Dynamics and the Navigation of Time and Space in Cognition and Social Change

[rough draft soliciting comments]


Frederick David Abraham


Dynamics (aka “dynamical systems theory”, popularly but improperly “chaos theory”, and “nonlinear dynamics”) is a mathematical discipline based on a vector calculus that describes the behavior of a system of interdependent variables over time using usually nonlinear differential equations. It was first applied to the momentum and position of objects in the study of the motion on earth and in the heavens. It has liberated all of science to deal with complex open systems (versus isolated features as closed systems), along with the computers that help us represent this behavior geometrically. It is one of a number of related approaches that attempts to integrate holistic with analytic ontological views. It empowers an interdisciplinary point of view and the interconnectedness of things.


Its most basic concepts are (1) the dynamical system whose geometric representation is the portrait of temporal trajectories into a view of the interaction of the variables in a state space under the influence of a vectorfield which in turn is represented by differential equations. The patterns displayed by the trajectories are called attractors. The portrait also displays other features, such as basins, separatrices, saddles and repellors. (2) The dynamical scheme, which are changes in the patterns generated by the system as a function of parameters in the equations; the geometric representation we call the response diagram. (3) Bifurcations, which are major changes in the portrait when critical values of control parameters are crossed. That is major changes rather than gradual ones. (4) Self-organization, which is when the system itself changes those control parameters and thus dramatically alters its own behavior.


We each have described these features (Abraham, Abraham, & Shaw, 1990), and in abbreviated form speculated about them in the analysis of time in cognition (fda) and history (rha) in Macey, Encyclopedia of Time (1994). Here we attempt an update of those views, emphasizing the difficulty of such modelling, and emphasizing the role of the psychophysics of the metrics to allow an emphasis on vectors of change in social systems. Briefly, we suggested a dual concept of time in cognition, where the mind navigated more rapidly in the portrait, and more slowly in decision-making in the parameter space (fda), and that theories of histories could be found that had properties of (a) point, cyclic, or chaotic attractors, (b) bifurcations, and (c) fractal properties—the replication of space-time behavior over different scales of observation (rha).


The problem is obvious. How does a mathematic that deals with continuous Cartesian variables and symmetrical space-time representation deal with the world of nonmetric (or a different metric, Stevens; Combs) and symmetry-breaking found in the more complex worlds of the mind and cultural change? We do not answer this question, but it is our concern. A recent debate on the metric of time is currently raging on CHAOPSYC, the listserver of the Society for Chaos Theory in Psychology and the Life Sciences. My friend, Gus Koehler offers one classification of time:


I think the parameter space is more complex as is the state space.  First, many time theorists see at least five different levels of time, each having its characteristic associated processes.  For example, the time of particles and fields, particularly in relativistic settings, is very different from biological time from organism to organism (internal clocks, diffusion processes, morphogenesis, etc.), is very different from phenomenological or socially constructed time (variation from culture to culture).  In my opinion, organizational time--that is the construction of social institutions and processes-is very dynamic and driven by at least three different temporal states: agent (biological involving aging, the formation of the social body, etc.), clock (mechanical), and nonotemporal time (or prositioning relative to the past, present and future often a simultanity). Each has its own characteristic control parameters, each its own state space.  The interesting issue is how these systems nest together in the process of organizational formation. (Koehler, CHAOSPSYC communication on the “orderings” thread, 1/15/2002).


Another friend, Jeff Pressing replies:


I don't really think that these different types of time have a lot of theoretical foundation.  One time is enough, as time is an emergent property of events.  Rather, model the events, in my view. (Pressing, reply to Koehler on the CHAOPSYC “orderings” thread, 1/15/2002).


It may (or may not) be that the various “levels of time” mentioned by Koehler could be given a functional relationship to the “one time” of Pressing, e.g., by psychophysical experiments. Both of their statements suggest the difficulty of establishing those relationships, or of even establishing the one “emergent” time.


