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- Brief professional biography
- Commitment to education
- Essential bibliography
- Essential secondary bibliography
- Websites

Henri Paul Cartan was born in Nancy on July 8, 1904, the son of élie Cartan, one of the most important mathematicians of the 20th century, and Marie-Louise Bianconi. In 1909, when élie, professor at the University of Nancy, was appointed as a lecturer at the Sorbonne, the family moved from Nancy to Paris. Then Henri studied at the Lycée Buffon and the Lycée Hoche in Versailles. In 1923 he was admitted at the école Normale Supérieure (ENS) in Paris. In those years, among the students at ENS there were some future participants in the early meetings of the Bourbaki group: André Weil and Jean Delsarte (admitted in 1922), René de Possel and Jean Coulomb (admitted in 1923), Jean Dieudonné and Charles Ehresmann (admitted in 1924), and Claude Chevalley and Jean Leray (admitted in 1926). Among Henri Cartan’s teachers at the ENS were Gaston Julia and his father élie, with whom he wrote some early papers. The students at the ENS also had to attend general courses at the Sorbonne, so Henri studied there too. Admitted to the concours d’agrégation in 1926, in 1928 he defended his dissertation of Doctorat ès sciences entitled Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications under the supervision of Paul Montel. After being awarded his doctorate, Cartan taught at the Lycée Malherbe in Caen from 1928 to 1929, at the University of Lille from 1929 to 1931 and then at the University of Strasbourg. In 1935 he married Nicole Antoinette Weiss. They had two sons and three daughters. In 1940 Cartan joined the University of Sorbonne in Paris, with the exception of two years (1945-1947), when he was stationed at the University of Strasbourg. From 1940 to 1965 he taught at the ENS in rue d’Ulm, and after at the Faculté des Sciences de Paris until 1969 when he became professor at the Faculté des Sciences d’Orsay, embryo of the new Université Paris-Sud created in 1970. He retired in 1975 and died August 13, 2008.

At the ENS, Cartan started the Séminaire Cartan. Fifteen ENS-Seminars written by Cartan were published between 1948 and 1964; these publications became a fundamental reference.

Cartan’s history as a mathematician is strongly linked to the life of the Bourbaki group. After a first informal meeting at the café Capoulade in Paris (December 10, 1934) and successive encounters, the group was finalized in the meeting at Besse-en-Chandesse (July 10-17, 1935). The participants were: Claude Chevalley, René de Possel, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, Szolem Mandelbrojt, and André Weil. A main aim of this group was to renew French mathematics that suffered the loss of a generation of mathematicians dead during the First World War. In his interview to the Notices of AMS (JACKSON 1999) Cartan claims “Almost all I know in mathematics I learned from and with the Bourbaki group.” (p. 785) Nevertheless he maintains that the Bourbaki style is no longer dominant in France, but in the past had an enormous influence in France and abroad.

Through his role as teacher and mentor Cartan contributed remarkably to the renaissance of French mathematics and created a generation of leading mathematicians. His students Jean-Pierre Serre and René Thom were each awarded the Fields medal. In his interview to the Notices of AMS (JACKSON 1999) Cartan describes his way of teaching at ENS: “You have to respect [the student’s] personality, to help [the student] find his own personality, and certainly not impose somebody else’s ideas.” (p. 786) His teaching was very individual.

Cartan influenced the development of mathematics all around the world. He believed in international cooperation and was an active promoter of university exchanges. Already in May 1931 he was invited by Heinrich Behnke (with Cartan in the photo below) to visit Germany and delivered a series of lectures at Muenster University in Westphalia. Cartan’s younger brother, a member of the French Resistance fighting against the occupying German forces, was deported by Nazis and executed in 1943. Nevertheless Cartan renewed contacts with his German colleagues in November 1946 when he visited the Research Institute in Oberwolfach and contributed to the reconstruction of mathematics research in Germany.

In the years 1950 and 1960, Cartan was an active member of the French and international community of mathematicians: President of Société Mathématique de France in 1950, member of the Fields Medal Committee in 1954 and 1970, and president of IMU (International Mathematical Union) from 1967 to 1970. The final act of this presidency was the organization of the sixteenth International Congress of Mathematicians in Nice (September 1-10, 1970). In 1991 he was appointed as a president of the committee established in France to organize the first congress of the European Mathematical Society.

