## Nicolas Bourbaki – An Author Who Never Existed

August 15, 2020 by anon

## History and Emergence

When a student takes up Mathematics as a career for research, he (she) surely comes across this name Nicolas Bourbaki very soon. Mathematics is built on some basic laws which include definition of Sets, Numbers, their addition and much more. These laws are called Axioms (**Peano Axioms** included). Some researchers suggest their students to read the first book of Nicolas Bourbaki which is Basic Set Theory, to get a strong basic foundation of Mathematics.

Back in the days in 18th century, when communication was possible only through letters, it was hard for mathematicians of different regions in the world to exchange opinions and understand the other fellow mathematician’s point of view. Mathematicians then still didn’t have the fundamentals well defined to rely upon; many people had questions which nobody could answer (like the **Axiom of Choice**). Some mathematicians realized the requirement of such axioms, which is exactly when Nicolas Bourbaki came into action.

**Nicolas Bourbaki** was born on January 14th 1935, as the collective identity of a group of several highly talented young French mathematicians. While there was no one person named Nicolas Bourbaki, the Bourbaki group, had an office in **Paris**. Their aim was to reformulate mathematics on an extremely abstract and formal but self-contained basis in a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strove for **Rigor** and **Generality**. Their work led to the discovery of several concepts and terminologies still used, and influenced modern branches of mathematics.

## Who Actually?

Some of the active founding members of the Nicolas Bourbaki group were Henri Cartan (1904-2008) (son of the geometer Elie Cartan), Claude Chevalley (1909-1984), Jean Delsarte (1903-1968), Jean Alexandre Eugène Dieudonné (1906-1992) and André Weil (1906-1998). The official list of founders includes four members who were less active, were physicist Jean Coulomb (1904-1999), Charles Ehresmann (1905-1979, who left in 1950 for unknown reasons), René de Possel (1905-1974) and Szolem Mandelbrojt (1899-1983). The rule was that all members would have to retire from the group at the age of 50 (Grothendieck and Lang left early, in anger). All the above Mathematicians are thus retired.

One of the first items on the original agenda was to understand the general Stokes Theorem, which unifies great results of vector calculus. However, it sparked a search for rigorous settings which would delay by many years the publication of the final presentation by Bourbaki of that particular topic. Mathematicians have made a plethora of important contributions under Bourbaki’s name. To name a few, the group introduced the null set symbol-∅; the ubiquitous terms injective, surjective, bijective and generalisations of many important theorems, including the Bourbaki-Witt theorem, the Jacobson-Bourbaki theorem and the Bourbaki-Banach-Alaoglu theorem.

The original goal of the group had been to compile an improved mathematical analysis text; it was soon decided that a more comprehensive treatment of all of mathematics was necessary. There was no official status of membership, and at the time the group was quite secretive and also fond of supplying disinformation. Regular meetings were scheduled (totaling about 4 weeks a year), during which the group would discuss vigorously every proposed line of every book. All their books were written and published in French and many translations have been made into different languages.

## Their books

Bourbaki, Nicolas (1939) | Livre I: Théorie des ensembles [Book I: Set theory] |

Bourbaki, Nicolas (1942) | Livre II: Algèbre [Book II: Algebra] |

Bourbaki, Nicolas (1940) | Livre III: Topologie [Book III: Topology] |

Bourbaki, Nicolas (1949) | Livre IV: Fonctions d’une variable réelle [Book IV: Functions of one real variable] |

Bourbaki, Nicolas (1953) | Livre V: Espaces vectoriels topologiques [Book V: Topological vector spaces] |

Bourbaki, Nicolas (1952) | Livre VI: Intégration [Book VI: Integration] |

Bourbaki, Nicolas (1961) | Livre VII: Algèbre commutative [Book VII: Commutative algebra] |

Bourbaki, Nicolas (1960) | Livre VIII: Groupes ET algèbres de Lie [Book VIII: Lie groups and algebras] |

Bourbaki, Nicolas (1967) | Livre IX: Théories spectrales [Book IX: Spectral theory] |

Bourbaki, Nicolas (1967) | Livre X: Variétés différentielles et analytiques [Book X: Differentiable and analytic manifolds] |

Bourbaki, Nicolas (2016) | Livre XI: Topologie algébrique [Book XI: Algebraic topology] |

The (still incomplete) volume on spectral theory (Théories spectrales) from 1967 was for almost four decades the last new book to be added to the series. After that several new chapters to existing books as well as revised editions of existing chapters appeared until the publication of chapters 8-9 of Commutative Algebra in 1983. A long break in publishing activity followed, leading many to suspect the end of the publishing project. However, chapter 10 of Commutative Algebra appeared in 1998, and after another long break a completely re-written and expanded chapter 8 of Algèbre was published in 2012. More importantly, the first four chapters of a completely new book on algebraic topology were published in 2016.

**Bourbaki’s direct influence has decreased over time. This is partly because certain concepts which are now important, such as the machinery of category theory, are not covered in the books. Bourbaki showed that passion is all above fame and money. Bourbaki should be a stand-alone example for researchers of all the generations.**

Guest post by Guru Sharan N.

All a rainbow needs is water (as in rain) and sunlight, so can we get a rainbow at home with a bucket of water on a sunny day?

Guru Sharan has finished his Masters in Mathematics. He is passionate to learn, understand and teach science at all levels. He is a science enthusiast and has been into pure science and mathematics since his high school. He always likes to think unconventional and out of the box.

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