ProvingAgency · Proving Agency in Mathematical Practice: Practical, Social, and Mental Aspects

Proving Agency in Mathematical Practice: Practical, Social, and Mental Aspects

Mathematical reasoning is essential to the development of mathematical and scientific knowledge, and is a crucial skill in our science- and technology-based European societies. Yet, despite the revolutionary advances in formal logic of the past century, the nature of reasoning and proofs in ordinary mathematical practice remains unclear. The ProvingAgency project aims to move forwards on this general question by shifting the focus from mathematical proofs themselves to the activity of proving that gives rise to them. The approach to be developed proposes to structure the inquiry around the different dimensions of proving agency—the capacities of proving agents necessary to the realization of the activity of proving. The general idea is then to attend to what mathematicians do rather than the end products of their activity. This immediately raises a number of issues: What are the necessary characteristics that agents need to possess to prove theorems? What are the conceptual, metaphysical, and normative resources required to account for the activity of proving? In which sense should proving be considered as a mental activity? Such a perspective would structure the inquiry around the different aspects of proving agency, that is, around the different dimensions of the proving agent and of the proving activity itself. Furthermore, it may help identify the most appropriate theoretical and empirical methods to model the different aspects of proving agency. The aim of the ProvingAgency project is to implement this approach concretely, within a restricted yet substantial perimeter, in order to investigate three central aspects of proving agency: (i) practical aspects which concern the different kinds of practical knowledge or know-how required by mathematical agents to prove theorems; (ii) social aspects which concern the issues that arise when mathematical agents are interacting in proving; (iii) mental aspects which concern the nature of proving taken as a mental activity. The main objective of the ProvingAgency project can be stated as follows: to implement a systematic, multi-disciplinary approach to the study of proofs and proving in mathematical practice, centered around the notion of proving agency, in order to account for some of the key practical, social, and mental aspects of proving agents and the activity of proving. Practical aspects are concerned with the practical knowledge required by mathematical agents to prove theorems. The objective here is to theorize the kind of mathematical understanding required to engage in the activity of proving, with a specific focus on practical knowledge and practical reasoning. Social aspects are concerned with the issues that arise when mathematicians are interacting in the activity of proving. This happens whenever several mathematical agents are proving a theorem together or when a mathematical proof is submitted by one or more agents for examination by the mathematical community. Using resources from social epistemology, we aim here to identify the social mechanisms at work when the activity of proving is conceived as a social endeavor. Mental aspects are concerned with the mental dimension of the activity of proving. We will aim here to provide a conception of proving as a mental activity that does justice to the fact that proving often requires to rely on mathematical artifacts, which we will do by building on recent developments in the emerging field of extended epistemology.

Data: CORDIS, © European Union