Theorems as Constructive Visions

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 Mathematical 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