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Scientific & Mathematical Roots of Complexity Science

Jeffrey Goldstein, Ph.D.
School of Management and Business
Adelphi University
Garden City, New York

The following list describes each of the scientific and mathematical disciplines displayed on the accompanying "whale" diagram. After each description are three items

  1. Specific themes from complexity science running through or arising from each discipline;

  2. The names of prominent researchers/theorists in each field (admittedly incomplete due to the many people who have made contributions in each field); and,

  3. related terms from the glossary in this volume.

Algorithmic Complexity Theory:

One of the important sources of contemporary conceptions of what is complex about complex systems. Specifically, algorithmic complexity is a measure of complexity developed by the mathematician Gregory Chaitin based on earlier work in Information Theory founded by Claude Shannon and work on probability and information conducted by the by the Russian mathematicians Kolmogorov and Solomonoff. Algorithm complexity theory defines and measures complexity in terms of a computer algorithm (or computer program) which could generate the data coming from a particular complex system. In other words, the degree of a system's complexity is a matter of how large a computer program would be needed to generate a bit string derived from the system under question (sequence of 0's and 1's, or the binary code at the core of computer languages). Measures of complexity utilized in the study of Artificial Life and similar cellular automata have been heavily influenced by Algorithmic Complexity Theory.

Themes: Definition and measure of complexity; relation of complexity to both randomness and order; recognition of the novelty of emergent structures; predictability and unpredictability of complex systems.

Researchers/Theorists: Gregory Chaitin, Charles Bennett, Murray Gell-mann

Glossary: Algorithm; Complexity (and Algorithmic Complexity); Logical Depth

Artificial Intelligence (AI):

The design of "smart" machines and robots which, obviously, have tremendous ramifications in our "Information Age." By exploring what intelligence means to humans in order to mimic it in machines, AI has been instrumental in the recent explosion of research in the cognitive processes of human beings. In addition, the development of intelligent machines has important implications for computational theory. AI has facilitated the search for basic structures of a complex system complex enough to be able to think. Consequently, AI has explored such themes as the hierarchical relationship of cognitive mechanisms, devices for simplifying or complexifying the dynamics of systems, and the elaboration of how interconnectivities effect the functioning of a complex system. Artificial Intelligence was partly spawned from earlier work in Cybernetics with servo-mechanisms, and has been influential in modern Computational Theory.

Themes: How complex systems process information; insight into cognitive processes occurring within and between human beings; the role of hierarchy in complex systems

Researchers/Theorists: Herbert Simon, Marvin Minsky, Roger Shank, Douglass Hofstadter, Danny Hillis. A significant and vociferous critique of some of AI's conclusions applied to human cognition has been the philosopher John Searle.

Glossary: Complexity; Hierarchy

Artificial Life:

The study of the life-like patterns emerging in cellular automata and related electronic networks. Pioneered by the computer scientist Chris Langton, and experimented with extensively at the Santa Fe Institute. The study of Artificial Life is promising insights into natural processes leading to the build-up of structure in self-organizing, complex systems. It is closely allied with research into Random Boolean Networks (Stuart Kauffman) and Emergent Computational Theory.

Themes: Computer simulations exhibiting self-organizing processes and emergent structures

Researchers/Theorists: Chris Langton; Doyne Farmer; Norman Packard; Thomas Ray; William Sulis

Glossary: Artificial Life; Cellular Automata; Boolean Networks; Emergence; Self-organization


A theory concerning what accounts for the essence of a living organism as opposed to a nonliving entity. Developed by the Chilean scientists Humberto Maturana and Francisco Varela, the theory of autopoiesis suggests that a living organism can be understood as a circular, autocatalytic-like process having its own survival as its main goal. The phenomenon of self-organization has sometimes been understood in terms of autopoeisis. The theory's emphasis on the circular "closure" of the living organism can be seen as a "remedy" for the over emphasis on "openness" found in "open systems" theory. Theories of autopoeisis have been used in discussions of the emergent structures in Artificial Life and other cellular automata.

Themes: How self-organizational processes require some kind of boundary or containment; the self-referential aspects of complex systems

Researchers/Theorists: Humberto Maturana; Francisco Varela

Glossary: Autopoiesis; Boundaries; Self-organization

Boolean Networks:

Electronic arrays developed by the medical researcher and evolutionary biologist Stuart Kauffman. These arrays are used to study self-organizing processes and the emergence of new, unexpected structures. The nodes in these arrays are connected to other nodes according to certain "boolean" or logical rules. Using the N/K Model of Boolean Networks yields insights into how manipulating the rules, the number of traits, and the number of inputs, leads to various self-organizing, emergent patterns. Of particular importance is the use of the construct of "fitness landscapes" which are graphical representations of the adaptive or fitness values of various modifications of genetic (and analogous) materials. The study of random, Boolean networks has provided important insights into how natural adaptive may occur, i.e., how innovations arise and the conditions needed to facilitate innovation.

