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## Interdisciplinaria

*On-line version* ISSN 1668-7027

#### Abstract

CATSIGERAS, Eleonora. Deterministic dynamics and chaos: Epistemology and interdisciplinary methodology.* Interdisciplinaria* [online]. 2011, vol.28, n.2, pp. 279-298. ISSN 1668-7027.

We analyze, from a theoretical viewpoint, the bidirectional interdisciplinary relation between Mathematics and Psychology, focused on the mathematical theory of deterministic dynamical systems, and in particular, on the theory of chaos. On one hand, there is the direct classic relation: the application of Mathematics to Psychology. On the other hand, we propose the converse relation which consists in the formulation of new abstract mathematical problems appearing from processes and structures under research of Psychology. The bidirectional multidisciplinary relation from - to pure Mathematics, largely holds with the ‘hard' sciences, typically Physics and Astronomy. But it is rather new, from the social and human sciences, towards pure Mathematics. Summarizing, the problem we focusing in this paper, is not only the application of the mathematical theory of dynamical systems to Psychology, but mainly the following questions: Which psychological processes are involved in the development of pure Mathematics? How can a multidisciplinary space be organized to activate the converse relation, from Psychology towards pure Mathematics? How may Psychology provide a rich field of new mathematical questions to be investigated, not only by applied mathematicians, but also by researchers on pure Mathematics? Even if large advances had been achieved, the application of the mathematical theory to Psychology is still mainly developed by mathematical psychologists and applied mathematicians, in the absence of pure mathematicians. Conversely, the development of the pure Mathematics is now a days mainly developed in the absence of applied scientists, particularly of human and social researchers. This is the opposite situation to the antique posture, in which theoretical Mathematics and Philosophy, for instance, were almost a single science. Along this paper we aim to found how the potential strength of the mathematical tools can be more fully exploited in the interdisciplinary space, and how the necessary development of new abstract and adequate tools in pure Mathematics, may be detected while immersed into an interdisciplinary discussion. This discussion does not need to be ‘applied', in its restricted sense. In fact, Mathematics may still remain abstract and theoretical, bust just break its apparent isolation from other sciences, in particular to those related with the human thinking, like Philosophy and Psychology. The methodology of our analysis along this paper follows three steps: First, we present a partial review, focused in several aspects of the mathematical research, in their interdisciplinary relation with Psychology. Then, we state and analyze epistemologically, the mathematical abstract definitions of dynamical systems, and in particular of deterministic chaos. Finally, we suggest a general meta-theory in the organization of the interdisciplinary space between Mathematics and Psychology, which we illustrate with an hypothetical example. This paper is organized in six sections: At the first one, we briefly introduce the discourse. At the second section, we present a partial survey of the knowledge in the interdisciplinary fields among Mathematics, Psychology and other sciences. That survey is focused on the theory of dynamical systems, and is very partial respect to the whole abundant development in this interdisciplinary field. The third section states the mathematical definitions of dynamical and autonomous system, and of deterministic chaos, and analyze them epistemologically. Among other properties, we revisit the argument of self-organization of deterministic chaos. At the fourt hand fifth sections, we propose a method and a metatheory, according to which, the interdisciplinary space between Mathematics and Psychology may organize its purposes and actions. We consider the epistemological objection of Nowak and Vallacher (1998). They observe that the traditional notions of causality holds in social psychological research, and oppose to (some of) the mathematical models of dynamical systems, which feedback the same variable from one time to the next. In fifth section too, arguing on a particular hypothetically example, we propose a method to model mathematically such systems with causal transitions, provided that the system is deterministic. The modeling method that we propose in this metatheory, solves the epistmological objection of Nowak and Vallacher, in some particular cases. Finally, the last section states the conclusions.

**Keywords
:
**Dynamical systems; Chaos; Mathematical psychology; Interdisciplinary methodology; Epistemology.