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Sabelli
HC.
Process Theory, a General Theory of Natural and Human
Systems .
Proc of
the Internat Soc for the Systems Sciences. 1991:3:168-174.
PROCESS THEORY, A GENERAL THEORY OF NATURAL AND HUMAN SYSTEMS
H.C.Sabelli
Rush University, Chicago, Illinois, USA
Abstract
Process theory integrates dynamics, thermodynamics, evolutionary
biology, sociodynamics, psychodynamics and process philosophy.
Here we present an abstract formulation of the theory as a set of
three postulates, asymmetry, opposition, and bifurcation, which correspond
to the three basic fields of mathematics, and may represent universal
forms of nature and thought.
Key
words:
Process theory, dynamics, thermodynamics, asymmetry, opposition, chaos,
bifurcation.
A process is an
evolving system open to the exchange of energy, information and matter.
Everything is a process, even if it appears to be, in a limited
context, a purely random fluctuation, a closed system, a structure, or an
idea. In recent decades, the science of processes has been enriched by the
development of non-linear dynamics [25;27;28;1] and the thermodynamics of
processes far from equilibrium [13]. In search for a general theory of
natural and human systems, we have interpreted dynamics via the concepts
of the union of opposites (Heraclitus, Lao-Tzu, Hegel, Engels, Freud,
Bohr) and cosmic asymmetry [11]. This
is process theory (PT) [15;17].
PT provides concrete hypotheses in a wide range of fields: thermodynamics
[18;19], social history and action [15;20;22;23], medicine [21],
psychology and psychiatry [17;18;19), experimental methodology [3;4], and
dynamic mathematical logic [14]. PT also suggested the concept of God as
the Attractor of evolution, rather than as its creative origin [15]. Born
within the context of medicine and psychology, PT derives its basic
concepts from biology [16], and has been empirically tested through its
clinical applications. Here we shall formulate PT in terms of three universal forms,
which, in the Pythagorean sense, are both natural and mathematical: the
asymmetric ordering relation of lattice theory, the inverse of group
theory, and bifurcation from topological theory.
Asymmetry,
opposition and hierarchy
The group of French mathematicians who, under the collective name
of Bourbaki, were the cutting edge of mathematical research earlier on
this century, considered lattice theory, group theory and topology as the
pillars of mathematics. This is the mathematical basis for the hypothesis that every
process embodies three fundamental and universal forms: asymmetry,
opposition and bifurcation. For instance, the periodic table of elements
manifests three different and overlapping forms: the linear ordering of
atomic number, the cyclicity of spiral growth, and two bifurcations (the
lanthanides and the transition metals). According to PT, (1) every process
is part of a process of asymmetric flow of energy; (2) every flow of
energy creates opposition which convey information and leads to patterns
of greater symmetry and stability (attractors); and (3) the interaction of
processes concentrates energy flow and information, producing
bifurcations, chaos and structures (from matter to psychological
patterns).
Asymmetry:
As Pasteur postulated, asymmetry seems to be a universal feature of
processes and structures [16]. Whereas traditional algebraic-like models
may depict particle and systems as point-like, modern physics portrays
particles as one-dimensional strings, and mathematical dynamics represents
processes as vectors (asymmetric arrows) creating trajectories in a phase
space [1]. Asymmetry is modeled by the ordering relation < which is
non-reflexive (for no a, a<a), asymmetric (if a<b, then b<a is
not true) and transitive (if a<b and b<c, then a<c).
This simple relation is the foundation for a rich mathematical
field, lattice theory [2], the basis for mathematical logic.
"Becoming" satisfies the postulates of <.
According to PT, all processes are ordered sets. This lattice
structure is apparent in the sequence of atomic elements in the periodic
table, in the tree-like structure of biological evolution and zoological
and botanical classifications, in the connections of neurons in the
central nervous system. Negation in mathematical logic is both two-valued
and asymmetric: 0 < 1. A
is not symmetric with no-A, because at each point in space-time either A
or no-A predominates in a given respect.
Extending this concept of asymmetric negation, Sabelli [14] defined
sublation as a mathematical model for dialectic negation (overcoming).
