A Practice-Based Critique of Reverse Mathematics
Jan W. Gronwald

Abstract:
The present thesis studies the programmatic and formal choices made in the development of Reverse Mathematics (RM), a framework for the analysis and extraction of foundational assumptions underlying ordinary-mathematical theorems. It offers a critique of RM, based on its unfaithful representation of the latter. Among other issues, the main two problems preventing the classical RM to fulfil its foundational ambitions are the arbitrary techniques of encoding the informal mathematics, and the lack of distinction between a theorem and its proof that in practice leads to the possibility of calibrating a single theorem with different set-existence principles. Then, the Constructive RM is tried against the same questions. The thesis concludes that the constructive frameworks for RM offer a more fine-grained analysis together with a more faithful representation of some significant portions of informal mathematics, but they cannot analyze theorems whose constructive version are viewed within classical mathematics as inequivalent to their classical counterparts.