Éléments de mathématique

Éléments de mathématique is a treatise on mathematics by the collective Nicolas Bourbaki. Begun in 1939, the work has run to several volumes, and remains in progress. The first volumes were published by Éditions Hermann from 1939 in the form of booklets, and later as bound volumes. Following a legal dispute with the editor, 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 (or: "installments") and has published a new volume in 2016, treating algebraic topology.

First volume of the 1970 edition.

The unusual singular "mathématique" in the title is deliberate, to convey the authors' belief in the unity of mathematics.[1][2] A companion volume, Éléments d'histoire des mathématiques, collects and reproduces several of the historical notes which previously appeared in the work.

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

Development

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, 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.[3] The Éléments de mathématique remains unfinished to this day.

Early versions are available online.[4] Most of the books published were out of print for years. The publisher Springer started their republication in 2006.

Structure

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.[5]

Volumes

Éléments de mathématique is divided into books, volumes, and chapters. A book refers to a broad area of investigation or branch of mathematics (Algebra, Integration); a given book is sometimes published in multiple volumes (physical books) or else in a single volume. The work is further subdivided into chapters with some volumes consisting of a single chapter.

Typically of mathematics textbooks, the Élément's chapters present definitions, mathematical notation, proofs of theorems and exercises, forming the core mathematical content of the work. The chapters are supplemented by historical notes and summaries of results. The former usually appear after a given chapter to contextualize the development of its topics, and the latter are occasionally used sections in which a book's major results are collected and stated without proof. Eléments d'histoire des mathématiques is a compilation volume of several of the historical note sections previously published in the Éléments proper, through the book on Lie groups and Lie algebras. The first installment of the Éléments to be published was the Summary of Results in Set Theory in 1939; the first proper chapter of content on set theory-with proofs and theorems-did not appear until 1954.

The volumes of the Éléments have had a complex publication history. Material has been published chronologically out of order of its intended logical sequence, revised for new editions, and compiled and partitioned differently in subsequent reprints. The large majority of the Éléments has been translated into an English edition, although this translation is incomplete. Currently the complete French edition of the work consists of 12 books printed in 28 volumes, with 70 chapters. The English edition completely reproduces seven books and partially reproduces two, with three unavailable; it comprises 14 volumes, reproducing 58 of the original's 70 chapters. [6][7][8][lower-alpha 1]

