Intuitive Foundations for Geometry: Euclidean geometry and beyond – Geometries

A wealth of research has demonstrated that humans possess intuitions about number from infancy on; A wealth of research has demonstrated that humans possess intuitions about number from infancy on; later, these intuitions guide children’s learning of formal arithmetic. Much less is known however of the intuitive foundations of another branch of mathematics: geometry. In my past research, I started investigating this question focusing on Euclidean geometry (the so-called “natural geometry”), and in particular angle, a central Euclidean concept. Results however showed little to no sensitivity to angle in young children and infants, and systematic errors persisting even in adults. The present project addresses the question of the foundations of geometry further, in four different axes of research. First, we will conduct an in-depth analysis of the mechanisms underlying angle perception in adults and children. Indeed, with a description of the mechanisms involved, we will be in a better position to understand the failures we have observed, especially in children. In the second axis of research, we take a broader perspective to probe intuitions beyond Euclidean geometry: if Euclidean geometry, and angles, are not intuitive, is there a ‘natural’ geometry? We will test infants’, children’s and adults’ perception of geometric properties at different levels of invariance: from the low-level property of shape orientation, to properties belonging to affine or projective geometry. In a third axis, we will depart from perception and assess the functional properties of children’s geometric representations, namely whether infants can engage in geometric reasoning. Lastly, the fourth axis of research will probe processes of conceptual learning, using a test case known to be counter-intuitive: spherical geometry. We will ask whether conceptual learning involves processes of “incubation” that operate subconsciously, learning being experienced instead as a sudden flash of insight (“I get it!”).