Another Case of Epistemological Obstacles: the principle of permanence

Abstract

The negative numbers constituted a conceptual problem for mathematics as long as quantities and numbers had not been separated epistemologically and as mathematics was understood to be the science of quantities. The solution of the mathematical problem was achieved in the 19th century in a part of the mathematical community, as an element of the rise of the new paradigm of mathematics, overcoming the traditional substantialist ontology and establishing the relationist epistemology, based on the algebrisation of mathematics. The group of mathematics teachers at secondary schools was not prepared in its majority, however, to accept the new paradigm. It was in particular the principle of permanence, which proved to be an epistemological obstacle for them. They continued to adhere to the Platonist view, relying on geometrical justifications, maintaining that any mathematical statement should be capable of being demonstrated. Disguising their own obstacles to be those of the students who would accept nothing arbitrary in mathematics but rather absolute logical consistency, these teachers turned the principle of permanence to constitute an “obstacle didactogène” as Brousseau had called those obstacles caused by characteristics of teaching. Keywords: negative numbers, epistemological obstacles, principle of permanence