Even the relation of identity must be asymmetric because becoming
is unidirectional, At is not equal to At+i but
rather At < At+i.
This is a dynamic form of identity in which we recognize the
asymmetry of the precursor and the successor, the cause and the effect,
rather than postulating a two-way reflexivity. A dynamic logic is by
necessity an asymmetric logic.
Opposition:
The term opposition refers to both synergic and antagonistic
interactions. This apparent ambiguity of language reflects the fundamental
fact that synergic interactions and antagonistic ones are closely
associated. This is the
concept of the union of opposites.
For instance, supply and demand are not antagonistic forces that can come
to a balance --as assumed without evidence by economic theory-- but rather
they are opposing forces which synergistically increase the energy of the
system, and thereby create bifurcations and chaos, such outcomes depending
also on the structure of the social hierarchy. As reviewed in a companion
article [16] opposites coexist in all natural processes. Oppositions are
also fundamental in mathematics. A
group is a closed set in which every element a has and inverse a-1
such that a x a-1
= E, the identity element (for all a, E x a = a). Mathematical groups
model cyclic and non-cyclic forms of transformation, symmetries such as in
the organization of subatomic particles, crystals, and the coexistence of
opposites in thought processes [12]. Group theory clearly captures
fundamental properties of processes, namely the existence of specific
pairs of complementary processes. As group theory is a fundamental field
of pure mathematics, the mathematical theory of information pioneered by
Shannon [24] is fundamental in the computer sciences, and potentially in
genetics and psychobiology. Information is an asymmetry or difference
between two opposite alternatives. For information to exists, one of two
opposing values must predominate at any given time and place, and its
opposite must predominate in another point in time-space, or in some other
respect. The temporal
alternation ranges from the small oscillations of quantum uncertainty to
the wave form of electromagnetic waves, material waves, biological
rhythms, etc, but in all cases, it represents the closed curve in the
phase plane. Yet the traditional Western concept of rationality, from
Aristotle to mathematical logic and psychological theory, views opposites
as mutually exclusive classes. Opposites cannot coexist (principle of
no-contradiction), nor can they both be absent (principle of excluded
third). As black and white partitions are obviously wrong, modern
scientists tends to view opposites as polarities, allowing for gradations
such as probabilities or rank orders, but this approach denies the
existence of partitions between classes while still failing to portray the
mutual implication of opposites. Process philosophy recognizes that each
category contains its opposite in a diminished form; this is represented
symbolically by the yin/yang of Taoism, and it actually occurs in fractals
[9]. This coexistence of
opposites is the core of Hegel's dialectic principle of contradiction,
which is compatible with Aristotle's restricted formulation of the
principle of no-contradiction --namely
that opposites cannot jointly predominate at a given time, in a given
place and in the same respect.
PT operationalizes the union of opposites by representing opposites
as mutually orthogonal axes as in Cartesian geometry. This is in contrast
to linear conceptions of opposites as linear (vectors at 1800
angle). The orthogonal representation reflects the fact that opposites are
not only partially antagonistic but also partially synergic. One can thus
decompose each force into these two components, one parallel and one at
1800 angle with its complementary. We label the two opposite
and mutually orthogonal vectors as p, representing a positive scale from 0
to absolute truth, absolute good, or ideal choice, and n representing a
negative vector from 0 to absolute false, absolute bad, or total
rejection. These two vectors determine a square grid in which we can
represent both the positive and negative aspects of any given event or
entity as a point. For instance, buying, choosing a partner or a course of
action, involves consideration of benefits and costs. To profit from the
psychological implications of verticality, we stand our grid on its zero.
In this diamond, the bottom vertex where the two scales originate,
represents the neutral case, neither one value nor its opposite. The top vertex represents the contradictory case, both good
and evil, true and false, wanted and rejected. The plane of opposites may
also be useful as a more accurate way to poll opinion, to vote, or to
measure interpersonal choices as discussed in a companion article [3]. The
plane of opposites implies a logical 2 x 2 table, as we can divide the
plane in 4 quadrants, positive, negative, neutral, and contradictory.
