Hypothesis 1: Everything is asymmetric in time, quality, and space, and hence everything changes, and is a source of change. Symbolically:

(i) for all x, x_t < x_t+i (temporal asymmetry)

(ii) x_i,t < x_j,t and x_i,t > x_j,t if and only if i = j and t=t' (dynamic antisymmetry)

(iii) if x < y and y < z, then x < z (transitivity).

          Nothing is symmetric, nor it remains identical to itself. An entity x is identical to itself only when examined from the one perspective (i=j) and at one instant of time (t=t'). Time is a universal asymmetry. The universe has the properties of a partially ordered set (lattice theory). The simplest process is action (energy x time), as illustrated by the dimensions of the Planck constant.

          Based on the asymmetry of biomolecules, Pasteur proposed the existence of a fundamental asymmetry in all forms of matter [18]. Current research has confirmed his hypothesis at all levels of integration [12, 24]: the non-conservation of parity in beta decay; the optical rotation of atoms; non-equilibrium in thermodynamics; the asymmetric preponderance of matter over anti-matter; the time-asymmetric collapse of the wave function in quantum mechanics; the asymmetry of crystals; the violation of gauge symmetry by superfluids; the lack of time symmetry in magnets; ionic asymmetry across plasma membranes; anatomical asymmetries, such cerebral lateralization; social asymmetries of class, sex, race, nationality; and the obvious asymmetry of time and of causation.  Pasteur's cosmic asymmetry may represent the fundamental order that Einstein postulated to exist, underlying the probabilistic structure of statistical and quantum mechanics. This fundamental determinism, however, does not exclude creativity and uncertainty, as complex deterministic systems can create chaos.  Quantum phenomena may be deterministic chaos. We propose that unidimensional asymmetry is the fundamental form of energy, as reflected by time and cause. The asymmetry of energy becomes embodied in the information it carries, and the structures it forms. Information, implies a difference between opposites. In this sense, the asymmetry is truly cosmic (which means ordering as well as universal), as it may explain the evolution from simple to complex. Asymmetry is the dynamic oneness of the universe, the shared asymmetry of all processes reflected by the universal asymmetries of time and energy. Asymmetry is a union of unequal opposites, the coexistence of two different, opposite poles in a unity.

                              Union of opposites

     Hypothesis 2:  Opposition is universal. For all x, there is a P(x_i,t) if and only if there is a x_j,t' such that no-P(x_j,t'); and P(x_i) - no-P(x_j) tends to 0 as t tends to infinity. 

          If a property P applies to an entity x at times t with respect to i, then there is another time t', or another perspective j, in which P is not true regarding x. Every system contains opposites [positive (P) and negative (no-P) forces]. Every type of interaction coexists with its opposite: harmony and conflict, synergy and antagonism, attraction and repulsion, union and separation. Fundamental entities such as subatomic particles coexist with anti-particles; and social roles coexist with complementary others. (Obviously, chairs do not coexist with anti-chairs, but are complementary opposites to the seated human form.)                   

          The existence of an inverse for every member of a set defines group theory.  The universe is a group in which every entity has an inverse. Whereas group inverses are symmetric, actual oppositions are always asymmetric; one of the opposites predominates over the other at particular moments of time, at particular points of space, and in particular respects. Matter and anti-matter exemplify the asymmetry of opposition, as for every type of "elementary" particle there is an anti-particle, but the former is much more abundant. The ambiguous figures presented to us by Gestalt psychologists, make us aware of the fact that, under normal circumstances, one form carries the information, while the other represents the background, because it is not differentiated from the more or less uniform environment.

          Oppositions are embodied in energy and in matter, and they communicate information.  Information is the "news of a difference" [1]; information thus implies the coexistence and the difference (asymmetry) of opposites. In catastrophes, the difference of opposites functions as an asymmetric factor.  Opposition is based on asymmetry.  There always is a quantity or lineal order upon which the duality of opposition is based; for instance, attraction and repulsion are constructed upon a continuum of distance. This is a particular case of Hegel-Engels law relating quantity and quality [24].  Hence, instead of opposite vectors departing from a common origin, both opposites grow in the same direction, from negative to zero to positive; for instance, time runs from the remote to the immediate past, to the present (0 origin) to the future.

