Two Ways to Think About (Implicit) Structure
Year: 2026
Type: article
Abstract: The article compares two ways to think about the implicit structure of systems studied in pure mathematics. The first approach specifies implicit structure in terms of formal languages, relying on a model-theoretic notion of definability. The second approach defines the notion in terms of invariance criteria. The structural features of a given system are often considered to be properties which are invariant under specific transformations of the system. In the paper, I analyze these two approaches to implicit structure in terms of invariance and definability conditions by drawing to the field of finite geometry. I then explore their formal relationship in the framework of relational algebra, based on the concepts of Galois closures and Krasner algebras. Finally, I examine the equivalence of mathematical systems with respect to their implicit structure in terms of the model-theoretic notion of bi-interpretability.
Source: Erkenntnis
Author: Georg Schiemer
Institution: University of Vienna
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,290
OpenAlex ID: W7223183272