Nicola Bonatti M.Phil

2023 April, 1 - September, 30

Affiliation: MCMP/LMU Munich

Research for a study about:

A Unified Approach of Extremal Assumptions

The project aims to provide a comprehensive account of extremal assumptions in mathematical theories, as well as their relevance for the epistemology of mathematics. Extremal axioms impose a condition of either minimality or maximality on the admissible models of an axiomatic theory. For example, while the axiom of Induction enforces the requirement of minimality on the models of Peano arithmetic, the axiom of Line Completeness imposes the condition of maximality over the models of Hilbert geometry. The first part of the project investigates how the restrictions of minimality and maximality can be implemented in precise and general terms. Specifically, two main questions will be addressed: (i) How to formulate extremal assumptions in a suitable logical language? (ii) Under what conditions do extremal assumptions entail the categoricity of the theory? It is claimed that the minimality and maximality constraints can be understood with respect to either the size of the individual domain or according to the structure-preserving maps between models. The second part of the project considers the role played by extremal axioms in securing knowledge of the intended model of foundational theories. In each of the cases considered an extremal axiom is being put forth to the effect that a particular formal notion provides a satisfactory characterization of a vaguely conceived intuitive idea. Thus the justification of an extremal axiom has to depend on a combination of intrinsic and extrinsic considerations.