     Interacting forces can be either synergic or antagonistic. In reality, most interactions have both synergic and antagonistic components. We thus represent opposites as two axes creating a phase plane. In the case of complementary opposites, these axes are mutually orthogonal. This phase plane of opposites (see figures in [6]) provides a method to study how opposites coexist --the union of opposites of process philosophy and dialectic logic. Every point in the plane (or space) determined by the orthogonal coordinates represents the intensities of each of the two or more complementary and opposite forces coexisting at a given moment. This is in contrast to traditional and mathematical logic that, interpreting reality in an extensional or substantialist manner, view opposites as mutually exclusive systems separated by a sharp and impermeable boundary, such as is illustrated in a Venn diagram. It also differs from the continuous model that assumes opposites to be inversely related. The phase plane of opposites provides more complete information than dichotomous questionnaires or linear scales in the study of choice [4], and presumably also in psychological testing and opinion polls. Processes often include three or more opposite alternatives (trifurcations) such as harmony, conflict or separation in interpersonal relations, in which case we use a tridimensional phase space [24, 9, 33].

     The flow of energy, whether mechanical, electrical, social, or psychological, creates resistance and opposition. As a result, opposites become more symmetric. If the opposing forces are relatively weak, they may neutralize each other. If they are both intense, they co-create qualitative change --bifurcations and the formation of structures and systems.  Whereas information is lost in transmission [36], it is increased by bifurcation and by system formation.  In parallel with the increase in entropy, processes tend to increase the rate of information flow at focal points [3]; thereby processes evolve in time, and are locally controlled by their most complex structure (supremacy of the complex). Evolution from simple to complex (for which there is overwhelming empirical evidence in physics, biology, history and psychology) co-exists with the maximization of entropy (second law of thermodynamics). This coexistence of opposite processes represents a thermodynamic enantiodromia. PT expands the second law of thermodynamics, postulating a more general hypothesis: processes tend to symmetry --including not only the point attractor of entropy, disorder and rest, but also complex attractors, chemical structures, biological organisms, etc [24, 25, 26, 34]. The asymmetric flow of energy creates symmetric material structures. This asymmetric flow from prior and simpler asymmetry to greater symmetry and complexity spontaneously creates structures that catalytically direct further evolution (supremacy of the complex).

Co-creation of structures via the interaction of opposites

          Whereas mechanics (classic, statistical, relativistic, or quantic) postulate  stationary and reversible processes, thermodynamics and dynamics postulate that processes asymmetrically tend to attractors. But why the pendulum of the clock is kept inside a box?  Because only closed, isolated processes maintain their attractors. Open processes interact, and thereby change their attractors (bifurcation). Although attractors are defined as forms of stability towards which processes flow when left undisturbed, a wide variety of mathematical and physical systems become more, rather than less, organized after being perturbed; for instance, low-frequency oscillations may control the multiple components of the system. To describe such phenomena, Haken [17] has introduced the concept of synergetics.  Most significantly, increasing energy facilitates order; for instance, when the energy of a lasser is increased beyond a critical point, the apparent atomic randomness changes into sinusoidal periodicity [20]. We define as co-creations those interactions that generate bifurcations; within the context of an interaction, or a sequence of them, the interacting processes are complementary opposites. The simplest interactions are additive, creating linear sequences with positive and negative numerical values.  For instance, information regarding the direction of change is provided by the algebraic sum of opposites.  Additive combinations may also create spirals (e.g. the Fibonacci series). Next, interactions may be multiplicative, amplifying; in this manner, opposites enhance each other, and hence their combination is always positive.  For instance, opposite processes are synergic in providing energy to a system.  Synergic and antagonistic relations between opposites define the asymmetric and bifurcating parameters in catastrophes (see above). Combinations of additive and multiplicative interactions may generate non-linear functions, which often are folded rather than monotonic, i.e. a single value of one variable may be associated with two values of the other (ambiguity or contradiction).  In this manner, bifurcations occur. 

          Processes are cascades of bifurcations. Far from remaining in stable patterns (attractors), processes change (bifurcations) and create new emergent attractors [38] and dissipative structures [23] as result of multiple interactions with other processes. Through a similar process, physical structures (including matter itself) may form and evolve. As structures, and matter itself, are stable forms generated by equi-librium, (the equality of two opposing forces), the asymmetric factor in the process of creation may indeed correspond to the difference between opposites. The bifurcating factor may be the flow of energy and information, as the creation of complexity appears to be associated with increasing free energy flow density: 1 erg gm^-1 s^-1 for the galaxy as a totality; 2 for the sun; 500 for plants; 17,000 for animal bodies; 150,000 for human brain [10]. 

