Nicolas Bourbaki is the collective
pseudonymA pseudonym is a fictitious name used by a person, or sometimes, a group.Pseudonyms are often used to hide an individual's real identity, as with writers' pen names, graffiti artists, resistance fighters' or terrorists' noms de guerre and computer hackers' handles. Actors, musicians, and other...
under which a group of (mainly
FrenchFrance , officially the French Republic , is a country located in Western Europe, with several overseas islands and territories located on other continents. Metropolitan France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean...
) 20th-century
mathematicianA mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts...
s wrote a series of books presenting an exposition of modern advanced
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, beginning in 1935. With the goal of founding all of mathematics on
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
, the group strove for
rigourRigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism...
and generality. Their work lead to the discovery of several concepts and terminologies still discussed.
While Nicolas Bourbaki is an invented personage, the
Bourbaki group is officially known as the
Association des collaborateurs de Nicolas Bourbaki (Association of Collaborators of Nicolas Bourbaki), which has an office at the
École Normale SupérieureThe École Normale Supérieure is a French grande école...
in
ParisParis is the capital of France and the country's most populous city. It is situated on the river Seine, in northern France, at the heart of the Île-de-France region...
.
Aiming at a completely self-contained treatment of the core areas of modern mathematics based on set theory, the group produced Elements of Mathematics (
Éléments de mathématique) series, which contain the following volumes (with the original French titles in parentheses):
The book
Variétés différentielles et analytiques was a
fascicule de résultats, that is, a summary of results, on the theory of
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
s, rather than a worked-out exposition.
Nicolas Bourbaki is the collective
pseudonymA pseudonym is a fictitious name used by a person, or sometimes, a group.Pseudonyms are often used to hide an individual's real identity, as with writers' pen names, graffiti artists, resistance fighters' or terrorists' noms de guerre and computer hackers' handles. Actors, musicians, and other...
under which a group of (mainly
FrenchFrance , officially the French Republic , is a country located in Western Europe, with several overseas islands and territories located on other continents. Metropolitan France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean...
) 20th-century
mathematicianA mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts...
s wrote a series of books presenting an exposition of modern advanced
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, beginning in 1935. With the goal of founding all of mathematics on
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
, the group strove for
rigourRigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism...
and generality. Their work lead to the discovery of several concepts and terminologies still discussed.
While Nicolas Bourbaki is an invented personage, the
Bourbaki group is officially known as the
Association des collaborateurs de Nicolas Bourbaki (Association of Collaborators of Nicolas Bourbaki), which has an office at the
École Normale SupérieureThe École Normale Supérieure is a French grande école...
in
ParisParis is the capital of France and the country's most populous city. It is situated on the river Seine, in northern France, at the heart of the Île-de-France region...
.
Books by Bourbaki
Aiming at a completely self-contained treatment of the core areas of modern mathematics based on set theory, the group produced Elements of Mathematics (
Éléments de mathématique) series, which contain the following volumes (with the original French titles in parentheses):
| I |
Set theory The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
|
(Théorie des ensembles) |
| II |
Algebra Algebra is the branch of mathematics concerning the study of the rules of operations and the things which can be constructed from them, including terms, polynomials, equations and algebraic structures...
|
(Algèbre) |
| III |
TopologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
|
(Topologie générale) |
| IV |
Functions of one real variable Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
|
(Fonctions d'une variable réelle) |
| V |
Topological vector space In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
s |
(Espaces vectoriels topologiques) |
| VI |
Integration Integration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
|
(Intégration) |
| and |
later |
|
| VII |
Commutative algebra Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
|
(Algèbre commutative) |
| VIII |
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s |
(Groupes et algèbres de Lie) |
| IX |
Spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. The name was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many...
|
(Théories spectrales) |
The book
Variétés différentielles et analytiques was a
fascicule de résultats, that is, a summary of results, on the theory of
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
s, rather than a worked-out exposition. A final volume IX on
spectral theoryIn mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. The name was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many...
