What is Complexity? - The philosophy of complexity per se with application to some examples in evolution
"That property of a language expression which makes it difficult to formulate its overall behaviour, even when given almost complete information about its atomic components and their inter-relations.".
This is a very general definition, which is intended to have different interpretations in different contexts, especially with respect to the phrase "difficult to formulate its overall behaviour". Here "language" is meant in a general sense, to include diagrams*1. "Atomic components" are defined by what are irreducible signs in the chosen language of representation; this would correspond to undefined signs, functions, predicates and constants in a formal logic*2. "The formulation of the overall behaviour" is assumed to be of much the same type as that which specifies the atomic components and their inter-relations.
This definition relates the difficulty in formalisation of the whole compared to that of its fundamental parts (from the point of view of the language). It is only applicable in cases where there is at least a possibility of gaining almost complete information about the components, thus clearly separating ignorance from complexity. You will get different concepts of complexity depending on the base language chosen, the type of difficulty focused on and the type of formulation desired within that language.
For example, consider a hypothetical herd of animals, where the flight behaviour of each is fairly well understood in terms of its reaction to danger, its wish to follow others of its kind, etc., where we are trying to understand the flight behaviour of the whole herd in terms of the direction and path. If it turned out that all the animals always followed one specified leader, or all went in a direction represented by the average of the separate directions they would have gone individually then we would be justified in calling the flight behaviour of the herd simple. If the behaviour turned on the precise configuration of the herd at the time of attack, so that the animals followed different individuals at different instances as the configuration developed, we would be justified in calling the behaviour more complex.
To illustrate how the language can make a difference consider a case where the language did not take into account the direction of travel (of individual or herd) but only the average speed. In this case, the behaviour might be simple, despite the fact that the behaviour is complex from the point of view of direction.
The type of "difficulty" can also be crucial. Given a language which includes direction and speed, a predator might be concerned with the difficulty of predicting in which direction the herd will initially set off, an ecologist might only be concerned with the eventual direction and distance a herd travelled until it settled down. The first might be difficult to predict, the other easy; in the first case the behaviour of the herd would be complex while in the second it would be simple. The types of difficulty considered are obviously limited by the base language chosen.
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