Laboratório da Complexidade

Research program                                                                                   

 

 

Concerning Complexity                                                                           principles and methods

 

 

The study of the so called complex phenomena has become increasingly important for many scientific areas. After profound conceptual transformations, through the XX century, in physics (relativity, quantum mechanics), in biology and life sciences (Darwin , DNA), the development of computers, an enormous development in the social sciences and an exponential growth of technological diversity it is clear the current difficulty to understand and deal with these type of phenomena. Its diversity is enormous so is the difficulty of systematization, existing a dispersion of models and approaches. In part, this dispersion is due to the fact that the topic has been developed more or less simultaneously by different researchers in different specialized sciences. Although the referred researchers generally had mathematical knowledge there is a lack of abstraction ability to capture the essence and what is common in the complex phenomena, beyond solving particular problems. Therefore, there is a lack of unifying conceptual tools for the formation of a coherent scientific paradigm. There are those who defend the absence of a general theory of complex systems. Here, however, we assume the existence of general laws, for now unknown, and seek its clarification and formalization.

The different approaches to complex systems ranges from those based on agents, (computation point of view), on statistical physics, on game theory, to those based on chemistry or on biology. All these partial effort are very important, however if isolated or related through empirical evidence it can only lead to more models and more phenomena. It is necessary a conceptual synthesis. From mathematics it is possible to initiate this synthesis and to achieve the conceptual clearness needed. As in the beginning of the modern scientific paradigm, with Galileu and Newton, where the interplay between the experimental sciences and mathematics were remarkable.

In contrast to the explosive development in the applied areas, the mathematics of nonlinear science and complexity has been developed slowly but constantly, since Poincaré. From the mathematician side the obstacles may be some insensibility to the natural sciences, incomprehension of the experimental method and to be caught in details around mathematical rigor in the initial stages of the developing theories.

 

 

The complex systems has the feature of having many interacting parts. The traditional application of analysis will lead to taking, at some point, a limit which in a certain sense is a simplifying step in the description of the system. This was useful for the linear systems or to the use of a purely statistical description (as in statistical physics and quantum mechanics). However, in complex and nonlinear systems this conceptual process leave the essential aspects of the systems unknown.

It is now clear that are necessary methods (mathematical methods) to describe systems composed of many, not negligible parts, each part as to be taken into account. It is not a question of more calculus power with more powerful computers. The problem is on one hand to develop new mathematical framework in which scale, coarse graining and refinement in the description is taken into account in a fundamental way. On the other hand, dimension must not be an explicit concept or parameter. Hierarchy of systems, types of coupling, growing and vanishing of systems, must be also a cornerstone of the mathematical framework. Moreover, completely new approaches are probably necessary to deal with turbulence, equilibrium in biology or economy, intelligence, emergence - origin of life, to clearly enunciate and define measure(s) of the complexity of the systems.

What is underlying, in the change of techno-scientific paradigm, is a slow process transforming the energy based society to the information based society. It is essential to develop mathematical-physics methods for information processing, in the same way the classical physics was developed within the energy (industrial age) paradigm. The analogue to the heat engine is now the computer

It can be accomplished with the articulation of known techniques and the development of new ones. The full power of matrix and operator algebras, in an algebrization (quantization) process. The symbolic dynamics and topological dynamics, in which the point (in a metric space) is replaced by a more complex mathematical entity (an operator-observable or a function). One decisive discipline is combinatory, and its relation to differential geometry and topology. The relation between algorithms and dynamical systems. To see a nonlinear dynamical system as an analogical computer, through symbolic dynamics. To form networks of dynamical systems. Compare with the process which lead to the scientific development in XVIII and XIX centuries, around energy.

Simultaneously, speculative thought, testing, practice and artistic reflection, will be essential in the creative process, especially at a turning point of scientific and cultural paradigm, in which it is necessary to look for new directions to explore.