ABSTRACT: Between 1906 and 1911, as a response to Betrand’s Russell’s review of La Science et l’Hypothèse, Henri Poincaré launched an attack on the movement to formalise the foundations of mathematics reducing it to logic. The main point is the following: the universality of logic is based on the idea that their truth is independent of any context including epistemic and cultural contexts. From the free context notion of truth and proof
it follows that, given an axiomatic system, nothing new can follow. One of the main strategies of Poincaré’s solution to this dilemma is based on the notions of understanding and of grasping the architecture of the propositions of mathematics. According to this view mathematic rigour does not reduce to “derive blindly” without gaps from axioms, mathematical rigour is, according to Poincaré, closely linked to the ability to grasp the architecture of mathematics and contribute to an extension of the meaning embedded in structures that constitute the architecture of mathematical propositions. The focus of my paper relates precisely to the notion of architecture and to the notion of understanding. According to my reconstruction, Poincaré’s suggestions could be seen as pointing out that understanding is linked to reason not only within a structure but reasoning about the structure.