The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics, Structuralism, and the Oulipo in France

Abstract

The ArgumentThe group of mathematicians known as Bourbaki persuasively proclaimed the isolation of its field of research — pure mathematics — from society and science. It may therefore seem paradoxical that links with larger French cultural movements, especially structuralism and potential literature, are easy to establish. Rather than arguing that the latter were a consequence of the former, which they were not, I show that all of these cultural movements, including the Bourbakist endeavor, emerged together, each strengthening the public appeal of the others through constant, albeit often superficial, interaction. This codependency is partly responsible for their success and moreover accounts for their simultaneous fall from favor, which, however, can clearly be seen as also stemming from different internal problems. To understand this dynamics, I argue that Bourbaki's role can best be captured by using the notion of cultural connector, which I introduce here.

Mathematics, PISA, and culture: An unpredictable relationship

“…Also, the sequence of activities she undertook, all of which exploited number base materials, seemed designed to "help children... make the transition from intuitive experiences towards abstract understanding" (Vanpaemel et al, 2012, p.425). Significantly, despite the explicit real-worldness of the topic, the task, and those that preceded it, was located in a mathematical world of multiplicative structures, very much in keeping with the Bourbakian belief about the primacy of the pure over the applied (Aubin 1997;Weintraub and Mirowski 1994), and the subordination of the applications of mathematics (Klein 2003;Maasz and Schlöglmann 2006). …”

“…Such activities underpinned Papy's mandated curriculum, the main features of which were "the unity of mathematics, the progressive introduction of fundamental mathematical structures, the use of (pseudo) concrete situations, the student's ability to construct mathematical demonstrations, and a basic familiarity with logical concepts" (Vanpaemel et al 2012). In this respect, while Bourbaki generally eschewed emotions with respect to mathematics, it was acknowledged that intuition, based on the deep and immediate knowledge of the logic of the structures familiar to mathematicians, had a role to play (Aubin 1997).…”

“…Commentary: During this episode, as she did throughout her lesson sequence, Karin's teaching was located within the mathematical world of the Bourbakists (Aubin 1997;Weintraub and Mirowski 1994), with no concessions to the application of what she taught (Klein 2003;Maasz and Schlöglmann 2006). It is known that the hierarchical nature of quadrilaterals is problematic for learners (Fujita and Jones 2007) and throughout this episode Karin focused attention not only on the conceptual basis of this hierarchy but also the logic underpinning it.…”

“…However, her frequent public discussion did not, in this or any other case, privilege, or even encourage, students' non-standard solution processes, as would be expected in an RMEfocused classroom (Van Putten et al 2005). Her aim, throughout this and other episodes, was the creation of a collective understanding of the same Bourbakian structural relationship (Aubin 1997), which was unrelated to any obvious application of mathematics (Klein 2003;Maasz and Schlöglmann 2006). In sum, this particular episode demonstrated various characteristics, some of which were clearly RME and some of which were clearly Bourbakian.…”

Structuralism, Attitude and the Computer: Questioning the Notion of “Cultural Computing”

“…"I cannot say," he simply wrote, "to what extent this [tendency] is conditioned by the internal dynamics of the development of mathematics, or by ideological currents like the degradation of science's superior image in public opinion and scientists' questioning of the social status of their practice." 20 This turning away from the importance of structural ideas, or a turning away from the idea that the observation and study of emergent properties might be a fascinating thing in itself, has been replaced, in some peoples' minds, by a renewed fascination with the qualities of surface. In this view, it's the surface of a work of art (or work from other intellectual areas) that creates interest.…”