La rigueur en mathématiques. Une approche critique et pratique (original) (raw)

2018

The concept of rigor plays a key role in mathematics. On the one hand, from a conceptual point of view, our understanding of mathematics relies heavily on the concept of 'rigorous proof', which alone is able to establish a theorem. On the other hand, rigor is a crucial notion in the historical evolution of mathematics. The so-called 'arithmetization of analysis' shaped the history of twentieth-century mathematics. This whole movement is often described as an improvement of mathematical rigor. However, at first sight, and despite its importance, rigor appears as a puzzling concept. As opposed to concepts such as truth, proof, theorem, etc. rigor has no meta-mathematical definition. It thus appears that rigor does not seem to be a purely logical notion. Hence, rigor exemplifies a key concept that plays a great role in the history of mathematics, without being itself a merely technical notion. The goal of this Dissertation is accordingly to enhance our understanding of such puzzling concept. In the first part, I study the available conceptions of rigor. I address in particular Bolzano's, the 'intuitionistic' as well as the 'formalist' conceptions of rigor, and I show that none of these conceptions is able to provide a necessary and sufficient condition that matches with the concept of rigor in use in current mathematical practice. In the second and third part, I establish a new framework that aims at gaining a positive understanding of the concept. Such framework is built upon a practical and sociological conception of mathematics. This work is my 2nd year of Master dissertion.