Complexity is a specific condition of a collection of interconnected elements: the elements cannot be connected to one another at the same time.
Therefore, complexity means that a selection is necessary in order to create relations between the elements.
Consequently, the distinction between element and relation is fundamental to this definition of complexity, which requires selective connectivity between the elements. To observe complexity, we must distinguish a situation of selective connectivity from one that is not selective.
Complexity can thus be defined as a Form, the two sides of which are the selective connectivity and the complete connectivity of the elements.
Complexity can be observed in a system or from the point of view of a system if it is observed in the system environment [⇒ System/Environment (System/Umwelt)], as well as in the →World. Only the complexity of a system is organized complexity. It consists of the selective connectivity of the elements of the system; it is the selective organization of →Autopoiesis.
The number of abstractly possible relations between the elements of a system increases exponentially with the increase in the number of elements: two elements form four relations, three elements nine, and so on. When the number of elements in a system is very large, the number of relations reaches orders of magnitude that cannot be directly controlled by the system itself. This implies that, within the system, not everything can be actualized and simultaneously connected to everything else; each of the system operations refers to a further domain of potentialities.
Complexity describes the fact that there are more possibilities than can be actualized, i.e., more communication in social systems and more thought in psychic systems. For what concerns →Social Systems, a specific communication (“What do you think about that?” or “the exchange rate has gone up”) can only connect directly to a limited number of further communications. Since every actualized communication refers to a domain of alternative possibilities, every connection must choose between numerous possibilities: for instance, the answer to the question “what do you think about that?” is only one of many possible answers, the comment on the news that the exchange rate has changed is only one of many possible comments. Here we observe a compulsion to selection: something is realized as a datum and the rest remains in the background as a domain of possible references. Selection constitutes the temporal dynamic of complexity [→Time]; the basis of selection is the fact that actualizations occur sequentially since no system can actualize everything at the same time.
The observation of complexity emerges alongside →Meaning. A meaning-constituting system observes the complexity of the world from its own point of view. Since the world is conceived as a Unity of the difference between system and environment, complexity is also relative to this difference and is dependent on the observation of this difference through meaning-constituting systems.
Complexity (including that of the environment) exists only when it is observed by a system. The relevance of observation for the construction of complexity is particularly emphasized in the concept of hypercomplexity, which means that complexity also includes the consequences of its observation. Hypercomplexity is the result of a second-order observation [⇒ Operation/Observation (Operation/Beobachtung), i.e., the result of including the observing system in the observation: a society, for instance, is hypercomplex if it observes the consequences of its observations on its environment.
A system observation does not determine the complexity of that system environment. The complexity of the environment is constituted independently of the system, since the system can only grasp it through observational operations, which can irritate the system itself. This is the paradox of the system which is unable to control environmental complexity, even if this complexity only exists as observed by the system.
The difference between system and environment marks a difference in the level of complexity: the environment is always more complex than the system, because the system draws a boundary which limits the domain of the possible within the system itself. The difference in level of complexity between system and environment appears as relating of relations, whereby the abstractly possible relations between the elements in the system (system complexity) are limited from compatibility with its environment (environmental complexity). In a (social) system, it is not possible to actualize everything in the operations (in the communication) at the same time, for the reason that complexity is structured according to a perspective of compatibility with the environment.
Reduction of complexity
In such a system, reductions are necessary to realize and maintain an internal complexity that renders the system compatible with the environment. Reduction of complexity means that the abstract possibility of relations between elements is structured into a specialized system including a limited number of possible relations.
Complexity reduction means selectively maintaining a domain of possibilities based on structures. The →Structures determine how much internal complexity a system can generate and tolerate. Complexity is only realized and maintained in a system through reduction: reduction and maintenance of complexity do not contradict one another, but are rather mutual conditions. The maintenance and reduction of complexity depend on structures which preselect the possibilities available to connect the elements to one another.
The complexity level of a system changes with the change of the selectivity of the relations made possible by the system structure. A system can increase its complexity in connection with an increase in the complexity of its environment—not because it conforms to the environmental complexity, but because it operates autonomously on the basis of its own structures. The increase in complexity in a system triggers an increase in complexity in the systems observing it, because their environments become more complex. Under these conditions, an Evolution of the system is possible; however, this evolution is not a simple increase in complexity, but rather a change in structure.
Complexity / Differentiation
The concept of complexity must be distinguished from that of →Differentiation. Differentiation refers to the distinction system/environment and not to the distinction element/relation. An increase in complexity does not mean an increase in differentiation; whilst complexity varies continuously, differentiation varies discontinuously (as in the case of change in the form of society differentiation).
However, the concepts of complexity and differentiation can be linked together. The increase in complexity—i.e., in the relations between the elements—entails limits to the widening of the system: no system can withstand an arbitrary and undefined increase in its complexity. Thus, limits are drawn within the system, which generate subsystems. The form of differentiation among subsystems determines the limits of complexity that can be attained in the system. The changes in the level of complexity can trigger changes in the system form of differentiation. Changing the system differentiation criteria has decisive consequences for the level of complexity that the system can tolerate. This is particularly important for the system of society; today’s society, for instance, generates far greater complexity than earlier societies due to its forms of differentiation [→Differentiation of Society]. [C.B.] – (Unlocking Luhmann, p 49–52)
Social Systems (1995: 23-28); Theory of Society (2012: Ch. 1.9); Temporalization of Complexity (1978); Introduction to Systems Theory (2012: Ch. II.8).