Our suggestion of a dual as well as infinite cognitive temporal representations is also evident in the theoretical approaches of Michon (1994), with his distinction of implicit (non consciousness) and explicit (conscious) representations. Our navigation of parameter space versus trajectories in the state space might map into this distinction, but that could be an oversimplification. Michon states that the implicit does not require “symbolic representation of the temporal parameters of the situation. But formal representation would:


. . . the temporal structure of the situation may be mapped onto or into formal representations, that is, logical, numerical, or mathematical structures. In such cases we speak of models, or even theories, of time: temporal logics and the motion equations of Newtonian or relativistic mechanics are examples. J. T. Fraser’s hierarchy of temporalities for instance, may mapped a series of psychophysical measurement scales. J. I have shown that these scales can be applied to the world as seen from several perspectives: atemporality, protemporality, ecotemporality, biotemporality, and nootemporality do correspond to nominal, ordinal, interval, ratio, and absolute measurement, respectively. (Michon, 1994, p. 512.)


As is now well known, the development of relativity, quantum electrodynamics, and cognition has seen not only a considerable interaction of these fields, but also the emergence of a new ontology in which the dynamical interactions of variables at all levels and places of observation in the universe have become better appreciated. A complexity of mathematics is required to represent the dynamics of these processes, most of which were beyond understanding via ordinary experience. This ontology led to a view of infinite potentials, reduced to one in the singularity of the present moment, which in turn becomes a set of infinite potentials for the future. This ontology also led to the view in which the fractal boundary of the observer/observed, objective/subjective practically dissolves.


We might argue that the same might be said for the implicit as well as the explicit, but the “models” also remain implicit! Michon also reflects the infinite potential idea in a discussion that also conveys our sense of cognitive navigation in nonlinear dynamical representations of time:


Evidently, a keen sense of time, that is, a detailed repertoire of representations of time, has survival value. It allows its owner to review past experience and to anticipate future events effectively and efficiently. It expands the temporal horizon of the organism, allowing it to attune itself in intricate ways to the requirements of its dynamic environment. Given this biological significance of representing time, it is not surprising that humans possess a rich repertoire of temporal representations. (ibid, p. 511).


Probably the most sophisticated nonlinear mathematical modelling in psychophysics is that of Robert Gregson. He also resolves the Koehler-Pressing distinction and well articulates the hierarchical and network approaches to modelling:


The idea of time features in experimental psychology in different senses, as a central variable in processes, as merely a counter-variable for tracking processes, and as a phenomenological experience which is the intended psychophysical focus of experimental studies. . . (Gregson, 1995, p.73.)


One of the ideas to which theoreticians in mathematical psychology come back again and again, in one form or another is that of hierarchical dominance of subprocesses by a higher-level process. By hierarchy is meant that there are at least two levels, one of elementary processes and another which has its own dynamics and acts as a controller, switching the elementary processes in and out and selecting between them. Such selection can arise externally as a result of feedback from the consequences of previous system outputs. Note that this is not the argument about the minimum number of levels in a distributed network which are necessary and sufficient for learning, but to something evden more abstract. (Ibid, p. 74.)


So we are left with the difficulties of metrics for the dynamics of cultural bifurcations. We consider this could be important if we are to understand how to help create conditions in society that allow them to evolve new potentialities. There seem to be two varieties of exploration, roughly equivalent to Michon’s implicit/explicit distinction. We will briefly discuss two representatives, In the first, the emphasis involves the role of time in psychodynamics leading to discontent in individuals that lead them to transcend or struggle against the constraints society places on them. The second emphasizes explicit concepts of time between subcultures come into conflict.


Kristeva (1980) provides an example of the implicit approach. Julia Kristeva has an approach to social theory that is her own unique blend of linguistic and psychoanalytic theory. Her basic psychoanalytic distinction between pre-Oedipal and Oedipal aspects of personality development lies at the foundation of her theorizing. Narcissistic identification and maternal dependency, anarchic component drives, polymorphic erotogenicism, and primary processes characterize the pre-Oedipal. Paternal competition and identification, specific drives, phallic erotogenicism, and secondary processes characterize Oedipal functioning.

