“Apenas setas”: a teoria das categorias como linguagem para uma matemática estruturalista

Abstract

For some years now, there has been a philosophical debate about the relationship between two distinct intellectual movements. Within mathematical practice, the Theory of Categories emerged in the 1940s. Initially without any major foundational pretensions, little by little the theory gained in scope and came to be considered, at the very least, a very useful language for characterizing and studying abstract mathematical structure. Almost simultaneously, philosophers concerned with questions about the nature of mathematical objects proposed a structuralist program, based on the idea of shaping an understanding of the nature of mathematics that takes into account that each area of this science describes the formal factors common to various structured systems. The affinity between Category Theory and a structuralist philosophy of mathematics is almost obvious, leading naturally to the question of the possibility of the theory being employed as an autonomous conceptual framework, capable of articulating a peculiar view of mathematics, without any kind of dependence on other foundational approaches, such as Set Theory or Type Theory. This possibility has been denied by some philosophers of mathematics, giving rise to a dispute without a unanimous solution. This thesis presents a panoramic view of this whole scenario and tries to show that those who reject the autonomy of category theory have epistemological presuppositions that are not unanimously accepted.