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Cambridge Journal of Economics 2000, 24, 311–324
Aspects of dialectics and non-linear
dynamics
J. Barkley Rosser Jr*
Three principles of dialectical analysis are examined in terms of non-linear dynamics
models. The three principles are the transformation of quantity into quality, the
interpenetration of opposites, and the negation of the negation. The first two of these
especially are interpreted within the frameworks of catastrophe, chaos and emergent
dynamics complexity theoretic models, with the concept of bifurcation playing a
central role. Problems with this viewpoint are also discussed.
Key words: Dialectics, Non-linearity, Catastrophe, Chaos, Complexity
JEL classification: B00, B14, B24, B49
1. Introduction
Among the deepest problems in political economy is that of the qualitative transformation
of economic systems from one mode to another. A long tradition, based on Marx, argues
that this can be explained by a materialist interpretation of the dialectical method of
analysis as developed by Hegel. Although Marx can be argued to have been the first clear
and rigorous mathematical economist (Mirowski, 1986), this aspect of his analysis
generally eschewed mathematics. Indeed some (Georgescu-Roegen, 1971) argue that the
dialectical method is in deep conflict with ‘arithmomorphism’, or a precisely quantitative
mathematical approach, that its very essence involves the unavoidable invocation of a
penumbral fuzziness that defies and defeats using most forms of mathematics in political
economy.
However, this paper will argue that non-linear dynamics offers a way in which a mathe-
matical analogue to certain aspects of the dialectical approach can be modelled, in
particular, that of the difficult problem of qualitative transformation alluded to above. This
is not the entirety of the dialectical method, which remains extremely controversial and
redolent with remaining complications. We shall not attempt either to explicate or to
defend the entirety of the dialectical approach, much less resolve its various contradictions,
although we shall note how some of its aspects relate to this more specific argument.
© Cambridge Political Economy Society 2000
Manuscript received 3 March 1997; final version received 22 April 1998.
Address for correspondence: J. Barkley Rosser Jr, Department of Economics, MSC 0204, James Madison
University, Harrisonburg, VA 22807, USA; email: rosserjb@jmu.edu
* James Madison University. I wish to thank the following individuals for making research materials or use-
ful comments available to me: Peter M. Allen, William A. Brock, Steven N. Durlauf, Carla M. Feldpausch,
Masahisa Fujita, Stephen J. Guastello, Cars H. Hommes, Heikki Isomäki, Andrew Kliman, Blake LeBaron,
Hans-Walter Lorenz, Walter G. Park, Tönu Puu, Marina Vcherashnaya Rosser, Chris M. Sciabarra, Mark
Setterfield, Ajit Sinha, John D. Sterman, Wolfgang Weidlich and two anonymous referees. The usual caveat
applies.
312 J. B. Rosser Jr
In particular, we shall discuss certain elements of catastrophe theory, chaos theory and
complex emergent dynamics theoretical models that allow for a mathematical modelling
of ‘quantitative change leading to qualitative change’, one of the widely claimed foun-
dational concepts of the dialectical approach, and a key to its analysis of systemic political
economic transformation. These approaches are all special cases of non-linear dynamics,
and their special aspects, which allow for this analogue, depend on their non-linearity. We
note that there are some linear models that generate discontinuities and various ‘exotic
dynamics’, e.g., models of coupled markets linked by incommensurate irrational
frequencies, some of which may arguably also be subject to a dialectical interpretation.
However, we shall not investigate these examples further in this paper. In most linear
models, continuous changes in inputs do not lead to discontinuous changes in outputs,
which will be our mathematical interpretation of the famous ‘quantitative change leading
to qualitative change’ formulation.
Section 2 of this paper briefly reviews basic dialectical concepts. Section 3 discusses
how catastrophe theory can imply dialectical results. Section 4 considers chaos theory
from a dialectical perspective. Section 5 examines some emergent complexity concepts
along similar lines, culminating in a broader synthesis. Section 6 will present conclusions.
