Is This Object Mathematical, Para-, or Proto-? A Didactic Application of the Epistemological Profile

We propose here an epistemological tool for creating and elaborating didacticians’ own reference models of different objects of knowledge involved in mathematical activities. It is based on a fundamental distinction of objects of mathematics introduced by the didactic transposition theory: the mathematical, the paramathematical, and the protomathematical. According to our initial hypothesis, any object in mathematics can possess multiple statuses for instances observing it who can be persons, positions, or institutions, e.g., “mainly mathematical but paramathematical just a little”. Such possible multiplicity of the degree of mathematization of different objects can be clarified through an application of Bachelard’s idea of epistemological profile.