Cartan’s strong interest in cooperation and internationalism was not confined to mathematics. He was involved in the European Federalist movement and in 1984 he was a chief of an electoral list called “Pour les états-Unis d’Europe“ [For the United-States of Europe], which had less than 1% of votes. In the interview mentioned above, he explains that he became a European Federalist by applying the mathematical way of reasoning to politics:

“You see, a mathematician thinks: “What is this question? What happens exactly? Why is it so, and not so? What is the reason? What is the logical consequence of all this?” I am applying this to politics. I have tried to analyze situations and to draw logical consequences. This was the way I became a European Federalist because I understood that there is no other way.” (JACKSON 1999, p. 788)His political involvement was also devoted to supporting human rights. In 1974, together with the American mathematician from Latvia Lipman Bers, Laurent Schwartz, and Michel Broué, he set up the Comité des mathématiciens, an informal organization aimed at supporting the dissident mathematicians persecuted in their countries for political reasons. For this activity in favor of dissidents, Cartan received the Pagels Award from the New York Academy of Sciences.

Cartan was a member of the Académie des Sciences of Paris and of other academies in Europe, the United States, and Japan. He has received honorary doctorates from several universities. He was awarded with the Gold Medal of the National Centre for Scientific research (1976), the Wolf Prize in Mathematics in 1980 and was made Commandeur de la Légion d’Honneur in 1989.

In Cartan’s works, direct reference to the school world is very rare, as evidenced by the 100 titles in the three volumes edited by Remmert and Serre (1979). This is not surprising; in his paper on the new math programs, Cartan himself acknowledged that he had very little experience with secondary teaching:

“Je n’ai guère de titres à prendre la parole, car c’est la première fois que je participe aux travaux de la C.I.E.M [ICMI (International Commission on Mathematical Instruction)], et j’ai vraiment très peu d’expérience de l’enseignement secondaire.” (CARTAN, 1963, p. 84). [I do not have much to contribute here, since it is the first time I am participating in the works of the C.I.E.M [ICMI (International Commission on Mathematical Instruction)], and I really have very little experience in mathematics teaching at secondary level.]As we have seen in the biographical notes, Cartan taught as a secondary teacher only in 1928-1929. Afterwards his entire career was devoted to mathematical research and university teaching. Nevertheless he had some indirect influence in the school world of his times concerning both the mathematical contents to be taught and political issues

Mathematical contents for teaching

The changes in mathematical research taking place in the second half of twentieth century made the issue of updating the mathematical knowledge of schoolteachers a main concern both of teachers and mathematicians. Important initiatives were organized in France for in-service teacher education, see (BARBAZO, 2009; DUBREIL, 1959). For example, the teacher Yves Crozes, at that time president of the French association of Mathematics teachers, organized some successful talks addressed to teachers on modern theories, mainly on axiomatics, in his Lycée Henri IV. In the same vein, talks for teachers were delivered at the Centre International d’études pédagogiques de Sèvres, during the Journées internationales d’information sur l’Enseignement des Mathématiques [International information days on mathematics teaching]. At the beginning of 1956, the Société mathématique de France (with Paul Dubreil as a president) and the Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP) planned a more permanent initiative consisting of cycles of talks delivered at the Institut Henri Poincaré by some important French mathematicians. The first cycle lasted from February 9, 1956 to June 6, 1957. The texts of the lectures were published in the Bulletin de Association des Professeurs de Mathématiques de l’Enseignement Public: n. 176 (Mach 1956), n. 177 (May 1956), n. 179 (October 1956), n. 180 (December 1956), n. 183 (January 1957), n. 184 (March 1957), n. 185 (June 1957), n. 187 (October 1957), n. 188 (January 1958), n. 191 (March 1958). Afterwards, the texts were gathered in a volume, see (CARTAN et al. 1958). A second cycle of talks was organized by the Société mathématique de France and the Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP) at the Institut Henri Poincaré from November 15 1957 to May 22 1958. The texts were published in the Bulletin de Association des Professeurs de Mathématiques de l’Enseignement Public: n. 192 (June 1958), n. 194 (October 1958), n. 196 (January 1959), n. 198 (March 1959), n. 199 (June 1959). Afterwards, the texts were gathered in a volume, see (CARTAN, DIXMIER, DUBREIL, LICHNéROWICZ, & REVUZ, 1962). Cartan was one of the contributors in both cycles, see (1958a; 1958b; 1962). In her book on the history of CIEAEM, Lucienne Félix (1986, pp. 82-84) stresses the impact of this initiative on teachers’ mathematical knowledge.

