers to the state-space, becomes a new state-variable axis of a new, thereby expanded, post-bifurcation state-space.  Concurrently, a new control-space is born.  New control axes or dimensions, representing the new control qualities or metrics, extend from 0, replacing the old control parameter-space, now extinct or accrued to the state-space, with a new one, constituted of metrics measuring qualitatively different control attributes.

Stellar [meta-]evolution exemplifies this meta-dynamic. Partial differential equations, not total differential equations, are the usual language for stellar evolution models.  However, our context is that of a hypothetical finite dimensional state/control meta-space model, a total-differential model, of stellar [meta-]dynamics. During the Hydrogen-burning phase of a star's life-process, stellar core relative Hydrogen mass-concentration is a key state-variable. Helium is a "waste product" or 'entropy' of the Hydrogen burning process. Relative Helium concentration, at this stage, in the stellar core, is the key self-bifurcation control-parameter. The key state-process, Hydrogen fusion, converts more and more core Hydrogen to Helium.  That state-process thus also progressively shifts the value of the core Helium-density control-parameter higher, as it depletes more Hydrogen, and accumulates more Helium, in the stellar core.
 
When the Helium parameter crosses a critical threshold, the expansive force of the Hydrogen fire wanes in the stellar core. Accelerated self-gravitational self-re-contraction thus ensues. This contraction compressively heats the stellar core. Depending upon the star's initial conditions, the temperature threshold for Helium ignition may thereby be breached. 

Helium ignition may be modeled as a self-bifurcation, and as a metafinite conversion-singularity, of the star's state-trajectory. The star's core life-process, hence its external appearance and outer behavior, transforms quanto-qualitatively. A core-process founded on Hydrogen fusion transitions to a core-process founded on Helium fusion. The former '[self-]pollutant' of the Hydrogen-burning star, Helium, becomes its new vital resource. That former 'entropy' of the star becomes its new 'negentropy', or "free energy" resource.  Relative Helium mass-concentration, former control-variable, becomes new state-variable. Metrics of the relative mass-concentrations of the "wastes" of Helium fusion become the new control-variables. Most of the star's mass is still Hydrogen.  Hydrogen fusion, continuing peripherally and intermittently, mainly outside the core, continues to co-determine the states and meta-states of the star. The metric of relative Hydrogen concentration thus continues to function as a state variable. The state-space has expanded to incorporate a former control-axis. A new control-space [axis / dimension / metric] has emerged.

The vantage of self-bifurcation, of dialectics or meta-dynamics, sees neither state-space nor control-space as static. The state-space itself, as a totality, is a dynamical self-variable -- not only in its basin/attractor contouring or flow structure, but even in its fundamental geometry, its very dimensionality. Likewise control-space. We see a unified or unitary and [self-meta-]evolving state/control metaspace, combining state-space and control-space axes.

These meta-dynamical processes are not captured, not modeled, by standard integrodifferential equation models of such self-reflexive, self-refluxive meta-systems. These standard equations generally track no further than the boundaries between the sub-critical and critical values of control parameters, at best. The meta-evolutionary drive by which such systems propel themselves across their critical thresholds in control-space and beyond is not rendered in them.  Coupling of state-variables and control-variables is usually omitted. Cumulative movement of control-point in response to the self-movement of the state-point is neglected.  It is usually tacitly assumed that control parameter settings can be reset only by forces external to the system itself. The possibility of internal control, self-determination, self-transformation is usually not considered.
  
Yet it is the very way of things.  Self-bifurcative metadynamism is ubiquitous in nature, including 'human nature'.

Consider an 'onto-dynamic' cosmos-model which identifies the following succession of ontos, plus their various hybrids, as forming the prime gradient of cosmic meta-evolution: (1) sub-nuclear 'nonlinear waves', "quantum fields" or "particles", (2) sub-atomic "particles" ['meta-sub-nuclear "particles" 'made of' sub-nuclear "particles"', 'meta-fields made of fields', or 'meta-waves made of waves']; (3) atoms ['meta-sub-atomic "particles" 'made of' sub-atomic "particles"]; (4) molecules ['meta-atoms made of atoms'], (5) prokaryotic 'pre-cells' or  'proto-cells' ['meta-molecules made of molecules']; (6) eukaryotic cells ['meta-prokaryotes 'made of' prokaryotes']; (7) "multi-cellular organisms", i.e. plant and animal 'meta-biota' [[eukaryotic] 'meta-cells made of [eukaryotic] cells']; (8) animal societies ['meta-organisms made of organisms'], and; (9) human [or humanoid] 'meta-societies' ['meta-animal-societies made up of animal societies' via 'social endosymbiosis' or 'social symbiogenesis']

We omit from this onto-dynamical cosmos-model both the 'multi-ontic cumulum' of 'hybrid' micro-formations and the macro-cosmic and meso-cosmic 'vessels' of these micro-ontos, galaxies, stars, "solar" systems, intra-"solar"-systemic planets, intra-planetary oceans, lithospheres, atmospheres, biospheres, noospheres, etc., but only for the moment.

. . .

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