Part
III. Bipolar feedback in creative evolution.
As models for physical or biological
feedback processes, mathematical developments are too simple to be realistic.
But as mathematical entities, they instantiate
creative developments generated by various types of feedback.
The process equation proves, with mathematical
certainty, that bipolar feedback creates periodicity, chaos, bios, and infinitation, a sequence of patterns of
increasing statistical entropy.
Wherever there is bipolar feedback, such results
must obtain. In bipolar feedback, complexity results from bifurcation
into opposites, not from dialectic synthesis.
Biotic organization occurs after chaos, not
in the edge between order and chaos.
Bios often emerges from chaos, but chaos is
not necessary to create bios.
Bipolar opposition generates both chaos and
bios. This offers a model for natural creative processes as partially
determined by the past and open to the future.
The import of deterministic
mechanism for creating novelty and complexity cannot be overstated.
Creative processes are currently explained
as the result of chance: the origin of life was a unique and rare accident according to Monod; biological diversification
is attributed to random mutation.
According to statistical mechanics, we are
faced with a natural tendency of any state of order to turn on its own into a less ordered state, not the other
way around. This is of course not an intrinsic property of shuffling,
as it would be perfectly thinkable that shuffling could actually create order. Yet nobody expects this course to
take place. Indeed one would have to wait a long time for it to happen
by chance. But evolution does occur in the physical universe, and
in the biological realm.
It seems likely that bipolar and mutual feedback
that so effectively generates pattern in mathematical recursions, may also be responsible for some creative natural
processes.
Bipolar
feedback in biological and economic processes
Finding biotic patterns in natural processes suggests that bipolar feedback may be one of
the universal processes responsible for the creative features of evolution. Bipolar feedback is the necessary consequence of repeated actions and diverse, synergic
and antagonistic, interactions, which occur at all levels of organization. For instance, the biotic pattern of heart rate variation reflects the antagonistic action
of the sympathetic accelerating and the parasympathetic decelerating nerves. Given
the dire prognostic implication of heart rate regularity, it is significant that physiological regulation generates
bios rather than homeostatic equilibrium.
Similarly,
the biotic and parabiotic patterns
observed in economic processes indicate continual creation of novelty and complexity, at variance with the equilibrium
models that dominate standard economics. Contrary to their hypothetical equilibrium, supply not only satisfies demand but also increases it by creating
new possibilities, needs and wants. Conversely, demand not only consumes supplies but also fosters production by providing a
market. General equilibrium theory investigates what prices, production rate and consumption rates
are consistent with the overall pattern in the market –and hence pose no incentive for change. It also supports
the status quo by contending that nothing needs or can be done. A necessary equilibrium between supply and demand is often adduced to justify as unavoidable
policies that are unjustifiable from a humanitarian perspective. However the economy is a process that shows long-term progress, perpetual novel behavior,
emergent phenomena, and large momentary jumps, such as major price revolutions, “shocks” and major crises.
Neither
equilibrium nor random walk models can explain novelty or discontinuity. The process equation model suggests an explanation for the observed price revolutions, namely
infinitations generated by a critical increase in the bipolar feedback gain. This is consistent with historical evidence: the four waves of price increased documented in the Western world since the 12th century accompanied an increase
in population, commerce and industrial production.
Economic systems may be expected
to operate best when including multiple and diverse feedback.
This is congruent with the intuitive notion
that planning alone, no matter how well conceived, is insufficient, because actions always have unforeseen, and
unwanted, consequences.
Such feedback needs to include both positive
and negative components; recent history illustrates how, free from risk, financial institutions engage in irresponsible
speculation, enterprises fail to improve productivity, and economies fall into stagnation.
That economic processes display biotic patterns
implies that they are highly modifiable by human intervention, neither determined by fixed laws nor the product
of random chance.
Natural and human processes
generate novelty and complexity, not equilibrium.
The concept of equilibrium, of great value,
and narrow domain, in the case of mechanical processes, has been unduly extended to other fields.
Thermodynamic equilibrium is a state of uniformity
and rest; most processes occur far from equilibrium [Prigogine, 1980].
Health is not equilibrium; chemical equilibrium
begins to be approached only with death.
“Social equilibrium” is a misnomer for processes
of great complexity, hopefully far from stagnation.
“Mental equilibrium” is likewise metaphorical,
and misleading; longitudinal studies of emotions reveal that equilibrium patterns predominate among criminals,
while healthy individuals show wide variations of mood [Sabelli et al, 1997].
