Dr. Ron Eglash                                     

Published in Numc Sumus Vol. 1 No. 1 Summer 2000



Chaos, Utopia, and Apocalypse: ideological readings of the nonlinear sciences


It has long been noted that concepts from new scientific paradigms are recruited in the service of specific ideologies.  The family of mathematical systems referred to as "chaos theory" or "nonlinear science" has been extraordinary in the degree to which this recruitment has occurred.  This essay will survey some of these attempts at mathematical ideology, and show how specific features of the mathematical systems are deployed in this discourse. 


Following an introduction to chaos theory, we will examine its use in relation to ideological categories.  In the first case we will look at the romantic humanists, starting with cybernetics and the counter-culture in the 1960s.  The second case will examine postmodernists, primarily from literary theory.  The third will describe the use of chaos in military institutions, where romanticist ideology is also supported.



A Brief Introduction to Chaos

Chaos theory can be divided into three areas of mathematics.  In fractal geometry, whose now-familiar graphs of wild tangles appear on everything from rock videos to academic journals, simple figures or numeric relations are allowed to accumulate increasing complexity through recursive iterations.  The resulting figures can be quantitatively distinguished by an analytically derived measure, the fractal dimension.  Hence fractals offer a revolutionary method for the analysis of natural features of the world, and a striking advance in the synthesis (either virtual or material) of natural systems.


In the second area, dynamical systems theory, simple ("low-dimensional") difference equations or differential equations are shown to be capable of either short-term uncertainty in the relation of initial conditions to final conditions (fractal basin boundaries), or long-term (in fact infinite) aperiodic fluctuations (chaotic attractors).  In both cases of this deterministic chaos there is extreme sensitivity to initial conditions, popularly known as the "butterfly effect."  Thus the practical applications include the possibility of simple deterministic equations to describe what were thought to be the complexity of random noise (e.g. models of neural signals), as well as a new appreciation for the complexity hidden in simple system (e.g. the non-periodic behavior of a healthy heartbeat).


The third area consists of aggregate self-organizing systems.  Here a large number of homogenous individual units exhibit   coherent collective behavior, most importantly spatial versions of deterministic chaos.  While the addition of random ("stochastic") noise can be used in all three areas, it is most common here, since emergent phenomena often require repeated varieties of interactions between individual units.  Applications include neural nets, the prediction and control of fluids (turbulence, chemical mixing, high-temperature plasma), many natural systems (forest fire, avalanche, earthquake), and some social systems (economics, group behaviors).



Romantic Humanism and Chaos Theory


The "counter-culture" of the 1960s offered the opportunity for a convergence of romantic humanism and mathematical modeling in the context of cybernetics.  In addition to Norbert Wiener's left political leanings, there was Margaret Mead, Gregory Bateson, Kurt Lewin, Warren McCulloch, Francisco Varela, C.H. Waddington, Karl Pribram, Magoroh Maruyama, Heinz von Forester, Hazel Henderson, Tolly Holt, and Barry  Commoner, all of whom linked the politics of romantic humanism with specific attributes of cybernetic systems.  These links came from both directions.  Mead, for example, was drawn to the technical material through her ideology, while Bateson was drawn into the ideology through his technical interests.  Boulding, one of the four founders of general systems theory, was instrumental in the political activism of the SDS (Wright 1989), whose founders were inspired by the communications theory of Paul Goodman (Widmer 1980), who lived next door to Bateson during their stay in Hawaii.