(
Théories spectrales) from 1983 marked the presumed end of the publishing project; but a further commutative algebra
fascicleA fascicle is a bundle or a cluster.Fascicle may also refer to:In anatomy:* Muscle fascicle, a bundle of skeletal muscle fibers surrounded by connective tissue* Nerve fascicle,a larger bundle of axons enclosed by the perineurium...
was produced in 1998.
While several of Bourbaki's books have become standard references in their fields, some have felt that the austere presentation makes them unsuitable as textbooks. The books' influence may have been at its strongest when few other graduate-level texts in current pure mathematics were available, between 1950 and 1960.
Notations introduced by Bourbaki include: the symbol for the
empty setIn mathematics, and more specifically set theory, the empty set is the unique set having no members; its size is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
and a
dangerous bend symbolThe dangerous bend symbol was created by the Nicolas Bourbaki group of mathematicians and appears in the margins of mathematics books written by the group. It looks like a street sign that indicates a "dangerous bend" in the road ahead, and is used to mark passages tricky on a first reading or...
, and the terms
injective,
surjective, and
bijective.
It is frequently claimed that the use of the
blackboard boldBlackboard bold is a typeface style often used for certain symbols in mathematics and physics texts, in which certain lines of the symbol are doubled. The symbols usually denote number sets. Blackboard bold symbols are also referred to as double struck, although they cannot actually be produced by...
letters for the various sets of
numberA number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number...
s was first introduced by the group. There are several reasons to doubt this claim.
Influence on mathematics in general
The emphasis on
rigourRigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism...
may be seen as a reaction to the work of
Henri PoincaréJules Henri Poincaré was a French mathematician and theoretical physicist, and a philosopher of science...
, who stressed the importance of free-flowing mathematical intuition, at a cost of completeness in presentation. The impact of Bourbaki's work initially was great on many active research mathematicians world-wide.
It provoked some hostility, too, mostly on the side of classical analysts; they approved of rigour but not of high abstraction. Around 1950, also, some parts of
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
were still not fully axiomatic — in less prominent developments, one way or another, these were brought into line with the new foundational standards, or quietly dropped. This undoubtedly led to a gulf with the way
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics in an attempt to explain natural phenomena. Its central core is mathematical physics,[Sometimes mathematical physics and theoretical physics are used synonymously to refer to the...]
is practiced.
Bourbaki's direct influence has decreased over time. This is partly because certain concepts which are now important, such as the machinery of
category theoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
, are not covered in the treatise. The completely uniform and essentially linear referential structure of the books became difficult to apply to areas closer to current research than the already mature ones treated in the published books, and thus publishing activity diminished significantly from the 1970s. It also mattered that while especially
algebraic structureIn algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
s can be naturally defined in Bourbaki's terms, there are areas where the Bourbaki approach was less straightforward to apply.
On the other hand, the approach and rigour advocated by Bourbaki have permeated the current mathematical practices to such extent that the task undertaken was completed. This is particularly true for the less applied parts of mathematics.
The Bourbaki seminar series founded in post-WWII Paris continues. It is an important source of
survey articleIn academia, a survey article is a paper that is a work of synthesis, published through the usual channels...
s, written in a prescribed, careful style. The idea is that the presentation should be on the level of absolute specialists, but for an audience which is
not specialized in the particular field.
The group
Accounts of the early days vary, but original documents have now come to light. The founding members were all connected to the
Ecole Normale SupérieureThe École Normale Supérieure is a French grande école...
in
ParisParis is the capital of France and the country's most populous city. It is situated on the river Seine, in northern France, at the heart of the Île-de-France region...
and included
Henri CartanHenri Paul Cartan was a son of Élie Cartan, and was, as his father was, a distinguished and influential French mathematician.-Life:...
,
Claude ChevalleyClaude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups...
, Jean Coulomb,
Jean DelsarteJean Frédéric Auguste Delsarte was a French mathematician.He was born in Fourmies, France and died in Nancy, France....
,
Jean DieudonnéJean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
,
Charles EhresmannCharles Ehresmann was a French mathematician who worked on differential topology and category theory. He is known for work on the topology of Lie groups, the jet concept , and his seminar on category theory....
,
René de PosselLucien Alexandre Charles René de Possel was a French mathematician, one of the founders of the Bourbaki group, and later a pioneer computer scientist, working in particular on optical character recognition....