2. Basic dialectical concepts
In a famous formulation, Engels (1940, p. 26) identifies the ‘laws’ of dialectics as being
reducible to three basic concepts: (1) the transformation of quantity into quality and vice
versa, (2) the interpenetration of opposites, and (3) the negation of the negation, although
Engels’s approach differs from that of many others on many grounds (Hegel, 1842;
Georgescu-Roegen, 1971; Ilyenkov, 1977; Habermas, 1979). Whereas Marx largely used
these concepts to analyse historical change, Engels drew on Kant and Hegel to extend this
approach to science. Although his discussion in The Dialectics of Nature was reasonably
current with regard to science for the time of its writing (the 1870s and early 1880s),
much of its content is seen to be scientifically inaccurate by today’s standards, and many
of its examples thus hopelessly muddled and wrong-headed. Furthermore, the arguments
of this book would later be used to justify the ideological control and deformation of
science under Stalin and Khrushchev in the USSR, most notoriously with regard to the
Lysenkoist controversy in genetics.
1
For both Marx and Engels (1848), the first of these was the central key to the change
from one mode of production to another, their historical materialist approach seeing
history unfolding in qualitatively distinct stages such as ancient slavery, feudalism and
capitalism. Engels (1954, p. 67) would later identify this with Hegel’s (1842, p. 217)
example of the boiling or freezing of water at specific temperatures, qualitative (discon-
tinuous) leaps arising from quantitative (continuous) changes. In modern physics, this is a
phase transition and can be analysed using spin glass or other complexity type models
(Kac, 1968). In modern evolutionary theory, this idea has shown up in the concept of
‘punctuated equilibria’ (Eldredge and Gould, 1972), which Mokyr (1990) and Rosser
(1991, ch. 12) link with the Schumpeterian (1934) theory of discontinuous technological
1
Although agreeing that Lysenkoism involved serious scientific errors, especially with regard to breeding
for disease resistance, and also led to deep injustices towards individual Soviet scientists, Levins and
Lewontin (1985, ch. 7) note that not all of Lysenko’s ideas and proposals were silly and that in fact Soviet
agriculture performed reasonably well in terms of increasing productivity during the period of the greatest
Lysenkoist control of agroscience (1948–62).
Aspects of dialectics and non-linear dynamics 313
change. Such phenomena can arise from catastrophe theoretic, chaos theoretic and com-
plex emergent dynamics models.
The interpenetration of opposites leads to some of the most controversial and difficult
ideas associated with dialectical analysis. Implicit in this idea are several related concepts.
One is that of contradiction, and the argument that dynamics reflect the conflict of
contradicting opposites that are simultaneously united in their opposition. According to
Ilyenkov (1977, p. 153), ‘We thought of a dynamic process only as one of the gradual
engendering of oppositions, of determinations of one and the same thing, i.e., of nature as a
whole, that mutually negated one another.’
Setterfield (1996) notes that contradictions may be logical in nature or between real
conflicting forces, with Marx probably favouring the latter view, although it is difficult to
distinguish genuine dialectical contradictions from mere differences. For Marx and
Engels (1848) these real conflicting forces were the classes in conflict over control of the
social surplus and of the means of production, although they also argued, as is laid out
more fully in Marx (1977), that a crucial contradiction is between the forces and relations
of production, united in the mode of production. This in turn fundamentally arises from
the evolution of the contradiction between use-value and exchange-value within the
commodity itself, yet another union of conflicting opposites.
Another interpretation is that this ‘unity of opposites’ implies a negation of the idea of
the ‘excluded middle’ in logic. Thus, both ‘A’ and ‘not A’ can simultaneously be true.
Georgescu-Roegen (1971) makes much of this aspect in his denigration of ‘arithmo-
morphism’, and interprets this as meaning that, between two opposites, there is a
‘penumbra’ of fuzziness in their boundary in which they coexist and interpenetrate, much
as water and ice coexist in slush (Ockenden and Hodgkins, 1974). Such an approach can
be dealt with using fuzzy logic (Zimmermann, 1988), which in turn ultimately relies on a
probabilistic approach. Georgescu-Roegen (1971, pp. 52–9) further argues that the
probabilistic nature of reality itself is evidence of the fuzzily dialectical nature of reality in
that truth criteria in a probabilistic world are simply arbitrary. This leads him to argue that
there is a deeper contradiction between continuous human consciousness and discon-
tinuous physical reality, discrete at the quantum level. Rosser (1991, ch. 1) argues that
this is a matter of perspective or the level of analysis of the observer.
