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Bourbaki: a secret society of mathematicians. Transl. from the French by Anna Pierrehumbert. (English) Zbl 1099.01022

Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3967-5). 168 p. (2006).
The book under review addresses the general mathematically interested public. Its aim is to help to popularize, uncover, and analyze one of the most secretive, nevertheless most incisive occurrences in the development of modern mathematics, namely the myth of Nicolas Bourbaki and “his” monumental work “Les Éléments de Mathématique”.
It is certainly known to every mathematician that Nicolas Bourbaki is not a specific individual but an invented personage, serving as the collective pseudonym for a group of mainly French mathematical researchers. The official name of this secretly composed Bourbaki group reads “Association des Collaborateurs de Nicolas Bourbaki”. Founded in 1935, this secret society of mathematicians is situated at the renowned École Normale Supérieure in Paris, France, and its activities have been surrounded by mysteries, legends, and speculations within the mathematical community all along. On the other hand, Bourbaki’s fundamental treatise in several volumes, published over the last 70 years, has always been ubiquitous as one of the most influential standard texts on the foundations of modern (pure) mathematics in the second half of the 20th century.
Just as public and important was (and is) the famous, “Séminaire Bourbaki” (founded in 1948), whose nearly 900 published lectures on recent developments in mathematics decisively engraved the face of mathematical research in our days. As the author of the present book points out in his preface, the full story of Bourbaki is one of shadows and dazzling light, surrounded with folklore, enthusiastic admiration, hostile criticism, success and failure, rise and decline, immortality and damnation, idiosyncrasies, and diverse other contradictions. Based on many first-hand accounts, including official documents, testimonies by former Bourbaki members and their relatives or fellows, photographs, and related historical research articles, the author tries to tell the full story of Bourbaki in all its fascinating, multifarious aspects. In doing so, thereby maintaining a high degree of objectivity and balance throughout, he contributes to lifting the veil from the myth Bourbaki, and to explaining its essence to a wider public in a generally intelligible manner.
As to the contents, the book is divided into eleven chapters titled as follows:
1. A group forms; 2. The story of a name; 3. Young turks against stubborn priests; 4. Bourbaki’s “Éléments de Mathématique; 5. Towards axioms and structures; 6. A snapshot of Bourbaki’s work: filters; 7. The Bourbaki seminar; 8. Subtle and austere schoolboys; 9. “For the honor of human spirit”?; 10. New Math in the classroom; 11. An immortal mathematician?
Each chapter comes with its own brief introductory summary, expressed by a few characteristic sentences, and on virtually every page the reader finds illustrating photographs, facsimiles, mathematical explanations, biographical epitomes, or additional clarifying remarks.
Chapter 1 depicts the beginning of the legendary enterprise “Bourbaki” in December 1934, the original goals of its young founding members (H. Cartan, C. Chevalley, J. Delsarte, J. Dieudonné, and A. Weil), and how the modest project of writing a new textbook on analysis soon turned into the ambitious plan of refounding modern (pure) mathematics in a systematic way based on set theory, axiomatic rigour, and the concept of mathematical structures. Also, the reader learns here many things about the founding members themselves, the internal working principles and rules of the group, its strict recruting and retirement policy, and what the specific method of collaboration between the individual members looked like.
Chapter 2 recounts how and why the name Nicolas Bourbaki was adapted by the group. Originating in École Normale Supérieure folklore, based on a student prank, and referring to the former real French general Charles Bourbaki (1816–1897), this pseudonym has always been a central part of the myth surrounding the group. Chapter 3 describes the scientific circumstances that triggered the creation of Bourbaki just in France after World War I, ranging from the extinction of a whole generation of young French scientists, during that devastating war, up to the pioneering work of German algebraists in the first quarter of the 20th centurye.
Chapter 4 discusses Bourbaki’s monumental treatise “Les Éléments de Mathématique”, published between 1939 and 1998 in ten books with over sixty chapters, in a down-to-earth language accessible also to non-mathematicians. Bourbaki’s new coherent terminology and notation, its deductive, structural viewpoint for reforming essential parts of pure mathematics from scratch, and the impact that these volumes actually evoked – all this is examined in this chapter.
Chapter 5 analyses more closely Bourbaki’s view of mathematics. The author characterizes Bourbaki’s work as an intellectual construction of profound unity, appearing as a hierarchy of abstract structures built on a strict axiomatic approach à la D. Hilbert.
Chapter 6 reflects upon the very fact that Bourbaki mainly focused on scrutinizing current mathematics rather than on inventing revolutionary new mathematics. Nonetheless, Bourbaki’s approach did produce several new fruitful concepts, including the so-called topological filters, and the latter ones are illustrated in more detail as a representative mathematical snapshot.
Chapter 7 describes another organizational invention of the Bourbaki group, namely the distinguished Bourbaki seminar, which is an outstanding mathematical institution in Paris since its foundation in 1948. The author analyzes the educational role that these seminars have played over the past half-century, and how the taste of the Bourbaki members is reflected therein.
Chapter 8 provides a humorous glimpse “inside Bourbaki”, mainly by showing the more private face of the group, which is far from being serious and barren. Be it the creation of funny new words, the writing of witty poems, the mutual irreverence among the Bourbaki members, their mutual schoolboy pranks, or their non-mathematical press releases – all this bespeaks the duality of Bourbaki’s subtle, creative spirit that underlies its enormous work.
Chapter 9 turns to the fact that Bourbaki did not only receive praise. The author reports on the harsh criticism that Bourbaki also had to encounter, on the other side, particularly for its sophisticated style, its choices of themes, its academic power in France, its alleged elitary arrogance, and its refusal to acknowledge anything outside of its own vision of pure mathematics, where some of this criticism even came from the group’s own (former) members, including its founders.
Chapter 10 investigates to what extent Bourbaki’s structural point of view in (pure) mathematics really affected the wave of “New Math” that surged into secondary education during the 1970s, and how far the power of Bourbaki was to hold responsible for the revolution, and the counterrevolution, that took place in teaching mathematics back then.
Chapter 11 finally ventures a conclusion concerning Bourbaki’s achievements, merits, survival chances, and its possible role in the future. Although Bourbaki continually replaces its members, it seems that the group has not been able to maintain its erstwhile mathematical youth and pace-making character. The change of the general mathematical atmosphere since the 1980s, the increasing lack of time also for the leading mathematicians in our days, the naturally declining enthusiasm for such a never-ending, gigantic project after seven decades, and the fact that mathematics is now moving much too quickly – all these circumstances make Bourbaki’s prospects for the future rather uncertain. However, Bourbaki has had a lasting influence on mathematics that can barely be overestimated, but now “its work is done and done well”, as Laurent Schwartz, a prominent former Bourbaki member, once put it. Intellectually, Bourbaki profoundly modernized (pure) mathematics and clarified its language and concepts. Some mathematical disciplines are indebted to Bourbaki more than others, above all the algebraically imbued ones, but in general Bourbaki did add quite a bit to “the honour of the human spirit”, despite some of its unquestionable (but excuseable) scholastic mistakes and rigidities.
Altogether, the present book provides a highly interesting and elucidating account on the once very mysterious Bourbaki group, with numerous new facts, original quotations, propaedeutical explanations, historical reminiscences, biographical sketches, and many rare photographs. This book makes the personage of Nicolas Bourbaki vividly palpable for a wide public, thereby turning a modern fable into an authentic report.

MSC:

01A72 Schools of mathematics
01A60 History of mathematics in the 20th century
01-02 Research exposition (monographs, survey articles) pertaining to history and biography
01A65 Development of contemporary mathematics

Biographic References:

Bourbaki, Nicolas
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