While some of these cybernetic-political constructs, such as "holism" or "interconnectedness," were rather vague, two dimensions stand out as both analytically specific and fundamental to the argument.  One dimension concerns information structure, the other is the physical representation of that information.  The most fundamental characterization for an information structure is its computational complexity, which is a measure of its capacity for recursion (i.e. self-reference, reflexivity).  This mathematical result agrees nicely with our intuition about the crucial role of reflexive awareness in our own "information structure."  The most fundamental characterization for a representation system is the analog-digital distinction.  Digital representation requires a code table (the dictionary, the Morse code, the genetic code, etc.) based on physically arbitrary symbols (text, numbers, flag colors, etc.).  Sassure specified this characteristic when he spoke of the "arbitrariness of the linguistic signifier." Analog representation is based on a proportionality between physical changes in a signal and changes in the information it represents (e.g. waveforms, images, vocal intonation).  For example, as my excitement increases, the loudness of my voice increases.  While digital systems use grammars, syntax, and other relations of symbolic logic, analog systems are based on physical dynamics -- the realm of  feedback, hysteresis, and resonance.  The dichotomy is fundamental to current cybernetic debates concerning, for example, the type of representation used by neurons in human brains, or the type which should be used in artificial brains.

In the first few years of American cybernetics, analog and digital systems were seen as epistemologically equivalent; both considered capable of complex kinds of representation (c.f. Rubinoff 1953).  But by the early 1960s a political dualism was coupled to this representation dichotomy.  The "counter culture" side of the cybernetics community made the erroneous claim that analog systems were more concrete, more "real" or "natural," and (according to this romantic cybernetics) therefore ethically superior.  In social domains, this converged with Rousseau's legacy of the moral superiority of oral v.s. literate cultures.[1]


Recursion was also politicized, following Norbert Wiener's work, as both metaphor and essence of humanism.  The feedback loops of homeostasis became self-government, autonomy, self-actualization, and other varieties of personal and social self-organizing frameworks.  Hence both analog systems (now no longer representation but rather a concrete Real or Natural) and recursion became at least ideological signifiers, if not entire political practices.


These two elements were carried into similar works in the 1970s and early 1980s by a series of New-Age forerunners that included Capra's Tao of Physics, Toffler's The Third Wave, and Ferguson's Aquarian Conspiracy.  Correctly noting the moves away from centralization (e.g. selfmanagement techniques) and digital formalisms (the increase in image and audio for the human-machine interface) in mainstream institutions, they concluded that the utopia promised by romantic humanism was around the corner.


At the same time, the mathematical modelling community began to re-examine the computational power of analog systems.  Despite the technological determinist claims for chaos theory as a direct result of digital computers (Franks 1989), many of the discoveries that led to chaos were based on analog systems.  This includes Ueda in Japan, Rossler in Germany, and the University of California "Chaos Cabal" (Shaw, Crutchfield, Farmer, and Packard).  Together with the discovery of universal (Turing-machine) computability in self-organizing systems (Wolfram 1984), it was clear that some of the technical terrain which had been abandoned in the 1960s would now be revised, and here too a utopic vision re-emerged.  In a 1985 issue of Scientific American, A.K. Dewdney casually mentioned that he had devised an analog Turing machine which could out-perform the digital version.  Researchers at Princeton University (Vergis et al 1985) announced an analog machine that could solve an NP-complete problem in less than exponential time -- a performance theoretically impossible for any digital system. Prominent physicists such as Pagels, Feynman, Penrose and others claimed that quantum indeterminacy, together with these new outlooks on chaos and complexity, proved that there are physical (i.e. analog) systems (especially the human brain) which have computational powers superior to digital Turing machines.


While none of these claims were ever proved correct,[2] it is not surprising that chaos theory's legitimation for both recursion and analog representation led to its incorporation by romantic humanists.  Although Capra, Toffler, Jantsch and others had made some intimations toward self-organization theory, the first specific use was in the work of mathematics professor Ralph Abraham, who had started making social connections in pre-chaos analog dynamics (Abraham 1976), and began a "General Evolution Research Group" which drew on general systems theory, catastrophe theory and nonlinear dynamics in social applications (c.f. Abraham 1981, 1986.)  The most important outcome of this group was Riane Eisler's Chalice and the Blade in 1987, in which chaos theory was combined with eco-feminist narratives on the utopic past (and possible futures).    