Engels (1940, pp. 18–19) confronted the contradiction between the apparently simul-
taneous acceptance of discontinuity arising from the idea of qualitative leaps and of con-
tinuity arising from the ‘fuzziness’ implied by the interpenetration of opposites in the
dialectical approach. He dealt with this by following Darwin (1859) in accepting a gradual-
istic view of organic evolution in which species continuously change from one into
another, while arguing that in human history, the role of human consciousness and choice
allows for the discontinuous transformation of quantity into quality as modes of pro-
duction discontinuously evolve.
Finally, there is the idea of wholes consisting of related parts implied by this formu-
lation. For Levins and Lewontin (1985) this is the most important aspect of dialectics and
they use it to argue against the mindless reductionism they see in much of ecological and
evolutionary theory, Levins (1968) in particular identifying holistic dialectics with his
‘community matrix’ idea. This can be seen as working down from a whole to its inter-
related parts, but also working up from the parts to a higher order whole. This latter
concept can be identified with more recent complex emergent dynamics ideas of self-
organisation (Turing, 1952; Wiener, 1961), autopoesis (Maturana and Varela, 1975),
emergent order (Nicolis and Prigogine, 1977; Kauffman, 1993), anagenesis (Boulding,
314 J. B. Rosser Jr
1978; Jantsch, 1979), and emergent hierarchy (Rosser et al., 1994; Rosser, 1995). It is
also consistent with the general social systems approach of the dialectically oriented post-
Frankfurt School (Luhmann, 1982, 1996; Habermas, 1979, 1987; Offe, 1997).
Indeed, even some Austrian economists have emphasised self-organisation arguments,
with Hayek (1952, 1967) developing an emergent complexity theory based on an early
version of neural networks models and eventually (Hayek, 1988, p. 9) explicitly acknowl-
edging his link with Prigogine and with Haken (1983). Lavoie (1989) argues that markets
self-organise out of chaos. Sciabarra (1995) argues that Hayek in particular uses a funda-
mentally dialectical approach.
Finally, the ‘negation of the negation’ has also been a very controversial and ideo-
logically charged concept. It represents the combining of the previous two concepts into a
dynamic formulation: the dialectical conflict of the contradictory opposites driving the
dynamic to experience qualitative transformations. Again, there would appear within
Marx and Engels to be at least two incompletely integrated ideas. On the one hand, there
is the idea of a sequence of ‘affirmation, negation and the negation of the negation’ or
‘thesis, antithesis, synthesis’, as described by Marx (1992, p. 79). This implies a historical
sequence of alternating stages, with Engels (1954, p. 191) suggesting the alternation of
communally owned property in primitive societies, followed by privately owned property
later, with a forecast return to communally owned property under socialism in the future.
1
On the other hand, in Marx and Engels (1848) this takes the form of one class being the
thesis, the opposed class during the same period and mode of production being the antithesis,
and the new mode of production with its new class conflict being the synthesis. We shall
not attempt in this paper to resolve this contradiction, nor shall we attempt to model this
explicitly in our mathematical approach.
3. Catastrophe theory and dialectics
The key idea for analysing discontinuities in non-linear dynamical systems is bifurcation,
and was discovered by Poincaré (1880–90), who developed the qualitative theory of
differential equations to explain more-than-two-body celestial mechanics. Consider a
general family of n differential equations whose behaviour is determined by a k-
dimensional control parameter , such that
dx/dtf
(x); xR
n
, R
k
(1)
with equilibrium solutions given by
f
(x)0 (2)
Bifurcations will occur at singularities where the first derivative of f
(x) is zero and the
second derivative is also zero, meaning that the function is not at an extremum, but is
rather at a degeneracy. At such points structural change can occur, as an equilibrium can
bifurcate into two stable and one unstable equilibria.
Catastrophe theory involves examining the stable singularities of a potential function of
(1), assuming that there is a gradient. Thom (1975A) and Trotman and Zeeman (1976)
1
This discussion follows directly on a somewhat contradictory and muddled discussion by Engels (op. cit.,
p. 190) in which he attempts to interpret calculus dialectically. Thus, a function is a thesis, its derivative is its
antithesis, and taking the integral of the derivative is the negation of the negation, which reproduces the
original function. This does not show the evolutionary or historical development emphasised by Engels on
the next page, where the negation of the negation is something new at an entirely different level, albeit
reproducing some aspect of the original thesis.
determined the set of such stable singularities for various dimensionalities of control and
state variables. Arnol’d et al. (1985) generalised this analysis to higher orders of dimen-
sionalities. These singularities can be viewed as points at which equilibria lose their
stability with the possibility of a discontinuous change in a state variable(s) arising from a
continuous change in a control variable(s).
