Éléments de mathématique


Éléments de mathématique is a treatise on mathematics by the collective Nicolas Bourbaki, composed of twelve books. The first volumes were published by Éditions Hermann from 1939 initially in the form of booklets and then as bound volumes. Following a disagreement with the editor, the publication was resumed in the 1970s by the CCLS, and then in the 1980s by Éditions Masson. Since 2006, Springer Verlag has republished all the fascicles, and has published a new volume in 2016.
The strange singular "mathématique" in the title is deliberate, to convey the authors' belief that the material is a unity, contrary to what conventional form of the title might suggest. Conversely, the title Éléments d'histoire des mathématiques by the same authors employs the plural to indicate that before Bourbaki, mathematics was a set of scattered disciplines, and that the modern notion of structure has allowed their unification.
The first six volumes follow a logical sequence. The subsequent volumes are dependent on the first six, but not on each other.

Volumes

Éléments de mathématique is divided into books, volumes, and chapters. A book refers to an area of investigation or branch of mathematics ; a given book is sometimes published in multiple volumes or else in a single volume. The work is further subdivided into chapters; some volumes consist of a single chapter. Currently, the complete original French edition of the work consists of 12 books printed in 28 volumes, with 97 chapters.
The large majority of Éléments de mathématique has also been translated into an English edition, although this translation is incomplete. Further, single volumes of the English edition sometimes collect smaller volumes of the French with the result that chapters are partitioned differently between the two editions' volumes. Currently, the English edition consists of 14 volumes. In it, seven books have been completely reproduced, three books are unavailable, and two books are partially available. On a chapter-by-chapter basis, the English edition reproduces 84 of the original's 97 chapters.

Inter-relationship of the books

In the first six books, every statement in the text assumes as known only those results which have already been discussed in the same chapter, or in the previous chapters ordered as follows:
  1. Set theory
  2. Algebra chapters 1 to 3
  3. General topology chapters 1 to 3
  4. Algebra chapter 4 onwards
  5. General topology chapters 4 onwards
  6. Functions of a real variable
  7. Topological vector spaces
  8. Integration
Later books assume knowledge of the first six books and their relationship to the other books in the series will be indicated at the outset.

Evolution of the project

The first volume, published in 1939, was the Fascicule de résultats of Théorie des ensembles. The publication of subsequent volumes did not follow the order of the Treatise. Publication continues intermittently - the tenth chapter of Algèbre commutative was published in 1998, an expanded second edition of the eighth chapter of Algèbre in 2012, and the first four chapters of a new book Topologie algébrique in 2016. This latest book was initially planned as the eleventh chapter of Topologie générale. The Éléments de mathématique remains unfinished to this day.
Early versions are available online. Most of the books published were out of print for years. The publisher Springer started their republication in 2006.