Extended cognition, dynamics, and algorithms. A turing machine based approach to the study of arithmetical skills

The fields of philosophy of mind and cognitive science have been characterized, in the last few decades, by a growing interest for explanations of mind's activity in terms of interaction between brains, bodies and the world. Embodiment, embeddedness, situatededness are key words that most often can be found in contemporary cognitive studies. However, some cognitive activities seem recalcitrant to this kind of treatment. Mathematical thinking is one of them. Explanations of human computational competencies, indeed, focus typically on representational issues, while giving less importance to the role of mind/body/environment interaction for the performance and development of algorithmic skills, namely, those capacities which are essential in order to operate with numbers and carry out symbolic transformation. The significance of these skills for a general understanding of computational activities is explicitely recognized in Alan Turing's theory of computation, which is focused on the construction of idealized models of the mechanisms at work in a real cognitive system, namely the one consisting of a man performing calculations with paper and pencil. In the present thesis I take seriously Marco Giunti's proposal to use a Turing machine (TM)-based computational architecture, namely the Bidimensional Turing Machine (BTM), in order to study human algorithmic skills. This work consists of two main parts. The first part, philosophically-oriented, deals with Andrew Wells' ecological interpretation of the TM's architecture and its relations with a set of philosophical and psychological positions such as classic computationalism, the extended-mind hypothesis and the dynamical approach to cognition; the second, more technical part, sets up a theoretical and methodological framework for the development and justification of BTM-based models of human algorithmic skills.