As a member of the Bourbaki group established in 1935, Cartan launched new methods in mathematics that after some decades affected the teaching in school through the movement of Modern Mathematics. One member of the Bourbaki group, Dieudonné, was an active promoter of the reform of mathematics teaching. He was a leading participant in the important meeting held in Royaumont (November 23 to December 4 1959) with the concrete aim of producing a curricular reform that was not limited to Europe. After Royaumont other meetings continued the discussion. Cartan was invited in the meeting promoted by ICMI in Bologna (4-8 October 1961) on the teaching of geometry in secondary schools. University professors, school inspectors, and delegates from UNESCO and OECD attended the meeting, see (UMI, 1962). The texts of the contributions were published in L’Enseignement Mathématique, the official organ of ICMI (1963, volume 9 of the second series). In his contribution, Cartan (1963) starts by complaining that the results of the previous meetings are still under discussion. He suspects that some people are not really wishing to change the current mathematics teaching. He advocates taking care of those students who will go to university as well as those who will stop their school career. He also stresses the need to prepare teachers for the new mathematics contents. He is convinced that, when planning a radical reform, it is necessary to focus first on the mathematical contents and after on mathematical objectives. About this point he is polemic with Hans Freudenthal who in his report claims that didactical problems have to be considered first, see (FREUDENTHAL, 1963). Cartan advocates that axiomatics is reduced to a minimum and that the algebraic ideas are stressed. In mathematics the teaching of algebra and geometry have not to be separated, but instead taught as follows

“En résumé, je souhaiterais que la géométrie classique (affine ou euclidienne) fût exposée avec le minimum d’axiomatique, et le maximum d’explicitations algébriques. Ces explicitations algébriques n’excluent nullement le langage géométrique ; elles le justifient! Elles n’excluent pas davantage la solution des problèmes par voie géométrique; il y aura toujours intérêt à ce qu’un même problème soit traité de deux manières, par voie géométrique et par voie analytique.” (Cartan, 1963, p. 90, Italic in the original) [Summarizing, I would hope that classical geometry (affine or Euclidean) is presented with the minimum of axiomatic, and the maximum of algebraic explanations . These algebraic explanations do not at all exclude geometric language; they justify it! Furthermore, they do not exclude the solution of problems using geometric means; it will always be interesting that a problem can be treated in two manners, geometrically and analytically.]In a message to Unione Matematica Italiana (1978, NUMI , 5(3), 23-28) Leray reported on a document prepared together with Cartan, Dieudonné, Leray, Lichnérowicz, published in the column Comité secret of Comptes Rendus de l’Académie des Sciences de Paris (1977, t. 285, Vie académique, deuxième semestre, 56-60). This document analyzes some aspects of mathematics teaching in the grades fourth and third of the French Collèges (students’ ages 13-14 years).

Political issues in mathematical instruction

The two contrasting positions about mathematics teaching held by Cartan and Freudenthal emerged also when political issues were to be managed in the years 1967-1970, with Freudenthal president of ICMI and Cartan president of IMU and as such ex officio member of ICMI Executive Committee. The point was the controversial relationship of the bodies they were chairing. The article (FURINGHETTI & GIACARDI, 2010) illustrates the struggle of the two bodies for finding agreement on this relationship. At the end Freudenthal (without consulting the IMU) launched two initiatives that marked the achievement of the autonomy of ICMI from IMU: a new journal on mathematics education, Educational Studies in Mathematics, was founded and the tradition of holding ICMI congresses independent of the International Congresses of Mathematicians was established. In this story, Cartan’s and Freudenthal’s actions epitomize the spirit of an epoch of changes in society and reflect two different views of the relationship of school and research.

H. CARTAN 1958a, Structures algébriques , in H. Cartan et al., Structures algébriques et structures topologiques. Monographies de L’Enseignement Mathématique, 7, pp. 5-15.

H. CARTAN 1958b, Sur la notion de dimension, in H. Cartan et al., Structures algébriques et structures topologiques. Monographies de L’Enseignement Mathématique, 7, pp. 163-174.

