The Cognitive Foundations of Mathematics: Human gestures in proofs and mathematical incompleteness of formalisms

In this text we will focus on some "geometric" judgements, which ground proofs and concepts of mathematics in cognitive experiences. They are "images", in the broad sense of mental constructions of a figurative nature: we will largely refer to the well ordering of integer numbers (they appear to our constructed imagination as spaced and ordered, one after the other) and to the shared image of the width less continuous line, an abstracted trajectory, as practice of action in space (and time).