Jenni Rytilä

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.

The paper shows how to use the Husserlian phenomenological method in contemporary philosophical approaches to mathematical practice and mathematical ontology. First, the paper develops the phenomenological approach based on Husserl's writings to obtain a method for understanding mathematical practice. Then, to put forward a full-fledged ontology of mathematics, the phenomenological approach is complemented with social ontological considerations. The proposed ontological account sees mathematical objects as social constructions in the sense that they are products of culturally shared and historically developed practices. At the same time the view endorses the sense that mathematical reality is given to mathematicians with a sense of independence. As mathematical social constructions are products of highly constrained, intersubjective practices and accord with the phenomenologically clarified experience of mathematicians, positing them is phenomenologically justified. The social ontological approach offers a way to build mathematical ontology out of the practice with no metaphysical magic.

The core idea of social constructivism in mathematics is that mathematical entities are social constructs that exist in virtue of social practices, similar to more familiar social entities like institutions and money. Julian C. Cole has presented an institutional version of social constructivism about mathematics based on John Searle’s theory of the construction of the social reality. In this paper, I consider what merits social constructivism has and examine how well Cole’s institutional account meets the challenge of accounting for the characteristic features of mathematics, especially objectivity and applicability. I propose that in general social constructivism shows promise as an ontology of mathematics, because the view can agree with mathematical practice and it offers a way of understanding how mathematical entities can be real without conflicting with a scientific picture of reality. However, I argue that Cole’s specific theory does not provide an adequate social constructivist account of mathematics. His institutional account fails to sufficiently explain the objectivity and applicability of mathematics, because the explanations are weakened and limited by the three-level theoretical model underlying Cole’s account of the construction of mathematical reality and by the use of the Searlean institutional framework. The shortcomings of Cole’s theory give reason to suspect that the Searlean framework is not an optimal way to defend the view that mathematical reality is socially constructed.