H. CARTAN, G. CHOQUET, J. DIXMIER, P. DUBREIL, R. GODEMENT, P. LELONG, L. LESIEUR, & A. LICHNéROWICZ, CH. PISOT, A. REVUZ, L. SCHWARTZ, J.-P. SERRE 1958, Structures algébriques et structures topologiques. Monographies de L’Enseignement Mathématique, 7.

H. CARTAN 1962, Volume des polyèdres , in H. Cartan et al., Problèmes de mesure.. Monographies de L’Enseignement Mathématique, 10, pp. 8-20.

H. CARTAN, J. DIXMIER, P. DUBREIL, A. LICHNEROWICZ, & A. REVUZ, 1962, Problèmes de mesure.. Monographies de L’Enseignement Mathématique, 10.

H. CARTAN 1963, Réflexions sur les rapports d’Aarhus et Dubrovnik, L’Enseignement Mathématique, s. 2, 9, pp. 84-90.

Retrieved 30 September 2015.

M. AUDIN 2012, Henri Cartan & André Weil. Du vingtième siècle et de la topologie , in P. HARINCK, A. PLAGNE, & C. SABBAH (Eds.), Henri Cartan & André Weil mathématiciens du XXe siècle. Journées mathématiques X-UPS (pp. 1-61), Palaiseau, éditions de l’école Polytechnique.

E. BARBAZO 2009, La formation continue des enseignants: phénomène naturel? ”, in F. PLANTEVIN (Ed.), Actes en ligne du séminaire “Formation continue” http://www.univ-irem.fr/spip.php?rubrique106 (retrieved 12 July 2015).

P. DUBREIL 1959, Rapport sur les bases scientifiques des mathématiques dans l’enseignement du second degré Structures algébriques , L’Enseignement Mathématique, s. 2, 5, pp. 273-277.

L. FéLIX 1986, Aperçu historique (1950-1984) sur la Commission Internationale pour l’étude et l’Amélioration de l’Enseignement des Mathématiques , Bordeaux, IREM http://www.cieaem.org/?q=node/18 (retrieved 12 July 2015).

H. FREUDENTHAL 1963, Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?. L’Enseignement Mathématique, s. 2, 9, pp. 28–44.

F. FURINGHETTI & L. GIACARDI 2008, The first century of the International Commission on Mathematical Instruction (1908-2008). The history of ICMI http://www.icmihistory.unito.it/ (retrieved 12 July 2015).

F. FURINGHETTI & L. GIACARDI 2010, People, events, and documents of ICMI’s first century, Actes d’Història de la Ciència i de la Tècnica, nova època, 3(2), pp. 11-50.

L. GIACARDI 2008, Documents, in F. FURINGHETTI & L. GIACARDI 2008, The first century of the International Commission on Mathematical Instruction (1908-2008). The history of ICMI

A. JACKSON 1999, Interview with Henri Cartan. Notices of AMS, 46, pp. 782-788.

OECE (1961a). Mathématiques nouvelles, OECE, Paris.

OECE (1961b). Un programme moderne de mathématiques pour l’enseignement secondaire , OECE, Paris.

R. REMMERT & J.-P. SERRE (Eds.) 1979, Henri Cartan, Oeuvres (3 vols.), Berlin-Heidelberg-New York, Springer.

J.-P. SERRE 2009, Henri Paul Cartan. 8 July 1904 - 13 August 2008, Biographical Memoirs of the Fellows of the Royal Society, 55, pp. 37-44.

UMI 1962, Il convegno di Bologna promosso dalla Commissione internazionale dell’insegnamento matematico. Bollettino dell’Unione Matematica Italiana, s. 3, 17, 199-214.

smf.emath.fr/VieSociete/Rencontres/JourneeCartan/NoticeCartan.html (retrieved 12 July 2015. This website contains a rich list of references, works, films, interviews, and articles on Henri Cartan). http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Cartan_Henri.html (retrieved 12 July 2015)

Photos from Archives of the Mathematisches Forschungsinstitut Oberwolfach

Signature from Académie des sciences, Archives et patrimoine historique

Author

Fulvia Furinghetti

Department of Mathematics

University of Genova – ITALY

furinghetti@dima.unige.it