Friday, 19 August 2011

upload

Bourbaki Group

Amir Aczel
Amir D. Aczel (born November 6, 1950) is a lecturer in mathematics and the history of mathematics and science, and an author of popular books on mathematics and science.
http://wn.com/Amir_Aczel
Andre Lichnerowicz
http://wn.com/Andre_Lichnerowicz
André Weil
André Weil (6 May 1906 – 6 August 1998) () 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. He was a founding member and the de facto early leader of the influential Bourbaki group. The philosopher Simone Weil was his sister.
http://wn.com/André_Weil
Armand Borel
Armand Borel (21 May 1923 –11 August 2003) 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. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups.
http://wn.com/Armand_Borel
Bernhard Riemann
(; September 17, 1826 – July 20, 1866) was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity.
http://wn.com/Bernhard_Riemann
Claude Chevalley
Claude Chevalley (11 February 1909, Johannesburg, South Africa – 28 June 1984, Paris) 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. He was a founding member of the Bourbaki group.
http://wn.com/Claude_Chevalley
David Hilbert
David Hilbert (; January 23, 1862 –
http://wn.com/David_Hilbert
Emil Artin
Emil Artin (March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian-Armenian mathematician.
http://wn.com/Emil_Artin
Emmy Noether
Amalie Emmy Noether, , (23 March 1882 – 14 April 1935) was a German American mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields, and algebras. In physics, Noether's theorem explains the fundamental connection between symmetry and conservation laws.
http://wn.com/Emmy_Noether
Euclid
Euclid ( ; Eukleidēs), fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.
http://wn.com/Euclid
Henri Cartan
Henri Paul Cartan (July 8, 1904 – August 13, 2008) was a son of Élie Cartan, and was, as his father, a distinguished and influential French mathematician.
http://wn.com/Henri_Cartan
Henri Poincaré
Jules Henri Poincaré (29 April 1854 – 17 July 1912) () was a French mathematician, theoretical physicist, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.
http://wn.com/Henri_Poincaré
Hyman Bass
Hyman Bass (born October 5, 1932, Houston, Texas) is an American mathematician, known for work in algebra. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus. He is currently the Roger Lyndon Collegiate Professor of Mathematics and Professor of Mathematics Education at the University of Michigan.
http://wn.com/Hyman_Bass
Jacques Hadamard
Jacques Salomon Hadamard (December 8, 1865 – October 17, 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.
http://wn.com/Jacques_Hadamard
Jean Delsarte
Jean Frédéric Auguste Delsarte (October 19, 1903 - November 28, 1968) was a French mathematician.
http://wn.com/Jean_Delsarte
Jean Dieudonné
Jean Alexandre Eugène Dieudonné (1 July 1906 – 29 November 1992) 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 mathematics, particularly in the fields of functional analysis and algebraic topology. His work on the classical groups (the book La Géométrie des groupes classiques was published in 1955), and on formal groups, introducing what now are called Dieudonné modules, had a major effect on those fields.
http://wn.com/Jean_Dieudonné
Jean Leray
Jean Leray (7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.
http://wn.com/Jean_Leray
Jean-Pierre Serre
Jean-Pierre Serre (born 15 September 1926) 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.
http://wn.com/Jean-Pierre_Serre
Laurent Schwartz
Laurent-Moïse Schwartz (5 March 1915 in Paris – 4 July 2002 in Paris) 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. For a long time he taught at the École polytechnique.
http://wn.com/Laurent_Schwartz
Lennart Carleson
Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis.
http://wn.com/Lennart_Carleson
Louis H. Kauffman
http://wn.com/Louis_H_Kauffman
Paul Dubreil
Paul Dubreil (March 1, 1904 – March 9, 1994) was a French mathematician.
http://wn.com/Paul_Dubreil
René de Possel
Lucien Alexandre Charles René de Possel (1905–1974) 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.
http://wn.com/René_de_Possel
Roger Godement
Roger Godement (born 1921) is a French mathematician, known for his work in functional analysis, and also his expository books.
http://wn.com/Roger_Godement
Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire (now in Poland) and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.
http://wn.com/Samuel_Eilenberg
Serge Lang
Serge Lang (May 19, 1927 – September 12, 2005) 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.
http://wn.com/Serge_Lang
Szolem Mandelbrojt
Szolem Mandelbrojt (10 January 1899 – 23 September 1983) 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.
http://wn.com/Szolem_Mandelbrojt

Email this Page

Make changes yourself !
Login or register to EDIT and SAVE any of these pages.

Into His Countenance (flute and string orchestra) - Live Performance

Order: Reorder
Duration: 6:18
Published: 16 Aug 2009
Uploaded: 19 May 2011
Author: PhillipLWilcher

The Bourbaki Ensemble David Angell, conductor Amanda Muir, flute Performance date : August 2nd, 2009 I wrote this piece during the immediate weeks following my mother's passing, mostly out of the belief that if I could give to her noble and courageous life a profile through my music, I would have my profile too. Her medical history was long and debilitating. She had first undergone surgery in the 1950s to have part of her thyroid removed. In the early 70s she underwent a hysterectomy. Ten years later she lost a kidney to cancer. Then came Parkinson's Disease and Osteoporosis. In 2004 she lost her other kidney to cancer and then in June 2005 her life to multiple tumors in her liver and lungs. She never complained. "Into His Countenance" represents my mother's soul moving towards the countenance of God. During the last days of her life, something was happening, something beyond life as we know it was preparing her way. The night before she died, I whispered into her ear a promise that I would see to it she would not be forgotten. I do something every day to honour the memory of her quietly courageous life. Please visit the Tribute Pages of my website - www.phillipwilcher.com - where you will find a tribute to her life titled "An Unforgettable Joy" Mum : I will never forget you. I love you! The year 2005 proved to be very difficult for me. I had also lost my beloved mentor and friend, the composer Miriam Hyde. You can also read about Miriam by visiting the same Tribute Pages <b>...</b>