Homeostasis and autopoiesis, the maintenance
and reproduction of organisms, are components of larger creative processes of individual development and biological
evolution. The planet is not a self-regulating living system, as envisioned by the
Gaia hypothesis; otherwise we would not suffer from environmental destruction.
Similarly, social systems are not
homeostatic and autopoietic; otherwise there would be no history.
Bipolar feedback in
development and evolution
Bipolar
and mutual feedback may play a role in biological development.
There is a gross
similarity between biological development and the time series generated by the process equation, from a single
"egg", through a cascade of divisions to biotic processes. Each forking in a cascade of
bifurcations is a differentiation
of opposites. In
embryological development, cell division leads to cell differentiation.
Similarly, evolution is a tree
of bifurcations.
Classical evolutionism focuses on chance as
the origin of innovation, and on conflict as determining selection.
Yet cooperation is just as important as competition,
as best illustrated by endosymbiosis, which is responsible for photosynthesis and for respiration. Biological
systems involve both synergic and antagonistic relations (bipolar feedback) that can generate innovation in a nonrandom
manner. Evolution is both determined and creative. It
is determined insofar as bipolar and mutual feedback represents the unavoidable consequence of repeated interactions. It
is creative insofar as it generates temporal transformation of pattern, novelty, diversification, nonrandom complexity,
and irreversibility.
Evolutionary innovations, and even the origin
of life, are often attributed to chance, but Lady Luck seldom is that generous.
Natural feedback is bipolar and hence creative,
rather than negative and homeostatic.
A
tree of bifurcations also describes cosmological development from a single Big Bang event followed by a sequence
of “symmetry-breakings” (bifurcations), with early inflation, continual expansion, and increasing complexity. The time series generated by
the process equation also evolves from a single state through multiple bifurcations, includes transient infinitations,
and a continual increase in amplitude and complexity.
Plots of the cosmic background radiation and
of galactic distribution show that radiation and matter are neither periodic nor uniformly or randomly distributed. Certainly
their aperiodic patterns must reflect creative processes.
A mathematical generator of physical reality?
Embodying
feedback, mathematical recursions generate complexity.
Do similar processes occur in nature?
The fact that numbers such as p and forms such as Mandalas emerge from both natural and mental processes points to
a profound relation between the two. It suggests to me that mathematical order is the simplest initial level of organization
in a creative but determined development:
Mathematical à Physical à Chemical àBiological à Social àPsychological,
in
which complexity increases with time, while, conversely, extension and duration decrease (figure 38).
Figure 38 Schematic representation
of the hierarchy of levels of organization. The most extensive level comprises mathematical relations, which are
universal and certain.
They are embodied in a physical universe that
is fundamentally uncertain.
Biological processes are physical. Mental
processes are biological.
At the psychological level of complexity,
human mathematics is isomorphic with the mathematical organization of nature.
Mathematical science reveals to us fundamental
forms that are embodied in action, opposition, and spatial structure.
The simplest and most extensive
level is the mathematical because mathematical relations are universally valid, whether or not they are embodied
in physical entities and/or perceived/conceived by intelligent beings.
Two plus two is four regardless of whether
or not there is an observer that can add, and whether or not there are four physical objects.
We need some form of physical representation
to understand an arithmetic operation, but the result does not depend on any particular one.
Mathematical relations necessarily apply even
when no one has taken the precaution to prove them, just as physical reality exists even when no has taken the
precaution to perceive it.
Logically, mathematical form has priority
over all other entities.
Thus physical entities, wherever they exist,
by necessity obey mathematical law.
As there is no mathematics without a physical
embodiment, there is no physical world without mathematical patterning.
Mathematical laws are universal.
But
if mathematics implies certainty, physics demonstrates uncertainty.
The uncertainty principle, by excluding absolute
values, excludes 0 and absolute rest.
Everything is in flux.
Flux permeates space and matter.
Consider
then the possibility that the physical universe arises by the necessary shaping of flux by necessary mathematical
forms. How can we learn which mathematical forms are necessary? Human
mathematics provides a credible indication.
Mathematical science, a product of human thought,
is “unreasonably effective” in describing physical reality, in Wigner’s felicitous phrase. After a very limited
period of mathematical development, we can calculate with surprising accuracy interplanetary travel and atomic
orbitals. Even more, we can learn much mathematics in relatively
few years! Undoubtedly our learning abilities must be connected with
the mathematical form of the fundamental laws of the cosmos.
