# Dieudonné, Jean

# DIEUDONNé, JEAN

(*b*. Lille, France, 1 July 1906;

*d*. Paris, France, 29 November 1992), *analysis, algebra, history of mathematics, Bourbaki*.

Dieudonné was distinguished as much for vast mathematical knowledge as for his own innovations. He influenced twentieth-century mathematics through his role in the Bourbaki group, his nine-volume *Treatise on Analysis*, his four-volume collaboration with Alexander Grothendieck, and his historical writing. He wrote tens of thousands of pages. More than any other mathematician in Bourbaki, Dieudonné stressed the simplifying role of axiomatics, which David Hilbert also stressed, as opposed to the generalizing role. Fundamental, classical theorems often use far less than the classical assumptions. Dieudonné would drop the irrelevant assumptions and prove the theorems from just the relevant axioms. Of course the results are also more general, but Dieudonné aimed more to organize and unify than to generalize.

**Youth** . Dieudonné deeply admired and respected his father, Ernest, who had supported a family from the age of twelve and rose from a modest employee to become general director of a textile manufacturing group. Ernest valued education, strove to make up for what he had missed, and married Léontine Lebrun who taught grade school until Jean was born. A few years later came a sister, Anne Marie. Jean’s mother taught him to read before he went to school. He favored dictionaries, encyclopedias, and universal histories. He began school in Lille, but in 1914 when the city declared itself indefensible and surrendered to the Germans he went to Paris to the Lycée Condorcet.

He spent 1919–1920 as a fellow of Bembridge School on the Isle of Wight to learn English. The school was founded in 1910 on the principles of John Ruskin, who would address social ills by “making a carpenter … happier as a carpenter” and making the elite a better elite (Hicks, 1974, p. 57). Dieudonné encountered algebra there and found his calling as a mathematician.

With the war over, he returned to Lille and the Lycée Faidherbe. In 1924 he was accepted at both the École Polytechnique and the École Normale Supérieure (ENS) and chose the latter. There he took courses from great mathematicians at the Sorbonne and the Collège de France, notably including C. Emile Picard and Jacques Hadamard. These were older men; France had lost the intermediate generation in World War I. Hadamard’s seminar raised topics from recent mathematics, but most instruction was in nineteenth-century analysis. Dieudonné graduated and took first place in the mathematics agrégation examination for teaching at a lycée. Accepted to the doctoral program at the ENS, Dieudonné did his military service from 1927 to 1928 and began research.

Princeton University made him a Proctor Visiting Fellow for 1928–1929. He studied with Hermann Weyl, and with Godfrey H. Hardy who visited for that same year. A Rockefeller Foundation grant for 1930–1931 let him study with Ludwig Bieberbach in Berlin and George Pólya in Zurich.

In 1931 Dieudonné completed his thesis at the ENS supervised by Paul Montel. He calculated bounds on the locations of zeros of a complex meromorphic function *f*(*z*) or its derivative *f'*(*z*) given specified bounds for values of *f*(*z*) on specified domains. This was the subject of virtually all his publications prior to Bourbaki.

With his record at the ENS, Dieudonné was extremely employable even in hard times. His role in Bourbaki made him a sought-after professor, although he had no interest in and small gift for teaching. He would eventually hold professorships at five universities in the United States and France, plus visiting professorships at eleven universities in Europe, Asia, and North and South America. He began as an instructor at Bordeaux in 1932 and went to Rennes from 1933 to 1937, first as an instructor and then in a teaching-research position (maître de conférences).

**Bourbaki** . In the fall of 1934, Odette Clavel dropped her program at a Sunday afternoon concert. Dieudonné picked it up, handed it to her, and married her on 22 July 1935. Fifty-six years later he described the marriage as fifty-six years of happiness. They had a son and a daughter: Jean-Pierre and Françoise. Dieudonné admitted to taking too much time away from them for work.