French edition				English edition
Book	Volume	Ch. no.	Chapter	Book	Volume	Ch. no.	Chapter
Théorie des ensembles	Théorie des ensembles[9][10]	1	Description de la mathématique formelle	Theory of Sets	Theory of Sets[11][12]	1	Description of Formal Mathematics
		2	Théorie des ensembles			2	Theory of Sets
		3	Ensembles ordonnés, cardinaux, nombres entiers			3	Ordered Sets, Cardinals, Integers
		4	Structures			4	Structures
Algèbre	Algèbre: Chapitres 1 à 3[13][14]	1	Structures algébriques	Algebra	Algebra I: Chapters 1-3[15][16]	1	Algebraic Structures
		2	Algèbre linéaire			2	Linear Algebra
		3	Algèbres tensorielles, algèbres extérieures, algèbres symétriques			3	Tensor Algebras, Exterior Algebras, Symmetric Algebras
	Algèbre: Chapitres 4 à 7[17][18]	4	Polynômes et fractions rationnelles		Algebra II: Chapters 4-7[19][20]	4	Polynomials and Rational Fractions
		5	Corps commutatifs			5	Commutative Fields
		6	Groupes et corps ordonnés			6	Ordered Groups and Fields
		7	Modules sur les anneaux principaux			7	Modules over Principal Ideal Domains
	Algèbre: Chapitre 8[21][22]	8	Modules et anneaux semi-simples		Unavailable in English	8	Semi-simple Modules and Rings
	Algèbre: Chapitre 9[23][24]	9	Formes sesquilinéaires et formes quadratiques			9	Sesquilinear and Quadratic Forms
	Algèbre: Chapitre 10[25][26]	10	Algèbre homologique			10	Homological Algebra
Topologie générale	Topologie générale: Chapitres 1 à 4[27][28]	1	Structures topologiques	General Topology	General Topology: Chapters 1-4[29][30]	1	Topological Structures
		2	Structures uniformes			2	Uniform Structures
		3	Groupes topologiques			3	Topological Groups
		4	Nombres réels			4	Real Numbers
	Topologie générale: Chapitres 5 à 10[31][32]	5	Groupes à un paramètre		General Topology: Chapters 5-10[33][34]	5	One-Parameter Groups
		6	Espaces numériques et espaces projectifs			6	Real Number Spaces and Projective Spaces
		7	Les groupes additifs ${\displaystyle \mathbb {R} ^{n}}$			7	The Additive Groups ${\displaystyle \mathbb {R} ^{n}}$
		8	Nombres complexes			8	Complex Numbers
		9	Utilisation des nombres réels en topologie générale			9	Use of Real Numbers in General Topology
		10	Espaces fonctionnels			10	Function Spaces
Fonctions d'une variable réelle	Fonctions d'une variable réelle[35][36]	1	Dérivées	Functions of a Real Variable	Functions of a Real Variable: Elementary Theory[37][38]	1	Derivatives
		2	Primitives et intégrales			2	Primitives and Integrals
		3	Fonctions élémentaires			3	Elementary Functions
		4	Équations différentielles			4	Differential Equations
		5	Etude locale des fonctions			5	Local Study of Functions
		6	Développements tayloriens généralisés, Formule sommatoire d'Euler-Maclaurin			6	Generalized Taylor Expansions, The Euler-Maclaurin Summation Formula
		7	La fonction gamma			7	The Gamma Function
Espaces vectoriels topologiques	Espaces vectoriels topologiques: Chapitres 1 à 5[39][40]	1	Espaces vectoriels topologiques sur un corps valué	Topological Vector Spaces	Topological Vector Spaces: Chapters 1-5[41][42]	1	Topological Vector Spaces over a Valued Field
		2	Ensembles convexes et espaces localement convexes			2	Convex Sets and Locally Convex Spaces
		3	Espaces d'applications linéaires continues			3	Spaces of Continuous Linear Mappings
		4	La dualité dans les espaces vectoriels topologiques			4	Duality in Topological Vector Spaces
		5	Espaces hilbertiens (théorie élémentaire)			5	Hilbert Spaces (Elementary Theory)