,
Szolem MandelbrojtSzolem Mandelbrojt was a Jewish-Polish mathematician. He worked mainly in classical analysis; he was a student of Jacques Hadamard, and became Hadamard's successor as Professor at the Collège de France....
and
André WeilAndré Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
. There was a preliminary meeting, towards the end of 1934.
Jean LerayJean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology....
and
Paul DubreilPaul Dubreil was a French mathematician.He was born in Le Mans, Maine, France and died in Soisy-sur-École, France. Dubreil was married to Marie-Louise Jacotin....
were present at the preliminary meeting but dropped out before the group actually formed. Other notable participants in later days were
Laurent SchwartzLaurent-Moïse Schwartz was a French mathematician. Alumnus of the École normale supérieure, he was awarded the Fields medal in 1950 for his works on the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function...
,
Jean-Pierre SerreJean-Pierre Serre is a French mathematician in the fields of algebraic geometry, number theory and topology. He has received numerous awards and honors for his mathematical research and exposition, including the Fields Medal in 1954 and the Abel Prize in 2003.-Early years:Born in Bages,...
,
Alexander GrothendieckAlexander Grothendieck is considered one of the greatest mathematicians of the 20th century.He is most famous for his revolutionary advances in algebraic geometry, but he has also made major contributions to algebraic topology, number theory, category theory, Galois theory, descent theory,...
,
Samuel EilenbergSamuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from Warsaw University in 1936. His thesis advisor was...
,
Serge LangSerge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He was a member of the Bourbaki group....
and
Roger GodementRoger Godement is a French mathematician, known for his work in functional analysis, and also his expository books.He started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan...
.
The original goal of the group had been to compile an improved
mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
text; it was soon decided that a more comprehensive treatment of all of mathematics was necessary. There was no official status of membership, and at the time the group was quite secretive and also fond of supplying disinformation. Regular meetings were scheduled, during which the whole group would discuss vigorously every proposed line of every book. Members had to resign by age 50.
The atmosphere in the group can be illustrated by an anecdote told by Laurent Schwartz. Dieudonné regularly and spectacularly threatened to resign unless topics were treated in their logical order, and after a while others played on this for a joke. Godement's wife wanted to see Dieudonné announcing his resignation, and so on one occasion while she was there Schwartz deliberately brought up again the question of permuting the order in which measure theory and
topological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
s were to be handled, to precipitate a guaranteed crisis.
The name "Bourbaki" refers to a French general
Charles Denis BourbakiCharles Denis Sauter Bourbaki was a French general.He was born at Pau, the son of Greek colonel Constantin Denis Bourbaki, who died in the War of Independence in 1827...
; it was adopted by the group as a reference to a student anecdote about a hoax mathematical lecture, and also possibly to a statue. It was certainly a reference to
Greek mathematicsGreek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. The word "mathematics" itself derives from the ancient Greek μάθημα , meaning "subject of...
, Bourbaki being of Greek extraction. It is a valid reading to take the name as implying a transplantation of the tradition of
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician and is often referred to as the "Father of Geometry." He was active in Hellenistic Alexandria during the reign of Ptolemy I...
to a France of the 1930s, with soured expectations.
Appraisal of the Bourbaki perspective
The underlying drive, in Weil and Chevalley at least, was the perceived need for French mathematics to absorb the best ideas of the Göttingen school, particularly
HilbertDavid Hilbert was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry...
and the modern algebra school of
Emmy NoetherAmalie Emmy Noether, , was a German-born mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields, and...
,
ArtinEmil Artin was an Austrian mathematician of Armenian descent. His father, also Emil Artin, was an Armenian art-dealer, and his mother was the opera singer Emma Laura-Artin. He grew up in Reichenberg in Bohemia, where German was the primary language...
and van der Waerden. It is fairly clear that the Bourbaki point of view, while
encyclopedic, was never intended as
neutral. Quite the opposite: it was more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy, with emphasis on formalism and axiomatics. But always through a transforming process of reception and selection — their ability to sustain this collective, critical approach has been described as "something unusual".
