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__**, 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".**

*A Contribution to the Immanent Critique of Arithmetic*To date, only excerpts from

**have been approved to be made public by the General Council of**

__Dialectical Ideography__**F**.

**.**

__E__**., and by its Special Council of Psychohistorians, whose [elected] chairperson is Karl Seldon himself.**

__D__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

**can rightly be published in book form.**

__Dialectical Ideography__Regards,

Miguel

__Dialectical 'Meta-Systems' as__

__via-Conversion Singularity__**. 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**

__Self-Bifurcating 'Meta-Systems'__*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***, etc. This sub-section introduces connexions of Dynamical Systems Theory to '**

*telos**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

**in terms of "infinitesimal" variations in**

*S***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

**, is an algorithm that "predicts", i.e., a 'recipe' that tells the user how to compute, the***s**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**{(x**,_{o}**y**,_{o}**z**, as measured "at" initial time_{o})}**t**._{o}
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**t**._{o}
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 '

**' or, more generally, the '***slope-invariant***' of the function-values,***change-invariant***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 "**" if their expression of that change-"law" contains terms of degree***nonlinear***>****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 '**' of those values, terms containing***self-reflexions***x(t)**,^{n}**n****>****1**, or any terms containing '**' 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 "***flexions***". The equation containing such (a) term(s) is also said to be a "***nonlinear***" differential, integral, or integrodifferential equation.***nonlinear*
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**of the***state*__S__ystem**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**, the State-Space Trajectory will be rather simple. The solution-geometry of***linear***x(t)**must be dominated by a*single*"**", or "equilibrium" point, essentially***fixed point***[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**total differential equations.***linear*
If the integrodifferential equation solved by

**x(t)****=****(****T(t), P(t), H(t)****)**is**, the repertoire of possible State-Trajectories is vastly richer. The ultimate or "asymptotic", "***nonlinear***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***of vast shape-variety, and/or various multiplicities of so-called "chaotic", asymptotically***orbits**aperiodic,**"**strange attractor*"**of even vaster shape-variety. The latter represent fractal, never-repeating but ever self-similar, not "random" but***orbits**deterministic*patterns of state-flow, surrounded by complex flow-fields. 'Non-pointal', that is, 'orbital' attractor solution-geometries describe various kinds of*sustained***regular or irregular, of the state-variables or measured aspects of the modeled nonlinear systems. Especially the irregular "***self-oscillations,***" orbits analogize to business "cycles", climate "cycles", and myriad other "imperfect" or "never exactly repeating", 'fluctuatory' processes in nature.***self-oscillator**Orbital**attractors, orbital repellors, and orbital saddles cannot arise in***linear***dynamical systems*. Neutral orbits*arise in*__can__*, but only in cases of systems with*__linear__differential systems**-"***pure*__i__*maginary*" eigenvalues,**l****=****ar****+****bi**,**a****=****0**.
Closed form solutions have been discovered only for special cases,
usually "barely"

**total differential equations. However, those solved special cases have yielded great treasure, both theoretically and practically.***nonlinear*
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***" of the system. In engineered environments, such parameters can be "shifted" or adjusted by agents operating external***Parameter-Space**controls*, such as thermostats. The "Parameter-Space" of a dynamical system is thus often also termed its "**". 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 "***Control-Space**bifurcation diagram*" of the system-behavior, hence the term "bifurcations".__State-Space, Control-Space [Parameter-Space], and__. 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 "

*State/Control-Metaspace***", apparently infinite values of state-metrics, attained "at" finite values of the time parameter. The**

*singularities**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

**which that self-growth entails.**

*self-bifurcation*
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

**. The resulting unified***Course Of Development***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***metaspace**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 '**' of that [***metastate**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*

**, mediately, when the control parameter**

*changes itself**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

**, 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,**

__0__*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**values, signifying the**__0__*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

**, 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.**__0__
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***. 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.***resource*
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**, combining state-space and control-space axes.***metaspace*
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*across their critical thresholds in control-space and beyond is not rendered in them. Coupling of state-variables and control-variables is usually omitted. C**themselves***umulative*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" '

**' 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 '***made of**social endosymbiosis*' or '*social**symbiogenesis*'].
We omit from
this

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

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