Eisler used two aspects of chaos theory.  In aggregate self-organization (citing Prigogine and Stengers), she saw a way to eliminate cultural difference: as interactions of identical units, all of human existence could be grouped in a single phase-space.  From dynamical systems theory (Abraham and Lorenz), she then translated these homogenous social changes in terms of bifurcations to various dynamical behaviors: "...androcracy first acted as a 'chaotic' attractor and later became the well-seated 'static' or 'point' attractor" (pg 137; here perhaps confusing chaotic attractor with transient chaos).  Thanks largely to the publication of Gleick's Chaos -- making of a new science in the same year, these were soon followed by a number of publications espousing a similar mathematical politics (e.g. Thompson 1989, Briggs and Peat 1989, Goerner 1989). 


Similar incorporations were made by Carolyn Merchant (1989), who also used the analog systems of chaos theory to argue for an ecological/bodily emersion in pre-modern human societies.  Her later work (1992) indicated that she had encountered some of the problems with analog romanticism; here she de-emphasized her earlier portrait of Native Americans as feeling rather than thinking beings.  Francisco Varella's updating of analog realism, "The Re-enchantment of the Concrete,"  made use nonlinear dynamics in neural systems, but specifically warned against any taint of artificiality suggested by chaos theory, maintaining that "cognition consists not of representations but of embodied action."


Briggs and Peat's Turbulent Mirror was a crucial work for many romantic humanists.  This was in part because it made clear, simple graphical presentations of several equations and concepts which Gleick, writing for a less specialized audience, had to leave to narrative metaphor.  But, more importantly, it used  elements of eco-feminism (e.g. Lovelock and Margulis' Gaia hypothesis), literary motifs from new-age subculture, and connections to the zen quantum mechanics lineage of Capra (D. Bohm, R. Sheldrake, D.R. Griffin).  The text also stressed Prigogine's work, despite its steadily decreasing scientific importance. 


Porush (1989, 1991) was the first to turn this from disadvantage to a positive technical stance, arguing that Gleick had left Prigogine out of his text because the stochastic nature of self-organizing systems went against an authoritarian appeal to control implied by the "deterministic chaos" of dynamical systems theory and fractal geometry.  Similar claims were made by Bey (1989, 1992) and Hayles (1990).  Allen (1993) attempted to provide empirical proof of the point by showing how simple prediction systems could plot short-term futures for the time series of strange attractors, but not for stochastic noise.[3]



The claim for maintaining relevance for Prigogine's work was technically problematic. Pre-chaos mathematics had defined complexity (the formulation of Komolgorov) as equivalent to randomness (the complexity of a number string was equal to the length of its shortest description; since a random string could not be compressed into an algorithm, it has the longest description possible).  But after chaos, complexity was redefined as computation: there is nothing computationally complex about a white noise (random signal), nor a purely periodic signal.  Rather, it is those patterns which combine order and chance -- fractals -- which are computationally most complex (in fact, fractal dimension can be used as an index of complexity).  Prigogine's title, "Order out of Chaos," was really about orderly structures emerging from white noise randomness; it was not a reference to the kind of chaos described by fractals or strange attractors (although it did concern certain nonlinear systems which were later given that analysis).


In addition to this technical lag, many romantic humanists began to feel the pressure to update their social theory, particularly in the face of a rising popularity of postmodernist works.  This task was taken on by Griffin (1988), who coined the phrase "Constructive Postmodernism" for this purpose.  Declaring deconstruction to be nihilistic, they framed modernity as rationalist reductionism, and offered their scientific holism as its post-replacement.[4]  Chaos theory (with appropriate nods toward Prigogine) was seen as a perfect vehicle for this stance. 


In summary, chaos theory in the hands of romantic humanists was seen as laudable in its legitimation of recursion and analog systems, particularly the holistic emphasis of self-organization in stochastic systems, but suspect in terms of its emphasis on determinism, and the artificiality (i.e. representational status) of analog systems.  



