Certain mathematical theories are about a single, so-called intended, structure (e.g., arithmetic is about natural numbers) while others investigate general properties of all structures they axiomatize (e.g., group theory). The former theories are known as non-algebraic, while the latter as algebraic. One of the problems in the philosophy of mathematics concerns a systematic explanation of this phenomenon, ideally providing a theoretical notion that could explicate the concept of intendedness. I will briefly review some of the existing philosophical approaches regarding arithmetic, including the work of Halbach and Horsten [1], Button and Smith [2] and Walter Dean [3], among others. In the main part of my talk I will suggest a novel perspective on this problem, based on the measures of complexity of models, such as Scott ranks. I will try to explain basic technicalities involved in this notion and illustrate how it deals with the problem at hand with a few examples of algebraic as well as non-algebraic theories, including PA (by leveraging results from [4]) and weaker systems like Robinson’s or Presburger’s arithmetic.

Bibliography

1. Volker Halbach and Leon Horsten,Computational Structuralism,Philosophia Mathematica,vol. 13 (2005), no. 2, pp. 174–186.
2. Tim Button and Peter Smith,The philosophical significance of {Tennenbaum}’s theorem,Philosophia Mathematica,vol. 20 (2012), no. 1, pp. 114–121.
3. Walter Dean,Models and computability,Philosophia Mathematica,vol. 22 (2014), no. 2, pp. 143–166.
4. Antonio Montalbán and Dino Rossegger,The structural complexity of models of arithmetic,The Journal of Symbolic Logic,(2023), pp. 1–17.