Mathematical Practice and Social Ontology

The relationship between mathematics and social ontology is often guided by the question of the possibility of applying mathematics to social sciences, especially economics. As interesting as these questions may be, they neglect the inverse possibility of applying a conceptual analysis derived from social ontology to mathematics. This issue will be devoted to the question of whether the distinction between social object and social fact, on the one hand, and between different theoretical approaches to the notion of social fact, on the other, can be successfully applied to mathematical practice. The project of this special issue is not a renewal of David Bloor's (1976) research, aimed at a sociological study of mathematics. It is rather a study of the possibility of applying philosophical theories of social objectivity to mathematical objects. This is a new topic that requires the search for adequate mathematical examples to satisfy the objectivity constraints proposed by the philosophy of social ontology. Tendencies in this direction can be traced, but no general survey has been offered. For example, Salomon Feferman (2011) characterized mathematical objectivity as a special case of intersubjective social objectivity. Two other authors included in this issue had already contributed to an analysis of mathematics that approaches or explicitly connects mathematical practice with social ontology. José Ferreirós (2016) defined mathematical practice as an activity supported by individual and social agents and characterized by stability, reliability, and intersubjectivity. Julian C. Cole (2013, 2015) described mathematical objects as institutional rather than mental objects, referring to Searle's theory of collective intentionality. The special issue Mathematical Practice and Social Ontology offers a survey of different point of views explaining the intersubjective origin of mathematical practices: Peirce's semiotics and pragmatism, Husserl's phenomenology, Wittgenstein's philosophy of language, Searle's social ontology, Brandom's inferentialism, embodied cognitive science and enculturation theory are all taken into account. The articles collected here explain how objectivity comes in different degrees, and how practices based on some intersubjective social constitution might account for mathematical objectivity. The papers address the question of whether and to what extent mathematics can be explained as a social intersubjective activity that produces objective knowledge. In doing so, they investigate the relationship between sociology, history of mathematics, the social ontology of processes and conversations, metaphysical theories of social ontology, and formal ontology, showing advantages and difficulties that emerge from an analysis of mathematics that takes into account social, historical and formal language elements.