Zoom Talk: Heather Burnett (CNRS, University of Paris-Diderot)
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(joint work with Andrew Arana, philosophy, Université de Lorraine, Archives Henri-Poincaré)
The goal of this talk is to bring together research from two apriori unrelated academic domains: the study of purity of methods in the philosophy of mathematics and the study of verbal hygiene in sociolinguistics. Verbal hygiene is, broadly speaking, “the urge to meddle in matters of language” (Cameron, 1995); more specifically, this term 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. The most widely discussed example of verbal hygiene is standard language prescriptivism, whose advocates argue that the pronunciations, words or phrases belonging to the “standard” language are superior to those found in non-standard varieties or registers. However, as Cameron argues, anti-prescriptivists (or descriptivists) also engage in verbal hygiene when they argue that all linguistic practices should be considered equally good.
Our main proposal is that advocacy of (im)purity in mathematics is structurally parallel to (anti)prescriptivism in language. At a basic level, in the same way that prescriptivism and anti-prescriptivism are ways of regulating linguistic practice, we argue that purity and impurity are ways of regulating mathematical practice. As such, in the same way that the study of (anti)prescriptivism is part of the study of verbal hygiene, we argue that the study of (im)purity in the philosophy of mathematics should be part of a broader study of the social forces regulating mathematical practices, which we call "mathematical hygiene".
To clarify the proposal, we will pursue as a case study of mathematical hygiene Descartes’ algebraic geometry and Newton’s subsequent criticism of it. We will show that both Descartes and Newton’s attitudes can be seen as shaped by their social context. Newton saw Descartes’ novel algebraic method as a challenge to his quest for “elegancy” in geometry, which itself was fueled by a deep adherence to a prisca geometria cum prisca scientia. We will also situate this dispute within the Querelle des Anciens et des Modernes that took place in late seventeenth-century France and Britain, and show how this social context helps make further sense of this case of mathematical hygiene.