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
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
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
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
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.
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
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
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.
Conclusion
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
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
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
GMK: We
handed out leaflets on
References
Abraham, RH.
"Vibrations and the realization of form," in Evolution in
the Human World, ed. Erich Jantsch and CH
Waddington, pp 134-149,
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.
Argyros,
AJ. A Blessed Rage for Order:
deconstruction, evolution and chaos.
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.
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
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.
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,
Eglash, R.
A Cybernetics of Chaos.
PhD. Dissertation in History of Consciousness,
Eisler,
R. The Chalice and the Blade.
Franks, J.
Review of Gleick (1987). The Mathematical Intelligencer
11:65-71.
Gibson, J.
The Perfect War. Atlantic
Monthly Press 1986.
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.
Hayles,
K. Chaos Bound.
Hayles,
K. Chaos and Order.
Lilienfeld,
R. The Rise of Systems Theory. NY: John Wiley 1978.
Lyotard,
JF. The Postmodern Condition: a
report on knowledge.
Mandelbrot, Benoit. The Fractal Geometry of Nature.
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.
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,
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.
Varela, FJ.
"Re-enchantment of the concrete." in Incorporations, ed J. Crary and
Vergis,
A., Steiglitz, K., and Dickinson, B. "The complexity of analog computation." Tech
report No. 337, Dept of C.S.,
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.