O estruturalismo e o debate ontológico em Filosofia da Matemática

Abstract

This doctoral dissertation aims at arguing for a modal commitment towards truth in mathematical structures, i.e., Mathematics commits itself to demonstrable propositions. This task involves presenting and rejecting mathematical platonism about mathematical Objects, that is, the ontological commitment to mathematics’ abstract Objects. The denial of platonism is founded on Benacerraf’s argument for structuralism, as presented in What Numbers Could Not Be. Structuralism is then presented as a view which does not demand ontological commitment in order to achieve a working concept of objective truth in mathematics, as commitment to demonstrable sentences in instances of structures is itself sufficient. The thesis is divided into three chapters: the first one includes a discussion of key concepts for discussing ontology in mathematics; the second one presents the platonist account, the indispensability argument and Benacerraf’s argument for structuralism; the third chapter contains a discussion about the nature of axiom choices, as well as the difficulty in conciliating ontology and epistemology for a satisfactory definition of mathematical truth, as also defended by Benacerraf. This last chapter is then concluded with an argument for the proposed thesis.