Kristeva also characterizes the pre-Oedipal feminine phase by a type of space for which she borrows Plato's term, chora (Timeaus), an enveloping, amorphous, nonmetric space that both nourishes and threatens. It also defines and limits self-identity. She characterizes the Oedipal male phase by a more metric space, for which one could correspondingly use Aristotle's term, topos. The self and the self-to-space are more precise and well defined in topos

Kristeva attributes to each of these spaces, a differing view of time. Pre-Oedipal time is a more phenomenological, subjective time, which Kristeva calls 'monumental' time, and is somewhat akin to Gorgias's Kairos, Bergson's durée, Loye's timeless and spatial times (Loye, 1984). Oedipal time, Kristeva's 'cursive' time, is a linear time akin to Hesiod's Kronos, Bergson's chronological time, and Loye's serial time. These are like our time for trajectory evolution in state space, and like our parameter space, which can be metric but to which the application of the psychological metrics discussed above would most apply.

Kristeva relates linguistics to these basic distinctions. Semiotics is associated with the pre-Oedipal realm. The speaking subject is divided, decentered, and process-oriented. Semiotic process is rooted in feminine libidinal, pre-Oedipal energy which needs channeling for social cohesion. Infantile drives are indeterminate and multifaceted. A phallically perceived mother (male maternity fantasy) dominates a feminine phase.

"If the semiotic is pre-Oedipal, based on primary processes and is maternally oriented, by contrast the symbolic is an Oedipalized system, regulated by the secondary processes and the Law of the Father. The symbolic is the domain of positions and proposition. The symbolic is an order superimposed on the semiotic. The symbolic control of the various semiotic processes is, however, tenuous and libel to break down or lapse at certain historically, linguistically and psychically significant moments. It results in an upheaval in the norms of the smooth understandable text. The semiotic overflows its boundaries in those privileged 'moments' Kristeva specifies in her triad of subversive forces: madness, holiness and poetry." (Surap, 1993.) 

It would be interesting to try to make her approach more detailed with the psychophysics of time and space, and with the nonlinear modelling of dynamical systems theory. We just pose this task, which is left for later development. Kristeva invoked the relevance of Thom’s catastrophe theory, which is a species of bifurcation theory, and largely responsible for the development of dynamical systems theory as a superset of itself. With that challenge, we now turn attention to an example where cultural concepts of time themselves came into conflict that led to social change.


Atkins “explores changes in the perception of time — the shift from peasant to industrial time — as they were experienced by these northern Nguni speakers on coming into contact with a society undergoing early stages of capitalist growth. . .” and how they influenced “Black/white colonial labor relations.” (Atkins, 1994, pp. 122-3. The Nguni tribes, Bantu speaking, principally Zulu, were mainly in Natal, now within the northern province of KwaZulu of South Africa. Atkins is exploring 19th century Natalan culture).


Time was at the nexus of the “Kafir labor problem.” No sooner was a work agreement mde than confusion arose from the disparate notions of the white employer and his African employee regarding the computation of time. Otherwise said, the record of persistent desertions from service was in many instances related to tehe fact that the terms of master-servant contracts, which were based on Eurpean units of measure, did not accord with the African mode of temporal reckoning. (Ibid, p. 123.)


The Zulu used the moon and stars, the sounds of different animals and birds, the weather, the seasons, and so on, to keep track of time. Fear of spirits in the forest and fields tended to keep people inside at night. Habitually the Zulu reckoned a day as from an hour after dawn to an hour before dusk. The appearance of isiCelankobe (evening start) signaled evening meal time. During the winter sugarcane manufacturing process (June-September), the Kafir day thus averaged a short 10.5 hour day. The processing required night work night work, since crushing had to be done immediately after harvesting to maintain the quality of the sugar.


Their inyanga, the word for moon and moon period, was lunar based and there were 13 inyanga a year, which led to confusion with the European month. Inyanga file (the moon is dead, signified the end of a month for the Kafir (apparently another term for the Nguni workers.) They expected to be paid then, and would not work longer.


