Complexity is generally used to characterize something with many parts where those parts interact with each other in multiple ways. Etimologically, complex refers to a system made of many intertwined parts, and that's still the definition we use in science, although a precise measure hasn't been agreed upon. See Complexity.
But how intertwined, i.e. how many and what kind of the interactions does a system need to be called complex? I think a complex system should be defined as one in which the interactions significantly alter the behaviour of the system, relative to the one with no interactions. The primary example of interactions that qualitatively affect the behavior of a system are nonlinear interactions (see Nonlinear systems).
Definition by Cosma Shalizi: a complex system is a high-dimensional systems where the variables are strongly interdependent. Complex systems are ones with a large effective number of strongly-interdependent variables. This excludes both low-dimensional systems, and high-dimensional ones where the variables are either independent, or so strongly coupled that only a few variables effectively determine all the rest. Since the 1980s, an interdisciplinary movement of physicists, mathematicians, economists, computer scientists, biologists, anthropologists and other scientists has explored techniques for modeling a broad range of such systems, and their common features and inter-connections. These techniques rely heavily on intensive, sophisticated computer simulations, and notions of information, search and adaptation feature prominently in the theories. (The Statistical Analysis of Complex Systems Models)
See also: How do I explain to non-mathematical people what "non-linear and complex systems" mean?
Furthermore, Warren Weaver posited in 1948 two forms of complexity:
- disorganized complexity
- organized complexity
The way I interpret this, is that organized or disorganized refers to the behaviour of the system, at some scale and coarse-graining level. If the system at some coarse-graining level has a behaviour that could be described by a less complex system (for example, as formalized by Kolmogorov complexity in AIT) than the original description, we say it displays organized complexity, and that new simpler behavior has emerged (see Self-organization). This may also be called complexity reduction. One can see that coarse-graining will produce less complex descriptions, pretty much by definition. However, to get emergence, the system must allow some coarse-graining procedure that produces reasonable descriptions, in the first place
Disorganized complexity refers to some scale which does not allow a simpler coarse-grained description.
For example, a gas of particles represent a complex system (as the particles interact with each other in complex ways, i.e. ways that change the behavior of the system significantly relative to a system of non-interacting particles). At the scale of particles, we have disorganized complexity, as there is no coarse-grained description that can simplify the dynamics while still talking of all the particles. We may then use probabilistic descriptions. At larger scales, we can talk about large groups of particles, and using, for instance averages from the probabilistic descriptions, we can construct coarse-grained descriptions in terms of "infinitesimal" volume elements interacting on less complex ways. We can say that that "hydrodynamic behvaior has emerged".
| Actually here I am referring to "complexity" as used in Complex systems theory. As Wiki says, Complexity theory can also refer to Computational complexity or Descriptional complexity (a fundamental concept in Algorithmic information theory). |
Complexity and Self-organization
Universality-Complexity Classes for Partial Differential Equation Systems (from xmorphia) Taking ideas of universality and complexity classes of cellular automata from Stephen Wolfram (c.f. A New Kind of Science).
https://en.wikipedia.org/wiki/Complexity_theory
Kolmogorov Complexity – A Primer
The First Law of Complexodynamics
Well the complexity follows that pattern in the macroscale at least. Also:
Non-equilibrium is more complex; I think: because equilibrium can be described simply: the long time behaviour of the simple dynamical system; while non-eq has many more possibilities
https://jeremykun.com/2012/04/21/kolmogorov-complexity-a-primer/
See also the related: Computational complexity, and also Descriptional complexity, and Complex systems.
Complexity theory may be seen as part of complexity science, or they may be seen as equivalent disciplines. In any case, this page includes complexity science.
http://www.complexity.ecs.soton.ac.uk/
People
http://turing.iimas.unam.mx/~cgg/
Norbert Wiener, cybernetics
Heinz von Foerster, Second-order cybernetics
Francis Heylighen, cyberneticist