O termo ‘axioma’ no tempo, considerando a relação entre a filosofia e a matemática alicerçada no pensamento sobre complementaridade ‘otteano’

Abstract

This research aimed to find out, regarding Mathematics Knowledge and its constitution, what means the fact that the term ‘axiom’ until the XIX century was understood as the antonym to hypothesis and, nowadays, to be considered its synonym. Thus, our goal with it was to highlight by means of a historical, philosophical approach as well as of a semiotics perspective, the meaning taken by the term ‘axiom’ that appear in the writings of philosophers and mathematicians since Plato’s time up to the modern times, revealing oscillations that occurred to such meaning and both reflect upon and interpret the relationship created between Philosophy and Mathematics starting from the point of view that focuses on the relationship between Language and Mathematics. Then, showing the factors/aspects that promote and/or imply such oscillations, in a way to conjecture over such movement between ‘meaning of the term/concept’ and the ‘relationship between Language and Mathematics’. Therefore, more specifically, we intended to highlight the conceptual transformation that happened in the Mathematics field, enhanced in the very analysis of the term ‘axiom’, when treating it in isolation, for its special status in the concepts of Mathematics. We got engaged in developing a reflexive-interpretative research following theoretical-bibliographical characteristics, to which we added, as a theoretical and methodological assumption, a vision in the sense of Complementarity of Michael F. Otte’s reasoning (or reasoning over Ottean Complementarity) among aspects that, traditionally, are treated in a dichotomic way when related to Mathematics Knowledge (intension and extension, descriptive/contemplative thought and operating/instrumental thought, quantity/quality). We identified the relationship between Language and Mathematics as our object of study to comprehend the relationship between Philosophy and Mathematics, which we aimed to observe, interpret and highlight in and through the analysis of oscillations of the meanings attributed to the term ‘axiom’. Our thesis shows that the change in the meaning of such term, evidenced in the relationship between Language and Mathematics, may have been moved from descriptive-contemplative aspects to 14 operational-instrumental ones, in such a way that the objectivity of Mathematics starts to reveal itself in activity and future uses and not in terms a priorifoundations anymore (OTTE, 2011). This presupposes that the meaning of the term ‘axiom’ can be found now in formal deductions and theory developed as an entity in itself, and its meaningful applications for such theory. Thus, among the results of Mathematics Philosophy, comes in evidence the discovery that we need to recognize that a semantics analysis of its concepts, as customary in Humanities, is not enough in Mathematics and we should take into account pragmatic aspects of the representations. This research observed that the more Language and Mathematics get close to each other the more the term axiom tend to be interpreted as a synonym to hypothesis, and that such proximity has evolved every time the philosophical thought got close to the mathematical one. The theoretical basis that supports the reflexive-interpretative analysis is the reasoning over the Ottean Complementarity (1993, 2001, 2003, 2008, 2012), based on Kant’s philosophy (1987, 2000, 2001) and Charles Sanders Peirce’s semiotics (1970, 1990, 1997, 2003). We got to the conclusion that this study offers its reflections over the genesis and historicity of Mathematics Knowledge to Mathematics teachers/educators as well as presents the possibility of a new way of didactics approach: reasoning over the Ottean Complementarity.