The plane of opposites thus represents the PT alternative to the
Venn diagrams of logic. In contrast to Venn diagrams, it allows to
represent the ever present contradiction of opposites, and it can be used
to study processes of change. By
giving a logical interpretation to the phase plane of opposites, one can
thus derive a dynamic logic which combines the process features of
dialectics with the mathematical rigor of mathematical logic.
The plane of opposites may be used to study the temporal course of
complex processes. The trajectory of the process is followed by connecting the
points representing its states at different times. For instance, we obtain
daily plots of the positive and negative feelings a person has for his
significant other to study the pattern of his emotional processes (psychogeometry)
[4]. In other words, we interpret the plane of opposites as a phase plane.
Poincaré's
phase space portrays a process by plotting its trajectory along two or
more dimensions. PT suggests
to choose opposites as these two dimensions. One can represent harmony and
conflict, the two basic forms of interaction, as the two axes of the phase
plane, allowing to represent their coexistence in various proportions,
ranging in intensity from the absence of both harmony and conflict
(separation = 0 communication) to the coexistence of both intense conflict
and intense cooperation (oppression, the aggressive dominance of one over
the other), and draw the trajectory of interpersonal feelings or social
relations in this phase plane. The outcome of such interactions defines a
third dimension; for instance, fight, flight or surrender are three
mutually exclusive outcomes of interactions between opposing feelings of
anger, fear, hunger, etc. Similarly,
a hierarchical order of choices and rejections is the outcome of positive
and negative feelings, and perceptions of benefit and cost. Hence the
trajectory of a process draws a shape in a tridimensional phase space. The
study of such phase portraits
has been shown to be useful in the analysis of complex processes [1].
Whereas each instantaneous process results from the interaction of
internal and external forces, processes
are characterized by their attractors,,
i.e. the pattern which it spontaneously follows once disturbances caused
by external factors die away. A
process may tend to equilibrium and rest (static
or point attractor) or may tend to cycle (periodic
attractor) or may tend to highly ordered geometric patterns which are
not periodic but have a fractal structure (chaotic
attractor).
Co-creative
evolution: A bifurcation is a change in the system's attractors.
Bifurcations result from interactions between processes, and
produce differentiation, i.e. the creation of information and structure.
Whereas function is reflected in the type of attractor present at a given
time, evolution results from a sequence of bifurcation events. A catastrophe
is a bifurcation in which an attractor suddenly appears or disappears.
Catastrophe theory describes discontinuous changes through a hierarchical
set of topological forms [25;28]. Catastrophes
are governed by a fixed number of parameters which are called control
parameters. In simple
catastrophes, changes between two equilibrium states is a function of two
control parameters, asymmetric
(a), and bifurcating (b). Change
is smooth at low values of b, and discontinuous at high values of b.
Small changes occur around one attractor at low values of a, and
around the other attractor at high values of a.
At middle values of a, changes occur between modes, and are
relatively large. In this manner, the behavior of the process determines a
tridimensional surface which features a fold.
In other words, the pattern of the process is non-planar and the
relation between control parameters and responses are non-linear. PT
offers a concrete interpretation to catastrophes as interactions of
opposites: when the qualitative
change or bifurcation is a simple catastrophe, the intensity of energy is
the bifurcating control parameter and is a function of sum of the opposite
vectors; and the difference between the opposite vectors constitutes the
information of the system, and represents the asymmetry control.
Opposites are antagonistic regarding the direction they impart upon the
outcome. When the two
opposites have equal intensity, that is to say in the symmetric cases
extending as a line from the neutral to the contradictory vertex, there is
no information. Information is an asymmetry or deviation from the central
axis of symmetry. Opposites
are synergic in providing energy to the process, and hence are additive in
the vertical axis representing the bifurcation factor. In this manner, the
interaction of opposites may account for the elementary catastrophes and
contribute to more complex forms of bifurcations, attractors and
structures. Although this
interpretation of catastrophes originates with us [3], it adheres to the
spirit of the theory which was inspired by Heraclitean philosophy [25]. It
includes the concept of inseparability of energy, information and matter
[15;16] and the notion of universal asymmetry. Thus energy is an asymmetry
in the unidirectional axis of synergy between opposites, while information
is an asymmetry in the bidirectional axis in which opposites are
antagonistic. Processes are
tridimensional organizations of matter controlled by, and embodying, two
parameters, energy and information. Matter is thus organized in a
hierarchical manner according to the density of information; the concept
of priority of the simple and supremacy of the complex is discussed in a
companion article [16].
