CONSTRUCTIVIST LOGIC AND EMERGENT EVOLUTION IN ECONOMIC
COMPLEXITY
J. Barkley Rosser, Jr.
James Madison University
May, 2008
Presented to K. Vela Velupillai for his festschrift
“The ‘fallacy of composition’ that drives a felicitous wedge between micro and
macro, between the individual and the aggregate, and gives rise to emergent phenomena
in economics, non-algorithmic ways – as conjectured originally by John Stuart Mill…,
George Herbert Lewes … , and codified by Lloyd Morgan … in his popular Gifford
Lectures - may yet be tamed by unconventional models of computation.” --K. Vela
Velupillai (2008, p. 21)
Acknowledgement: I wish to thank Cars Hommes, Roy Weintraub, Vela Velupillai, and
Stefano Zambellli for providing useful materials or comments.
1
I. Introduction: A Plethora of Complexities
This paper will extend further a discussion regarding the relationships between
different views of economic complexity, or complexity more generally, that has been
going on for some time, with Velupillai and some of his associates (2005) broadly on one
side and Rosser (2009) on the other. This earlier round tended to focus on the relative
usefulness of various versions of computational complexity as contrasted with what
Rosser labeled dynamic complexity. Here we shall seek to understand more deeply the
roots of each of these perspectives in logic and evolutionary theory, particularly the
concept of emergence associated with certain versions of evolutionary theory. While this
discussion will highlight more sharply the contrasts in these views, it will also lead to the
position of their possible reconciliation in actually existing economic systems, with this
perhaps a fulfillment of the desire stated by Professor Velupilllai in the opening quotation
provided above.
Following Day (1994), Rosser (1999, 2009) defined dynamic complexity as being
determined by the deterministically endogenous nature of the dynamics of a system: that
it does not converge on a point, a limit cycle, or simply expand or implode exponentially.
This “broad tent” definition includes the “four C’s” of cybernetics, catastrophe theory,
chaos theory, and the “small tent” complexity of heterogeneous agent models in which
local interactions dominate the dynamics. Curiously, while Horgan (1997, p. 303)
considers chaotic dynamics to be complex (and coined the term “chaoplexology” to link
the two concepts together), the list he provides of 45 definitions gathered by Seth Lloyd
2
does not clearly include anything that fits this view that is so widely used by economists.1
Indeed, many of the definitions appear to be variations of algorithmic or computational
complexity, many of them being information measures of one sort or another. This last is
not surprising, as Shannon’s information measure as Velupillai (2000) has argued that it
is the foundation for the Kolmogorov-Solomonoff-Chaitin-Rissanen varieties of
algorithmic and computational complexity. Most of these measures involve some
variation on the minimum length of a computer program that will solve some problem.
While there are several of these measures, a general argument in favor of all of
them is that each of them is precisely defined, and can provide a measure of a degree of
complexity. This has been argued to be favorable aspect of these definitions (Markose,
2005; Israel, 2005), even if they do not clearly allow for a distinguishing between that
which is complex and that which is not. At the same time they allow for qualitatively
higher degrees of complexity. In particular, a program is truly computationally complex
if it will not halt; it does not compute. The bringing in of the halting problem is what
brings in through the back door the question of logic with the work of Church (1936) and
Turing (1937) on recursive systems, which in turn depends on the work of Gödel.
1
The list with the name of the person most associated with the definition in parenthesis is information
(Shannon); entropy (Gibbs, Boltzmann); algorithmic complexity, algorithmic information content (Chaitin,
Solomonoff, Kolmogorov); Fisher information; Renyi entropy; Self-delimiting code length (Huffman,
Shanno-Feno); error-correcting code length (Hamming); Chernoff information; minimum description
length (Rissanen); number of parameters, or degrees of freedom, or dimensions; Lempel-Ziv complexity;
mutual information, or channel capacity; algorithmic mutual information; correlation; stored information
(Shaw); conditional information; conditional algorithmic information; metric entropy; fractal dimension;
self-similarity; stochastic complexity (Rissanen); sophistication (Koppel, Atlan); topological machine size
(Crutchfield); effective or ideal complexity (Gell-Mann); hierarchical complexity (Simon); tree sub-graph
diversity (Huberman, Hogg); homogeneous complexity (Teich, Mahler); time computational complexity;
space computational complexity; information-based complexity (Traub); logical depth (Bennett);
thermodynamic depth (Lloyd, Pagels); grammatical complexity (position in Chomsky hierarchy);
Kullbach-Liebler information; distinguishability (Wooters, Caves, Fisher); Fisher distance; discriminability
(Zee); information distance (Shannon); algorithmic information distance (Zurek); Hamming distance; longrange order; self-organization; complex adaptive systems; edge of chaos. These last three potentially fit
parts of the dynamic complexity concept, and self-organization in particular suggests emergence.