On Monday, 10 December 1934, Dieudonné joined a handful of ENS graduates called by André Weil to the now-vanished Café Capoulade to plan a thoroughly collaborative, definitive new analysis textbook. It would not have separate chapters by separate experts, but the whole group would write every part of it using the latest tools and the latest standards of rigor out of Hilbert’s school in Germany. It would reestablish French preeminence in mathematics. The others at the café were Henri Cartan, Claude Chevalley, Jean Delsarte, and René de Possel.

The mutation of their project foreshadowed their eventual impact on mathematics. They aimed to replace the text they had all studied and taught, Edouard Goursat’s 1902–1905 *Cours d’Analyse Mathématique*. Goursat reached classical problems, notably envelopes of families of curves or families of surfaces, the Dirichlet problem, the heat equation, and Fredholm’s equation, by sketchy methods of real and complex algebra, real and complex integration, and formal power series. The young mathematicians quickly saw that rigor would mean vast prerequisites. The projected analysis text turned into an entirely self-contained multivolume work on the methods most widely used across mathematics.

They formed a society under the fictitious name of Nicolas Bourbaki and began the *Elements of Mathematics* with volumes on set theory, algebra, topology, functions of one real variable, topological vector spaces, and integration. Dieudonné always insisted that the *Elements* are not encyclopedic because they select only the most useful generalities and reach no serious theorems, but they became the encyclopedia of a new conception of mathematics organized around methods rather than theorems. Despite many critics then and now, the new organization of mathematics became the worldwide norm for graduate training as it proved to be more accessible to students and finally more productive of new great theorems.

The methodical axiomatic style spread so quickly to other authors that a story has grown that the *Elements* themselves never worked as textbooks. But there were circles in the 1950s where, as Pierre Cartier recalled, “every time that Bourbaki published a new book, I would just buy it or borrow it from the library, and learn it. For me, for people in my generation, it was a textbook. But the misunderstanding was that it should be a textbook for everybody” (Senechal, 1998, p. 25).

Dieudonné personified Bourbaki. He was a powerful personal force within the group: He worked as a kind of sergeant at arms and did much of the writing. Sections of the *Elements* went through repeated drafts by different members and were critiqued by all, but Dieudonné wrote every final draft as long as he was an active member. The only works under the name of Bourbaki not approved by the group were the conceptual papers by André Weil (Bourbaki, 1949) and by Dieudonné (Bourbaki, 1950), and Weil and Dieudonné’s historical notes to the *Elements*(Bourbaki, 1960). Pierre Cartier says: “When Dieudonné was the ‘scribe of Bourbaki’ every printed word came from his pen. With his fantastic memory he knew every single word. You could say ‘Dieudonné what is the result about so and so?’ and he would go to the shelf and take down the book and open it to the right page. After Dieudonné retired no one was able to do this” (Senechal, 2005, p. 28).

Dieudonné often praised the way collaboration reshaped his research: “if I had not been submitted to this obligation to draft questions I did not know a thing about, and to manage to pull through, I should never have done a quarter or even a tenth of the mathematics I have done” (1970, p. 144). He published a little more on zeros of functions, but Bourbaki took him into abstract algebra and point-set topology and a modicum of the new logic.

With Henri Cartan, Dieudonné wrote a series of notes on *teratopology* or counterexamples to plausible guesses in point-set topology. He began work that he would later extend with Laurent Schwartz on topologies for infinite dimensional vector spaces with applications to functional analysis.

Dieudonné’s innovations were often extremely useful without being deep or hard. He gave the idea of *paracompact* topological spaces, where every open cover has some open locally finite refinement—that is, a cover by open subsets of sets in the original cover and such that any point lies in just finitely many of these subsets; he proved every separable metrizable space is paracompact. He defined *partitions of unity* for covers. That is, on suitable spaces (which, depending on details, are basically the paracompact spaces) given any locally finite cover of the space by open subsets *U*_{i,}, each set *U*_{i} of the cover can be assigned a smooth function *f*_{i} which is 0 outside *U*_{i} and bounded between 0 and 1 inside it, and at every point the
sum of the values of the functions is 1. A partition of unity on a cover gives a systematic way to take local constructions on each set of the cover and add them together to get smooth constructions on the whole space.

