Bi-interpretability is a widely accepted standard of equivalence for first-order theories, understood both semantically via definable inner models and syntactically via mutually inverse (up to isomorphism) interpretations. However, in model-theoretic practice, interpretations frequently employ parameters, yielding bi-interpretability with parameters as an induced standard of equivalence. Despite their prevalence, interpretations with parameters remain understudied from both logical and philosophical perspectives, with Visser characterizing their category as "terra incognita."

This paper addresses this gap through three main contributions. First, I examine key examples including the Klein-Beltrami model and the geometrization of arithmetic. Second, I develop a systematic semantic analysis of interpretations with parameters using uniformly definable families of models, showing how they induce relation-valued profunctors between categories of models. This perspective yields a novel characterization of bi-interpretability with parameters and a new categorical invariant.

Finally, I analyze the semantic status of parameters, arguing against their treatment as additional constants, since this yields a coarser notion of equivalence. Instead, I propose understanding parameters as well-behaved Hilbertian ε-terms, developing a notion of interpretation in the ε-calculus. The main technical conjecture is that bi-interpretability with parameters coincides with bi-interpretability in the ε-calculus, providing another perspective on this notion of equivalence.