Incomensurabilidade, antifairese e equivalência de áreas

Abstract

In this work, we address some events of the historical moment in which the discovery of incommensurability took place. From the combination of historical fragments, we were able to speculate some characteristics of Greek society about the mathematical development of the time. We highlight the Greek geometric skill and its independence from arithmetic resources. We looked for more recent studies and experiments that reaffirm the important role of the use of pictographic resources in understanding abstract concepts and the use of logic in solving problems, even if intuitively. The concept of reason, defined from the antifairesis mechanism, is inserted as one of the main tools in understanding incommensurability. We approach definitions, properties, correlate the antifairesis mechanism with some more modern perspectives and divide the mechanism into three structures: numerical, geometric relating segments and geometric relating areas. We conclude the work with some important results from the study of antifairesis involving areas.