Andrew Arana (Université de Lorraine - Archives Henri Poincaré) & Heather Burnett (Université de Paris, LLF, CNRS)
This paper aims to bring together the study of (dis)unity in the philosophy of mathematics and verbal hygiene in sociolinguistics. Unity is the normative idea that mathematical knowledge (instantiated as theorems) can be arrived at through a wide range of methods belonging to different domains of mathematics; whereas, disunity is the idea that ``proofs that draw only on what is ``close" or ``intrinsic" to what is proved" (Arana & Detlefsen 2011:3) should be preferred to proofs that employ methods from outside the particular domain of the theorem. Verbal hygiene (Cameron 1995) refers to the set of normative ideas that language users have about which linguistic practices should be preferred, and the ways in which they go about encouraging or enforcing others to adopt their preference. We introduce the notion of mathematical hygiene, which we define parallely as the set of normative discourses regulating mathematical practices and the ways in which mathematicians promote those practices. To clarify our proposal, we present two case studies from 17th century France. First, we exemplify a case of mathematical hygiene proper: Descartes' algebraic geometry and Newton's subsequent criticism of it. Then, we compare Descartes' mathematical hygiene with verbal hygiene from this period, as exemplified by the work of the grammarian Claude Favre de Vaugelas (Ayres-Bennet 1987). We argue that both Descartes' and Vaugelas' normative discourses on mathematics and language respectively can be seen as emanating from a common socio-political project: the development of a new bourgeois intellectual class. We conclude that the study of mathematical hygiene has the potential to yield new understandings of the social aspects of mathematical practice, and that similarities between mathematical and verbal hygiene at certain time periods, such as 17th century France, open up a new area of inquiry at the borders of sociolinguistics and the philosophy of mathematics.