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"Science has explored the microcosmos and the macrocosmos; we have a good sense of the lay of the land. The great unexplored frontier is complexity." - Heinz Pagels in "The Dreams of Reason"

A system has high complexity or is very complex, if there is no simple description or definition, i.e. if it is hard to define or if it can not be described in a simple way. Complexity is the opposite of simplicity and uniformity, complex is everything that is not simple. A complex system is difficult to understand, because it can not be fully explained by an understanding of its isolated parts and components. As Sunny Y. Auyang says in her book Foundations of Complex Systems Theories, "simplicity may have a unified form, but complexity has many varieties" (p.9). In this sense complexity can be viewed simply as variety or variation.



Complexity means there is no simplicity. Something is complex if it is *not* simple. The opposites of simple things are always hard to describe. As it is common for many buzzwords, it is not only hard to describe, but also very fascinating and interesting. Complexity is a typical buzzword. It is inherently difficult to give a clear and concise definition of complexity. Maya Paczuski argues that "There seems to be a trade-off between precision in the definition and generality of the definition" [1]. Complexity is a term which is very difficult to define, because it describes something which is hard to define. There are roughly two ways to define complexity: first, as the amount of difficulty in the process of understanding, describing and explaining systems ("how much effort is needed in the description process" or the length of the description). This dynamic view considers complexity as a part of a description and explanation process. Second, we can define complexity independent of the description process as a system property ("how is a system organized and structured" or the length of the blueprint) related to organization and structure. This organizational view considers complexity as a static property of a system, and depends on the number of parts, interactions, structures on the different scales of the system.

Origin and Creation

Basic Principles
Natural Selection
Red Queen Effect
Swarm Intelligence
Self-Organized Criticality
Butterfly Effect
Control Loop
Lever Point
Frozen Accidents
Path Dependence
Complexity, Simplicity
Edge of Chaos
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Complexity can arise from simplicity in different ways, for example through the interaction of opposite forces, by repeated application and iteration of simple rules, or by evolution.

Unification of Opposites

Complexity arises and emerges at the most basic level from the clash, combination or interaction of opposite and antagonistic forces, for instance an attractive and a repulsive force, a contracting and an expanding influence, a squeezing and a stretching effect, a splitting and a merging component. It is characterized by consonance in dissonance, stability in instability, regularity in irregularity, tractability in intractability, predictability in unpredictability, order in chaos, simplicity in intricacy and most of all by unity in diversity. It is said to be maximal at the edge of chaos.

Complexity arises from the sunshine in the rain as rainbows, from the sense in nonsense in form of insights, from the collision of reason and emotion in social relations, leading to a combination of opposite feelings, as we know well since Shakespeare's epic work. Complicated situations arise from hesitation in action (Hamlet), from love in hate and despair in love (Romeo and Juliet), from pleasure in displeasure, etc.

The Oxford Dictionary defines something as "complex" if it is "made of (usually several) closely connected parts. As Heylighen says, here we find "the basic duality between parts which are at the same time distinct and connected", the unity (connected parts) in diversity (distinct parts) which is characteristic of many complex systems.

Simplicity and Complexity

Simple computational rules can like simple mathematical equations generate complex, seemingly unpredictable random-looking behavior. It is perhaps the most fundamental idea that unifies science. Without this fundamental fact, science would be impossible, because we would never be able to explain, to describe or to predict complex phenomena in terms of simple rules or equations.

In mathematics and physics it is a well-known fact that simple systems or equations result in very complex behavior, especially in "Chaos Theory". Wolfram observed it in his NKS book for complex system that can be described by CA (Cellular Automata): "a simple program can produce output that seems irregular and complex".

Yet it's difficult to determine the degree of simplicity of the rules responsible for a particular phenomenon. It is hard to say if a certain complex observed phenomenon is caused by a set of simple rules or by other more complex rules.

CA (Cellular Automata), IFS (Iterated Function Systems), fractals and classical recursion are based on "feedback". The heart or core of these systems is the recursive, repeated or "iterative" step, rule or transition function which describes the next state of the system in terms of the system's current state. Of course you need also to specify the initial conditions.

Repeated application of simple rules can lead to complex results, and repeated application of non-random, regular and predictable rules can lead to random-looking, irregular and unpredictable patterns that are only predictable on certain scales through stochastic and probabilistic methods. Deterministic laws on the microscopic scale do not imply deterministic laws on the macroscopic scale. Although the immediate short-term evolution of the system is governed by and predictable in terms of definite laws, the long-term evolution can be unpredictable (in chaos theory this is known as sensitivity to initial conditions).

Complexity is related to unpredictability, irregularity and intractability: it is observed when known contemporary methods of analysis fail to reduce the behavior of a system to a program or set of equations that can predict the state of a system at a given moment with less effort than just describing it detailed bit by bit. Again, this does not mean there is no predictability or regularity at all, often only at a different level or scale, for instance regularity, tractability and predictability on a microscopic level and irregularity, intractability, unpredictability on a macroscopic level.