Bourbaki is especially identified with the idea of a mathematical *structure*. Dieudonné was clear: “I do not say it was an original idea of Bourbaki—there is no question of Bourbaki’s containing anything original” (1970, p. 138). But Dieudonné and Weil led the group in codifying ways that a few kinds of structure recur throughout mathematics; for example, the addition of real numbers, and of vectors, and multiplication of matrices are all associative binary operations. Or, for another example, divisibility of integers and inclusion of subsets are both transitive relations. Bourbaki from the first volume of the *Elements* in 1939 sought a general theory of all the ways a set can be structured: by operations on the set, or relations among its members, or a topology on the set. But the theory they produced was not general enough to apply to all the mathematical objects they needed, and it was too complicated to use when it did apply. After working with Grothendieck, Dieudonné concluded that Bourbaki’s theory of structures “has since been superseded by that of category and functor, which includes it under a more general and convenient form” (1970, p. 138).

**Early Years at Nancy** . The University of Nancy made Dieudonné an instructor in 1937 and then promoted him to maître de conférences. He held that post until 1946, although he was mobilized for the war in September 1939 and many university jobs were relocated to Clermont-Fer-rand when the Germans made France north of the Somme a *zone interdite*, forbidden to the French, under Belgian administration. He returned to Nancy by 1943 while it was still *zone interdite* (Eguether, 2003, p. 25). After the war, Dieudonné spent 1946–1948 as a professor at the University of São Paulo, Brazil, and returned to Nancy as a professor from 1948 to 1952. Fellow founder of Bourbaki Jean Delsarte was then dean of the Science Faculty and was assembling a brilliant collection of Bourbaki members or future members at Nancy.

Delsarte and Dieudonné brought Laurent Schwartz to Nancy. Schwartz was working on his distributions, which made rigorous the generalized functions used by physicists, such as the Dirac delta function. His basic tool was to set up a relation between, roughly speaking, the space of all real-valued smooth functions *f* on the real line and the space of all generalized functions *ϕ* on the same line. Exploring the foundations of his idea, he coauthored a paper on topological vector spaces with Dieudonné; the two of them recommended certain open questions from that paper to their student Grothendieck. Grothendieck in response created the idea of a nuclear space, and Dieudonné later described his student’s answers as “the greatest advance in functional analysis after the work of Banach” (1981, p. 220).

**Work on Groups** . The *classical groups* are certain groups of matrices with clear geometric sense—at least, they have a clear geometric sense when the matrices have real or complex numbers as entries. For example, the real general linear group GL_{n(}(*R*) consists of all invertible *n* by *n* matrices of real numbers and is geometrically the group of all linear maps from the *n*-dimensional real vector space to itself. Mathematicians since the mid-nineteenth century had studied these, and also analogous groups with entries in fields other than the real or complex numbers. From the late 1940s into the 1950s Dieudonné used the relatively new axiomatic theory of vector spaces, in its geometrical interpretation, to simplify the proofs and clarify the subject and solve some fundamental problems in it.

This led to Dieudonné’s deepest and most imaginative personal work in mathematics, his work on *formal groups*. The spaces of classical algebraic geometry are defined by polynomial equations, as for example *x ^{2}*+

*y*-1=0 defines the unit circle. Roughly speaking, a coordinate function on the unit circle is any polynomial

^{2}*P*(

*x,y*) in these same variables

*x,y*, with the proviso that polynomials

*P*(

*x,y*) and

*Q*(

*x,y*) count as the same function on the circle if their difference

*P*(

*x,y*)-

*Q*(

*x,y*) is a multiple of the defining polynomial

*x*+

^{2}*y*-1=0. When the polynomial coefficients are taken as real or complex numbers, then all the techniques of classical analysis apply. When they are taken in any field

^{2}*k*, some analysis still applies since there is a familiar formal rule for the derivative of a polynomial. But classical techniques using limits or convergent power series do not apply when the field

*k*has no topology (or no suitable topology) to support a notion of convergence. This happens in particular for fields of characteristic

*p*, for a prime number

*p*where multiplication by

*p*counts as multiplication by 0. The algebraic geometry of these fields was ever more central to number theory following work by Bourbaki members Claude Chevalley and Weil, among others.

