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

• Théorie des ensembles, chapitres 1 à 4, 352 p., 1970 (reprint in 1998)

English edition

• Theory of Sets, 2004

2. Algebra

French edition

• Algèbre, chapitres 1 à 3, 654 p., 1970
• Algèbre, chapitres 4 à 7, 432 p., 1981
• Algèbre, chapitres 8, 489 p., 2012
• Algèbre, chapitre 9, 212 p., 1959 (new edition in 1973)
• Algèbre, chapitre 10, 224 p., 1980

English edition

• Algebra I, chapters 1-3, 1973 (Addison-Wesley), 1989
• Algebra II, chapters 4-7, 1990

3. Topology

French edition

• Topologie Générale, chapitres 1 à 4, 376 p., 1971 (reprint in 1990)
• Topologie Générale, chapitres 5 à 10, 334 p., 1974

English edition

• General Topology, chapters 1-4, 1989
• General Topology, chapters 5-10, 1989

4. Functions of a Real Variable

French edition

• Fonctions d'une Variable Réelle, chapitres 1 à 7, 336 p., 1976

English edition

• Functions of a Real Variable, 2004

5. Topological vector spaces

French edition

• Espaces Vectoriels Topologiques, chapitres 1 à 5, 400 p., 1981

English edition

• Topological Vector Spaces, chapters 1-5, 1987

6. Integration

French edition

• Intégration, chapitres 1 à 4, 284 p., 1965 (new edition in 1973)
• Intégration, chapitre 5, 154 p., 1967
• Intégration, chapitre 6, 106 p., 1959
• Intégration, chapitres 7 et 8, 222 p., 1963
• Intégration, chapitre 9, 134 p., 1969

English edition

• Integration I, Chapters 1-6, 2004
• Integration II, Chapters 7-9, 2004

7. Commutative algebra

French edition

• Algèbre commutative, chapitres 1 à 4, 364 p., 1968 et 1969 (reprint in 1985)
• Algèbre commutative, chapitre 5 à 7, 352 p., 1964et 1965 (reprint in 1985)
• Algèbre commutative, chapitres 8 et 9, 208 p., 1983
• Algèbre commutative, chapitre 10, 187 p., 1998

English edition

• Commutative algebra, chapters 1-7, 1989

8. Differential manifolds (summary of results)

French edition

• Variétés différentielles et analytiques (fascicule de résultats), 198 p., 1971 (reprint in 1998)

9. Lie groups and algebras

French edition

• Groupes et algèbres de Lie, chapitre 1, 146 p., 1971
• Groupes et algèbres de Lie, chapitres 2 et 3, 320 p., 1972
• Groupes et algèbres de Lie, chapitres 4 à 6, 288 p., 1968 (reprint in 1981)
• Groupes et algèbres de Lie, chapitres 7 et 8, 272 p., 1975 (reprint in 1998)
• Groupes et algèbres de Lie, chapitre 9, 144 p., 1982

English edition

• Lie groups and Lie algebras, chapters 1-3, 1989
• Lie groups and Lie algebras, chapters 4-6, 2002
• Lie groups and Lie algebras, chapters 7-9, 2005

10. Spectral theory

French edition

• Théories spectrales, chapitres 1 et 2, 168 p., 1967

11. History of mathematics

French edition

• Éléments d'histoire des mathématiques, 376 p., 1974 (reprint in 1984)

English edition

• Elements of the history of mathematics, 1994

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

1. Mashaal (2006) p. 55
2. Mashaal (2006) p. 56
3. Ouvrages de N. Bourbaki at the Bourbaki site
4. Eléments de Mathématique series in Springer
5. Elements of Mathematics series in Springer
6. Archives de l'Association des Collaborateurs de Nicolas Bourbaki

• Mashaal, Maurice (2006). Bourbaki: A Secret Society of Mathematicians. Translated from the French by Anna Pierrehumbert. American Mathematical Society. ISBN 0-8218-3967-5. Zbl 1099.01022.