          Poincaré's phase space allows the study of complex processes by examining the form of their trajectory. The choice of axes is a critical and yet unsolved problem, but process philosophy suggest the axes should be opposites. This phase space of opposites (see figure N [6]) provides a mathematical formulation to study how coexisting opposites co-create structures. We currently use the phase space of opposites in our empirical studies of mood [6; 28] and interpersonal relations [5, 24]. The interaction of opposite motivations (energy-information) creates choices (material interactions between persons). The choice or rejection of a particular person does not depend exclusively on the corresponding attraction and repulsion except in extreme cases; in all others, the outcome depends on the entire system rather than on the interaction of opposite motivations at each point. In the case of interpersonal choices, the interaction between opposite motivations seldom is additive or linear, but more often creates a distribution which can be described by one or another of Thom's elementary catastrophes [7, 8]. Thus the interaction of opposites in a plane generates an asymmetry (opposition) in a third dimension, creating a structure in space. More generally, the non-linear interaction of forces at one level of N-dimensions creates a new opposition at the Nth+1 dimension.

     Catastrophes are the simplest forms of bifurcations, i.e. the transitions from one point attractor to another (e.g.choice to rejection). The simplest catastrophes are governed by only two control variables: (a) asymmetric, which at mid values is associated with large changes between the opposite modes, while at extreme values is associated with small changes around the modes; and (b) bifurcating, which at low values is at low values is associated with a continuous outcome, and at high values, with a discontinuous outcome. In the case of interpersonal choices, we found a and b to be functions of attraction and repulsion (P and N). Hence: b = f ( P + N); and a = g (P - N) [7]. PT postulates that b represents the synergy of opposites (energy and shared information) while a represents the antagonism between opposites, that provides information regarding the direction of the outcome. Catastrophes are hence functions of the difference and the sum of the interacting opposites, and represent the simplest case of how the interaction of opposites creates structure. We speculate that these concepts apply not only to catastrophes but also to more complex bifurcations, including the formation of structures. The asymmetric and the bifurcating factors governing all creative processes would then be functions of the sum and differences between interacting opposites. The generation of bifurcations, and the formation of structure would be enhanced by the flow of energy and information, and decreased by the difference between opposite forces. 

          We further speculate that the model might also apply to physical processes, as the distribution of matter and void is determined by processes of attraction and repulsion that concentrate and separate energy-carrying particles. The distribution of matter in physical space reflects the trajectory of energy-carrying particles in phase space.

          Hypothesis 3: Structures of matter are formed when opposite forces in a system are both of high and similar intensity.  As processes tend to symmetry, they form patterns and structures.  At the most fundamental level, energy forms matter, which is an equi-librium or symmetry between opposing energies.

     Inspired by living systems theory [21], process theory views energy, information and matter as three inseparable aspects (forms) of each system:

                                                 E = M x I   

E is physical energy, but social and psychological energy are forms of physical energy. M stands for pattern and structure, as matter is the fundamental structure of the macroscopic world. Information is provided by the difference [asymmetry] between oppositely directed forces of attraction and repulsion. The quantity of information that can communicated is limited by the frequency of the energy carrier. Hence I < c², as the velocity of the light c is the maximal attainable. Einstein's famous formula E=mc² represents the extreme case in which the rate of communication of information has reached its maximum, and hence equality, in both directions; energy can then convert into mass (m). More generally, the formation of structure is fostered by the E of the system, and by the equality of the mutually opposing forces (I = 0). 

          Whereas Einstein imagined a stable universe in which time was a fourth space-like dimension, current cosmologies postulate an evolutionary universe in stages; near the origin of the universe, only energy would have existed, matter being formed only later.  Shifting from the developmental model of stages to a process view of coexisting processes, suggests an alternative possibility: that E, I, and M co-exist in all phases of the evolution of the universe, and that at every stage, the conversion of energy into mater is greater than the conversion of matter into energy:  [E-->M]  > [M-->E]

          In more general terms, E, I and M are three related aspects of systems of the form M = f ( M/I), manifested in many different physical relations such as Einstein's law E = m c ², where mass m represents matter and c² represents the maximum rate of two way communication; Coulomb's law: V = i x R, where current i carries information and resistance R is a function of the material circuit. It is also manifested in human processes, such as the relation between labor (E), wealth (M) and technology (I) in economic value.

     As E is a function of both matter and information, neither matter nor ideas have absolute primacy over the other. equilibrium may be attained when a low-mass but informationally complex process matches the power of a high-mass simpler process.  PT postulates the priority of the simple and the supremacy of the complex [24]: Simple and abundant processes have priority (e.g. the predetermination of beliefs and feelings by health and economics), while complex but rare processes have local supremacy (e.g. mind controls bodily functions).  This translates as the priority of the biological, the mediation by the social, and the supremacy of the psychological in human processes [29], implying a clinical approach different from Engel's biopsychosocial model [30].  

