Abtraccionismo matemático y filosofía de la geometría

Abstract

Mathematical abstractionism is the position that claims that the method of abstraction must play a central role in the attempts to provide a philosophical foundation for modern mathematics. In this paper, we aim to explore the relevance of this position for the modern axiomatic foundation of Euclidean geometry. To this end, we focus on what we call the abstractionist conception of geometrical magnitudes, according to which the different kinds or species of magnitudes are obtained by a process of abstraction from a system of geometrical quantities, on the basis of an equivalence relation defined in strictly geometrical terms. After providing a brief historical overview of the development of this conception, we argue that a closer examination of the abstractionist approach to geometrical magnitudes brings to light philosophically significant issues concerning the ideal of purity of method in the foundations of geometry.