Remarks on the Geometry of Complex Systems and Self-Organization

In Vincenzo Fano, Enrico Giannetto, Giulia Giannini & Pierluigi Graziani (eds.), Complessità e Riduzionismo. © ISONOMIA – Epistemologica, University of Urbino. pp. 28-43 (2012)
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Abstract
Let us start by some general definitions of the concept of complexity. We take a complex system to be one composed by a large number of parts, and whose properties are not fully explained by an understanding of its components parts. Studies of complex systems recognized the importance of “wholeness”, defined as problems of organization (and of regulation), phenomena non resolvable into local events, dynamics interactions in the difference of behaviour of parts when isolated or in higher configuration, etc., in short, systems of various orders (or levels) not understandable by investigation of their respective parts in isolation. In a complex system it is essential to distinguish between ‘global’ and ‘local’ properties. Theoretical physicists in the last two decades have discovered that the collective behaviour of a macro-system, i.e. a system composed of many objects, does not change qualitatively when the behaviour of single components are modified slightly. Conversely, it has been also found that the behaviour of single components does change when the overall behaviour of the system is modified. There are many universal classes which describe the collective behaviour of the system, and each class has its own characteristics; the universal classes do not change when we perturb the system. The most interesting and rewarding work consists in finding these universal classes and in spelling out their properties. This conception has been followed in studies done in the last twenty years on second order phase transitions. The objective, which has been mostly achieved, was to classify all possible types of phase transitions in different universality classes and to compute the parameters that control the behaviour of the system near the transition (or critical or bifurcation) point as a function of the universality class. This point of view is not very different from the one expressed by Thom in the introduction of Structural Stability and Morphogenesis (1975). It differs from Thom’s program because there is no a priori idea of the mathematical framework which should be used. Indeed Thom considers only a restricted class of models (ordinary differential equations in low dimensional spaces) while we do not have any prejudice regarding which models should be accepted. One of the most interesting and surprising results obtained by studying complex systems is the possibility of classifying the configurations of the system taxonomically. It is well-known that a well founded taxonomy is possible only if the objects we want to classify have some unique properties, i.e. species may be introduced in an objective way only if it is impossible to go continuously from one specie to another; in a more mathematical language, we say that objects must have the property of ultrametricity. More precisely, it was discovered that there are conditions under which a class of complex systems may only exist in configurations that have the ultrametricity property and consequently they can be classified in a hierarchical way. Indeed, it has been found that only this ultrametricity property is shared by the near-optimal solutions of many optimization problems of complex functions, i.e. corrugated landscapes in Kauffman’s language. These results are derived from the study of spin glass model, but they have wider implications. It is possible that the kind of structures that arise in these cases is present in many other apparently unrelated problems. Before to go on with our considerations, we have to pick in mind two main complementary ideas about complexity. (i) According to the prevalent and usual point of view, the essence of complex systems lies in the emergence of complex structures from the non-linear interaction of many simple elements that obey simple rules. Typically, these rules consist of 0–1 alternatives selected in response to the input received, as in many prototypes like cellular automata, Boolean networks, spin systems, etc. Quite intricate patterns and structures can occur in such systems. However, what can be also said is that these are toy systems, and the systems occurring in reality rather consist of elements that individually are quite complex themselves. (ii) So, this bring a new aspect that seems essential and indispensable to the emergence and functioning of complex systems, namely the coordination of individual agents or elements that themselves are complex at their own scale of operation. This coordination dramatically reduces the degree of freedom of those participating agents. Even the constituents of molecules, i.e. the atoms, are rather complicated conglomerations of subatomic particles, perhaps ultimately excitations of patterns of superstrings. Genes, the elementary biochemical coding units, are very complex macromolecular strings, as are the metabolic units, the proteins. Neurons, the basic elements of cognitive networks, themselves are cells. In those mentioned and in other complex systems, it is an important feature that the potential complexity of the behaviour of the individual agents gets dramatically simplified through the global interactions within the system. The individual degrees of freedom are drastically reduced, or, in a more formal terminology, the factual space of the system is much smaller than the product of the state space of the individual elements. That is one key aspect. The other one is that on this basis, that is utilizing the coordination between the activities of its members, the system then becomes able to develop and express a coherent structure at a higher level, that is, an emergent behaviour (and emergent properties) that transcends what each element is individually capable of.
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