¿Qué significa demostrar?

Abstract

The purpose of this paper is to show the most relevant philosophical aspects that hang around the proof, concerning a method of  justification, of mathematical knowledge. Consequently, I try to point out which have been the most significant and revealing  moments in the history of the method of mathemati cal proof and hi ghlight its restrictions: from Pythagoras to the  formalist Hilbert`s program and Frege and Russell`s logicism. After that, I examine the limitations of the requirement of completeness of the for mali st program directl y related to Gödel ’s incompleteness theorem and dissociation between the  true and the provable. Following, I try to analyze, in its essential aspects, Lakatos’ criticism to the Euclidean proof theory as  unique method of mathematics and present, in contrast, this author`s empiricist stance on for mal sciences. Finally, I raise some  critical reflections by way of conclusion, expressing facts concerning the realm of the difficulty of reflection on mathematical  proof, understood as one of the most complex and abstract constructions of human cognition.