The complications arising from the two systems of time notations were enormous, as this agitated correspondence from “C.P.,” dated (and this is the pivotal clue) 29 October 1846, attests. “This afternoon, because I would not pay a kafir whose month is up on the last day of the month, I was abused like a thief. He shook his stick at me, and was so violent that if I could have got assistance, I would have sent him to the trunk (goal)” [emphasis added by Atkins, ibid., p. 124].


There was also a mail strike by ten men carring mail between Durban and Maritzburg hired to work six months (July 2nd to Jan 2nd) but “arrived at the conclusion that their engagement expired on the 28th of December.” (Ibid., p 124.)


What stands forth most clearly is that resorting to summary punishments (including such draconian measures as floggings and extended stretches in the goal) to discipline preindustrial workers around the question of time had the effect of driving labor from the market. (Ibid., p. 125.)


The problems were exacerbated by the contempt of the employers toward Zulu culture, and their stubbornness to adapt to the indigenous cognition of time or to their language and by their attempts to explain European time was considered as “an attempt to cheat them of their time” by the Kafir laborers.


We can see from this example from Colonial Africa, that Kristeva’s observations that language and the perception of reality are inextricably tied to psychological and social life and their evolution. We conclude that the perception and use of time is a dynamical process involved in the interaction of conscious and unconscious psychological social processes, but we end this all-to-brief introduction to the problem by reminding that the various binaries we have invoked, implicit/explicit, variable/parameter, semiotic/symbolic, preOedipal/Oedipal, mother/father, and others are convenient heuristics, that the fractal separatrices between them belie that dynamics patterns (attractors) constitute a reality that defies adequate description. Our modelling can only be an approximation to holistic open systems at all levels of discourse. Developing the psychological and social temporal metrics necessary for dynamical modelling will be a great challenge, especially as different metrics and vastly different time scales will have to be involved in the integrative models.




[EoT is S.L. Macy (Ed.), Encylopedia of Time. New York: Garland.]


Abraham, F.D. (1994a). Chaos and Dynamical Navigation of the Cognitive Map. EoT., pp. 89-91.


Abraham, F. D. (1994b). Chaos, bifurcations, & self-organization: Extensions of neurological positivism and ecological psychology. Psychoscience, 1, 85-118.
Abraham, F.D., Abraham, R.H., & Shaw, C.D. (1990) A Visual Introduction to Dynamical Systems Theory for Psychology. Santa Cruz: Aerial.


Abraham, R.H. (1994). Dynamics. EoT., pp. 171-173.

Abraham, R. H. (1994). Chaos, Gaia, Eros. San Francisco: Harper.

Atkins, K.E. (1994) “Kafir Time”: Preindustrial Temporal Concepts and Labor Discipline in Nineteenth-Century Colonial Natal. In J. Adjaye (Ed.), Time in the Black Experience. Westport: Greenwood.

Gregson, R.A.M. (1994). Cascades and Fields in Perceptual Psychophysics. Singapore: World Scientific.


Koehler, G. (2002). Comments in the “orderings” thread, on CHAOPSYC, the listserver of the Society for Chaos Theory in Psychology and the Life Sciences, January, 15. URL: www.societyforchaostheory.org

Kristeva, J. (1980). Desire in Language. (L. S. Roudiez, ed.; T. Gora, A. Jardine, & L. S. Roudiez, trans.) New York: Columbia. A collection of ten papers spanning from 1966 to 1976, with an excellent introduction by Roudiez, including a guide and discussion of terminology.

Kristeva, J. (1984). Revolution in Poetic Language. M. Waller (trans.). New York: Columbia.

Loye, D. (1984). The Sphinx and the Rainbow. New York: Bantam.

Michon, J.A., (1994). [all in EoT.]
Psychological Present. Pp. 504-505.
Psychology of Time. Pp. 507-511.
Psychology: Representations of Time. Pp. 511-512.
Psychopathology and Time. Pp. 513-514.

Pressing, J. (2002). Ibid. as in Koehler above.

Sarup, M. (1993). An Introductory Guide to Post-structuralism and Postmodernism (2nd ed.). Athens: Georgia