                  One, two, three as cosmic forms in systems and processes

          Every entity may be considered as a form in a complex space.  A space is a set of qualities, including time and physical distance, represented as dimensions. Thom [39] defines logoi as forms of nature in abstract, mathematical spaces. PT adopts the view that forms are shared by all levels of organization. Because they are forms in phase space, they are forms in physical space; and, for the same reason, they are biological, social, psychological and logical forms. (Philosophically, this represents the union, or identity, of materialism and idealism.) A biological rhythm is also a physical rhythm and a psychological one. In a system, rhythms by necessity are isomorphic at all levels of organization.

          PT postulates that all systems are organized by three universal and asymmetric forms which are homologous to the simple numbers.  As qualities, unity, opposition and triadic space-like structures are homologous to 1,2,3 as quantities. These forms repeat in every respect, and at each level of organization, thereby generating complexity, novelty, and fractal self-similarity. Regarding the composition of processes, the flow of energy is uni-directional; information requires the existence of at least two alternatives; structures of matter are three-dimensional. Unidirectionality, asymmetry, energy, unity, are expressions of oneness as a universal form of processes. Opposition, information, and symmetry are particular forms of duality. Likewise triadicity is manifested in a number of ways: in the three dimensions of physical space, the tridimensional phase space of catastrophes and chaos, the intercourse of two sexes to create a new life, the balance of opposite forces to erect material structures, the coexistence of energetic, informational and material aspects in all systems, etc [24]. Triadicity is characteristic of creative reasoning [13].

     Supporting the hypothesis that these are three cosmic forms, Bourbaki [2] considered lattice, group and topology theories as the three foundations of mathematics. Lattice theory explore the formal properties of asymmetry (<). The existence of an inverse for every member of a set defines group theory, which in this manner explores the multiple forms of opposition. Topology studies the tri- and n-dimensional forms of continuously modified structures, and the bifurcations between forms. PT assumes that simple and universal forms such oneness, opposition and space-like tridimensionality, are topologically nested within the more complex structures of nature and thought.

                                                        Discussion

          The phase space of opposites provides a practical method to study the creation of patterns and structures at all levels of organization, as we illustrate elsewhere [4-9, 24, 33, 34].  The fact that non-linearities occur such as catastrophes and chaos in simpler processes, and ambivalence and contradiction in psychological processes, occur when opposite forces are both intense and nearly equal, suggests a strategy to promote creativity, namely to increase the density of energy flow and the symmetry of the opposites.  For instance, we propose that creative social solutions may be found by the combination of right and left perspectives, rather than by the adoption of one or the other of these two views [5, 32].  Likewise, creative solutions to a marital conflict are often more readily achieved by a joined marital session in the midst of a crisis than by individual therapy, or by dealing with issues of low emotional intensity.

          Pasteur's discovery of cosmic asymmetry exemplifies how one may learn about the simple by studying its expression in complex processes. This "inference from complexity" [24] complements the more common analytic approach that attempts to understand the complex by decomposing it into its simpler parts. In Pasteur's spirit, we have considered what can be learned about the formation of systems through studies on psychological processes.

     PT is a scientific formulation of the process philosophy born as physiology in ancient Greece. Biological roots also underlie modern developments such as Pasteur's cosmic asymmetry, Miller's living systems theory, and Thom's catastrophes. Complex biological processes are taken as models for simpler physical processes, in a reversal of traditional reductionism. Neither approach has the solidity of mathematical proof, but both serve to generate interesting hypotheses to be tested. Progress in science is promoted by the generation of opposing hypotheses, which need not be true to be fruitful. In this philosophical spirit, PT interprets Pasteur's cosmic asymmetry as change, and makes it the basic order of the universe, as contrasted to the probabilistic approach of quantum and statistical mechanics; postulates that the universe tends towards a symmetry of opposites, and hence towards both simplicity and complexity (enantiodromia) as opposed to the statistical mechanical view of entropy; and interprets Einstein's formula as an net flow from energy towards information and matter. Thus processes create novelty, and are not reversible; time flows, and determinism is not absolute. Hence the possibility of human freedom, action and creativity, and of error, evil, and uncertainty. Philosophy --the love for knowledge-- does not aspire to the certainty that some claim for science, and others for religion, but it contributes to human progress by promoting alternative perspectives, and reasserting the Socratic-Cartesian methodological doubt. It is in this spirit that we formulate a process view of physical and human systems. Formulating its hypotheses mathematically allows one to apply and test them empirically.

Acknowledgement:  To María McCormick of the Society for the Advancement of Clinical Philosophy for her support of our work, and to the members of the Society for their discussions and suggestions.

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