Themes: The dynamics of adaptation, innovation, and learning; understanding the emergence of order (Kauffman's "order for free") out of the nonlinear dynamics of the networks

Researchers/Theorists: Stuart Kauffman; William Macready

Glossary: N/K Model; Random Boolean Networks

Catastrophe Theory:

A mathematical theory in the field of topology formulated by the French mathematician Renee Thom. A catastrophe is a discontinuous change during the evolution of a system modeled by structural equations and topological folds. Catastrophes are governed by control parameters whose changes of values leads either to smooth transition at low values to abrupt changes at higher, critical values. Catastrophes indicate points of bifurcation in dynamical systems. Catastrophe theory provides critical insights into occurrences of abrupt change in complex systems.

Themes: Insight into abrupt changes in complex systems

Researchers/Theorists: Rene Thom; Christopher Zeeman; Stephen Guastello

Glossary: Bifurcations; Catastrophes

Chaos Theory:

The study of dynamical systems characterized by sensitivity to initial conditions so that although the behavior is constrained within a particular range, the future behavior of the system is largely unpredictable. Unlike a random system which is also unpredictable, chaos is brought about by deterministic rules. Such systems are constituted by nonlinear, interactive, feedback types of relationships among the variables, components, or processes in the system. Chaos was first glimmered by the great French mathematician Henri Poincare a century ago. However, it wasn't until 1963 that the metereologist Edward Lorenz "discovered" chaos in data runs on a computer program he was using to model the dynamics of the weather. The term "chaos" was coined by the mathematicians Li and Yorke a decade later for a kind of aperiodic but bound behavior in mathematical systems of coupled differential equations. Chaos Theory has become an umbrella term for the study of many types of nonlinear, complex systems.

Themes: How small changes can have a disproportionately large effect on a complex system; the role of attractors in understanding the behavior of complex systems; revising of the nature of the dichotomy between orderly and random

Researchers/Theorists: Edward Lorenz; Jim Yorke; Ralph Abraham; Fred Abraham; Robert May; Doyne Farmer; Norman Packard; Robert Shaw; James Crutchfield

Glossary: Attractors; Chaos; Sensitive Dependence on Initial Conditions

Complex, Adaptive Systems Theory:

The study of complex, nonlinear, interactive systems which have the ability to adapt to a changing environment. Such systems are characterized by the potential for self-organization and exist in a nonequilibrium environment. CAS's evolve by random mutation, self-organization, the transformation of their internal models of the environment, and natural selection. Examples include living organisms, the nervous system, the immune system, the economy, corporations, societies, and so on. The Santa Fe Institute is known as the major center in the world for the study of CAS's.

Themes: How complex, nonlinear, interactive systems adapt to a changing environment along with other complex, adaptive systems in a co-evolutionary manner

Researchers/Theorists: Murray Gell-mann, Brian Arthur, Chris Langton, Doyne Farmer, Norman Packard, Stuart Kauffman, John Holland, William Sulis

Glossary: Adaptation; Complex, Adaptive Systems; Complexity

Computational Theory:

Research into the functioning, capabilities, and limitations of computers. Pioneered by the work of the remarkable English mathematician Alan Turing (who helped break the famous Enigma Code used by the Germans during WWII), and John von Neuman (the Hungarian born but US based mathematical prodigy), computational theory investigates such issues as the nature of algorithms, computer languages, and the applicability and usefulness of various types of computation to difficult problems in mathematics, the sciences, and other practical work with real world complex systems. A major research agenda of computational theory has been to delineate the nature of the complexity of various complex systems. Included in this is research into what defines a computable versus a noncomputable problem. Moreover, computational theory has provided us with the crucial distinction between hardware and software.

Themes: Computability as a way of talking about the complexity of a system; a way of typing complex systems according to their ability to process information (whether in man-made computers or in the naturally-occurring systems like the brain, ecosystems, and the immune systems.

Researchers/Theorists: Alan Turing, John von Neumann, Douglass Hofstadter, John Holland, Danny Hillis (and countless others as this has become a dominant scientific and mathematical field)

Glossary: Church-Turing Thesis; Information; Turing Machines

Condensed Matter and Solid-state Physics:

That branch of physics having do with solid state or condensed matter exhibiting such phenomena as magnetism and modeled by such constructs as spin-glasses or how metal atoms can be modeled in terms of their electron "spins" having an influence like positive or negative feedback on neighboring atoms. The dynamics of "spin-glasses" have been influential in the formulation of Kauffman's N/K models used in his Random Boolean Networks and other complex, adaptive systems. Such models yield insight into the dynamics of interactive systems through the changing of connectivity rules and the exploration of the ensuing emergent phenomena.

