Found in Translation: at the limits of the Hudetz program

TL;DR

This paper explores the concept of theory equivalence through translation, extending the framework beyond first-order logic to include theories in broader mathematical languages, especially in sciences like physics.

Contribution

It generalizes the theory of interpretation to encompass theories not naturally expressed in first-order logic, broadening the scope of definability in mathematical and scientific contexts.

Findings

Proposes a generalized account of interpretability for non-first-order theories

Extends the concept of definability to broader mathematical languages

Addresses limitations of traditional translation-based interpretability in sciences

Abstract

This paper aims to provide an analysis of what it means when we say that a pair of theories, very generously construed, are equivalent in the sense that they are interdefinable. With regard to theories articulated in first order logic, we already have a natural and well-understood device for addressing this problem: the theory of relative interpretability as based on translation. However, many important theories in the sciences and mathematics (and, in particular, physics) are precisely formulated but are not naturally articulated in first order logic or any obvious language at all. In this paper, we plan to generalize the ordinary theory of interpretation to accommodate such theories by offering an account where definability does not mean definability relative to a particular structure, but rather definability without such reservations: definable in the language of mathematics.