Éléments de mathématique

Éléments de mathématique is a treatise on mathematics by the collective Nicolas Bourbaki, composed of twelve books (each divided into one or more chapters). 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.

The series is divided into books and each book into chapters. Below is the list of books in the series.

1. Set theory

2. Algebra

3. Topology

4. Functions of a Real Variable

5. Topological vector spaces

6. Integration

7. Commutative algebra

8. Differential manifolds (summary of results)

9. Lie groups and algebras

10. Spectral theory

11. History of mathematics

12. Algebraic Topology

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:

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.

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.

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