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The aim of this research is to identify the foundations of students' belief in the validity of an assertion in their mathematical activity: what they recognize in practice as a proof and how they treat a refutation. We focused this study on the relationships between students' proving processes, the knowledge they have, the language they can use, and the role of the situational context. The types of proof processes evidenced by the students do not intrinsically characterize what we might call their "rationality", in that different levels of proof could be observed in their problem-solving activity. The meaning of the proof processes cannot be understood without a careful analysis of the students' conceptions of the mathematical concepts involved and their reading of the situation in which they act. The characteristics of the situation seem to determine the level of proof, while the students' image of mathematics also plays an important role, especially in dealing with refutations. It is observed that the passage from pragmatic to intellectual proofs requires a cognitive and linguistic foundation. Disregarding the complexity of this passage may be one of the main reasons for the failure of teaching mathematical proof, since this passage is often considered only at the logical level. In geometry in particular, this teaching occurs in a conceptual field that, for students has not yet constituted itself as a theory; since geometry was for them essentially restricted to the observation and construction of geometric objects with no need for proof. Thus, the teaching of proof is associated with what could be described as a cognitive break in student activity, related to the didactic break represented by the new requirement for mathematical proofs. [ABSTRACT FROM AUTHOR]