A catastrophe form that shows most of the phenomena occurring in catastrophe models
is that of the three-dimensional cusp catastrophe, shown in Figure 1. In this figure J is the
state variable, and C and F are the control variables. Assuming that the ‘splitting factor’ C
is sufficiently large, continuous variations in F can lead to discontinuous changes in J. The
intermediate sheet in Figure 1 represents an unstable set of equilibria points. Behaviour
observable in such a dynamical system can include bimodality, inaccessibility, sudden
jumps, hysteresis and divergence, the latter arising from variations of the splitting factor
C.
For Thom (1975A) this becomes the mathematical model of morphogenesis, of qualitat-
ive transformation from one thing into something else, following the analysis of D’Arcy
Thompson (1917) of the emergence of organs and structures in the development of an
organism. Furthermore, Thom (1975B, p. 382) explicitly links this to dialectics, albeit of
an idealist sort:
Catastrophe theory . . . favors a dialectical, Heraclitean view of the universe, of a world which is the
continual theatre of the battle of between ‘logoi’, between archetypes.
There is a serious criticism that can be made of this view, although we tend to favour the
view in this paper. It is the ‘anti-arithmomorphic’ dialectic position as enunciated by
Georgescu-Roegen (1971), which would argue that all we are seeing in such models is
discontinuous changes in variables or functions and not a true qualitative change. The
Aspects of dialectics and non-linear dynamics 315
Fig. 1.
316 J. B. Rosser Jr
latter would presumably be something beyond the ability of mathematics to describe. It
would not be simply a change in function or values of existing state variables, but the
emergence of a completely new variable or even a new function or set of functions and
variables. But at a minimum such structural changes imply qualitatively different
dynamics, even if the variables themselves are still the same, in some sense.
Another variation on this latter point arises from considering the phenomenon of diverg-
ence associated with the change in the value of a splitting factor such as C in Figure 1. One
goes from a system with one equilibrium to one with three equilibria, one of them
unstable. The new equilibria themselves may actually represent new states or conditions,
the qualitative change or emergence of new ‘variables’ or ‘functions’ in some sense. This is
certainly the interpretation of Thom, who identified such structural changes with the
emergence of new organs in the development of organisms.
Ironically, in mainstream economics, most of the criticism of catastrophe theory has
come from the opposite direction, claims that it is too imprecise, too poorly specified,
unable to generate forecasting models with solid theoretical foundations, too ad hoc, and
so forth. Much of this criticism has probably been overdone, as discussions in Rosser
(1991, ch. 2) and Guastello (1995) suggest.
Another possible difficulty is that it is not at all clear that the control versus state
variable idea maps meaningfully onto the dialectical taxonomy. After all, it can be argued
that it is the control variables themselves that should be undergoing some kind of
qualitative change as a result of their quantitative changes, rather than some state variable
controlled by them.
Yet another issue that cuts across all non-linear dynamical interpretations of dialectics
is that catastrophe theory analyses equilibrium states and their destabilisation. There is an
old view among dialecticians that equilibrium is not a dialectical concept, indeed that
dialectics is necessarily an anti-equilibrium concept. However, drawing on the work of
Bogdanov (1912–22), Bukharin (1925) argued that an equilibrium reflects a balance of
conflicting dialectical forces and that the destabilisation of such an equilibrium and the
emergence of a new one is the ‘qualitative shift’. This view was criticised sharply by Lenin
(1967) and was viewed by Stalin as constituting part of Bukharin’s unacceptable ideology
of allowing market elements to persist as an equilibrating force in socialist society. Stokes
(1995) argues that Bogdanov’s views provided the foundation for general systems theory
as it developed through cybernetics (Wiener, 1961). These approaches would eventually
lead to non-linear complexity theories, some of them emphasising disequilibrium or
out-of-equilibrium phase transitions, as in the Brussels School approach (Nicolis and
Prigogine, 1977).
