Quel statut donner aux calculs et surtout aux figures proposées par Henri Bergson pour expliquer le fonctionnement de la reconnaissance dans Matière et mémoire ? Ce travail suggère que ces calculs et ces figures ne sont pas que des métaphores commodes ; la réalité n'est pas dépourvue d'un certain nombre d’éléments et de grandeurs plus ou moins importantes. Il semble donc possible de supposer qu'après avoir arithmétisé les faits de conscience à l'aide de nombres spéciaux (dans l’Essai), Bergson géométrise l’être spirituel et matériel (dans Matière et mémoire).

In this article, I wish to discuss in an informal way the motivations and the motifs of the constructivist approach to logic and mathematics and by a natural extension to the general field of science, particularly theoretical physics. Foundational questions in those domains are not ruled by philosophical principles, but a critical philosophy of foundations could be the leitmotiv to the extent that it can be used as a criterion to decide between the theoretical options of scientific practices that are often oblivious to their own doctrinal presuppositions. My objective is to provide the justificatory reasons for a constructivist option or posture in the field of scientific knowledge.

The arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group F and some of its generalizations interpret the arithmetic without parameters.