Intégration	Intégration: Chapitres 1 à 4[43][44]	1	Inégalités de convexité	Integration	Integration I: Chapters 1-6[45][46]	1	Convex Inequalities
		2	Espaces de Riesz			2	Riesz spaces
		3	Mesures sur les espaces localement compact			3	Measures on Locally Compact Spaces
		4	Prolongement d'une mesure. Espaces ${\displaystyle L^{p}}$			4	Extension of a Measure, ${\displaystyle L^{p}}$ Spaces
	Intégration: Chapitre 5[47][48]	5	Intégration des mesures			5	Integration of Measures
	Intégration: Chapitre 6[49][50]	6	Intégration vectorielle			6	Vectorial Integration
	Intégration: Chapitres 7 et 8[51][52]	7	Mesure de Haar		Integration II: Chapters 7-9[53][54]	7	Haar Measure
		8	Convolution et Représentation			8	Convolution and Representation
	Intégration: Chapitre 9[55][56]	9	Mesures sur les espaces topologiques séparés			9	Measures on Hausdorff Topological Spaces
Groupes et algèbres de Lie	Groupes et algèbres de Lie: Chapitre 1[57][58]	1	Algèbres de Lie	Lie Groups and Lie Algebras	Lie Groups and Lie Algebras: Chapters 1-3[59][60]	1	Lie Algebras
	Groupes et algèbres de Lie: Chapitres 2 et 3[61][62]	2	Algèbres de Lie libres			2	Free Lie Algebras
		3	Groupes de Lie			3	Lie Groups
	Groupes et algèbres de Lie: Chapitres 4 à 6[63][64]	4	Groupes de Coxeter et systèmes de Tits		Lie Groups and Lie Algebras: Chapters 4-6[65][66]	4	Coxeter Groups and Tits Systems
		5	Groupes engendrés par des réflexions			5	Groups Generated by Reflections
		6	Systèmes de racines			6	Root Systems
	Groupes et algèbres de Lie: Chapitres 7 et 8[67][68]	7	Sous-algèbres de Cartan éléments réguliers		Lie Groups and Lie Algebras: Chapters 7-9[69][70]	7	Cartan Subalgebras and Regular Elements
		8	Algèbres de Lie semi-simples déployées			8	Split Semi-simple Lie Algebras
	Groupes et algèbres de Lie: Chapitre 9[71][72]	9	Groupes de Lie réels compacts			9	Compact Real Lie Groups
Algèbre commutative	Algèbre commutative: Chapitres 1 à 4[73][74]	1	Modules plats	Commutative Algebra	Commutative Algebra: Chapters 1-7[75][76]	1	Flat Modules
		2	Localisation			2	Localization
		3	Graduations, filtrations et topologies			3	Graduations, Filtrations and Topologies
		4	Idéaux premiers associés et décomposition primaire			4	Associated Prime Ideals and Primary Decomposition
	Algèbre commutative: Chapitres 5 à 7[77][78]	5	Entiers			5	Integers
		6	Valuations			6	Valuations
		7	Diviseurs			7	Divisors
	Algèbre commutative: Chapitres 8 et 9[79][80]	8	Dimension		Unavailable in English	8	Dimension
		9	Anneaux locaux noethériens complets			9	Complete Noetherian Local Rings
	Algèbre commutative: Chapitre 10[81][82]	10	Profondeur, régularité, dualité			10	Depth, Regularity, Duality
Théories spectrales	Théories spectrales: Chapitres 1 et 2[83][84]	1	Algèbres normées	Spectral Theory	Unavailable in English	1	Normed Algebras
		2	Groupes localement compacts commutatifs			2	Locally Compact Commutative Groups
Variétés différentielles et analytiques	Variétés différentielles et analytiques[85][86]	0	—	Differential and Analytic Manifolds	Unavailable in English	0[lower-alpha 2]	—
Topologie Algébrique	Topologie Algébrique: Chapitres 1 à 4[87][88]	1	Revêtements	Algebraic Topology	Unavailable in English	1	Covering Spaces
		2	Groupoïdes			2	Groupoids
		3	Homotopie et groupoïde de Poincaré			3	Poincaré Homotopies and Groupoids
		4	Espaces délaçables			4	Unloopable Spaces
Eléments d'histoire des mathématiques[89][90]		0	—	Elements of the History of Mathematics[91][92]		0[lower-alpha 3]	—