Chaos and Postmodernism


While N. Katherine Hayles' seminal Chaos Bound was self-declared as "constructive rather than deconstructive," (and did indeed depend on the romantic humanist foundations of Bateson, Prigogine, and Kolmogrov complexity), hers was the first to deeply engage the comparison between postmodernism and chaos theory. But making that transition was a difficult task.  Neither romantic accounts of analog systems nor humanist recursion were championed by postmodernity -- in fact, they were specifically rejected as foundations for ethics or liberatory practices.  Foucault had clearly presented the irony of humanism's recursion in, for example, his history of prison reform, where the prisoner's rehabilitation as a "self-guided" citizen was simply a more effective means of normative social control.  As for analog systems, Derrida had begun a devastating critique of realism which, although it left no room for analog representation, kept digital representation standing as the essence of meaningful expression for all cultures (even oral traditions), and purported to eliminate ethical appeals to the real, concrete, or natural.


This is not to say that postmodernism held a complete antipathy towards chaos theory -- in fact, one of its foundational texts, Lyotard's The Postmodern Condition (1984), leaned heavily on fractal geometry as an exemplar of the "paradoxes" that purported to show how the rationalism of science would be its own undoing.  Hayles continued (more cautiously) in this vein, noting that the recursive nature of fractal algorithms and nonlinear dynamical systems increased uncertainty in ways that were analogous to the application of reflexive thinking in postmodern philosophy and its undermining of truth foundations.  Since the displacement of the Real was one of these undermined foundations, the appeal to any difference arising from the analog/digital dichotomy was disallowed (an unnecessary taboo, since the status of analog systems as representation rather than the Real is upheld in chaos theory).


A second analogy in Hayles concerned her critique of certain early postmodernist formulations.  She specifically doubted the claims that a focus on local knowledge is more liberating than modernist concerns with global or universalist knowledge claims. This was supported in her review of de Man (1982), which described how absolute prescriptions for local knowledge become yet another form of globalization.  The Mandelbrot set was then offered as a visualization for this type of "interplay between the local and global."    


The metaphor was stretched to its breaking point, however, in her attempt to use analogies to material structures, particularly for architecture.  Fractal structures can be seen in many of the harmonious constructions of classic architecture (Mandelbrot 1981), but postmodern architectural form has typically been described as aharmonious pastiche.  While Hayles made very deep exegeses into postmodern identities as fragmentation, the connection to fractals was gratuitous.  The latin root of "fractal" is fractus, meaning broken, but this was chosen because (in additional to the quantitative link to fractional) a rough broken edge will typically have a fractal pattern (jaggedness within jaggedness).  This is unrelated to postmodernity's ensemble of fragments, in which heterogeneity rather than similarity is emphasized, and thus typically visualized as collage.  The point is well illustrated in Briggs (1992), a coffee table book almost completely filled with beautiful images of self-similarity.  The only exception is the postmodern collage (pg 167), which is strikingly non-self-similar, and leaves the reader wondering why the artist claims to be a "fractalist."


The point has not been lost on romantic humanists, and at least two authors (Steenburg 1991, Argyros 1991) have argued that chaos theory has created a direct challenge to postmodernism, since it seems to integrate many of the disruptions that early postmodern theory leaned on.  Postmodernists have, however, not only continued to devise literary metaphors from chaos theory (c.f. the collection in Hayles 1991); they have also extended this to social and political realms (Appaduri 1990, Young 1991, Gilroy 1993).  Here the fractal metaphor has been more accurately applied to structural patterns, e.g. Gilroy's  discussion of the fractal structure of cultural boundaries.


In summary: postmodern theorists have exploited the aspects of uncertainty in chaos theory (e.g. sensitivity to initial conditions, the indeterminism of the fractal boundary) in ways that are conspicuously disregarded by the romantic humanists, and, conversely, they have greatly de-emphasized the ways in which chaos can support portraits of organic harmonies and global coherence. 