4. Chaos theory and dialectics
The study of chaotic dynamics also originated with Poincaré’s qualitative celestial
mechanics. As argued in Rosser (1991, ch. 1 and 2), catastrophe theory and chaos theory
represent two distinct faces of discontinuity, and hence arguably of dialectical ‘quantity
leading to quality’. The common theme is bifurcation of equilibria of non-linear
dynamical systems at critical values.
Although there remain controversies regarding the definition of chaotic dynamics
(Rosser, ibid.), the most widely accepted sine qua non is that of sensitive dependence on
initial conditions (SDIC), the idea that a small change in an initial value of a variable or of
a parameter will lead to very large changes in the dynamic path of the system. This is also
Aspects of dialectics and non-linear dynamics 317
known as the ‘butterfly effect’, from the idea that a butterfly flapping its wings could cause
hurricanes in another part of the world (Lorenz, 1963).
Figure 2 exhibits this divergent behaviour from small initial changes that occur when
SDIC holds. This shows the two distinct paths over time for one variable with and without
a perturbation to an initial condition equal to 0·0001 for a three-equation system of
atmospheric circulation deriving from Edward Lorenz (1963). Lorenz concluded that the
butterfly effect implies the futility of long-range weather forecasting. Truly chaotic
systems exhibit highly erratic, apparently random, yet deterministic and bounded
dynamics.
A sufficient condition for SDIC to hold is for the real parts of the Lyapunov exponents
of the system to be positive. Oseledec (1968) showed that these can be estimated for a
system such as (1), if f
t
(y) is the tth iterate of f starting from an initial point y, D is the
derivative, v is a direction vector. The Lyapunov exponents are solutions to
Llim ln(Df
t
(y)v)/t (3)
t0
Although there are systems that are everywhere chaotic, many are chaotic for certain
parameter values and are not for others. In such cases, there may be a ‘transition to chaos’
as a parameter value is varied and a system experiences bifurcations of its equilibria. A
pattern exhibited by many well-known systems is for there to be a zone of a unique and
stable equilibrium, then beyond a critical parameter value there emerges a two-period
oscillation, then beyond another point emerges a four-period oscillation, an eight-period
oscillation, and so forth, a sequence known as a period-doubling cascade of bifurcations
(Feigenbaum, 1978). According to a special case of Sharkovsky’s (1964) theorem, the
emergence of an odd-numbered orbit (1) is a sufficient condition for the existence of
chaos. In some systems, as the parameter continues to change, chaos disappears and
period-halving bifurcations return the system to its original condition, although in some
systems there is simply an explosion or a transition to yet other kinds of complex
dynamics.
Probably the most intensively studied simple equation that generates chaotic dynamics
in economic models is the difference logistic, given by
x
t+1
x
t
(k–x
t
) (4)
Fig. 2.
318 J. B. Rosser Jr
with the ‘tuning parameter’ whose variations change the qualitative dynamics of the
system. As increases, the period-doubling cascade of bifurcations from an initial unique
equilibrium described above occurs, leading to chaotic dynamics and culminating in
explosive behaviour. May (1976) studied this equation in the context of an ecological
population dynamics model, in which k has the interpretation of a carrying capacity
constraint, but he also first suggested the applicability of chaos theory to economic
analysis in this paper. Figure 3 shows the period-doubling transition to chaos pattern for
the logistic equation, with on the horizontal axis and the system’s state variable, x, on
the vertical axis.
At least two possible dialectical interpretations can be drawn from (4) and generically
similar systems. One is the already mentioned idea that the cascade of bifurcations can be
seen as representing qualitative changes arising from quantitative changes. A smoothly
varying , or control parameter, reaches critical points where there is a discontinuous
change in the nature of the dynamics. Now, an anti-arithmomorphic dialectician can
again deny that this is what is meant by qualitative change in the Hegelian sense. Yes,
variables are behaving differently, but they are just the same old variables, this argument
runs. But, we note that if chaotic dynamics herald a larger-scale catastrophic discon-
tinuity, then there may be a greater chance for a deeper-level qualitative change to
happen. Such instances may be ‘chaostrophes’ associated with the ‘blue-sky’ disappear-
ance of an attractor after a chaotic interlude (Abraham, 1985), or lead to ‘chaotic
hysteresis’ (Rosser, 1991, ch. 17; Rosser and Rosser, 1994). Although not labelled as
such, an example of such a chaotic hysteretic model is a modified Hicks–Goodwin non-
linear business cycle model (Puu, 1997) in which chaotic dynamics appear at points of
discontinuous jumps in a hysteresis cycle.