The thermodynamics of bifurcations
According to PT, creative bifurcations, as well as the complex
organizations they generate, result from the concentration of energy
produced by the interaction of intense opposites. This view derives from
the thermodynamics of processes far from equilibrium developed by
Prigogine, as discussed in previous publications [15;18;19]. In brief, we
reformulate the first law of thermodynamics as a definition of energy as
universal asymmetry; we reformulate the second law as postulating the flow
of all processes towards symmetry, including as such not only the
uniformity of entropy, but also the creation of complex attractors and
structures. An equality or symmetry of opposite forces not only creates
equi-librium but may also erect a cathedral or make an enzyme. Only low
energy processes tend to equilibrium (point attractor).
As
the energy of the system increases, processes oscillate (periodic
attractor), become turbulent (chaotic attractors), and in such processes
far from equilibrium, new ordered structures emerge [13]. There is, hence,
a co-existence of opposing processes (enantiodromia)
of evolution and entropy maximization. This co-existence of evolution and
involution implies an (at least partial) conservation
of information. Finally, we propose a third law of thermodynamics
according to which increases in informational complexity result from and
cause an acceleration in the rate of energy flux.
Free energy flux density (ergs s-1 gm -1)
increases with the complexity of processes: 1 for our galaxy, 2 for the
sun, 80 for the earth climasphere, 500 for the biosphere, 17,000 for the
human body and 150,000 for the brain [5].
Local concentrations of free
energy and of informational complexity are associated with each other,
illustrating one of the interrelations between quantity and quality
(Hegel-Engels).
Numbers and other universal forms of processes
Based on his studies of musical instruments, Pythagoras postulated
that simple numbers represented basic patterns or laws of nature.
Since then, a large number of adimensional constants of nature have
been found. There is hence
little doubt that quantities define qualities, an assumption we make
whenever we quantify processes to study them.
Bifurcations typically occur at critical points: Yorke and Tien-Yen
Li [26] discovered that order three implies chaos, and Feigenbaum ([7]
discovered critical numbers at which chaotic behavior is generated.
May [10] discovered similar numerical relations in simple
ecological and epidemiological models.
The physical basis for these numerical relations is not understood.
The dimensionality of a process is one of the determinants of its
complexity. Unidimensional
processes can only have point attractors.
Bidimensional processes can also have periodic attractors.
Chaotic attractors and bifurcations require at least three
dimensions. We thus propose
that the basic forms of nature correspond to the smaller numbers: one
reflected in the oneness and unidirectionality of energy, two embodied in
opposition, and three associated to the tridimensionality of creative
processes.
As an abstract representation of the universal laws of processes,
PT describes three basic forms which reoccur in each of the different
aspects and levels of organization. From
the viewpoint of composition, we speak of energy, information, and matter
--and conversely of equilibrium (no energy), uniformity (no information),
and void (no matter). From the viewpoint of mathematical structures, we speak of
lattice asymmetry, group inverse, and topological bifurcation.
From the viewpoint of dynamics, one recognizes point, periodic and
chaotic attractors. From a structural viewpoint, we speak of cosmic asymmetry
(Pasteur), universal opposition (Heraclitus), and co-creation.
From the viewpoint of forms, we view as fundamental the open and
unidirectional line, the cycle which is a closed curve, and the
bifurcation, or partition of one into two. Chaotic, fractal geometries
arise from multiple bifurcations. Flows toward static attractors are
linear, and their intensity is a function of their distance to
equilibrium, described by equations of exponential growth or decay. The
combination of both exponential growth and cyclicity
create the growing spiral which is both open and closed. Linear, cyclic and spiral forms appear in the three related
mathematical constants of nature: e, the exponential which describes
lineal processes of growth or decay; pi, which
describes the fundamental closed curve, the circle; and phi, the
"divine proportion" which describes spiral growth and that
occurs in a number of mathematical, biological and aesthetic structures
such as Euclid's golden cut, the division of bronchi, and the proportions
of the Parthenon
[6;8].
