A mathematician might be looking at or studying a discrete or finite counterpart of an existing theory just to see, e.g., what theorems remain provable in both cases. This in itself is interesting from the perspective of so-called reverse mathematics. The core question is to find out what is necessarily required to prove certain theorems? See, e.g., Simpson (2005) and Stillwell (2016). Proofs that also hold in a discrete geometry are thus independent from any assumption about discreteness or continuity.  One might however go deeper into the foundations of mathematics and study finite geometries from a foundational perspective. One such approach is strict finitism (although sometimes the terms ultra-finitism or ultra-intuitionism are used as well) that is not meant as a subtheory of other foundational theories but as an alternative on its own. It shares with the many forms of constructivism the fundamental view that mathematical objects and concepts have to be accessible to the mathematician in terms of constructions that can be executed or performed. The various forms are distinguished from one another as to how the concepts of “execution” or “performance” are to be understood. Most constructivists allow for the potentially infinite, i.e., if a procedure or algorithm will (provably) terminate at some moment in the future, then the outcome is accepted as constructable. See Bridges &amp; Richman (1987) for an overview and the entry on   constructive mathematics. Strict finitism wants to go one step further and argues that an indefinite outcome is not be accepted as an outcome, since, as all computational resources are finite, it could very well be that these resources have been used up before the outcome has been reached. The additional qualification serves to make the distinction with Hilbert’s finitism which, roughly speaking, can be seen as a form of finitism on the meta-level (e.g., although mathematical theories can talk about infinite structures, still the proofs in such theories must have a finite length). As might be expected, strict finitism is not a popular view in the philosophy of mathematics. Nevertheless a number of proposals have been put forward. A history and an account of the actual (though now somewhat dated) state of affairs can be found in Welti (1987). In section 2 more will be said about such proposals.