Two Ways to Think About (Implicit) Structure

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.