Éléments de mathématique

First volume of the 1970 edition.

Éléments de mathématique is a treatise on mathematics by the collective Nicolas Bourbaki, composed of ten 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 Editions Masson. Since 2006, Springer Verlag has republished all the fascicles.

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.[1] 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.[2]

The first six volumes follow a logical sequence. The subsequent volumes are dependent on the first six, but not on each other.[1]

Volumes in the series

The series is divided into books and each book into chapters. Below is the list of books in the series.[3] [4] [5]


1. Set theory

French edition
English edition


2. Algebra

French edition
English edition


3. Topology

French edition
English edition


4. Functions of a Real Variable

French edition
English edition


5. Topological vector spaces

French edition
English edition


6. Integration

French edition
English edition


7. Commutative algebra

French edition
English edition


8. Differential manifolds (summary of results)

French edition


9. Lie groups and algebras

French edition
English edition


10. Spectral theory

French edition


11. History of mathematics

French edition
English edition

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. Algebra chapters 1 to 3
  2. General topology chapters 1 to 3
  3. Algebra chapter 4 onwards
  4. General topology chapters 4 onwards
  5. Functions of a real variable
  6. Topological vector spaces
  7. 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.[1] Publication continues intermittently - the tenth chapter of Algèbre commutative was published, in 1998 and the eight chapter of Algèbre was published in 2012 . The Éléments de mathématique remains unfinished to this day.

Early versions are available online.[6] Most of the books published were out of print for years. The publisher Springer started their republication in 2006.

References

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