Military Institutions and Chaos Theory


Modernist researchers have often noted the parallels between centralized hierarchies of military decision-making, and the centralized organization in military information technology.  The isomorphism has been analyzed from the perspectives of military historical development (Chapman 1987), history of cybernetics (Edwards 1988, 1995), bureaucratic defense management (Gibson 1986) and military social psychology (Gray 1989, 1995).  The military adoption of chaos theory, however, is in direct contradiction to these analyses.


One of the first explicit attempts to move toward military applications from the earlier holistic cybernetics was an essay by James Miller (1979), who's "general theory of living systems" (here applied to military command and control) drew on Wiener and von Bertalanffy.  A similar approach was proposed by Gregory (1986), who cites Varela, Bateson, and even the zen physics of Zukav.


Practical application of holistic military technology has occurred at every scale.  At the smallest level, individual skill organization has used a neural net model.  The linkage of individuals into decentralized military units has also been formally introduced under the rubric of "distributed decision-making" (Levis and Boettcher 1983, Saisi and Serfaty 1987).  At the largest scale is the machine-mediated battle organization of C3I ("AirLand Battle Management" in DARPA's Strategic Computing Initiative).  The formulation of operations doctrine for C3I was based in part on the use of "chaos" by the German Army in World War II. Richey (1984) notes that many Allied failures were due to conceptions of "battle fighting as a problem of imposing order on chaos" and that the German use of decentralized organization (Auftragstatik) allowed them to "accept chaos as the natural substance of combat." 


The increase in military research on chaos theory indicates that the colloquial use of "chaos" from  C3I doctrine started to dovetail with its mathematical sense as early as 1988, where the strategic computing report from DARPA listed 14 projects dealing directly with nonlinear science.  Both analog and digital designs for self-organizing neural networks were funded, and North (1988) noted their application to organizational structures in C3I.  Application of both fractals and dynamical systems theory to battlefield management were mentioned by Gary Coe, chief of modelling and analysis of the Pentagon Joint Chiefs of Staff in the same year (Zorpette 1988).  Another area of semantic convergence is the frequent metaphor of "brush fires in the Third World."  Mathematical models for brush fires of the vegetative type have become a featured example for chaos theory (Bak et al 1990), and the frustration over low-intensity war has already been expressed in terms of the contradiction between linear hierarchical order promoted by current technology, and the "chaos" of these dynamic self-organized conflicts (Dean 1986).


The most interesting line of research to emerge has been the dynamical systems models of Gottfried Mayer-Kress (see interview in appendix), who was at the time a researcher at the famed U.S. nuclear physics lab in Los Alamos (technically under the civilian Department of Energy, but for all practical purposes a military institution).  His first model (Mayer-Kress and Saperstein 1988) predicted that Ronald Reagan's Strategic Defense Initiative would result in an arms race of both offensive missiles and anti missile defenses.  His later work (e.g. Mayer-Kress 1989) used holism arguments similar to those of C3I cited above, but for a very different purpose: condemning the "static and linear paradigm of a traditional approach" for the failures of US arms control.


In summary:  much of the use of chaos theory by military institutions has been through arguments strikingly similar to (or even originating from) those of the romantic humanists.  Although this is in direct conflict with the humanists supposition of an inherently ethical component to holistic thinking, the work on arms control, equally contradictory to typical assumptions about the political consciousness of military environments, seems to have been influenced by their thinking.








It is tempting to posit that chaos theory is merely an ideological rorschach, its counter-intuitive visual and narrative surprises allowing anyone to see their own ideology validated, and that of their competitor's challenged.  But specific patterns do emerge; there are some commonalities in which certain aspects of chaos are consistently emphasized or diminished for the same ideological ends.  The overlap between romantic humanists and the military, however, should make it clear that none of these connections between ideology and mathematical modelling are binding very far beyond our own communities of thought and action.  Eventually, some of the intellectual work we have asked chaos to do for us will have to be our own doing.


