In physical systems, complexity is a measure of the probability of the state vector of the system. This is often confused with entropy, but is a distinct analysis of the probability of the state of the system, where two distinct states are never conflated and considered equal as in statistical mechanics.
Complexity is often used as a shorthand for the field that developed in the late 1980s around the use of mathematical and computationalmodeling of biological, economic and technologicalsystems known as "complex systems" (sometimes complex adaptive systems). These systems tend to exhibit high-dimensionality, non-linearity, and often, sensitive dependence of initial conditions.
In the sense of how complicated a problem is from the perspective of the person trying to solve it, limits of complexity are measured using a term from cognitive psychology, namely the hrair limit.
"When I hear the word 'complexity', I don't exactly reach for my hammer, but I suspect my eyes narrow. It has the dangerous allure of an incantation, threatening to acquire the same blithe explanatory role that 'adaptation' once did in biology". Philip Ball , Nature Materials 3, 78 (2004), doi:10.1038/nmat1069
