Abstract

Articulating various scales of historical analysis is what I consider as the most significant open question in my areas of research. Inquiries that defined their scope over a long time period and/or a large geographical range were common from the 17th to the 20th century. A typical example is the conception of the evolution of algebra in the long run, from  procedures for solving equations to a discipline dealing with abstract structures. Yet, the more recent development of  archives-directed researches on short-term time-scales and local social spaces  has challenged the historical outcomes that large-scale history had  undertook to account for, e.g.  large structural characteristics, processes, and conditions.  To be sure, such issues are not limited to the field of history of mathematics: the distinction between the longue durée, short term events and medium-term conjunctures had especially been much discussed by the French Annales School in the 1930s, while the development of Microhistory in the 1970s has lead to an interest in the interaction between micro and macro-levels of history. « Mathematical fictions » provide an opportunity for playing with scales of historical analysis. In several epistemological approaches, « fictions » have been used as an analytical category to divide the history  of mathematics into several periods. This category has often encompassed various notions  and problems in the longue durée, e.g. the infinite, squaring the circle, negative quantities, imaginary numbers, infinitesimals, non euclidean geometry, or even mathematics as a whole. But « mathematical fictions » may also be investigated as an actor’s category, which meanings may change in various times and social spaces. In this talk I will contrast the longue durée historiography of mathematical fictions with investigations on the short term of the 1880s and the medium-term of 1860-1960. While philosophers have usually associated fictions with ontological issues raised by mathematics, we will see that this terminology was rather used by practitioners of mathematics to address the issue of the choice that one has to make for mathematizing a situation. Mathematical fictions thus call for a history of mathematicians’ choices.