Appendix: An interview with Gottfried Mayer-Kress


RE:  Could you tell us a bit about your early life?


GMK: Well, I grew up in an authoritarian catholic environment where I was an altar boy under a tyrant priest.  I talked back at him and got punished, but had my first strong "question authority" experience when I was around 10-12.  I had no problem resisting the strong social pressure form my motor-bike friends to drink and smoke; at the same time I had their respect for what I was doing in the group.  When I was about 15 I learned that a cousin of mine had started prostituting, and against the strong resistance of my parents I hitch-hiked to the big city and tried to find her. I had the experience to follow my family and peers.


RE:  What about political work before coming to the U.S.?


GMK: I was a supporter of the green party and involved in many of their actions (anti-nuclear/environmental/peace actions etc.), but I never was a person who could enjoy their party and strategy discussion/fights.  I was a member, however of several ecological/peace groups and committees, and before I left I was the head of the local branch of the BUND, the Bund fuer Umwelt und Naturschutz Deutschland.


RE:  Why the move to Los Alamos?


GMK: I met Doyne Farmer at a conference, and was very impressed with his work on nonlinear dynamics as well as his politics.  He told me it would be possible to work under a guarantee that none of our research would be classified.


RE:  And did you continue political activities during your research at Los Alamos?


GMK:  We handed out leaflets on Nagasaki day 1984 and got chased away by lab security, and we had regular meetings in Los Alamos to hand out leaflets from a fenced-off area near the fire station (the "mud pit").  We got mixed responses from lab people, some very positive ("I'm on your side but I work from within") up to more negative comments.  Los Alamos Women for Peace hosted a group of Japanese survivors of the atomic bombing, and they asked me to show them the lab.  I felt funny as a German, showing them the US center of atomic weapons research, and I volunteered to talk to the public affairs office.  I got a very negative response from them, and finally arranged a guided tour through the science museum.  Here the guide started off with solar energy research in Los Alamos until I reminded him that the visitors were also interested in the (dominant) historical section of the museum, but by then he had almost run out of time, so he rushed by the replicas of Little Boy and Fat Man, that typically grandmas use as a backdrop for pictures of their grandchildren.


Abraham, RH.  "Vibrations and the realization of form," in Evolution in the Human World, ed. Erich Jantsch and CH Waddington, pp 134-149, Reading: Addison-Wesley 1976.


Abraham, RH.  "The function of mathematics in the evolution of the noosphere." in The Evolutionary Vision, ed. Erich Jantsch, AAAS Selected Symposium, pp. 153-168, 1981.


Abraham, RH.  "Mathematics and Evolution: a manifesto."  International Synergy Journal, vol 1(3), pp 14-23, 1986.


Allen, PM, and Phang, HK.  "Evolutionary intelligence: learning to deal with uncertainity in financial markets."  New Developments in Technology Studies: Evolutionary Economics and Chaos Theory.  Amsterdam May 1993.


Argyros, AJ.  A Blessed Rage for Order: deconstruction, evolution and chaos.  Ann Arbor: University of Michigan Press, 1991.


Appadurai, A.  "The Global Cultural Economy."  Public Culture 2,2 Spring 1990.


Bak P, Chen K, Tang C.  "A forest-fire model and some thoughts on turbulence."  Physics letters vol 147, no 5-6, pp 297-300,       1990.


Bateson, M.C.  Our Own Metaphor.  NY: Knopf 1972.


Bey, H.  Chaos -- the broadsheets of ontological anarchism.  Grim Reaper Press 1988, 1991.


Blum, L., Shub, M., and Smale, S.  "On a theory of computation and complexity over             the real numbers." Bull AMS, v21 N1 1-46, Jul 1989.


Bohm, D.  Wholeness and the Implicate Order.  Boston: Routledge and Kegan Paul 1980.


