It is unclear if complexity has a single straightforward definition, such that even the science of complexity is itself ‘complex’. Complexity is often regarded as relative concept, one that is dependent upon the limits of our existing models and methodologies. Arising in the 1970s from an interdisciplinary assortment of fields (cybernetics, economics, computer science, physics, ecology, biology, artificial intelligence, and philosophy) the study of complexity spans a variety of domains, and has varied applications depending upon its deployment. Currently, it has strong applications in climate modeling, economic prediction, biology, and the computer sciences.

In the Literature

Ladyman, Lambert, and Weisner [hereafter, LLW] sketch an overview of the characteristics of complex systems (“What is a Complex System“, 2011). While they do not present the necessary or sufficient conditions of being a complex systems, they do try to arrive at as precise of a description as possible, based upon the existing literature. They identify the major properties associated with the science, such as nonlinearity, feedback, spontaneous order, robustness and lack of central control, emergence, hierarchical organization, and numerosity. From these properties, LLW derive the more succinct definition: “A complex system is an ensemble of many elements which are interacting in a disordered way, resulting in robust organization and memory.”

Computational Complexity

LLW’s survey of the literature is supplemented by a informational account of complexity, which attempts to define effective measures of complexity as it regards the statistical, mathematical, physical, and computational accounts of complexity. They settle on computability as the most effective measure we have:

In principle, there is no reason to suppose that there could not be some true property of systems that measures their complexity even though we cannot compute it. However, since the statistical complexity can be computed and used in practice to infer the presence of complex systems, it is the best candidate we have considered for a measure of the order produced by complex systems.

Computability not only provides a means of grasping features of a complex system, but also provides a measure as to what should be construed as a  complex system by pointing at the limits of the system’s computability.

In another paper (“Why Philosophers Should Care About Computational Complexity“), Scott Aaronson suggests that “effectiveness” is closely related to the efficiency of computability – that is we must be able to compute a problem in “reasonable amount of time”:

One might think that, once we know something is computable, whether it takes 10 seconds or 20 seconds to compute is obviously the concern of engineers rather than philosophers. But that conclusion would not be so obvious, if the question were one of 10 seconds versus 101010 seconds! And indeed, in complexity theory, the quantitative gaps we care about are usually so vast that one has to consider them qualitative gaps as well. Think, for example, of the difference between reading a 400-page book and reading every possible such book, or between writing down a thousand-digit number and counting to that number.

Computational questions and the computability of a problem in efficient amount of time – say prior to the heat-death of the universe, or less dramatically, prior to the advent of the problem under consideration in a complex space – must be of central to concern to any program which wishes to apply the methodologies of complexity in concrete or practical terms. Knowing that something is beyond the bounds of our current computational capacities, however, does not mean that this knowledge is worthless, as it provides us with information regarding our current limitations in examining complex systems.

Other Notes

We can understand a complex system as a ‘structure with variation’ that is characterized by an exchange of information, but this exchange of energies or flows of material operate at varying levels which differ in both nature and kind, exhibiting behaviors such as branching, nesting, and interfacing… Yet the above remains almost entirely descriptive—a wide net that includes intellectual and cultural production, the economy, contemporary geopolitical circumstances, the climate… Robust understandings of complexity synthesize the mathematical, the physical, the biological, and the social. These objects and systems (social, technological, institutional, biological) possess an extensive history that is also the result of a non-trivial set of trajectories. Understanding these trajectories impacts our ability to analyze future behaviors, and, in a sense, ‘mine’ the outer limits of accessible temporality.