What Is a Theorem (in Practice)? The Role of Metamathematics in the Making of Mathematics

This article advocates the benefits of a sociological perspective for the philosophy of mathematical practice. Drawing from the literature of the sociology of sciences, it defends a community-centered approach of the study of mathematical practice and assesses the role of the notion of metamathematics in mathematical change and in stabilized mathematical practices. It relies on the case study of the emergence of geometric control theory at the beginning of the 1970s and of the citational practices associated to the community of control theory since the mid-1990s. The case study shows that the introduction of geometric tools in control theory at the end of the 1960s induced a change in the metamathematical views that control theorists had on their objects. It is then demonstrated how membership to the community of control theory shapes the production and the reception of the theorems of Stefan, Sussmann and Nagano. Interpreting the historical development and citational practices of this community through the perspective of metamathematics, this paper concludes by discussing the role of the orbit theorem in control theory, both as a cognitive label and as a social marker of membership to this community.