A elaboração de uma "epistemologia da imaginação e da intuição" no campo da matemática e implicações para a educação matemática : diálogos com Henri Poincaré e Gaston Bachelard

Abstract

In this thesis, of a theoretical nature that was carried out in the Postgraduate Program in Education for Science and Mathematics of the State University of Maringá - UEM, there was the proposal gathering subsidies to place, in its epistemological basis, the problem of the elaboration of a "Epistemology of imagination and intuition" in the field of mathematics, aiming at (mathematical) education. This work was based on the epistemology and philosophy of science of Henri Poincaré and Gaston Bachelard, because both brought many relevant contributions in this field and for our research; his conceptions of the construction of science, his relation to mathematics, and the role of imagination and intuition establish dialogue in his works. Imagination and intuition, in addition to logic, are the engines of mathematical thinking, which, through its dynamicity, promote creativity in mathematics itself and in other sciences. In the thesis, much more than finding answers, we seek, from a coherent discussion, to put in its basis the question of the elaboration of an "epistemology of imagination and intuition" in the field of mathematics. This discussion, according to our understanding, must be made through examples from the field of mathematics. They are not mere illustrations of theory, but they allow to put in motion the notions that imagination and intuition do emerge in mathematical thought, constituting itself as a research methodology. When choosing this methodological path, we seek to give visibility and sustainability to the problem of the elaboration of this epistemology. We are considering imagination and intuition in its epistemological dimension, that is, as processes of access to mathematical knowledge, discovery and creation processes within the internal dynamics of mathematical knowledge. In the presented context, it is important to clarify that, for the beginning of the research, we elaborate some ideas as starting point of the discussion and we call them "working hypotheses". Thus, we associate, initially, the intuition (mathematics) with the discovery and the imagination (mathematics) with the creation, becoming this a constructive process. To imagine is to create and it is a free act, but it is not arbitrary. This thesis, besides bringing contributions to the field of philosophy and epistemology of science and mathematics itself, has implications for mathematical education by the formative potential that imagination and intuition give to the teaching of mathematics, which makes it possible to bring clarifications, reflections, arguments and ideas for teachers of mathematics in initial and continuing formation. This will allow them to develop innovative methodologies for teaching mathematics in which imagination, intuition and creativity are presented, in addition to the purely logical and algorithmic