3
In general, one advantage of the dynamic definition is that it provides a clearer
distinction between systems that are complex and those that are not, although there are
some fuzzy zones as well with it. Thus, while the definition given above categorizes
systems with deterministically endogenous limit cycles as not complex, some observers
would say that any system with a periodic cycle is complex as this shows endogenous
business cycles in macroeconomics. Others argue that complexity only emerges with
aperiodic cycles, the appearance of chaos, or discontinuities associated with bifurcations
or multiple basins of attraction or catastrophic leaps or some form of non-chaotic
aperiodicity. So, there is a gradation from very simple systems that merely converge to a
point or a growth path, all the way to fully aperiodic or discontinuous ones. In almost all
cases, some form of nonlinearity is present in dynamically complex systems and is the
source of the complexity, whatever form it takes.2 This sort of hierarchical categorization
shows up in the levels identified by Wolfram (1984; Lee, 2005) which arguably combine
computational with dynamic concepts.
The idea of either emergence or evolution, much less emergent evolution, is not a
necessary part of the dynamic conceptualization of complexity, and certainly is not so for
the various computational ones. However, it has been argued by many that evolution is
inherently a dynamically complex system (Foster, 1997; Potts, 2000), possibly even the
paradigmatic complex system of them all (Hodgson, 2006a). Curiously, even as complex
dynamics are not necessarily evolutionary, likewise emergence is not always present in
evolution or even arguably its core (which is presumably natural selection with variation,
mutation, and inheritability). But, the most dramatic events of evolution have been those
2
Goodwin (1947) and Turing (1952) studied coupled linear systems with lags that could generate complex
dynamics, although the normalized uncoupled equivalent forms of these are nonlinear.
4
involving emergence, the appearance of higher order structures or beings, such as the
appearance of multi-cellular organisms. While most evolutionary theorists reject
teleological perspectives that increasing complexity associated with successive
emergence events (Gould, 2002), the association of emergence with evolution goes back
to the very invention of the concept of emergence by Lewes (1875), who was influenced
by John Stuart Mill’s (1843) study of heteropathic laws, and who in turn influenced C.
Lloyd Morgan (1923) and his development of the concept of emergent evolution.
While many advocates of computational ideas find the idea of emergence to be
essentially empty, we shall see that it reappears when one considers the economy as a
computational system (Mirowski, 2007a). Whatever its analytic content, emergence
seems to happen in the increasingly computational based real economies we observe,
with this emergence being a central phenomenon of great importance. Thus, some effort
of reconciliation of these ideas is desirable, although this effort will first take us on a
journey through the deeper controversies within the logic of computability.
II. The Struggle for the Soul of Mathematics: Constructivism versus Formalism
In a discussion in which one side of the argument has been accused of ultimately
lacking mathematical foundation, the emergent evolution view, it may seem a distraction
to consider a debate within mathematics itself. However, Roy Weintraub (2002) has
argued quite convincingly that to understand the evolution of economic theory one must
understand the parallel evolution of mathematical theory as well, given the deep reliance
upon mathematics that economics indulges in. So, we shall consider a foundational
struggle that remains unresolved in mathematics, whose implications have only recently
5
begun to seep into the awareness of mathematical economists. This is the struggle
between constructivism and formalism.