Briggs, J. Fractals: the patterns of chaos.  NY: touchstone 1992.


Briggs, J and Peat, FD.  Turbulent Mirror.  NY:Harper and Row 1989.


Capra, F. The Tao of Physics.  Shambhala 1975.


Chapman, G.  "Computing, high technology weapons and the US-Soviet Confrontation in Europe."  in Computers in Battle, ed D. Bellin and G. Chapman, NY: Harcourt Brace Jovanovich 1987.

De Man, P.  "The Resistance to Theory."  Yale French Studies 63:3-20.


Dewdney, A.K.  "Analog gadgets."  Scientific American, June 1985 pp 18-29.


Dewdney, A.K. "Building computers in one dimension sheds light on irreducibly         complicated phenomena."  Scientific American, (Feb 1985) pp. 18-30.


Derrida, J.  Of Grammatology.  Baltimore: Johns Hopkins, 1974.


Edwards, P.  "Border Wars:  The Science and Politics of Artificial Intelligence."  Radical America vol 19 no 6 pp 39-50 1985.


Eglash, R.  "Postmodern cybernetics: holistic military technology."  Paper delivered at          Social Studies of Science Society, Irvine CA, Nov 1990.


Eglash, R.  A Cybernetics of Chaos.  PhD. Dissertation in History of Consciousness,           University of California, Santa Cruz,         1992.


Eisler, R.  The Chalice and the Blade.  San Francisco: Harper and Row.


Franks, J.  Review of Gleick (1987).  The Mathematical Intelligencer 11:65-71.


Gibson, J.  The Perfect War.  Atlantic Monthly Press 1986.


Gilroy, P. The Black Atlantic: modernity and double consciousness.  Harvard Univ Press 1993.


Goerner, S.J.  Chaos and Its Implications for Psychology.  The Saybrook Institute, 1989.


Goerner, S.J.  Chaos and the Evolving Ecological Universe.  Langhorne PA: Gordon and Breach 1993.


Gray, C.  Artifical intelligence and real war.  South End Press, 1988.


Gray, C.  Technohistory.  Malabar Fl: Krieger Pub 1995.


Gleick, J. Chaos - making of a new science. NY: Viking 1987.


Gregory, Dik.  "Delimiting Expert Systems."  IEEE Syst Man Cyber 16:6  pp 834-43 1986.


Griffin, D.R.  The Reenchantment of Science: postmodern proposals.  Albany: SUNY Press 1988.


Hayles, K.  Chaos Bound.  Ithaca: Cornell Univ Press 1990.


Hayles, K.  Chaos and Order.  Chicago: University of Chicago Press 1991.


Kerman, CE. Creative tension: the life and thought of Kenneth Boulding.  Ann Arbor: Univ Michigan Press 1974.


Levis, A and Boettcher K.  "Decision making organizations with acyclic information structures."  IEEE Syst Man Cyber 13 pp 384-391 1983.


Lilienfeld, R.  The Rise of Systems Theory.  NY: John Wiley 1978.


Lyotard, JF.  The Postmodern Condition: a report on knowledge.

Minneapolis: Univ of Minnesota Press, 1984.


Mandelbrot, Benoit.  The Fractal Geometry of Nature.  San Francisco: W.H. Freeman           1982


Mandelbrot, Benoit.  "Scalebound or scaling shapes:  A useful distinction in the visual arts and in the natural sciences."  Leonardo 14, 45-47, 1981.


May, R.  "Simple Mathematical Models with Very Complicated Dynamics." Nature 261        (1976), pp 459-67.


Mayer-Kress, G. and Saperstein, A. "A Nonlinear Dynamical Model of the Impact of SDI on the Arms Race."  Journal of Conflict Resolution, Dec 1988.


Merchant, C.  Ecological Revolutions.  Chapel Hill: Univ North Carolina Press 1989.