Dieudonné made up for a large part of the loss by abandoning topological convergence and working formally with arbitrary infinite power series. A *formal group* is roughly an algebraic space where the coordinate functions are not only polynomials but infinite series, and such that the space is also a group, like the classical groups. Through the 1950s Dieudonné made many classical analytic techniques apply in very useful ways without using the classical notion of convergence (later collected in Dieudonné, 1973).

**Scheme Theory** . In 1952 Dieudonné accepted a one-year professorship at the University of Michigan. That led to a professorship at Northwestern University from 1953 to 1959, where he gave the lectures on analysis that became *Foundations of Modern Analysis*(1960), the final result of Bourbaki’s original plan for a textbook on analysis. This book has been a strong, immediate influence on far more mathematicians than have ever read anything else that Dieudonné wrote. It violates a stereotype of Bourbaki as it is thoroughly geometrical, but it is typical Dieudonné: It is axiomatic, quite general, and uses that generality entirely to simplify the theory. It defines derivatives as linear approximations to functions between (finite or infinite dimensional) Banach spaces and yet proves not one nontrivial theorem on Banach spaces. It uses the Banach space axioms because they assume all and only the structure needed for the basic theorems of differential calculus. They are to the point. Deep considerations on Banach spaces have no place here—simple, general facts about derivatives do.

Dieudonné left Northwestern to become the first professor of Mathematics at the Institut des Hautes Études Scientifiques (IHES) near Paris, modeled on the Institute for Advanced Study in Princeton, New Jersey. He brought his student Grothendieck, instantly making the IHES a power in mathematics. Grothendieck in fact had abandoned analysis, though, and begun the sweeping recreation of algebraic geometry around his new notion of *scheme*.

Roughly speaking, a scheme is an algebraic space, defined like most kinds of manifolds by coordinate functions on patches of the space, with the astonishing innovation that these “functions” need not be polynomials or even functions in any set theoretic sense. Rather, any ring in the sense of abstract algebra can be the ring of “coordinate functions” on a patch of a scheme, with the ring elements treated as coordinate functions. This appalling abstraction and generality struck many mathematicians as impervious to geometric intuition. But Grothendieck and Dieudonné knew better. They saw it not as general but as simple: The apparatus of analysis and classical topology is dropped in favor of the mere ring operations of addition and multiplication.

Much fundamental geometric intuition does survive. Notably, an algebraic triviality says that a subset *S* of a ring *R* generates the unit ideal if and only if some finite subset *T* of *S* already does. In scheme theory this has two immediate, fundamental consequences: the basic schemes are compact, and they admit analogues to Dieudonné’s partitions of unity with the difference that the “functions” *f*_{i} are elements of arbitrary rings and are in no sense bounded between 0 and 1. At each point they do add up to 1.

Dieudonné took up another historic multi-volume collaboration, this time with Grothendieck as his single co-author. He did this “with the sole goal of bringing to the public the brilliant ideas of his young collaborator. One rarely sees such disinterested effort” (Cartan, 1993, p. 4). Dieudonné painstakingly organized a flood of Grothendieck’s notes into *Les Éléments de géométrie algébrique* (1960–1967), still the standard reference on schemes in the twenty-first century.

**Move to Nice** . In 1964 Dieudonné became the first dean of the faculty of science at the newly created University of Nice, where the Mathematics Institute is named after him. He held the deanship until 1968 and faced student unrest with his life-long respect for other people’s intentions yet rejection of all leftist politics. The professorship became honorary in 1969. He was elected to the Académie des Sciences of France in 1968, and quickly got a number of his comrades from Bourbaki into it.

