Formalism, abstraction, strictness and the difficulty in mathematics Formalismo, Abstracción, rigor y dificultad en la matemática

Abstract

We have this four concepts that we usually missunderstand as synonims when we try to make a characterization of the mathematics by no specialized people in the discpline.  The confusion comes to occur, even among people who have received some type of mathematical training because it is not uncommon to find disseminators of the subject that do not clearly establish the differences and relationships that exist between these terms. Those who have had the opportunity to approach mathematics by asking about the nature of their concepts and the structuring of mathematical knowledge, above the simple manipulation of the associated operative processes and / or derivatives of these concepts, know that there are relationships but also differences between the four terms of the statement that contribute, precisely, to the unitary configuration of diversity that is, very surely, one of the essential characteristics of mathematics. In order to stimulate this approach, which we must necessarily promote, to mathematics through behaviors other than traditional mechanical operability; I will try to present in a simple way (hopefully simplicity does not distort the concepts) each of these terms and clarify the most frequent confusions in the interpretation of them.