The archives are so populated with few fictional characters as Nicolás Bourbaki, the pseudonym under which a secret group of French mathematicians has been writing a general treatise on the discipline, the Elements d and mathematics , for almost ninety years. This invented mathematician is embodied in bills for office supplies, sonnets of his creation, an invitation to his daughter’s supposed wedding and even the angry reaction of the American Mathematical Society to his attempt to become a partner in 1950. Now, its presence also extends to the streets of Paris.
Those who go down from the Pantheon towards the Luxembourg Gardens via Soufflot Street will be able to find themselves, when they turn the corner with Boulevard Saint Michel , with a commemorative plaque. It says like this: “The group of mathematicians N. Bourbaki was spatially specified for the first time on 10 December 1934 at the Capoulade cafe that occupied this place ”. It was inaugurated a few weeks ago by the Paris city hall, at the initiative of the professor and disseminator Roger Mansuy. Unusual decoration for an angle through which all kinds of fast food establishments have paraded, since that day when the founding members began to meet with the idea of writing a textbook.
What has become of Bourbaki? Has it suffered the same fate as the café where it was born? Despite Pierre Cartier, the group’s secretary for decades, declaring in 1998 that “Bourbaki is dead”, the adventure continues . The most visible aspect of the group’s vitality is undoubtedly the Bourbaki seminary, which since its inception in 1948 has only ceased to be held during the first year of the pandemic. Not simply because the idea ceases to be groundbreaking: to share an outside look on the latest significant advances in geometry, analysis or number theory, to make them more accessible to the rest of the mathematical community.
How many real numbers are there?
In these one-hour talks, the long work of understanding and summarizing –sometimes transforming– the chosen theorems culminates. It will be reflected in a text of about thirty pages, which is distributed to the attendees on the day of the seminar and then published in the magazine Astérisque . The process is not without its risks: sometimes the way of thinking about certain mathematical objects has been changed forever, and sometimes fatal errors have been found in the original articles. For a few years, these Saturday meetings have been preceded by the Betty B. seminar, created in honor of Bourbaki’s supposed great-granddaughter, with the aim of facilitating the understanding of some of the talks the following day for master’s degree students or PhD.
After several remote sessions, the seminar will return in a big way in person on the first Saturday in October at its historic headquarters, the Henri Poincaré Institute. Four mathematicians will present what has kept them busy day and night in recent months: not their own research, but that of other colleagues.
Perhaps this desire to disseminate will surprise those who associate Bourbaki with the image of the author of austere treatises that changed the course of twentieth-century mathematics and – almost always to his regret – the way it was taught in school. Today, the influence of his books is much less than fifty years ago, perhaps paradoxically because his style has been completely imposed among mathematicians: symbols of such common use as the empty set or words like “injective” did not exist before. that Bourbaki invented them. Nor does the idea of organizing a text into independent sentences, each followed by its demonstration.
Ninety years later, the group continues its effort to find the definitive presentation of the most useful parts of the maths. In 2016, he published a new book, the first in twenty years: Algebraic Topology . In 2019, a revised edition of the first volume on Spectral Theories , to which soon will follow an unpublished second part focused on one of the key results of the compact group representation theory: the Peter-Weyl theorem.
To write these treatises, the ten active members of Bourbaki –In theory still secret– they meet in a “congress” every summer. According to the method adopted by their founders, they read aloud, word for word, the essays that those responsible for each project have carefully prepared during the rest of the year. Few are the sentences that are read at once, without finishing completely transformed. At the end of the congress, the manuscript looks like a trench. And start over. The process until the final version of the book can take more than ten years.
Does it make sense, in the context of current scientific practices, to spend all that time working on books whose impact is known in advance increasingly limited? Group members are the first to ask. They argue, they advance arguments for and against, they disagree –as in almost nothing–, and they keep writing.
Javier Fresán is Professor Hadamard at the École polytechnique (France).
Edition and coordination : Ágata A. Timón G Longoria (ICMAT).
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT) , in which the researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
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