These three constants are related in the simple and elegant formula
discovered by Euler [8]. Postulating the universality of asymmetric oneness,
opposition (two-valuedness) and tridimensional hierarchy implies that the
simplest and most fundamental level of organization is mathematical.
Psychological processes share on these forms, can construct --or
reconstruct--mathematical structures through reasoning, and give a measure
of subjectivity to all our ideas about the external world.
Physical and human processes thus exist in an intermediate level,
sharing at one extreme the universal and objective forms of mathematics,
and at the other the subjective forms of ideas, feelings, wishes and
beliefs. PT thus combines formalism or objective idealism, physical and
biological materialism, and subjective idealism.
This is not an eclectic mixture of philosophical postures, but
their replacement by a dynamic view centering on the concept of energy.
In its evolution, energy flows from simple physical processes to
the creation of the spirit. By
analogy with the organization of neurobiological processes [16], we
propose that in this process, there is a
priority of mathematical form, a supremacy of psychological or spiritual
creation, and a mediation between the two by physical processes.
The scientific structure of process theory
As the shore line of the Mississippi delta is as complex as
the shore line of the entire gulf of Mexico (fractal self-similarity), so
all processes from simple physical processes to complex psychological
processes, are self-similar in that the same universal forms of oneness or
asymmetry, duality or opposition, and tridimensional organization reoccur
at every level of organization. Thus
the abstract laws of PT may be applicable to, and testable as scientific
hypotheses in all fields, from thermodynamics
to logic, sociology and psychology.
Testability is what makes a hypothesis scientific, and the greater
the range of application for a hypothesis, the greater the opportunity to
refute it. Thus, the
generality of the hypotheses proposed by PT renders them more, rather than
less scientific even regarding the most strict Popperian view.
Of course, it is not possible now, or ever, to prove the three
hypotheses of PT, but they serve as heuristics to develop concrete
hypotheses in a wide diversity in fields of inquiry.
Undoubtedly many will find unscientific, and even distasteful to
say that there is a formal similarity between such different things as
asymmetry, lattice order, physical energy, oneness, and exponential decay;
or between group structure, periodic attractors, information, logical
negation, and opposition. But
such recognition of the existence of universal forms is not alien to
scientific thinking, from Pythagoras and Plato to Galileo and Thom.
Accustomed to observations and experimental testing, and imbued by
philosophical empiricism, most scientists forget that the most rigorous
and fundamental sciences, mathematics and theoretical physics, utilize the
axiomatic-deductive method. PT
may be understood as a set of three postulates:
1. everything is asymmetric, and therefore changes and produces
change; 2.
everything contains opposites, and therefore communicates
information; 3. at high intensities, the interaction of opposites is
nonlinear, and therefore creates novel tridimensional structures. There can be no proof for these postulates, as there can be
no proof for the postulates of geometry or algebra. The validity of an abstract theory resides in its internal
consistency and its ability to be generate models applicable to empirical
problems. PT does suggests a
number of concrete hypothesis in many different fields.
PT also has a heuristic value, as it indicates how to promote
creativity through the interaction of opposites. Creativity is maximal
when the rate of change is high and multiple opposites coexist in complex
equilibrium --such as in a dissipative structure or a chaotic attractor.
Creativity is minimal when the energy is low and both opposites are
avoided --such as in centrism and eclecticism.
Extreme black and white partitions represent catastrophes prone to
sudden switches to the opposing extreme. We thus propose the "golden
middle" of coexisting opposing views --not centrism or
eclecticism-- as the optimum strategy for scientific or political progress
[23]. The plane of opposites may thus be useful to develop strategies for
scientific research, psychological interventions, or social action.
PT, unfortunately, does not provide us with a recipe for
creativity. For every truth there is an opposite and greater truth, but it
requires imagination and knowledge to envision it.
We
thank the support given by Maria McCormick and by the Roger McCormick
foundation to the Society for the Advancement of Clinical Philosophy.
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