In Vincenzo Fano, Enrico Giannetto, Giulia Giannini & Pierluigi Graziani,
Complessità e Riduzionismo. ISONOMIA - Epistemologica Series Editor. 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.