See also

• Euclid's Elements
• Principia Mathematica

Notes

1. In both cases, the counts of each edition's books and volumes include the historical compilation Eléments d'histoire des mathématiques. The chapter count refers to the chapters of mathematical content in the Éléments proper, excluding sections (or chapters) of historical notes reproduced in Eléments d'histoire des mathématiques.
2. Differential and Analytic Manifolds first appeared as two volumes of summaries of results, later compiled into a single volume. As of 2021 no proper proof-based chapters associated with the book's subject have been published.
3. Eléments d'histoire des mathématiques is a compilation volume of several of the historical note sections previously published in the Éléments proper. Although the volume is internally organized with 26 chapters, its reproduced historical content does not count toward the chapters of mathematical content within the Éléments.

References

1. Mashaal (2006) p. 55
2. Aczel, Amir D. (2006). The Artist and the Mathematician: the Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed. Thunder's Mouth Press. p. 99–100. ISBN 9781560259312.
3. Bourbaki, Nicolas (2016). Topologie Algébrique: Chapitres 1 à 4. Éléments de mathématique. Springer. p. xiv. ISBN 9783662493601. French paperback edition.
4. Archives de l'Association des Collaborateurs de Nicolas Bourbaki
5. Algebra II, v-vi.
6. Ouvrages de N. Bourbaki at the Bourbaki site
7. Eléments de Mathématique series in Springer
8. Elements of Mathematics series in Springer
9. Bourbaki, Nicolas (1970). Théorie des ensembles. Éléments de mathématique. Springer. ISBN 9783540340348. French paperback edition.
10. "Théorie des ensembles". springer.com. Springer.
11. Bourbaki, Nicolas (2004). Theory of Sets. Elements of Mathematics. Springer. ISBN 9783540225256. English paperback edition.
12. "Theory of Sets". springer.com. Springer.
13. Bourbaki, Nicolas (1970). Algèbre: Chapitres 1 à 3. Éléments de mathématique. Springer. ISBN 9783540338499. French paperback edition.
14. "Algèbre: Chapitres 1 à 3". springer.com. Springer.
15. Bourbaki, Nicolas (1989). Algebra I: Chapters 1-3. Elements of Mathematics. Springer. ISBN 9783540642435. English paperback edition.
16. "Algebra I: Chapters 1-3". springer.com. Springer.
17. Bourbaki, Nicolas (1981). Algèbre: Chapitres 4 à 7. Éléments de mathématique. Springer. ISBN 9783540343981. French paperback edition.
18. "Algèbre: Chapitres 4 à 7". springer.com. Springer.
19. Bourbaki, Nicolas (1990). Algebra II: Chapters 4-7. Elements of Mathematics. Springer. ISBN 9783540007067. English paperback edition.
20. "Algebra II: Chapters 4-7". springer.com. Springer.
21. Bourbaki, Nicolas (2012). Algèbre: Chapitre 8. Éléments de mathématique. Springer. ISBN 9783540353157. French paperback edition. Original 1958 edition revised in a 2012 edition.
22. "Algèbre: Chapitre 8". springer.com. Springer.
23. Bourbaki, Nicolas (1959). Algèbre: Chapitre 9. Éléments de mathématique. Springer. ISBN 9783540353386. French paperback edition. Original 1959 edition revised in a 1973 edition.
24. "Algèbre: Chapitre 9". springer.com. Springer.
25. Bourbaki, Nicolas (1980). Algèbre: Chapitre 10. Éléments de mathématique. Springer. ISBN 9783540344926. French paperback edition.
26. "Algèbre: Chapitre 10". springer.com. Springer.
27. Bourbaki, Nicolas (1971). Topologie générale: Chapitres 1 à 4. Éléments de mathématique. Springer. ISBN 9783540339366. French paperback edition.
28. "Topologie générale: Chapitres 1 à 4". springer.com. Springer.
29. Bourbaki, Nicolas (1989). General Topology: Chapters 1-4. Elements of Mathematics. Springer. ISBN 9783540642411. English paperback edition.
30. "General Topology: Chapters 1-4". springer.com. Springer.
31. Bourbaki, Nicolas (1974). Topologie générale: Chapitres 5 à 10. Éléments de mathématique. Springer. ISBN 9783540343998. French paperback edition.
32. "Topologie générale: Chapitres 5 à 10". springer.com. Springer.
33. Bourbaki, Nicolas (1989). General Topology: Chapters 5-10. Elements of Mathematics. Springer. ISBN 9783540645634. English paperback edition.
34. "General Topology: Chapters 5-10". springer.com. Springer.
35. Bourbaki, Nicolas (1976). Fonctions d'une variable réelle. Éléments de mathématique. Springer. ISBN 9783540340362. French paperback edition.
36. "Fonctions d'une variable réelle". springer.com. Springer.
37. Bourbaki, Nicolas (2004). Functions of a Real Variable: Elementary Theory. Elements of Mathematics. Springer. ISBN 9783642639326. English paperback edition.
38. "Functions of a Real Variable: Elementary Theory". springer.com. Springer. (URL number refers to English hardback edition.)
39. Bourbaki, Nicolas (1981). Espaces vectoriels topologiques: Chapitres 1 à 5. Éléments de mathématique. Springer. ISBN 9783540344971. French paperback edition.
40. "Espaces vectoriels topologiques: Chapitres 1 à 5". springer.com. Springer.
41. Bourbaki, Nicolas (1987). Topological Vector Spaces: Chapters 1-5. Elements of Mathematics. Springer. ISBN 9783540423386. English paperback edition.
42. "Topological Vector Spaces: Chapters 1-5". springer.com. Springer.
43. Bourbaki, Nicolas (1965). Intégration: Chapitres 1 à 4. Éléments de mathématique. Springer. ISBN 9783540353287. French paperback edition. Original 1965 edition revised in a 1973 edition.
