The project aims to provide an interdisciplinary framework for understanding geometric cognition and geometric practices. The project comprises three main aspects.

The first aspect focuses on the methodological requirements of the proposed framework. In the context of the project, such a framework is called a mixed-constraint framework (encompassing cultural, historical, and cognitive/biological constraints). This framework is developed from methodological considerations concerning the empirical study of mathematical practices. First, the project examines how empirical sciences—history, anthropology, cognitive neuroscience, and educational sciences—relate (or should relate) to one another in their joint investigation of mathematical practices. Building on the premise that mathematical cognition is paradigmatically embedded in human mathematical practices, a criterion (a cluster of properties)—drawing on the classificatory practices of empirical fields studying mathematics—is proposed to determine what qualifies as a mathematical practice. Then, the project provides historical, philosophical, and practical justifications for selecting geometric practices and geometric cognition as a primary case study.

The second aspect consists of a critical examination of what is called the "Natural Geometry Hypothesis," which is meant to represent in the project the predominant conception of geometric cognition. According to this hypothesis, there exists a set of basic, domain-specific, innate, and (almost) universally shared cognitive systems whose function is to represent geometric properties of shapes and environmental layouts. The project argues against attributing genuinely geometric

content to the representations generated by these basic systems and revealed through standard cognitive tasks. Nevertheless, the basic practices supported by the underlying cognitive capacities may still be regarded as geometric in a weaker sense, provided that they can be ontogenetically related to genuine geometric practices in a significant way.

Finally, the project develops an alternative conception of geometric cognition, drawing both on developments in 4E cognition, developmental psychology, psychology of mathematics education, and historical accounts of cognitive evolution. The central aim of this account is to show how, over time, the symbolic systems of representation embedded in historically constituted practices have shaped and channelled the cognitive functions involved in geometric practice as much as basic cognitive features have constrained the production and use of such cognitive artifacts.