Themes: How to mode the behavior of interconnected systems in terms of coupling between components and various means for moving the system into and out of equilibrium states

Researchers/Theorists: Philip Anderson; Daniel Stein; Richard Palmer; Bernard Derrida; Gerald Weishbuch

Glossary: Coherence; Feedback; Parameters (Order); N/K Models


The study of control mechanisms such as thermostats, guided missile guidance systems, and other early "smart" machines. Pioneered by the mathematicians Norbert Wiener and John von Neumann, the term "cybernetics" comes from "cyber" or "steer" in Greek. Cybernetics is interested in how machines can be constructed to "steer" themselves such as in guided missiles. After World War II, Cybernetics was instrumental in the development of Artificial Intelligence and General Systems Theory. Cybernetics ideas spread to a host of other fields including physiology, neuroscience, operations research, various engineering disciplines and so on. Along the way, cybernetics became interested in machine learning and thus provided a foundation for Artificial Intelligence and was the first field where ideas of self-organization were conceived. Cybernetics has made much use of the concept of equilibrium and has conceived of self-organization in terms of the self-regulation of equilibrium-seeking systems. Furthermore, cybernetics posits the need for a "requisite variety" between the internal states of a system and the variation in its environment. In this way, cybernetics has laid important groundwork for the study of adaptation of complex systems to a complex environments.

Themes: How systems can be understood in terms of the dynamics of negative and positive feedback; how systems can regulate their own behavior; adaptational, tranformative, and learning processes in complex systems

Researchers/Theorists: Norbert Wiener, W. Ross Ashby, Heinz von Foerster, Arthur Burks, Gregory Bateson (applied to psychology and social systems), and Stafford Beer (applied to businesses)

Glossary: Equilibrium; Feedback; Information

Dynamical Systems Theory (and Nonlinear Dynamical Systems Theory)(NDS):

The mathematical discipline which studies how systems evolve over time according to the dynamics of their equations. Emerging from classical mechanics, the study of differential equations, and topology, dynamical systems theory utilizes the constructs of nonlinearity, attractors, bifurcations, and phase (state) space to talk about transformations of system behavior. Dynamical systems are usually considered deterministic systems, although they can be influenced by random events. Much of the early work was done by Russian mathematicians who had a head start on the study of nonlinear dynamics. Dynamical Systems Theory has conceptualized many of the fundamental principles on which complexity sciences depend. It is the grandparent of chaos theory.

A further development of dynamical systems theory bringing in research into systems modeled by nonlinear differential and difference equations. The mathematics of NDS were instrumental in the development of Chaos Theory, particularly the concepts of attractors, bifurcation, phase portraits, and measures of stability such as Lyapuonov Exponents. Prominent contemporary theorists include the mathematicians Steve Smale and Ralph Abraham.

Themes: Transitions systems through different attractor regimes; how systems can be influenced by very small changes (fluctuations or perturbations)

Researchers/Theorists: Henri Poincare; Steve Smale; Ralph Abraham; Leon Glass (biology); Ary Goldberger (medicine)

Glossary: Attractors; Bifurcation; Catastrophes; Chaos; Initial Conditions; Stability

Emergent Computation Theory:

Research into the computational capacities of emergent structures in complex, self-organizing systems that can be used to measure the complexity of these structures. It recognizes emergent phenomena by their information processing capacity. That is, one can understand the emergent phenomena found in complex, adaptive systems by their their innate potential for processing information. Growing-out of work in Chaos Theory and Artificial Life, Emergent Computation Theory has postulated that a way to measure the complexity of a system is to ascertain what specific type of Turing Machine can be most effectively model of a complex system's time series measurements.

Themes: Emergent structures as an intrinsic feature of complex systems to generate innovative structures

Researchers/Theorists: James Crutchfield; Melanie Mitchell; James Hanson

Glossary: Complexity; Information; Time Series; Turing Machines

Evolutionary Biology:

Biological theory of evolution originating in the work of Charles Darwin. It studies the process of evolution leading to the appearance and disappearance of species through the mechanisms of random mutation and natural selection of the fitter mutants. As such evolutionary biology has laid the foundation for our understanding of adaptation of living organisms to changes in their environment. These ideas of adaptation are providing a template in which to understand processes of adaptation in all complex, adaptive systems particularly the work of John Holland and Stuart Kauffman.

