Mathematical intuition and the cognitive roots of mathematical concepts

Giuseppe Longo, Arnaud Viarouge. Mathematical intuition and the cognitive roots of mathematical concepts. Invited paper, Topoi, Special issue on Mathematical knowledge: Intuition, visualization, and understanding (Horsten L., Starikova I., eds), Vol. 29, n. 1, pp. 15-27, 2010. (TopoiIntuitCogni.pdf)International audienceThe foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This « genealogy of concepts », so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis by this too often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to the philosophical frame as well as to the some recent advances in Cognition that support our claim, the cognitive origin and the constitutive role of mathematical intuition