The following is a list of some of the criticisms commonly made of the Bourbaki approach:
- algorithm
In mathematics, computing, linguistics, and related subjects, an algorithm is an effective method for solving a problem using a finite sequence of instructions. Algorithms are used for calculation, data processing, and many other fields....
ic content is not considered on-topic and is almost completely omitted
- problem solving
Problem solving is a mental process and is part of the larger problem process that includes problem finding and problem shaping. Consideredthe most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of...
, in the sense of heuristics, receives less emphasis than axiomatic theory-building
- analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
is treated 'softly', without 'hard' estimates
- Measure theory is developed from a functional analytic
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
perspective. Taking the case of locally compactIn topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...
measure spaces as fundamental focuses the presentation on Radon measureIn mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
s and leads to an approach to measurable functions that is cumbersome, especially from the viewpoint of probability theory. However, the last chapter of the book addresses limitations, especially for use in probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, of the restriction to locally compact spaces.
- combinatorics
Combinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
is not discussed
- logic
Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
is treated minimally
- applications
Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.-Divisions of applied mathematics:...
are not covered.
Furthermore, Bourbaki make only limited use of pictures in their presentation. In general, Bourbaki has been criticized for reducing
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
as a whole to
abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
and soft analysis.
Dieudonné as speaker for Bourbaki
Public discussion of, and justification for, Bourbaki's thoughts has in general been through
Jean DieudonnéJean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
(who initially was the 'scribe' of the group) writing under his own name. In a survey of
le choix bourbachique written in 1977, he did not shy away from a hierarchical development of the 'important' mathematics of the time.
He also wrote extensively under his own name: nine volumes on
analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
, perhaps in belated fulfillment of the original project or pretext; and also on other topics mostly connected with
algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
. While Dieudonné could reasonably speak on Bourbaki's encyclopedic tendency, and tradition (after innumerable frank
tais-toi, Dieudonné! ("Hush, Dieudonné!") remarks at the meetings), it may be doubted whether all others agreed with him about mathematical writing and research. In particular Serre has often criticised the way the Bourbaki works were written, and has championed in France greater attention to problem-solving, within
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
especially, not an area treated in the main Bourbaki texts.
Dieudonné stated the view that most workers in mathematics were doing ground-clearing work, in order that a future
Riemannwas an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.-Early life:...
could find the way ahead intuitively open. He pointed to the way the axiomatic method can be used as a tool for problem-solving, for example by
Alexander GrothendieckAlexander Grothendieck is considered one of the greatest mathematicians of the 20th century.He is most famous for his revolutionary advances in algebraic geometry, but he has also made major contributions to algebraic topology, number theory, category theory, Galois theory, descent theory,...
. Others found him too close to Grothendieck to be an unbiased observer. Comments in
Pal TuránPaul Turán was a Hungarian mathematician who worked primarily in number theory. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers....
's 1970 speech on the award of a
Fields MedalThe Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. The Fields Medal is often viewed as the top honor a mathematician can receive. It...
to Alan Baker about theory-building and problem-solving were a reply from the traditionalist camp at the next opportunity, Grothendieck having received the previous Fields Medal
in absentia in 1966.
Bourbaki's influence on mathematics education
In the longer term, the manifesto of Bourbaki has had a definite and deep influence. In secondary education the
new mathNew Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s. The name is commonly given to a set of teaching practices introduced in the U.S...
movement corresponded to teachers influenced by Bourbaki. In France the change was secured by the
Lichnerowicz CommissionAndré Lichnerowicz was a noted French differential geometer and mathematical physicist of Polish descent.-Biography:...
.
The influence on graduate education in pure mathematics is perhaps most noticeable in the treatment now current of
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s and
Lie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s. Dieudonné at one point said 'one can do nothing serious without them', for which he was reproached; but the change in Lie theory to its everyday usage owes much to the type of exposition Bourbaki championed. Beforehand
Jacques HadamardJacques Salomon Hadamard was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...
despaired of ever getting a clear idea of it.
External links
- Official Website of L'Association des Collaborateurs de Nicolas Bourbaki
- A long article about Nicolas Bourbaki, from PlanetMath
PlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is hosted by the Digital...
- 25 Years with Bourbaki, by Armand Borel
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...