Complexity can also increase in a system during the course of evolution. This increase is often neither steady nor uniform. Although complexity has increased dramatically in the course of evolution, and the universe indeed spins out intricate and complex structures, it doesn't do it uniformly or effortlessly, and the complexity does not increased everywhere in the universe during evolution. Our point of view as humans is very deformed and distorted, because we live on a small "island of complexity". Our planet earth is a small island with very high complexity in a large sea of void and simplicity. Our planet is the exception. More complex forms are often based on less complex structures. The more complex a system becomes in the course of evolution, the more localized it seems to be. Planets are rare in our galaxy, and a habitable planet like Earth is very rare. The most complex structures can often be found at the edge: at the surface of a planet, at the edge between icy and hot regions, fluid and dry areas, between permanent stagnation and constant change, in short: between order and chaos. An expansion of complexity (often at this famous edge of chaos) therefore seems to be correlated with an confinement to a limited space. It is at least clear that habitable zones suitable for life and organic life-forms are rare in our universe, they are found near at the edge of chaos in the habitable zones ("the islands of complexity").

The most complex systems we know are the result of evolution, and they have evolved over a long period of time: from tiny microorganisms living in the oceans (for example the anucleate green algae) to the yellow-foot kangaroo in Australia, nearly every complex thing on earth which is subject of interest is the result of evolution. Yet the increase of complexity during time was not steady. Reasons for sudden jumps of complexity during the course of evolution are:

  • The accumulation of more and more frozen accidents
  • The development of new codes and systems in systems
  • The crossing of a larger fitness barrier in evolution (bypass it, tunnel through it, or overcome it)

Irreducible Complexity

Irreducible complexity was originally a controversial concept invoked in support of intelligent design and creationism. An irreducibly complex system is defined as one that could not function if any of the parts would be removed. There is no way in which these systems could be broken down into smaller systems or components without loosing the function. Each part of an irreducibly complex system is indispensable in maintaining the system's function, and all the components must be in place before the system functions at all.

Michael J. Behe, an advocate of intelligent design, defines "irreducible complexity" as follows: "By irreducibly complex I mean a single system composed of several well-matched, interacting parts that contribute to the basic function, wherein the removal of any one of the parts causes the system to effectively cease functioning. An irreducibly complex system cannot be produced directly (that is, by continuously improving the initial function, which continues to work by the same mechanism) by slight, successive modifications of a precursor system, because any precursor to an irreducibly complex system that is missing a part is by definition nonfunctional. An irreducibly complex biological system, if there is such a thing, would be a powerful challenge to Darwinian evolution." (p. 39, M.J. Behe, "Darwin's black box: the biochemical challenge to evolution", 1996, The Free Press)

An irreducibly complex system has multiple interacting parts which are all required and indispensable to maintain the function of the system. The function of such a system is a true emergent property. Although such a system can hardly evolve through gradual adaptation and accumulation of parts, it can evolve through exaptation. The argument that an irreducibly complex system cannot be the result of Darwinian evolution and must therefore have been designed is obviously wrong, it is usually assembled from parts that had developed for other uses and functions. It is quite common for structures to change their functions during the course of evolution.

Another way of describing irreducible complexity is to say it is something that can only be adequately described by the production of the thing itself. An example is a cellular automaton whose outcome can be found effectively only by carrying it out explicitly. This is called computational irreducibility.

Relations and Connections

Complexity and Emergence

Just as a system is very complex, if there is no simple description, a property is emergent, if there is no suitable model to describe it (among those used for the definition of the system and its parts). The difficulty of complexity is to find a short and simple description, the difficulty of emergence is to find a short and simple model. In other words: complex means lack of description, emergence means lack of model. Complexity means we can not describe the phenomena completely, because we do not have a model or description. Emergence means we can not describe the phenomena completely, although we have a model and description of local rules and actions. Both phenomena, complexity and emergence are closely related hallmarks of complex systems. A system with high complexity can not be fully understood in terms of its isolated parts.

Science and Engineering

Complex systems and complexity are the essence of science and technology. The goal of engineers and technology/engineering is to hide and to master complexity. Mastery of complexity requires experience, skills and endurance in order to predict problems and fight against bugs, to create simple UI for complex devices and to conceal complex tools and instruments behind simple interfaces. E. C. Zeeman said in his book "Catastrophe Theory" from 1977: "Technical skill is mastery of complexity while creativity is mastery of simplicity". Science requires creativity in order to discover simple rules for complex phenomena, to recognize simple rules responsible for complex patterns and to find simplicity behind complexity.

The goal of scientists and science in general is to explain and to understand complexity. David Hilbert (1862-1943) said "The art of doing mathematics consists in finding that special case which contains all the germs of generality". In general, the art of science is to find that special theory which describes and explains as many complex phenomena and observations as possible: "One of the principal objects of theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity" (J. W. Gibbs). Albert Einstein formulated it in this way "The aim of science is, on the one hand, a comprehension, as complete as possible, of the connection between the sense experiences in their totality, and, on the other hand, the accomplishment of this aim by the use of a minimum of primary concepts and relations.."

In order to understand complexity, scientists who deal with complex systems in general often try to start a classification or categorization. Ernst Mayr, that grand old father of evolutionary biology, claimed that "classifications are necessary wherever one has to deal with diversity" (Ernst Mayr, The Growth of Biological Thought, Harvard University Press, 1982, p. 147). One could add complexity, because complexity is closely related to unity in diversity. Any attempt to understand complex systems or complexity in general must therefore start with a classification or categorization of the basic terms, principles and phenomena, as we could observe in early biology. This Wiki is an example for such an attempt.



  • Sunny Y. Auyang, Foundations of Complex Systems Theories: In Economics, Evolutionary Biology, and Statistical Physics, Cambridge University Press, 1998
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