Making up mathematical reality: a dual grounding model of mathematical social construction

A metaphysical view called mathematical social constructionism states that the abstract structures and objects studied in mathematics are socially constructed entities; they are produced by and depend on the human practices of mathematics that are social, historically developed, and shared among communities. Yet, it is not quite clear what it means for mathematical entities to exist in virtue of practices. This paper aims to answer this question by employing the notion of metaphysical grounding. I develop a dual grounding model that analyses the metaphysical relation between concrete mathematical practices and abstract mathematical entities as two relations of partial grounding: (1) specific patterns of using epistemic resources (axioms and theorems, proof methods, notations, etc.) ground the features of mathematical entities, and (2) social patterns of mathematical communities (learning, using shared meanings and aims, mutual responsiveness, etc.) ground the existence of mathematical entities. I argue that the grounding model where mathematical entities are grounded in social practices has important benefits, because it fits with actual mathematical practice and explains why mathematical social constructions are real entities. By offering a clarified, more detailed picture of the metaphysics behind mathematical social construction, the dual grounding model helps to explain how we can make up mathematical reality by doing mathematics together.