Themes: How processes of adaptation can be understood as the result of random mutations, recombination of genotypes, and natural selection; the crucial role of the "edge of chaos" as a zone where adaptive experimentation may be at is optimum.

Researchers/Theorists: Charles Darwin (and his followers); Later on Jacques Monod, Stephen Jay Gould, Richard Lewontin, Richard Dawkins, Stuart Kauffman

Glossary: Adaptation; Edge of Chaos; Genetic Algorithms; N/K Model

Evolutionary Systems Theory:

Syntheses of evolutionary biology with congruent concepts from Cybernetics, General Systems Theory, and Dynamical Systems Theory, serves to integrate many fields in terms of principles of system development and transformation. The journal World Futures features articles on Evolutionary Systems Theory.

Themes: General constructs from the theory of evolution applied across a great many complex systems; emphasis on evolutionary transformation

Researchers/Theorists: Ervin Laszlo, Vilmos Csanyi, Rod Swenson, Sally Goerner

Glossary: Bifurcation; Chaos; Complexity; Complex, Adaptive Systems

Far-from-equilibrium Thermodynamics:

The study of self-organization in physical systems founded by the Russian-born Belgian physical chemist Ilya Prigogine, winner of the Nobel Prize in chemistry. Self-organization has been studied from a thermodynamics perspective considering the relation between the build-up of structure seen in thermodynamics versus the supposed tendency of an increase of entropy (from the Second Law of Thermodynamics) to tear down form. Self-organizing, emergent patterns are termed "dissipative structures." Many of the ideas are revisions of earlier thermodynamic concepts applied to the build-up of organization in a physical systems.

Themes: Self-organization understood as a process occurring in a nonlinear system at a far-from-equilibrium system; how complex systems can take advantage of random events in the build-up of new forms

Researchers/Theorists: Ilya Prigogine; Gregoire Nicolis

Glossary: Dissipative Structures; Equilibrium; Far-from-equilibrium; Self-organization

Fractal Geometry:

A geometrical pattern or set of points which is self-similar on different scales. The geometry of this pattern does not fall within the normal whole dimensions one, two, or three. Instead, a fractal is "in-between" one and two or two and three and so on dimensions. For example, the coast of England can be understood as a fractal, because as you observe from closer and closer points of view (i.e., changing the scale) it keeps showing a self-similar kind of irregularity. Fractal dimensionality is one way to measure the complexity of a dynamical system. Furthermore, strange attractors have a fractal dimensionality. Fractals have become popular through the amazing imagery of graphical depicted Mandelbrot or Julia Sets. The study of Fractal Geometry has been a great aide in discovering universal principles in complex systems, scaling phenomena being one of these. Moreover, fractals can represent power law distributions.

Themes: Understanding aspects of complexity in terms of repeated irregularities on different scales; the benefits conferred on a system from having a fractal structure

Researchers/Theorists: Benoit Mandelbrot; Michael Barnsley

Glossary: Attractor; Chaos; Complexity; Fractal

Game Theory:

Originally developed by the great mathematician John von Neumann and the economist Oscar Morgenstern, Game Theory explores the various outcomes when interactive, semi-autonomous agents engage in either cooperative and noncooperative behavior. For example, in the famous Prisoner's Dilemma Game, two agents or "players" are arrested for armed robbery, and the different outcomes of their resulting cooperative or noncooperative strategies in the face of the district attorney's deal-making are assessed. Game Theory constructs are helpful in understanding the global effect of local "rules" (i.e., the various strategies used by the agents), and thereby, it is another complementary framework for understanding adaptation.

Themes: Emergence of global patterns in complex systems according to the rules or strategies followed by interactive agents

Researchers/Theorists: John von Neumann; Oscar Morgenstern; Bernardo Huberman; Natalie Glance; Robert Axelrod

Glossary: Adaptation; Emergence

General Systems Theory:

Following from earlier work in Cybernetics, Information Theory, and Evolutionary Biology, General Systems Theory attempted to search for general principles of system across diverse scientific disciplines. As such, it provides a precursor to the similar search for general principles in Complex, Adaptive Systems Theory. Key ideas include negative feedback, stability, equilibrium-seeking, self-regulation, and "open systems" referring to the need for vital systems to be in active exchange with their environments.

