Platonismo e naturalismo em matemática: os axiomas da teoria dos conjuntos

Abstract

Working with demonstrations is the main activity of a mathematician. Inmathematics, most propositions accepted as true ones are liable to demonstration, in other words, they are seen as theorems. But a demonstration needs axioms to start the proving process. The same process occurs in the Set Theory, since the Set Theory is a formaltheory. A set theoretic axiom can not be demonstrated, but it is accepted as true. Or it is simply accepted. This work evaluates the processes by which the axioms of Set Theory are accepted, or justified by Platonism and Naturalism in Mathematics. In this context, this work begins with a description of case studies, namely the non-constructive reasonings andthe notion of existence in Set Theory. To begin with our philosophical analysis we have chosen Platonism in Mathematics, which considers the existence of mathematical objects in a metaphysical context. We analyze Gödels Platonism in Mathematics and the epistemological problem it has, which is placed in an argument with a causal theory of knowledge bias. With the impossibility to have a justification for the axioms of the Set Theory based on metaphysics, through an intellectual intuition, the problem of justification for the set theoretical axioms remains. The problem is to find a justification for the set theoretical axioms appropriate for mathematical affairs. Therefore, we present Maddys Mathematical Naturalism as a plausible solution to the mathematical practice for thejustification of the axioms of the Set Theory, which constitutes a neglecting of Mathematical Platonism, in favor of a mathematical epistemology suitable for mathematical everyday use.