He was a visiting professor at the University of Notre Dame (United States) in fall 1966 and again for two years, 1969–1971. At this time he took up his analysis textbook again and expanded to the nine-volume *Éléments d’analyse*(1968–1982). No doubt the first volume, and then the first few volumes, had more direct influence on more mathematicians than the later volumes, but the whole was a fantastic achievement and shaped the general conception of analysis for decades. The capstone of his academic career was organizing the World Congress of Mathematics in Nice in 1970. He turned to writing the history of mathematics.

In common with Weil, Dieudonné believed the history of mathematics should be impersonal, not about anecdotes, and not about priority disputes, but about the development of the leading ideas. His three key works are the historical part of his course on algebraic geometry (1974; translated to English, and expanded, 1985); his expert history of functional analysis (1981); and the massive, detailed history of twentieth-century algebraic and differential topology (1989). Any one can be recommended to graduate mathematics students learning those subjects—and any one can be faulted in detail by specialist historians. Nevertheless, they are invaluable documents. He wrote numerous entries for the *Dictionary of Scientific Biography* (Dugac, 1995, p. 119).

Tall and impressive, though not given to physical exercise, Dieudonné had an energetic enthusiasm that was punctuated by explosive bursts of temper. Friends found him optimistic, generous, honest, and with a strong sense of responsibility—although he said history precluded optimism. He distrusted political reform and was pleased to be received into the generally conservative Légion d’Honneur. He inherited strong discipline from his parents but in no ascetic way. He enjoyed good food and great wine and conversation. He was a skilled pianist and played for an hour or two each morning. Five or six hours of sleep per night was enough.

Dieudonné called himself happy. In old age he said Socrates and Michel de Montaigne were his models for taking difficulties in the best possible way, and expressed confidence that with death, “like all animals, I will entirely disappear.” He died surrounded by his wife and children, keeping the attitude he expressed five years earlier: “Now I am ready to go. If one tells me ‘it will be in one month’ that is perfect. I ask no more. I have had everything that I wanted in life” (quoted in Dugac, 1995, p. 20).

**Honors** . Among the very many honors he received, the Académie des Sciences of Paris awarded Dieudonné their Grand Prize in 1944 and the Petit D’Ormoy Prize in 1966 and made him a member in 1968. In 1966 Dieudonné received the Gaston Julia Prize. He became a correspondent of the National Academy of Sciences (United States) in 1965 and a member in 1968, at the same time as he was elected to the Académie des Sciences of France. He became a foreign member of the Real Academia de Ciencias of Spain in 1970, and of the Académie Royale de Belgique in 1974. The American Mathematical Society awarded him the Steele Prize in 1971, and the London Mathematical Society made him an honorary member in 1972. In 1978 he was made an Officer of the Légion d’Honneur.

## BIBLIOGRAPHY

### WORKS BY DIEUDONNÉ

As Nicolas Bourbaki. *Éléments de Mathématique*, 27 vol. Paris: Hermann et Masson, 1939–1998. Translated into English as *Elements of athematics*. Paris: Hermann, 1974–. There are also translations into Japanese and Russian. Written under a pseudonym of a variable and purportedly secret group, Dieudonné wrote all the final drafts into the 1950s.

As Nicolas Bourbaki. “The Architecture of Mathematics.” *The American Mathematical Monthly* 57 (1950): 221–232.

As Nicolas Bourbaki, with André Weil. *Elements d’histoire des mathématiques*. Paris: Hermann, 1960. Reprints of the historical notes from Bourbaki’s *Eléments*. Translated into English, German, Spanish, Russian.

“Recherches sur quelques problèmes relatifs aux polynômes et aux fonctions bornées d’une variable complexe.” *Annales scientifiques de l’École Normale Supérieure* 48 (1931): 247–358.

“Les méthodes axiomatiques modernes et les fondements des mathématiques.” *Revue Scientifique* 77 (1939): 224–232.

*Théorie analytique des polynômes d’une variable*. Paris: Gauthier Villars, 1939.

*Sur les groupes classiques*. Paris: Hermann, 1958.

*Foundations of Modern Analysis*. New York: Academic Press, 1960.