Merchant, C.  Radical Ecology.  NY: Routeldge 1992. 


Miller, J.G.  "Potential applications of a general theory of living systems to the study of military command and control."  Decision Information ed CP Tsokos and RM Thrall, Academic Press 1979.


North, RL.  "Neurocomputing: its impact on the furute of defense systems."  Defense Computing 1988.


Pagels, H.R.  The Dreams of Reason: the computer and the rise of the sciences of   complexity. NY:  Simon and Schuster 1988.


Porush, D.  "Rewriting Chaos: a study in the popular making of a new science."  Society for Social Studies of Science, Nov 1989.


Porush, D.  "Fictions as dissipative structures: Prigogine's theory and postmodernism's roadshow."  in Hayles 1991.

Prigogine, I and Stengers, I.  Order out of Chaos.  NY: Bantam 1984.


Richey, SW.  "The Philosophical Basis of the AirLand Battle."  Military Review May 1984.


Rossler, O.  Z. Naturforsch 13a, 259 1976.


Rubel, L.A.  "Digital simulation of analog computation and church's thesis."  J Symbolic       Logic, v 34 n3, pp 1011-1017, sept 1989.


Rubinoff, M.  "Analogue and digital computers -- a comparison." Proceedings of the IRE,     Oct 1953, pg 1254-1262.


Ruelle, D. and Takens, F. "On the nature of turbulence."  Communications in Mathematical Physics 20 (1971) pp 167-92.


Saisi D and Serfarty D.  "Model-based interface design and evaluation for distributed decision making."  Proc IEEE CH2503-1 1987.


Schroeder, M.  Fractals, Chaos, and Power Laws.  NY: W.H. Freeman, 1991.


Steenburg, D.  "Chaos at the marriage of heaven and hell."  Harvard Theological Review 84 (1991) 447-66.


Thompson, WI.  Imaginary Landscape.  NY: St Martin's Press  1989


Varela, FJ.  "Re-enchantment of the concrete."  in Incorporations, ed J. Crary and S. Kwinter, Zone/MIT Press.


Vergis, A., Steiglitz, K., and Dickinson, B.  "The complexity of analog computation."              Tech report No. 337, Dept of C.S., Princeton University, Feb 1985.


Wolfram, S.  "Universality and complexity in cellular automata" Physica 10D (1984), 1-35.


Young, T.R.  "Chaos and social change: metaphysics of the postmodern."  The Social Science Journal, vol 28, no 3, pp 289-305 1991.


Zorpette, G.  "Fractals: not just another pretty picture."  IEEE Spectrum Oct 1988.


[1]           This was combated in different ways by structuralists and post-structuralists.  According to Levi-Strauss, the arbitrariness of non-western symbolics (e.g. a fox stands for stupidity in one mythology and cunning in another) proves that they are just as digital as Europeans, with the exception of the oral/literate dichotomy.  Derrida, while agreeing with this project, takes Levi-Strauss to task for retaining the oral/literate dichotomy, and details how speech is just writing in air instead of paper -- thus again using digitality as the justification for epistemological equivalence.  Tragically, poststructuralists have adopted Rousseau's assumption that analog representation is not as abstract as digital.

[2]           Vergis et al were disproved in Rubel (1989).  Dewdney, writing in response to my queries, said that he had lost the papers concerning this work.

[3]           Allen's proof used the time series generated by only a few oscillations of the strange attractor, so the prediction system was merely faced with a nearly-periodic waveform.  Had he utilized samples from higher frequencies (several thousand oscillations per sample point) he would have found the data as "noisy" as any stochastic source.  Allen, as well as Prigogine himself, have suggested that the declining use of Prigogine and his omission in historical accounts is due to an anti-European/pro-American bias.

[4]           For the remainder of this essay I'll continue to use "postmodernism" to refer to only the deconstructive kind, and "romantic humanism" to refer to the constructive analyses of Griffin and his colleagues.