Incommeasurable magnitudes and irrational numbers

Abstract

Some books on the history of mathematics argue that the discovery of incommeasurable quantities, around the 5th century, Before the Common Era, was followed by a crisis of logical foundations in the structures of mathematics. We noticed, from some publications and articles, that this fundamental crisis may have been a mistake or a misinterpretation of ancient writings, because, if it had occurred, it did not alter the rumors of Greek discoveries. Eudoxo's proposed solution stems from the evolution of studies that were ongoing and, two millennia later, served as the basis for the construction of Dedekind's cuts, basic to the conception of the rigorous concept of real numbers (rational and irrational). This bibliographical research work aims to revisit and resume, from the historical mathematical character, the main ideas that emerged with the emergence of incommensurable quantities and their particularities, from Greek antiquity, through their consummation that culminated in the formal idea of irrational numbers until we reach the present day. As the main contribution of this work, we highlight the presentation of new perspectives and approaches to the topic in vogue based on looks at the past through historical facts, making it possible to understand how the chains of ideas took place over time, enabling the expansion of understanding of mathematical objects and concepts today.