Theorems as Constructive Visions

Giuseppe Longo (1, 2)

1 CREA - Centre de recherche en épistémologie appliquée
2 LIENS - Laboratoire d'informatique de l'école normale supérieure

This paper briefly reviews some epistemological perspectives on the foundation of mathematical concepts and proofs. It provides examples of axioms and proofs, from Euclid to recent "concrete incompleteness" theorems. In reference to basic cognitive phenomena, the paper focuses on order and symmetries as core "construction principles" for mathematical knowledge. A distinction is then made between these principles and the "proof principles" of modern Mathemaical Logic. The role of the blend of these different forms of founding principles will be stressed, both for the purposes of proving and of understanding and communicating the proof

Mots clés: Cognition; Foundations of Mathematical Knowledge

Domaines: Mathématiques [math]; Histoire et perspectives sur les mathématiques [math.HO]; Sciences de l'Homme et Société; Histoire, Philosophie et Sociologie des sciences; Sciences cognitives

Giuseppe Longo. Theorems as Constructive Visions. Hanna, Gila; de Villiers, Michael. Proof and Proving in Mathematics Education. The 19th ICMI Study, Springer, pp.51-66, 2010, 978-94-007-2129-6. ⟨hal-03319485⟩