Theorems as Constructive Visions

Abstract

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. It then distinguishes between these principles and the “proof principles” of modern Mathematical Logic. It also emphasises the role of the blending of these different forms of founding principles for the purposes both of proving and of understanding and communicating the proof.