Formalist mathematics is also known as classical mathematics, and it is the
dominant view within mathematics. While the issues at hand trace back earlier, the first
major public disputes between the schools appeared in the mid-to-late 19th century, as
Leopold Kronecker (1968) attacked the increasing use of infinite sets in formal
mathematics, initially against Weierstrass, but eventually against Georg Cantor (1915),
who was the main developer of the idea of multiple levels of transfinite sets. Kronecker
opposed the use of completed infinities in proofs, and called for an emphasis on the
“intuitively valid” natural numbers in proofs with finite steps, constructive proofs in
which the observer can see exactly what is going on and how the proof is arrived at.3
The next round of this would be between Luitzen Brouwer, of fixed point theorem
fame, and David Hilbert. In addition to calling for constructive proofs, Brouwer (1908)
emphasized the intuitionistic idea that the excluded middle should not necessarily be
excluded. This fit in easily with the questioning of infinite sets as many of the key
theorems of Cantor involved proof by contradiction, including the one establishing that
the real numbers constitute a set of a higher level of infinity than the countably infinite
natural numbers, along with the theorem establishing that there is no highest level of
infinity at all. These theorems were thrown out the window by Brouwer and his
intuitionistic followers. This debate became most heated in the 1920s, with Hilbert
(Hilbert and Ackermann, 1950) supporting the newly emerging ZFC formal axiom
system that was developed by Zermelo (1908) and Fraenkel (1961). Hilbert’s removal of
3
See Barrow (1992, Chap. 5) for an account of the disputes and those between Brouwer and Hilbert, with
Kleene (1967, pp. 186-198) for a more detailed account of the mathematical issues involved.
6
Brouwer from a leading editorial board pushed intuitionism to the fringes until it would
be revived with constructivism by Bishop (1967), supported by the increasing use of
computers in mathematics and the sciences more broadly.
Besides criticizing the use of completed infinite sets and the Law of the Excluded
Middle in formal mathematical proofs, the constructivists also objected to the “C” part of
the ZFC, the Axiom of Choice, which allows for the ordering of infinite sets and indeed
of the real number continuum, and was first proposed by Zermelo. Substantial parts of
topology and such useful concepts as Lebesgue measure depend on at least a limited form
of the Axiom of Choice applicable to countable infinities to prove things about them,
there being different versions of the axiom applying to different levels of infinity (Zorn,
1935). Ironically, the ultimate axiom of choice allowing for the well ordering of all
infinite sets is false (Specker, 1953), although even more ironically this apparently
ultimate victory of constructivism depends on a proof by contradiction.4 Even more
ironically, the Axiom of Choice is much less necessary in nonstandard analysis
(Robinson, 1966) in which infinite real numbers are allowed, and in which one can
differentiate the Dirac delta function (Keisler, 1976, p. 230).5
III. The Struggle Moves to Economic Theory
It has been Velupillai and his associates (Velupillai, 2005; Doria, Cosenza, Lessa,
Bartholo, 2009) who have brought this constructivist critique of formal, classical
mathematics into economics to critique the foundations of much of mathematical
4
For a very neat proof of this falsity, see Rosser (1978, pp. 540-541).
Velupillai (2007) discusses the mathematical complications associated with the Feynman integral
approach to dealing with the difficulties of the Dirac delta function.
5
7
economic theory.6 A general thrust has been to uncover non-constructivist proofs of key
theorems underlying many widely accepted results in economic theory. Thus, Velupillai
(2006, 2009) focuses on the standard proofs of the existence of general equilibrium.
These proofs have generally used fixed point theorems, with that of Brouwer being the
most fundamental. However, this theorem and its relatives such as Kakutani’s, used in
the general equilibrium existence proofs, depend on Sperner’s Lemma, which in turn
depends on the Bolzano-Weierstrass Theorem, which is proven using a proof by
contradiction. Hence, it is non-constructive.7 Velupillai notes that this issue was
understood at some level by the key developers of modern computable general
equilibrium (CGE) models (Scarf, 1982; Shoven and Whalley, 1988), but that they in the
end basically glossed over the issue.
Especially for the latter, the emphasis has been on using CGE models to carry out
policy analyses, with presumably the approximate solutions one obtains given that one is
operating with strictly countably infinite rational numbers “good enough.” Velupillai
(2005, p. 164) has questioned this in practice, citing the peculiar finance function of
Clower and Howitt (1978), which has its rational number solutions distinctly different
from the solutions for the adjacent irrational numbers. He also argues (2005, p. 186) that
roundoff errors associated with these problems can sometimes lead to wildly divergent
outcomes that can lead to real problems, such as the serious misfiring of a Patriot missile
during the First Gulf War that missed its target by 700 meters, thereby killing 28 soldiers
in “friendly fire” in Dhahran, Saudi Arabia.
6
Although the first to specifically consider the implications of the Gödel theorems for economics was
Albin (1982).
7
Needless to say this constitutes yet another irony, that the great intuitionist leader’s most famous theorem
did not follow the constructs of constructivist intuitionism. Brouwer himself became aware of this and
eventually provided an alternative version that did so (Brouwer, 1952).