*Éléments d’analyse*, 9 vols. Paris: Gauthier-Villars, 1968–1982. Volume I is a translation of Dieudonné 1960. Volumes 1–8 translated as *Treatise on Analysis*, 8 vols. New York: Academic Press, 1969–1993.

“The work of Nicholas Bourbaki.” *American Mathematical Monthly* 77 (2, 1970): 134–145.

*Introduction to the Theory of Formal Lie Groups*. New York: Marcel Dekker, 1973.

Editor, and author with others. *Abrégé d’histoire des mathématiques: 1700–1900*, 2 vols. Paris: Hermann, 1978.

*Choix d’OEuvres Mathématiques*, 2 vol. Paris: Hermann, 1981.

*History of Functional Analysis*. Amsterdam: North Holland, 1981.

*Panorama des mathématiques pures: le choix bourbachique*. Paris: Gauthier-Villars, 1977. Translated into English by I. G. Macdonald as *A anorama of Pure Mathematics, as seen by N*. *Bourbaki*. New York: Academic Press, 1982.

*Cours de géométrie algébrique*, 2 vol. Paris: Presses Universitaires de France 1974. The first volume is translated into English, with an additional chapter on recent work, by Judith Sally as *History of Algebraic Geometry*. Monterey, CA: Wadsworth, 1985.

*A History of Algebraic and Differential Topology, 1900–1960*. Boston: Birkhäuser, 1989.

*Pour l’honneur de l’esprit humain—les mathématiques aujourd’hui*. Paris: Hachette, 1987. This popularization, published in English as *Mathematic—the Music of Reason*, was translated by Harold G. and H. C. Dales. Berlin: Springer-Verlag, 1992.

With Henri Cartan. “Notes de tératopologie,” I, II, III. *Revue Scientifique* 77 (1939): 39–40, 180–181, 413–414.

With Alexander Grothendieck. *Les Éléments de géométrie algébrique*. 4 vols. Bures-sur-Yvette, France: Publications Mathématiques de l’IHÉS, 1960–1967.

### OTHER SOURCES

Borel, Armand. “Twenty-Five Years with Nicolas Bourbaki, (1949–1973).” *Notices of the American Mathematical Society*45, no. 3 (1998): 373–380.

Bourbaki, Nicolas (pseud. of André Weil). “Foundations of Mathematics for the Working Mathematician.” *Journal of Symbolic Logic*14 (1949): 1–8.

Cartan, Henri. “Jean Dieudonné.” *Gazette des Mathematiciens* 55 (1993): 3–4. This consists largely of quotes from Dieudonné (1981).

Cartier, Pierre. *Jean Dieudonné (1906–1992): Mathematician*. Bures-sur-Yvette, France: Institut des Hautes Études Scientifiques, 2005. A keenly observed biographical and mathematical account by a member of Bourbaki during the 1950s. It is available online from the IHES.

Dugac, Pierre. *Jean Dieudonné: Mathématicien complet*. Paris: Gabay, 1995. A colleague in history of mathematics assembles quotes from Dieudonné’s less accessible publications and some unpublished writing, testimony by others, and photographs.

Eguether, Gérard. “Jean Delsarte.” *1903–2003: Un siècle de mathématiques à Nancy*, edited by Daniel Barlet. Nancy: Institut Élie Cartan, 2003.

Goursat, Édouard. *Cours d’Analyse mathématique*, 2 vol. Paris: Gauthier-Villars, 1902–1905.

Hicks, Judith. “The Educational Theories of John Ruskin: A Reappraisal.” *British Journal of Educational Studies* 22 (1, 1974): 56–77.

Mashaal, Maurice. *Bourbaki: Une Société Secrète de Mathématiciens*. Paris: Belin, Pour la Science, 2000.

Senechal, Marjorie. “The Continuing Silence of Bourbaki: An Interview with Pierre Cartier.” *Mathematical Intelligencer* 20 (1998): 22–28.

*Colin McLarty*

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**Dieudonné, Jean**