As some philosophers have argued, the philosophy of mathematics can become a microcosm of philosophy’s more general and central questions—questions of epistemology, metaphysics and the philosophy of language. Furthermore, the study of those parts of mathematics to which philosophers of mathematics most often attend (logic, set theory, arithmetic) seems designed to test the merits of extensive philosophical views about the existence of abstract entities or the tenability of a particular picture of human knowledge. In line with these ideas, I present the relation between a mathematical structure and its basis as a microcosmos from which to study metaphysical grounding. The analysis shows that formal resources from mathematics decrease ambiguity when introducing the concept of grounding in mainstream terms. 

In mathematics, results are often pursued in such a way as to minimize the resources explicitly invoked to derive them and maximize the generality of those results. To this extent, the notion of a basis for a particular structure is recurrent in many areas of mathematical practice. A basis B   for a space M, constituted by a set X, is characterized as a subset of X with two characteristic properties: 

1. B  generates X: Every element of X \ B   can be decomposed into elements  of B .  

2. B   is minimal: If b is an element of B   then it cannot be decomposed into other elements of B .

The methodological advantage of working with bases in mathematics is that relevant properties can be easily determined for B and then inferred for all M. The core of my proposal is that such inference occurs by virtue of the relation between facts aboutMand  facts about B . This relation is what I call Mathematical Grounding. As an illustration, consider the cases of a vector space V and a topological space τ . Linear properties can be derived from properties of a basis B V of V . Specific B V -facts ground linear-facts of V . Likewise, a specific topology on a space τ can be derived from properties of the open sets of a basis B τ . Topological B τ -facts ground topological τ -facts. 

The aforementioned relation models the notion of grounding in metaphysics. The Grounding Mathematical Model (GMM) accounts for grounding as a constitutive relation of determination or explanation that cannot be causal or probabilistic. The GMM specifications accurately capture the relevant properties of grounding as described in traditional accounts, incorporating new theoretical advantages. To provide an example, we admit that the relationship between a basis and its corresponding structure is generative. Although generativity is a property present in diverse contexts of knowledge, in the case of mathematical grounding, the absence of causal relations fosters a metaphysical (rather than physical) approach: the generation of linear facts from basis-facts cannot have causal elements. These results are convenient for the study of grounding when contrasted with causation. Multiple instances support the idea of an analysis of grounding from the GMM’s perspective.