8
A stream of related papers have shown that when one tries to put various standard
results of economic theory into computable forms, they turn out not to be effectively
computable in a general sense. These results have been found for Walrasian general
equilibria (Lewis, 1991; Richter and Wong, 1999), Nash equilibria (Prasad, 1991; Tsuji,
da Costa, and Doria,1998), more general aspects of macroeconomics (Leijonhufvud,
1993), and whether or not a dynamical system is chaotic (da Costa and Doria, 2005).
The more purely theoretical problems such as the lack of non-constructivity of
basic theorems often involves such issues the law of the excluded middle. However, for
these problems of effective non-computability, the issue often revolves around the
problem of the fact that computers operate with the countably infinite rational numbers
(as are presumably all actual economic data),8 while much of the theory has been proven
assuming the real number continuum. Efforts have been made to overcome the problem
of being able to compute with real numbers (Blum, Cucker, Shub, and Smale, 1998), but
these remain controversial and have not been generally accepted by the advocates of the
constructivist approach.
IV. Computing Emergence
The re-emergence of the emergence concept was associated with the general
emergence of the complexity concept during the 1980s and 1990s, with Kaufmann (1993)
in particular associating it with evolution and the emergence of higher order structures in
the evolutionary process, indeed pushing this process as potentially more important than
8
It is not necessarily the case that all economic data must be rational numbers. Thus, irrational numbers
were first understood from real estate, as in a value that might be based on the length of a diagonal of a
square piece of land. However, conveniently, the set of algebraic numbers is still countably infinite, and
the only transcendental numbers that might serve as the basis for economic data are probably a small finite
set such as л and e.
9
the more traditional Darwinian mechanisms associated with natural selection (although
presumably the higher order structures that survive and reproduce because they are
successful in the evolutionary process of natural selection). Cructchfield (1994) proposed
a more computational form of it, and Bak (1996) associated it with self-organization.
Some other approaches considered in Rosser (2009) include the hypercyle model of
Eigen and Schuster (1979),9 the multi-level evolution model of Crow (1953), Hamilton
(1972), and Henrich (2004), the synergetic model of entrained oscillations (Haken, 1983),
and the frequency entrainment model of Nicolis (1986) as the anagenetic moment
(Rosser, Folke, Günther, Isomäki, Perrings, and Puu, 1994), which draws on Boulding
(1978). McCauley (2004) has criticized these approaches due to their lack of possessing
invariance principles, although he recognizes that symmetry breaking may be a way to
approach emergence in a more formal manner.10 This becomes part of a broader debate
between physics and biology, which we are not going to resolve here, but recognize that
the point must be noted.
While emergence is usually argued to involve hierarchy and the appearance of a
new, higher level within a hierarchical structure, some discussion sees it as a pattern
appearing out of a more disordered state, somewhat along the lines of self-organization.
The frequency entrainment approach of Nicolis can be viewed as an example of this sort
of emergence. Examples of this may well be emergence of a common language (Niyogi,
2006) or of flocking in animals or herding by agents in a market. This latter has been
considered in its general form by Cucker and Smale (2007), drawing ultimately on
9
For further development of this idea as a basis of emergence see its application in the form of hypercyclic
morphogenesis (Rosser, 1991, 1992).
10
McCauley (2005) in particular identified Haken as providing a symmetry breaking approach to
emergence, ultimately depending on the work of Turing (1952). Chua (2005) argues that such symmetry
breaking (and the origin of true complexity) must be based on local interactions.
10
Vicsek, Czirók, Ben-Jacob, and Shochet (1995), in which a set of mobile agents converge
on the same velocity, with birds flocking being the canonical example.
This convergence depends on the elements of an adjacency matrix whose elements
indicate the strength of influences from one bird to another.
However, the more common perspective on emergence is that it involves
processes or events at a lower level of a hierarchy that generate the existence of or a
phenomenon at a higher level of that hierarchical structure. This is more difficult to pin
down in terms of conditions, especially computable conditions. Thus, Lee (2005)
considered a formulation in which emergence was said to occur when a transformational
function led to one state being changed into another that is more computationally
complex as measured by Kolmogorov (1965) complexity, which is the minimum program
length that will halt and describe a state on a universal Turing machine. However, Lee
notes that determining this minimum length of program is itself not computable (any
attempted program to solve it will not halt), a rather basic problem for any
computationally defined measure.