44. "Intégration: Chapitres 1 à 4". springer.com. Springer.
45. Bourbaki, Nicolas (2004). Integration I: Chapters 1-6. Elements of Mathematics. Springer. ISBN 9783642639302. English paperback edition.
46. "Integration I: Chapters 1-6". springer.com. Springer.
47. Bourbaki, Nicolas (1967). Intégration: Chapitre 5. Éléments de mathématique. Springer. ISBN 9783540353331. French paperback edition.
48. "Intégration: Chapitre 5". springer.com. Springer.
49. Bourbaki, Nicolas (1959). Intégration: Chapitre 6. Éléments de mathématique. Springer. ISBN 9783540353195. French paperback edition.
50. "Intégration: Chapitre 6". springer.com. Springer.
51. Bourbaki, Nicolas (1963). Intégration: Chapitres 7 et 8. Éléments de mathématique. Springer. ISBN 9783540353249. French paperback edition.
52. "Intégration: Chapitres 7 et 8". springer.com. Springer.
53. Bourbaki, Nicolas (2004). Integration II: Chapters 7-9. Elements of Mathematics. Springer. ISBN 9783642058219. English paperback edition.
54. "Integration II: Chapters 7-9". springer.com. Springer. (URL number refers to English hardback edition.)
55. Bourbaki, Nicolas (1969). Intégration: Chapitre 9. Éléments de mathématique. Springer. ISBN 9783540343905. French paperback edition.
56. "Intégration: Chapitre 9". springer.com. Springer.
57. Bourbaki, Nicolas (1971). Groupes et algèbres de Lie: Chapitre 1. Éléments de mathématique. Springer. ISBN 9783540353355. French paperback edition.
58. "Groupes et algèbres de Lie: Chapitre 1". springer.com. Springer.
59. Bourbaki, Nicolas (1989). Lie Groups and Lie Algebras: Chapters 1-3. Elements of Mathematics. Springer. ISBN 9783540642428. English paperback edition.
60. "Lie Groups and Lie Algebras: Chapters 1-3". springer.com. Springer.
61. Bourbaki, Nicolas (1972). Groupes et algèbres de Lie: Chapitres 2 et 3. Éléments de mathématique. Springer. ISBN 9783540339403. French paperback edition.
62. "Groupes et algèbres de Lie: Chapitres 2 et 3". springer.com. Springer.
63. Bourbaki, Nicolas (1968). Groupes et algèbres de Lie: Chapitres 4 à 6. Éléments de mathématique. Springer. ISBN 9783540344902. French paperback edition.
64. "Groupes et algèbres de Lie: Chapitres 4 à 6". springer.com. Springer.
65. Bourbaki, Nicolas (2002). Lie Groups and Lie Algebras: Chapters 4-6. Elements of Mathematics. Springer. ISBN 9783540691716. English paperback edition.
66. "Lie Groups and Lie Algebras: Chapters 4-6". springer.com. Springer.
67. Bourbaki, Nicolas (1975). Groupes et algèbres de Lie: Chapitres 7 et 8. Éléments de mathématique. Springer. ISBN 9783540339397. French paperback edition.
68. "Groupes et algèbres de Lie: Chapitres 7 et 8". springer.com. Springer.
69. Bourbaki, Nicolas (2005). Lie Groups and Lie Algebras: Chapters 7-9. Elements of Mathematics. Springer. ISBN 9783540688518. English paperback edition.
70. "Lie Groups and Lie Algebras: Chapters 7-9". springer.com. Springer.
71. Bourbaki, Nicolas (1982). Groupes et algèbres de Lie: Chapitre 9. Éléments de mathématique. Springer. ISBN 9783540343929. French paperback edition.
72. "Groupes et algèbres de Lie: Chapitre 9". springer.com. Springer.
73. Bourbaki, Nicolas (1968). Algèbre commutative: Chapitres 1 à 4. Éléments de mathématique. Springer. ISBN 9783540339373. French paperback edition.
74. "Algèbre commutative: Chapitres 1 à 4". springer.com. Springer.
75. Bourbaki, Nicolas (1989). Commutative Algebra: Chapters 1-7. Elements of Mathematics. Springer. ISBN 9783540642398. English paperback edition.
76. "Commutative Algebra: Chapters 1-7". springer.com. Springer.
77. Bourbaki, Nicolas (1964). Algèbre commutative: Chapitres 5 à 7. Éléments de mathématique. Springer. ISBN 9783540339410. French paperback edition.
78. "Algèbre commutative: Chapitres 5 à 7". springer.com. Springer.
79. Bourbaki, Nicolas (1983). Algèbre commutative: Chapitres 8 et 9. Éléments de mathématique. Springer. ISBN 9783540339427. French paperback edition.
80. "Algèbre commutative: Chapitres 8 et 9". springer.com. Springer.
81. Bourbaki, Nicolas (1998). Algèbre commutative: Chapitre 10. Éléments de mathématique. Springer. ISBN 9783540343943. French paperback edition.
82. "Algèbre commutative: Chapitre 10". springer.com. Springer.
83. Bourbaki, Nicolas (1967). Théories spectrales: Chapitres 1 et 2. Éléments de mathématique. Springer. ISBN 9783540353300. French paperback edition.
84. "Théories spectrales: Chapitres 1 et 2". springer.com. Springer.
85. Bourbaki, Nicolas (1971). Variétés différentielles et analytiques. Éléments de mathématique. Springer. ISBN 9783540343967. French paperback edition.
86. "Variétés différentielles et analytiques". springer.com. Springer.
87. Bourbaki, Nicolas (2016). Topologie Algébrique: Chapitres 1 à 4. Éléments de mathématique. Springer. ISBN 9783662493601. French paperback edition.
88. "Topologie Algébrique: Chapitres 1 à 4". springer.com. Springer.
89. Bourbaki, Nicolas (1974). Eléments d'histoire des mathématiques. Éléments de mathématique. Springer. ISBN 9783540339380. French paperback edition.
90. "Eléments d'histoire des mathématiques". springer.com. Springer.
91. Bourbaki, Nicolas (1994). Elements of the History of Mathematics. Elements of Mathematics. Springer. ISBN 9783540647676. English paperback edition.
92. "Elements of the History of Mathematics". springer.com. Springer.

• 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.