De hipaso de Metaponto a Eudoxo de Cnido: a história por trás da descoberta dos irracionais

Abstract

This study aims to analyze the historical, philosophical, and mathematical context of the discovery of irrational magnitudes in Ancient Greece, highlighting the crisis provoked by incommensurability and the contribution of Eudoxus of Cnidus’ theory of proportions to the reconstruction of the foundations of Greek mathematics. To this end, a qualitative research approach was conducted, based on a literature review of classic works in the History of Mathematics and Mathematics Education, drawing primarily on Boyer (1996), Eves (2004), Roque (2012), Miguel and Miorim (2019), D’Ambrosio (2012), Pommer (2018), as well as studies related to Euclid's Elements. The investigation discusses the Pythagorean conception of number and commensurability, the discovery of the irrationality of the diagonal of a unit square, anthyphaeresis as a method for identifying incommensurability, and the theory of proportions developed by Eudoxus as a response to the problem of irrationals. As a main result, it was found that the so-called Crisis of Incommensurables represented a major epistemological rupture by exposing the limits of the Pythagorean paradigm and driving the development of new conceptual frameworks for the treatment of magnitudes. Furthermore, the study points to the History of Mathematics as a relevant pedagogical resource for teaching irrational numbers, contributing to a more contextualized, investigative, and meaningful approach to mathematics learning.