From September 2025 I will be the Principal Investigator in the project Meta-Theoretic and Intertheoretic Reductions in Foundations of Mathematics (MIRFOM) funded by NCN (National Science Center OPUS grant). The grant is hosted by the University of Warsaw. Below you can find the short description of the research objectives. Do not hesitate to contact me, if you are curious about the project!

Metatheoretic and Inter-Theoretic Reductions in Foundations of Mathematics

Reductions are ubiquitous in studies of the foundations of mathematics. The proposed project aims to analyze two groups of such reductions: meta-theoretic and inter-theoretic reductions. Examples of the first kind are various formal explications of philosophical concepts (such as the explication of the concept of truth by means of axiomatic theories) and standpoints, such as so-called foundational equivalences between philosophical standpoints and formal theories. Examples of the second type consist of reductions between formal theories such as interpretability, feasible interpretability, proof-theoretic reduction, or definability. In the project, we want to analyze the epistemic significance of these two types of reductions.

Many natural formal theories occurring in the foundations of mathematics seem to have a designated subject matter. For example, arithmetical theories like Peano Arithmetic (PA) are often described as first-order theories of the arithmetic of natural numbers; certain subsystems of second-order arithmetic, like the theory of arithmetical comprehension (ACA_0), are designed to grasp the real number analysis; axiomatic theories of truth, like the theory of compositional truth (CT), or Kripke-Feferman theory of self-applicable truth (KF), are meant to formalize the notion of truth used in mathematical and informal reasoning; theories of sets, such as Zermelo-Franekel theory (ZF) are formal theories grasping the concept of set. Meta-theoretic reductions state that certain concepts or foundational standpoints are reducible to a formal theory. They are crucial for formal investigations into the foundations of mathematics: without them, there seems to be no connection between a formal theory and a (broadly construed) subject matter of an area of mathematics. They also provide a good sense of the conceptual content of a theory–theories formalizing an area of mathematics or a foundational standpoint are intuitively about the concepts used in the area of mathematics or the foundational standpoint. Our project aims at analyzing this intuitive idea philosophically and making it formally precise.

Inter-theoretical reductions are crucial tools in the proof-theoretic and model-theoretic analysis of formal theories. Despite their prevalence and utility, their epistemic status remains unclear. What philosophical consequences one can draw from the existence of a given reducibility relation between two formal theories? Our main objective in this part of the project is to analyze this question in the context of the conceptual content of a theory. How strong a reducibility relation do we need to preserve the conceptual content between theories? Also, it seems that in acceptance of a formal theory formalizing some concept (e.g. accepting ZFC as formalizing a concept of set), one seems to be implicitly committed to acceptance of concepts that are not explicitly formalized by the theory (e.g. a concept of a natural number). How strong a reducibility relation between a theory formalizing one concept and a theory formalizing a different concept should be to give rise to such conceptual commitments. This appears to be an interesting and insufficiently explored philosophical issue. Presumably, the target reducibility relation is rather strong. Hence, one can ask about the epistemic interpretation of weaker kinds of reductions between theories. In the project, we propose an interpretation of certain natural reducibility relations as a good measure of conceptual distance between theories.