Themes: The search for general principles of the dynamics of living and other complex systems

Researchers/Theorists: Ludwig von Bertalanffy

Glossary: Adaptation; Equilibrium; Self-organization

Genetic Algorithms:

A type of computer program developed by the computer scientist John Holland whose strategy of arriving at solutions is based on principles taken from genetics. Basically, the genetic algorithm utilizes the mixing of genetic information in sexual reproduction, random mutations, and natural selection at arriving at solutions. The use and study of genetic algorithms has been instrumental in the development of a more general Complex, Adaptive Systems Theory.

Themes: Inquiry into principles of learning and adaptation; designing evolving computer programs

Researchers/Theorists: John Holland

Glossary: Attractor; Chaos; Complexity; Fractal

Information Theory:

Formulated during and after World War II, Information Theory focussed on measurements of the amount of information a communications channel could contain. "Information" refers to the degree of variety versus redundancy capable of being transmitted electronically. Information Theory has been a keystone in the development of the study of self-organization and complexity as well as computational theory. Self-organization can be conceptualized in Information Theory in terms of the paradoxical nature of "noise" (or random fluctuations or perturbations) as either disorganizing or organizing. Complex systems can be understood as information-processing mechanisms. Information is now being used as a general concept linking all types of systems physical, social, computational.

Themes: Information can be seen as the cognate in a social system as what energy is in a physical system; search for general principles of information across many types of systems

Glossary: Information; Novelty; Redundancy

Researchers/Theorists: Claude Shannon; Norbert Wiener; Henri Atlan (biology)

Neural Nets:

An outgrowth of Artificial Intelligence, Neural Nets are electronic automata used to for machine learning that are based on associative theories of human cognition. Using various algorithms, they are often programmed to learn how to recognize a pattern. Changing the rules of interaction between the "neurons" in the network can lead to interesting emergent behavior, so in that way, neural nets are another tool for investigating self-organization and emergence. Many believe neural nets are a better model of the way the living brain works than the operation of digital computers. The investigation of neural nets is providing a great many insights into emergent patterns in complex systems. Moreover, the study of neural net pattern recognition is providing insight into how the brain may function in its perception of patterns in the environment.

Themes: Another example of complex systems composed of interacting semi-autonomous agents that can adapt and learn; insight into pattern recognition in complex system and the build-up of internal models

Researchers/Theorists: J.J. Hopfield; T. J. Sejnowski

Glossary: Adaptation; Genetic Algorithms; Neural Nets

Self-organized Criticality (SOC):

Research and theorizing about natural, abrupt changes formulated by the physicist Per Bak. Systems are viewed as evolving naturally, in a self-organizing manner, to a critical state at which abrupt changes can occur which abrupt changes can occur. Examples of such systems include plate tectonics leading to earthquakes, avalanches, sudden stock market dips or surges as well as crashes, and so on. By considering such systems "weakly chaotic" and exploring them in terms of power laws, Per Bak has contrasted them with "strongly chaotic" systems. Some of the themes of SOC have been incorporated by Stuart Kauffman into his ideas on the "edge of chaos."

Themes: Understanding of abrupt, cascading or "avalanche" type of change in a complex system; another picture of systems being in a poised state or readiness of change; Information can be seen as the cognate in a social system as the energy in a physical system.

Researchers/Theorists: Per Bak; Chao Tang; Kurt Wiesenfeld

Glossary: Bifurcation; Chaos; Power Law; Self-organization; Stability


The study of self-organizing systems initiated by the German physicist Hermann Haken, who did early research on the emergence of coherence in lasers and other emergent phenomena in physical systems. Synergetics emphasizes the exploration of order parameters which move the focus of studying complex systems from the lower level of components up to the level of the emergent structures. The term "Synergetics" has become roughly synonymous to complexity science in Europe and Russia.

Themes: Understanding emergent phenomena in terms of the order parameters determining their coherent structure

Researchers/Theorists: Hermann Haken

Glossary: Coherence; Parameter (Order); Self-organization

System Dynamics:

Understanding the dynamics of complex systems in terms of a network of interlocking negative and positive feedback loops, e.g., how the functions of production, inventory, ordering, and shipping are interrelated. Diagramming complex systems with the visual aides of System Dynamics can help in indicating how changes will effect other parts or subsystems of the system. System Dynamics also provides practice in thinking systemically about systems, i.e., conceiving of the overall "holistic" interaction of components. System Dynamics has been influenced by Cybernetics and General Systems Theory, and more recently has included some elements of Dynamical Systems Theory and Complex, Adaptive Systems Theory, Synergetics, and Far-from-equilibrium Thermodynamics.

Themes: Another way to conceptualize complex systems as interacting semi-autonomous units influencing one another through positive and negative feedback

Researchers/Theorists: Jay Forrester; George Richardson; Peter Senge

Glossary: Feedback


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