Mathematical structures
are both archetypes, both physical and mental.
The correspondence
is not surprising, because brain processes
are physical processes, and thereby
pre-adapted to portray reality. Further, brain has developed through evolutionary processes of adaptation and selection.
Evolution has
encapsulated the world as neurological order [Vandervert, 1988].
Thus brain processes,
from perceptions to mathematical constructions, provide us with a reasonably appropriate, albeit certainly not
perfect, picture of the real world.
We do not perceive
space as tridimensional because the labyrinth has three orthogonal semicircular canals, but our ear is so constructed
because space has three dimensions. We perceive time as flowing because processes are irreversible changes of energy.
The
human mind embodies the same fundamental processes as physical nature.
Mathematical science must thus be considered
as natural science.
Pure mathematical creations that do not attempt
to model natural processes often provide deep insights into nature.
Figure
39: The lattice
structure of mathematical foundations.
What
are the cosmic forms revealed by mathematical science?
The
historical and logical foundations of mathematics are arithmetic and geometry that abstract fundamental forms of
nature.
The
small integers abstract fundamental forms that are embodied at all levels of organization.
Flux
has zero organization, action is unidirectional, information is an opposition of two values, three appears in space,
color, Sarkovskii’s theorem, and many conceptual categories.
These
simple forms appear to be universal, and sufficient to generate higher dimensional organization.
Oneness
and twoness appear in many forms: line and circle, one dimensional and two dimensional infinitation, lattice order
and group inverse, continuity and bifurcation, one choice and two values of information, energy with two polarities.
One
of the oldest known numerical archetypes is ? that portrays the relation between linear and circular order.
In
the process equation, the sinusoidal form formed by linear action and circular opposition generates three-dimensional
patterns.
Bios,
as natural creative processes, differ from standard attractors (including chaotic ones) in that they expand their
phase space volume.
Notwithstanding,
bios is an attractor in the sense of a form generated by boundless recursion of a simple generator.
In
our times, mathematical forms are described at a more abstract level by lattice, group and topological theories,
which are regarded as the pillars of mathematics by leading mathematicians such as Bourbaki,
and MacLane [1986], and
by leading psychologists such as Piaget.
(most
people who have studied Piaget, however, seem to have missed the point).
Asymmetric
and transitive order defines lattices.
Inverse
(opposition) and symmetry define group.
Continuous
and discontinuous transformation is the subject matter of topology.
At
an even more abstract level, lattices, groups and topologies are sets that can in principle be constructed from
the empty set, and the forms abstracted by lattice, group and topological theories are investigated by category
theory.
I
regard these disciplines as organized into a lattice (figure 39).
These
abstract forms are archetypes embodied in physical processes.
Action, the conjoint
flow of energy and time,
is unidirectional and causative; action thus embodies
asymmetry and transitivity, the defining features of mathematical order (lattice theory).
Relations
embody group symmetry; they start with, and always include opposition,
i.e. the inverse that defines groups.
Physical space, within
matter is a deformation of space made out of the same stuff as the spreading radiation, embodies topological space. The
formless flux that permeates “empty” space is the physical counterpart of the empty set.
Physical
flux and the empty set are opposite and coincident.
This
indicates the need to develop abstract set theory in process terms, abandoning the static assumptions that lead
to paradoxes.
(At this time, the
theory of static sets has by far being developed more than the mathematics of evolving sets.)
It
is tempting to consider the similarity between the patterns generated by bipolar feedback and fundamental physical
processes. Given bipolar feedback, any initial value (including 0
if the equation is generated with the cosine rather than the sine function) initiates a chain of development, just
as physical reality may emerge from quantum flux.
The series converges to a steady state, such
as observed with inertial action.
Further interactions split the series into
new opposites, reminiscent of processes of differentiation that generate information (two values = 1 unit of information). Repeated
bifurcations lead to chaos, which is similar to turbulence.
Further interaction generates and biotic patterns,
similar to those of living processes. As living processes, these patterns are time limited (complexes) and indeed
the entire biotic phase terminates (infinitation) just as life always ends in death. New biotic patterns follow,
just as life renews itself.
Notably, also new pairs of opposites are generated
by the sequence.
These simple patterns repeat at all levels
of organization.
