Intuitive acceptance of proof by contradiction

The formal acceptance of a mathematical proof is based on its logical correctness but, from a cognitive point of view, this form of acceptance is not always naturally associated with the feeling that the proof has necessarily proved the statement. This is the case, in particular, for proof by contradiction in geometry, which can be linked to a loss of evidence in various ways, owing to its particular logical structure and to the difficulty in managing geometrical figures with contradictory properties. In this paper, we observe that students produce argumentation by starting with the assumption that the claim is false (indirect argumentation), and that they seem to accept this as more evident than the proofs by contradiction. On the basis of the notion of intuitive knowledge developed by Fischbein and through the analysis of task-based interviews, we investigate the intuitive acceptance of proof by contradiction and of indirect argumentation, underlining, in particular, that indirect argumentation can be produced as a compromise between a proof by contradiction and the need for a more evident argument.