La preuve en mathématique

Abstract: Proof and deductive method in mathematics have their origin in the classic model of exposition developed by Euclid in his famous book on Elements. The attitude of mathematicians towards this method has certainly evolved in the past centuries, but the relationship between understanding and acceptability of mathematical statements has not dramatically changed and still constitutes a characterising element of this discipline. This paper is aimed at explaining and discussing some aspects which may be considered at the origin of difficulties related to proof; in particular, it focusses on the tension between two poles, that of production and that of systematisation of mathematical knowledge. Some examples drawn from different research projects are presented with the aim of illustrating the complementarity of various aspects and problems concerning proof. In the first part, the theoretical construct of Cognitive Unity is used to analyse and interpret the relationship between the production of a conjecture and its proof. In the second part, we present two long-term teaching experiments that are part of a research project whose principal goal is to introduce pupils to theoretical thinking and to study the role of particular microworlds in the teaching/learning processes. Assuming a Vygotskian perspective, attention is focussed on the semiotic mediation accomplished through cultural artefacts; in the case of Geometry the microworld is Cabri-géomètr, in the case of algebra the microworld is the prototype "L'Algebrista" (designed and realized by our team).