The second such interpretation involves the concept of the interpenetration of oppo-
sites. This interpretation can be derived from considering the dual role of the x variable in
(4). It operates in both a positive and a negative way, tending both to push up and to
push down. Now, this may seem fairly trivial, as many such equations exist. But indeed,
at the heart of most chaotic dynamics is a conflict between factors pushing in opposite
directions. In effect, as increases, the strength of this conflict can be thought of as
intensifying.
Fig. 3.
Aspects of dialectics and non-linear dynamics 319
In the population ecology model of May (1976), represents the intrinsic growth rate
of the population, and the negative aspect represents the effect of the population crashing
into the ecological carrying capacity, k. One can view this system dialectically and holistic-
ally as a population with its environment. Conflicting forces operate through the same
variable, the population, hence the interpenetration of the opposites whose interaction
drives the dynamics. As this conflict heightens, bifurcations occur and quantitative
changes lead to qualitative changes in dynamics as the system transits to chaos.
5. Emergent dynamics complexity and dialectics
In contrast to the theories of catastrophe and chaos, there is no single criterion or model of
complex dynamics, but rather a steadily increasing plethora which we shall not attempt
to explicate in any detail here (Arthur et al., 1997; Rosser, 1998). Indeed Horgan (1997,
pp. 303–4) reports at least 45 different definitions of complexity, including some such as
algorithmic complexity in which we are not interested. Almost all involve some degrees of
stochasticity in their formulation, yet some are analytical equilibrium models involving
such phenomena as the spin glass models that imply phase transitions and hence could be
viewed as the modern versions of the Hegel–Engels boiling/freezing water example
(Brock, 1993; Rosser and Rosser, 1997). Some involve non-chaotic strange attractors,
fractal basin boundaries or other complicated non-linear phenomena, besides catastrophe
and chaos, although some of these can exhibit them as well (Lorenz, 1992; Rosser and
Rosser, 1996; Brock and Hommes, 1997; Feldpausch, 1997). Virtually all of these
models can be seen to exhibit the sort of dialectical dynamics associated with chaotic
dynamics in terms of bifurcation points generating qualitative dynamical changes and
conflicts between opposing elements driving the dynamics.
In contrast, there are dissipative systems models that imply either fully out-of-
equilibrium dynamics, as in the Brussels School models (Nicolis and Prigogine, 1977)
mode-locking entrainment models (Sterman and Mosekilde, 1994), the Santa Fe adap-
tive stock market dynamics models (Arthur et al., 1997) and ‘edge of chaos’ models
(Kauffman, 1993), or a temporary equilibrium that differs from a presumed long-run
equilibrium as with the self-organised criticality approach (Bak et al., 1993). Many of
these models involve large-scale equations systems and simulations with self-organisation
phenomena emerging from the dynamics of conflicting forces. Such self-organisation has
long been identified by many observers as constituting exactly the kind of qualitative
change that the dialecticians seek, and may represent overcoming the problem of the lack
of new variables or functions emerging associated with the catastrophe and chaos models.
All of these models can be united under the label emergent dynamics complexity.
However, at this point we need to step back a bit and consider how the currents
involving complexity and dialectics have developed. A central point that appears is the
gulf that exists between the analytic Anglo-American tradition and the Continental
tradition. Urban/regional models based on the Brussels School ‘order through fluctu-
ations’ approach (Allen and Sanglier, 1981) exhibited polarising outcomes and multiple
equilibria long before such models became popular in Santa Fe. In a survey of urban/
regional modelling, Lung (1988) attributes this to the tradition of ‘dialectical discourses
of French culture’ in contrast with ‘Anglo-American approaches’, the dialectical tendency
extending beyond the Germanic Hegelian base into Latin Europe as well. Indeed, we
have already seen this with Thom’s willingness to put a dialectical interpretation upon
catastrophe theory.