Furthermore, it is not at all clear that because state A is more computationally
complex than state B, that A is therefore at a higher level of a hierarchy. This problem
arises from the very fact that the emergence of a higher (supposedly more
computationally complex) level of a nested hierarchy may arise from some sort of
entrainment of behavior or oscillation at the lower level of the hierarchy. As with
Haken’s slaving principle and adiabatic approximation, the convergence or entrainment
at the lower level means that what were viewed as distinct elements requiring individual
modeling at the lower level can be aggregated and viewed as a single element at the
11
higher level. This can result in the higher level dynamics being describable by a shorter
length program due to this simplification through aggregation. Hence, it may well not be
that the higher level must describable by a program of shorter length than the shorter
level, even though in reality the lower level is embedded within the higher level.
Lee then proceeds to consider the hierarchy aspect more directly. This leads to
following Langton (1992) in comparing the four level Chomsky (1959) hierarchy with
the four level hierarchy of Wolfram (1984). Chomsky’s categorizes each of four levels
according to computational power are regular language, context-free language, contextsensitive language, and recursive enumerable language (that is, a Turing machine), which
Mirowski (2007a) sees as a natural way to distinguish market forms. That of Wolfram is
more closely linked to dynamic conceptualizations and refers to cellular automota
systems, moving from systems that converge on a homogeneous state (Class I), ones that
evolve toward simple separated structures or periodic structures (Class II), ones that
evolve towards chaotic dynamical structures (Class III) , and ones that evolve to complex
localized structures (Class IV).
Curiously, Langton’s formulation seems to invert the latter two categories. He
defines an order parameter λ that depends on the number of finite cell states, their
neighborhood size, and the number of transition paths. As this parameter rises one sees
the system passing from Class I to Class II to Class IV and finally to Class III, with the
sharpest jump in the parameter being between the second and third of these. He then
suggests that from the computational perspective the first two are halting, the third is
undecidable, and the fourth is non-halting. It is Crutchfield (1994) who links such a
passing through a hierarchy of computational classes as emergence, and Jensen (1998)
12
who ties all this to self-organized criticality, although these formulations have come
under much criticism from various quarters, such as McCauley, who suggests the idea of
complete surprise and unpredictability following Moore (1990) as a better foundation for
the idea of emergence. Returning to Lee’s argument, the highest level in which nonhalting takes place is that of the universal Turing machine.
V. Emergence in Market Dynamics
So, much of this has operated at a fairly abstract level. Does it have anything to
do with real markets? Vernon Smith (1962, 2007) emphasizes how the experimental
evidence (and actual evidence in many constructed markets) has shown that certain kinds
of market structures are consistent with very rapid approaches to equilibria, particularly
the double-auction form, a result strongly reinforced by Friedman and Rust (1993).
What, if anything, does all this discussion of problems of computability and emergence
have to do with real markets, where we can see mechanism designs that regularly
generate equilibrium outcomes?
A challenging view of this has been provided controversially by Mirowski
(2007a) in his theory of markomata, of markets as evolving forms of algorithms, that the
fundamental essence of markets is that they are algorithms, not merely that they can be
simulated by computers. This then suggests that computability issues are deeply inherent
in the structure and functioning of markets. The link with emergence is that he sees
market forms evolving, and that over time higher Chomskyian hierarchical levels emerge
that nest the lower levels, with these higher levels taking the form, for example, of futures
markets, markets for derivatives, and so forth.
13
This brings our discussion from the previous section into play rather forcefully.
There we saw the argument that lower levels of either Chomskyian or Wolframian
hieararchies tend to be both “better behaved” (less dynamically complex) and also more
computable. As one moves to the higher levels one eventually moves to the level of a
universal Turing machine, but one also approaches the zone of halting problems and
other matters of non-computability of one sort or another. The lack of effective
computability of general equilibrium begins to loom in a more serious fashion as one
considers the possibility that real world markets may be a series of unresolved hierarchies
with varying degrees of computability. Lower level spot markets constructed as double
auctions may function (compute) very well and rapidly achieve solutions. But, as these
become embedded in higher order markets involving derivatives made out of derivatives
that agents do not even understand when they are trading them, potential problems can
arise, and the difficulties observed in financial markets in recent years may well represent
exactly such phenomena playing themselves out, systems breaking down because of their
ultimate failure to compute at the highest levels.
