Polycephalic Euclid? Collective Practices in Bourbaki’s History of Mathematics Anne Sandrine Paumier and David Aubin * Historiography of Mathematics in the 19th and 20th Centuries Workshop, Bergische Universit¨ at Wuppertal, March 20–22, 2013 Abstract In this paper, we argue that Bourbaki’s historiography, which has been extremely influential among mathematicians and historians of mathematics alike, reflected the special conditions of its elaboration. More specifically, we investigate the way in which the collective writing practices of the members of the Bourbaki group in both mathematics and the history of mathematics help to explain the particular form taken by the Elements of the History of Mathematics (1960). At first sight, this book, which has been seen as an “internalist history of concepts,” may seem an unlikely candidate for exhibiting collective aspects of mathematical practice. As we show, historical considerations indeed stood low on the group’s agenda, but they nevertheless were crucial in the conception of some parts of the mathematical treatise. We moreover claim that tensions between individuals and notions related to a collective understanding of mathematics, such as “Zeitgeist ” and “mathematical schools,” in fact structured Bourbaki’s historiography. Introduction Incessant chronological difficulties, which arise when we suppose the physical existence of a single Euclid, lessen, without vanishing, if we accept to take his name as the collective title of a mathematical school [Itard 1962, p. 11]. 1 That the French historian of mathematics Jean Itard (1902–1979) retained the idea of Euclid as a collective mathematician, and more precisely as “the collective title of mathe- matical school,” was anything but innocent. In the early 1960s, the “polycephalic” author Nicolas Bourbaki and the members of this group of mathematicians were dominating the mathematical scene in France and beyond. Many, at the time, saw in Bourbaki the lat- est incarnation of what constituted a true “mathematical school” (or in French une ´ ecole math´ ematique ), wholly original in nature, with clear methods, objectives, and ideology. 2 Only in the context of the Bourbakist experience might the idea of a collective Euclid have had some sort of appeal. 3 As explained by the historian Fabio Acerbi, the lack of historical and biographical data about Euclid had produced yet another fantasy about ∗ Universit´ e Pierre et Marie Curie, Institut de Jussieu de math´ ematiques de Jussieu, 4 place Jussieu, 75252 Paris Cedex 05. paumier@math.jussieu.fr and daubin@math.jussieu.fr. 1 “Les difficult´ es qui surgissent `a chaque instant dans la chronologie lorsque l’on admet l’existence physique d’un seul Euclide s’att´ enuent sans disparaˆ ıtre lorsque l’on accepte de prendre son nom comme le titre collectif d’une ´ ecole math´ ematique.” Unless when taken from a published translation, all translations are our own. 2 For historical accounts of Bourbaki’s domination in the 1960s, see [Corry 1996] and [Aubin 1997]. It was in a paragraph referring to the “´ ecole Bourbaki” that Andr´ e Delachet first revealed to the world that Bourbaki was a “polycephalic” mathematician (his term) [Delachet 1949, p. 113–116]. 3 Note that following conventions established by Liliane Beaulieu, we will refer to members of the group as Bourbakis and their followers as Bourbakists [Beaulieu 1989]. 1