Hermann Günter Grassmann (1809-1877) e o desenvolvimento do pensamento geométrico : a complementaridade entre alguns aspectos da Die Lineale Audehnungslehre (1844)

Abstract

This is a theoretical research and aims to study the foundations of mathematics present in the ideas which permeate the work of Die Lineale Ausdehnungslehre (Extension Theory) (1844) by the German mathematician Hermann Günter Grassmann (1809- 1877). As the Ausdehnungslehre is originally published in German, we used in this study its Spanish translation – Teoria de la Extension (1947). The discussions in this research are based on the concept of complementarities related to the analysis of the cognitive and epistemological aspects of mathematical concepts studied in the algebraic-geometry context. This research aims to investigate the configuration of the dualities: intentionality-extensionality of mathematical concepts and analysis-synthesis in the development of the geometric thinking, verified since Euclid and, mainly, in Grassmann’s work. The Ausdehnungslehre is important not only for the new concepts explained, but also for presenting an extremely rich introduction to ideas which reflect the fundamentals of mathematics through an interesting and significant philosophy from a historical point of view. Therefore, an investigation into the origins of this thinking and its possible consequences, not only for Mathematics but also for Epistemology, Philosophy and Education is necessary. We can infer with this research that Grassmann’s epistemology may add new elements to the approaches of the development of the geometric thinking and highlight the importance of the epistemological questions in the evolution of mathematical knowledge. It is our judgment that the Mathematics education needs and should be close to the discussions of ideas that arise in the study of historical and philosophical evolution of mathematical concepts.