Is Categoricity Virtues of Mathematical Theories?

Project management: Dr. Gareth Rhys Pearce Ba, MA

Funded by: ÖAW Post-DocTrack

Projekt-Title:  
Is Categoricity Virtues of Mathematical Theories?

Research stay: 01.12.2025-31.05.2026

Affiliation: University of Vienna

Short Description: 
Categoricity is an unusual property of mathematical theories. In some contexts, such as arithemtic, it is a virtue of a theory. There is a unique intended model of arithemtic and, all else being equal, one might want an axiomatic theory of arithemtic which uniquely describes that model, though of Gödel's theorems put significant limitations on when this is possible. However, in other contexts, such as group theory or geometry, there is no intended model. A significant part of why these algabraic theories are useful is precisely because they are not categorical, so manage to describe a wide array of structures. In other contexts, it's unclear if categoricity should or shouldn't be a virtue. The disagreement between the Universe and Multiverse views of Set Theory can be cast as a disagreement over the role of categoricity. But what are the conditions under which cateogiricity is a virtue of a mathematical theory and why?
Applying work from my doctoral thesis, I attempt to answer this question.