The
universe is a creative development, generated by interactions governed by necessary mathematical relations (arithmetical,
algebraic, etc) materialized by quantum flux.
Lattice order, group symmetry and topological
continuity constitute bipolar feedback, which is sufficient to generate periodicity, chaos, bios, infinitation,
and new sets of opposites.
This model creates a “toy universe”. Additional
creative processes are of course involved in the generation of real processes. Even strings, the simplest physical
entities, are more complex than the patterns generated by simple feedback equations.
The notion that the physical universe arises by the necessary
shaping of flux by mathematical forms was already formulated in antiquity. It has been formulated by Slutzky after
discovering the generation of probabilistic cycles by weighted averaging of uncorrelated white noise. Tryon
suggested that the universe may have arise from a random quantum fluctuation.
My specific formulation of this view is that
physical reality arises from the materialization of lattice order, group opposition, and topological (continuously
transforming) space by quantum flux.
Whether or not the universe
is creative is not open to question --there is sufficient empirical evidence for novelty and complexity. The
question is whether we offer scientific or metaphorical accounts of creation.
Bipolar feedback is only one example of creative
processes. But shifting our focus from chaos and catastrophe to bios
serves to transfer our attention from unpredictability to creativity; from turbulence to life, from emergence to
generation, from chance to development; and from uncertainty to a dialectics of mathematical certainty and physical
uncertainty
Psychological implications
Having indulged in cosmological
speculation let me return to the human level to consider the human implications of these views.
Bipolar and mutual feedback is a concept that
intuitively emerges from consideration of natural human systems such as families.
It also invites application to complex environmental
and human systems, because it may have both explanatory and heuristic power.
Computer experiments with the
process equation show that the complexity of patterns generated by bipolar feedback increases with the intensity,
diversity, and symmetry of the bipolar feedback [Sabelli and Kauffman, 1999]. This suggests guidelines for promoting
creativity in social and personal development, not available from equilibrium, random, or chaotic models. Marriage,
for instance, may be expected to be most creative when the two spouses have similar power and intelligence, invest
intensely, and are capable of diverse, modulated responses, rather than extreme positive or negative responses.
Describing reality as creative
by both synergy and conflict evokes very different feelings and behavior than one-sided models stressing systemic
integration or biological struggle.
Bipolar feedback is a fundamental component
of learning, ranging from simple conditioning to the testing, proof and refutation in the development of scientific
hypotheses. Learning requires sustenance and correction, encouragement
and criticism.
Intellectual progress is fostered by the bipolar
feedback of support and opposition.
The best ideas, when held dogmatically, rapidly
decay into superstition, partisan creed, or cultural chauvinism.
On the other hand positive feedback is also
important, as nutriment is required for the flourishing of ideas, and of their creators.
Clinically, partial contradiction, rewarding
some aspects of the patient’s behavior and discouraging others, is more psychotherapeutic than either unconditional
emotional support (Carl Rogers) or critical psychoanalysis.
Political action likewise may be most effective
when it combines synergy and opposition.
Political systems may be most adaptive and
progressive when including bipolar and diverse feedback.
Focusing on creative processes
gives us tools to act.
Co-creation is readily applicable to personal
life as tool and as goal.
Hopefully it may also promote healthier social
behavior. To describe social and economic processes as creative,
rather than determined and tending to equilibrium, promotes progressive action.
Regarding socioeconomic processes as creative
rather than chaotic inspires healthier social behavior.
Also, the notion of ongoing creation implies
that each of us participates in his own creation and in the creation of our world.
This provides meaning to life, which it does
not spontaneously emerge from the notion of a determined universe, or one governed by chance.
In summary:
1. Bipolar feedback generates patterns of increasing complexity, from convergence to periodicity
to chaos to bios.
2. Bios is a new type of organization that can be empirically demonstrated by measures of novelty,
diversification and non-random complexity.
3. Biotic patterns are present in creative natural processes (biological, economic, meteorological).
4. Bipolar feedback may account for creative evolution in physical, biological, and human processes.
At a more speculative level,
mathematical certainty and physical uncertainty may be the creative pair of opposites that underlies physical processes.
Acknowledgements:
This work was supported
by the Society for the Advancement of Clinical Philosophy. It is a pleasure to thank A.
Abouzeid, L. Carlson-Sabelli, L. Kauffman, Jerry Konecki, M. Patel, A. Sugerman, and J. Sween for ongoing discussions
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