Many criticisms were leveled at Mirowski’s argument in the symposium
accompanying his article, with some focusing on institutional elements and the argument
that markets are social processes of exchange between humans within certain institutional
setups (Conlisk, 2007, Kirman, 2007; also separately, Hodgson, 2006b). This form of
criticism downplays or even denies the idea of a market as an algorithm. However,
another vein of criticism came from the other direction (Zambelli, 2007), asserting that
Mirowski had failed to properly account for issues of computability, in particular that he
did not understand that the market structure could well collapse down to universal Turing
14
machine, that he was wrong to asset the finite nature of lower level markets in the
hierarchy. While finite mechanisms are computable, there are an infinite set of these
finite mechanisms available to a universal Turing machine, and that even though it is
finite in a sense, the fact that in effect it is unbounded makes it as if is infinite (Odifreddi,
1989).
I find myself substantially agreeing with Mirowski’s (2007b) response to this
latter argument. He agrees with several critics that the Chomskyian hierarchy may be too
simplistic, too “flat” in its lower levels, and then suddenly jumping up to a much higher
level of computational complexity, sort of a “1,2, infinity” story. He states his desire for
a more nuanced and fully gradated hierarchy, but then contrasts his view with the usual
view that simply reduces all markets and market mechanisms to the same level,
especially the general equilibrium view that assumes one big central decisionmaker that
simultaneously solves all markets. This is surely a contrast with the world of Vernon
Smith and of Friedman and Rust with their separately solving markets, even if these do
quite well all by themselves. The problems arise when they interlink and aggregate and
become embedded in the higher-order markets. The lower level markets are simpler and
finite and easily solved. It is as one moves up the line to the higher levels, to the
potential general equilibrium that keeps racing out of one’s reach as the market
constantly evolves newly emerging forms at the higher levels of aggregations of
derivatives upon derivatives upon options upon futures upon spot markets, and so on.11
It must be recognized that despite Mirowski’s remark about “1,2, infinity” what is involved here is not a
leap to infinity and thus to true undecidability or non-computability. The problem is more one of an
increasing complexity that actually existing markets and computers are unable to keep up with in fact, if
not in principle.
11
15
Besides the evidence of market malfunctions in collateralized debt obligation
markets and other such esoterica, other theoretical work is appearing that is consistent
with this form of Mirowski’s argument, and especially of its potential for an open-ended
non-computing that can lead to all kinds of destabilizing and complex dynamics. In
particular, while they avoid the question of hierarchical levels as such, Brock, Hommes,
and Wagener (2008) establish that contrary to the usual view that increasing the number
of available assets in financial markets stabilize and improve efficiency by spreading risk,
they may destabilize markets with heterogeneous agents, especially if those agents
engage in trend extrapolating behavior, which has long been known to be destabilizing in
asset markets (Zeeman, 1974).
They discuss a variety of cases, including ones where the standard results hold,
but it would appear that it is possible for the two results to hold simultaneously with some
minor adjustments to their models. Thus, it may be the case under certain circumstances
that an increase in the number of assets could both increase efficiency and even local
stability in the markets, even as they threaten the broader resilience of the markets and
their global stability. This might be manifested in a particularly extreme form if one were
to simultaneously observe a decrease in variance coinciding with an increase in kurtosis,
increasingly obese fat tails coinciding with a hugging of the mean in the middle. Such an
outcome would be an economic equivalent of the “stability-resilience” tradeoff long
posited to hold in ecological systems by Holling (1973).
VI. Conclusion
16
This paper has reconsidered the argument over the nature of economic complexity
as to whether it is best thought of in computational terms or in dynamic terms, taking as
the accompanying idea of the latter the idea of emergence in evolutionary processes. The
analysis of computational complexity took us into the deeper, revived debates now
flowing into economics from logic and metamathematics over constructivism and
formalism. The latter was found to be on a somewhat weak footing from the standpoint
of computability theory, relying on various theories of hierarchies and the emergence of
new levels in them, none of these fully developed or satisfactory, despite the use of such
concepts as bifurcation via symmetry breaking.
A possible meeting point of these ideas is seen in the markomata idea of
Mirowski, with his vision of markets being simultaneously algorithmic systems that are
evolving higher order structures. This evolutionary process drags the system into
increasing problems of computability, and this idea may well be a key way of considering
the sorts of difficulties observed in recent years in global financial markets.
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