Chapter 10 - Summary and perspectives

Summary

The term “complex” is being used more and more frequently in science, often in a vague sense akin to “complication”, and referred to any problem to which standard, well-established methods of mathematical analysis cannot be immediately applied. The spontaneous, legitimate reaction of the careful investigator to this attitude can be summarized by the questions: “Why study complexity?”, “What is complexity?”.

In the first part of the book, we have illustrated several examples from various disciplines in which complexity purportedly arises, trying, on the one hand, to exclude phenomena which do not really call for new concepts or mathematical tools and, on the other, to find common features in the remaining cases which could be of guidance for a sound and sufficiently general formulation of the problem. While amply answering the former question, the observed variety of apparently complex behaviour renders the task of formalizing complexity, i.e., of answering the latter question, quite hard. This is the subject of the main body of the book.

Aware of the difficulty of developing a formalism which is powerful enough to yield meaningful answers in all cases of interest, we have presented a critical comparison among various approaches, with the help of selected examples, stressing their complementarity.