320 J. B. Rosser Jr
Without doubt the dialectical method/approach is in very ill repute in many Anglo-
American circles, where the emphasis is upon reductionism, positivism, a narrow version
of Aristotelian logic, comparative statics and forecastibility along Newtonian–Laplacian
lines. The dialectical method is viewed as unscientific, fuzzy-minded, and given to ideo-
logical mumbo-jumbo. This latter view has increased, especially in economics with the
increasing tendency for dialecticians in the Anglo-American economics world to be
Marxists. Of course, in Continental Europe, Marxist analysis tends to be more accepted,
but non-Marxist dialectical approaches or interpretations are more widespread, as the
discussions by Thom, Prigogine, and even the possibly dialectical element showing up
in Hayek indicate. Thus, Europeans in general are more willing to admit the dialectical
interpretations of emergent order and self-organisation in complex dynamical systems as
we have presented them above than are their American counterparts.
As a final frisson to this discussion, let us consider somewhat more closely the Stuttgart
School synergetics approach of Haken (1983), which is very closely related to Prigogine’s
Brussels School approach. We can see in this approach the integration of several of our
kinds of non-linear dynamics with their related dialectical interpretations. As with Allen
and Sanglier (1981) and the Brussels School approach, Weidlich and Haag (1987) use the
synergetics approach to model multiple equilibria and polarisation in urban/regional
models, followed by the analytical results of Fujita (1989) and the more recent simulation
modelling in Santa Fe by Krugman (1996). Unsurprisingly, Krugman completely ignores
any dialectical interpretation of the self-organisation phenomenon, reflecting the Anglo-
American bias.
Following Haken (1983, ch. 12), there is a division between ‘slow dynamics’, given by
the vector F, and ‘fast dynamics’, given by the vector q, corresponding respectively to the
control and state variables in catastrophe theory. F is said to ‘slave’ q through a procedure
known as ‘adiabatic approximation’, and the variables in F are the ‘order parameters’
whose gradual (‘quantitative change’) leads to structural change in the system.
A general model is given by
dq/dtAqB(F)qC(F) (5)
where A, B and C are matrices, and is a stochastic disturbance term. Adiabatic approxi-
mation allows this to be transformed into
dq/dt–(A B(F))
–1
C(F) (6)
which implies that the slow variables are determined by AB(F). Order parameters
are those with the least absolute values, and ironically are dynamically unstable in the
sense of possessing positive real parts of their eigenvalues in contrast to the fast ‘slaved
variables’.
This implies a rather curious possibility regarding structural change within the syner-
getics framework. Haken (ibid.) identifies the emergence of chaotic dynamics with the
destabilisation of a previously stable ‘slaved variable’ as the real part of its eigenvalue
passes the zero value and goes positive. Such a bifurcation can lead to a complete restruc-
turing of the system, a chaostrophic discontinuity with more substantial qualitative impli-
cations in terms of the relations between variables, if not necessarily for their existence.
The former slave can become an order parameter, and Diener and Poston (1984) call this
particular phenomenon ‘the revolt of the slaved variables’. If this is not a dialectical out-
come, then there are none in non-linear dynamics.
Aspects of dialectics and non-linear dynamics 321
6. Conclusions
We have reviewed the three main ‘laws of dialectics’ as presented by Engels in The
Dialectics of Nature (1940, p. 26). These are the transformation of quantity into quality
and vice versa, the interpenetration of opposites, and the negation of the negation. We
have seen how such non-linear dynamical models, such as those capable of generating
catastrophic discontinuities, chaotic dynamics and a variety of other complex dynamics
such as self-organisation can be interpreted as manifesting these laws, especially the first
two. In particular, the role of bifurcation is seen as central to implying the first of these
concepts, although we note that we have presented at best a very superficial overview of
these various non-linear dynamical models.
However, we must conclude with a caveat that has floated throughout this paper.
Dialecticians who oppose the use of mathematical modelling at all, who identify such
modelling with ‘arithmomorphism’ and a denial of essential dialectical fuzziness, will
remain unconvinced by all of the above. They will see the kinds of discontinuous changes
implied by the various bifurcations in these models as simply sudden changes in the values
or behaviours of already existing variables, rather than the true qualitative emergence that
cannot be captured mathematically. They might have a harder time maintaining such a
position with regard to complexity models with self-organising or emergent hierarchy
dynamics, but even with these they can make similar arguments that one is simply seeing
different behaviour of already existing variables, however new and different that behaviour
might appear.
Of course, this ‘hard core’ position is exactly that which is derided by the analytic
Anglo-American tradition that sees dialecticians as hopelessly fuzzy and unscientific. The
debate between these strongly held positions can itself be viewed as a dialectic that
remains unresolved.
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