The Flautist's Pavane - flute and string orchestra by Phillip Wilcher

Order: Reorder
Duration: 5:15
Published: 14 Feb 2009
Uploaded: 26 Feb 2011
Author: PhillipLWilcher

This is a computer simulation only of my piece The Flautist's Pavane. In writing this piece, for the most part I assigned a different melody or voice to each individual group of instruments so that what you hear here is 7 part writing. This piece along with my piece "Into His Countenance" will be premiered in a performance by the Bourbaki Ensemble in July. PLesae visit: www.phillipwilcher.com

Play next
List all Videos
Autoplay
Autorepeat
Shuffle

clear restore
list images
close sort

Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality. Their work led 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 Supérieure in Paris.

Books by Bourbaki

Aiming at a completely self-contained treatment of the core areas of modern mathematics based on set theory, the group produced the Elements of Mathematics (Éléments de mathématique) series, which contains the following volumes (with the original French titles in parentheses):

Set theory (Théorie des ensembles)
Algebra (Algèbre)
Topology (Topologie générale)
Functions of one real variable (Fonctions d'une variable réelle)
Topological vector spaces (Espaces vectoriels topologiques)
Integration (Intégration)

and later

Commutative algebra (Algèbre commutative)
Lie theory (Groupes et algèbres de Lie)
Spectral theory (Théories spectrales)

The book Variétés différentielles et analytiques was a fascicule de résultats, that is, a summary of results, on the theory of manifolds, rather than a worked-out exposition. A final volume IX on spectral theory (Théories spectrales) from 1983 marked the presumed end of the publishing project; but a further commutative algebra fascicle 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 $\varnothing$ for the empty set and a dangerous bend symbol, and the terms injective, surjective, and bijective.

It is frequently claimed that the use of the blackboard bold letters for the various sets of numbers was first introduced by the group. There are several reasons to doubt this claim.

Influence on mathematics in general

The emphasis on rigour may be seen as a reaction to the work of Henri Poincaré, 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 geometry 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 physics is practiced.

Bourbaki's direct influence has decreased over time. It also mattered that, while especially algebraic structures 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 articles, 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 Supérieure in Paris and included Henri Cartan, Claude Chevalley, Jean Coulomb, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, René de Possel, Szolem Mandelbrojt and André Weil. There was a preliminary meeting, towards the end of 1934. Jean Leray and Paul Dubreil were present at the preliminary meeting but dropped out before the group actually formed. Other notable participants in later days were Hyman Bass, Laurent Schwartz, Jean-Pierre Serre, Alexander Grothendieck, Samuel Eilenberg, Serge Lang and Roger Godement.

The original goal of the group had been to compile an improved mathematical analysis 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 spaces were to be handled, to precipitate a guaranteed crisis.

The name "Bourbaki" refers to a French general Charles Denis Bourbaki; 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 mathematics, Bourbaki being of Greek extraction. It is a valid reading to take the name as implying a transplantation of the tradition of Euclid 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 Hilbert and the modern algebra school of Emmy Noether, Artin 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:

algorithmic content is not considered on-topic and is almost completely omitted problem solving, in the sense of heuristics, receives less emphasis than axiomatic theory-building analysis is treated 'softly', without 'hard' estimates Measure theory is developed from a functional analytic perspective. Taking the case of locally compact measure spaces as fundamental focuses the presentation on Radon measures 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 theory, of the restriction to locally compact spaces.

combinatorics is not discussed

logic is treated minimally

applications are not covered.

Furthermore, Bourbaki make no use of pictures in their presentation. In general, Bourbaki has been criticized for reducing geometry as a whole to abstract algebra and soft analysis.

Dieudonné as speaker for Bourbaki

Public discussion of, and justification for, Bourbaki's thoughts has in general been through Jean Dieudonné (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 analysis, perhaps in belated fulfillment of the original project or pretext; and also on other topics mostly connected with algebraic geometry. 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 championed greater attention to problem-solving, within number theory 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 Riemann 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 Grothendieck. Others found him too close to Grothendieck to be an unbiased observer. Comments in Pal Turán's 1970 speech on the award of a Fields Medal 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 math movement corresponded to teachers influenced by Bourbaki. In France the change was secured by the Lichnerowicz Commission.

The influence on graduate education in pure mathematics is perhaps most noticeable in the treatment now current of Lie groups and Lie algebras. 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 Hadamard despaired of ever getting a clear idea of it.

See also

Bourbaki–Witt theorem

Bourbaki dangerous bend symbol

Notes

References

David Aubin, The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics. Science in Context, 10 (1997), p. 297-342.

Maurice Mashaal (2006). Bourbaki: A Secret Society of Mathematicians. American Mathematical Society. ISBN 0-8218-3967-5.

Amir Aczel (2007). The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed. High Stakes Publishing, London. ISBN 1-84344-034-2.

External links

Official Website of L'Association des Collaborateurs de Nicolas Bourbaki

A long article about Nicolas Bourbaki, from PlanetMath

25 Years with Bourbaki, by Armand Borel

Apology of Euclid, by S. Kutateladze

Category:Nicolas Bourbaki Category:Collective pseudonyms Category:French mathematicians Category:Nonexistent people Category:Collegiate secret societies

This text is licensed under the Creative Commons CC-BY-SA License. This text was originally published on Wikipedia and was developed by the Wikipedia community.