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.
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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