Background

René-Louis Baire was born on January 21, 1874, in Paris, Ile-de-France, France. The son of a tailor, Baire was one of three children from a poor working-class family in Paris.

Education

Baire won a scholarship competition for the city of Paris in 1886, he entered the Lycée Lakanal as a boarding student; there he completed his advanced classes in 1890 after having won two honorable mentions in the Concours Général de Mathématiques. In 1891 Baire entered the section for special mathematics at the Lycée Henri IV, and in 1892 was accepted at both the École Polytechnique and the École Normale Supérieure. He chose the latter, and during his three years there attracted attention by his intellectual maturity.

Although he placed first in the written part of the 1895 agrégation in mathematics, Baire was ranked third because of a mistake in his oral presentation on exponential functions, which the board of examiners judged severely. In the course of his presentation, Baire realized that his demonstration of continuity, which he had learned at the Lycée Henri IV, was purely an artifice since it did not refer sufficiently to the definition of the function. This disappointment should be kept in mind because it caused the young lecturer to revise completely the basis of his course in analysis and to direct his research to continuity and the general idea of functions. While studying on a scholarship in Italy, Baire was strengthened in his decisive reorientation by Vito Volterra, with whom he soon found himself in agreement and who recognized the originality and force of his mind.

On 24 March 1899 Baire defended his doctoral thesis, on the theory of the functions of real variables, before a board of examiners composed of Appell, Darboux, and Picard.

Career

Baire began his teaching career in the lycées of Troyes, Bar-le-Duc, and Nancy, but he could not long endure the rigors of teaching the young. In 1902, as a lecturer at the Faculty of Sciences of Montpellier, he wrote a paper on irrational numbers and limits. In 1904 he was awarded a Peccot Foundation fellowship to teach for a semester at the Collège de France. At that time this award went to young teachers to enable them to spend several months, free of routine duties, in developing their own specialties. Baire chose to work on a course in discontinuous functions, later edited by his pupil A. Denjoy and published in the Collection Borel, a series of monographs on the theory of functions.

Upon his return to Montpellier, Baire experienced the first violent attack of a serious illness that became progressively worse, manifesting itself in constrictions of the esophagus. After the crisis passed, he began drafting his paper “Sur la représentation des fonctions discontinues.” Appointed professor of analysis at the Faculty of Science in Dijon in 1905, to replace Méray, he devoted himself to writing an important treatise on analysis (1907-1908). This work revivified the teaching of mathematical analysis. His health continued to deteriorate, and Baire was scarcely able to continue his teaching from 1909 to 1914. In the spring of 1914, he decided to ask for a leave of absence. He went first to Alésia and then to Lausanne. War broke out while he was there, and he had to remain - in difficult financial circumstances - until the war ended.

Baire was never able to resume his work, for his illness had undermined his physical and mental health. He now devoted himself exclusively to calendar reform, on which he wrote an article that appeared in the Revue rose (1921). While still at his retreat on the shores of Lake Geneva, Baire received the ribbon of the Legion of Honor, and on 3 April 1922 was elected corresponding member of the Academy of Sciences. A pension granted him in 1925 enabled him to live in comparative ease, but the devaluation of the franc soon brought money worries. His last years were a struggle against pain and worry.

Thus Baire was able to devote only a few periods, distributed over a dozen years, to mathematical research. In addition to the already mentioned works, of particular importance are “Sur les séries à termes continus et tous de même signe” and “Sur la non-applicabilité de deux continus à net n+p dimensions.”

Baire’s doctoral thesis solved the general problem of the characteristic property of limit functions of continuous functions, i.e., the pointwise discontinuity on any perfect aggregate. In order to imagine this characteristic, one needed very rare gifts of observation and analysis concerning the way in which the question of limits and continuity had been treated until then. In developing the concept of semicontinuity - to the right or to the left - Baire took a decisive step toward eliminating the suggestion of intuitive results from the definition of a function over a compact aggregate. But in order to obtain the best possible results, one needed a clear understanding of the importance of the concepts of the theory stemming from aggregates.

Until the arrival of Bourbaki, his success greatly influenced the orientation of the French school of mathematics. Baire’s work, held in high esteem by Émile Borel and Henri Lebesgue, exerted considerable influence in France and abroad while its author found himself incapable of continuing or finishing the task he had set himself.

There can be no doubt that the progress of modern mathematics soon made obsolete Baire’s work on the concept of limit and the consequences of its analysis. But this work, written in beautiful French, has an incomparable flavor and merits inclusion in an anthology of mathematical thought. It marks a turning point in the criticism of commonplace ideas. Moreover, the class of Baire’s functions, according to the definition adopted by Charles de la Vallée Poussin, remains unattainable as far as the evolution of modes of expression is concerned. This model of a brief and compact work is part of the history of the most profound mathematics.

He died on July 5, 1932, at the age of 58 in Chambéry, France.

Achievements

• Since the beginning of the nineteenth century, mathematicians had been interested in the development of the theory of functions of real variables only incidentally and in relation to complex variables. Baire’s work completely changed this situation by furnishing the framework for independent research and by defining the subject to be studied.
  
  In appreciation to his contribution to the field of mathematics, Baire received the ribbon of the Legion of Honor, and on 3 April 1922 was elected corresponding member of the Academy of Sciences. Among Baire’s other most important works are Théorie des nombres irrationels, des limites et de la continuité (Theory of Irrational Numbers, Limits, and Continuity) published in 1905 and both volumes of Leçons sur les théories générales de l’analyse (Lessons on the General Theory of Analysis) published in 1907–08.

Works

More photos

• book

  • Théorie des Nombres Irrationnels, des Limites et de la Continuité

    (Mathematics is the study of such problems as quantity, st...)

    1907
  • Sur Les Fonctions De Variables Réelles 1905
  • Collection de Monographies sur la Théorie des Fonctions; Leçons sur les Fonctions Discontinués
  • Théorie des Nombres Irrationnels, des Limites et de la Continuite
  • Lecons Sur Les Fonctions Discontinues 1908
  • Leçons sur les Théories Générales de l'Analyse 1905

Membership

René-Louis Baire was elected a corresponding member of the Academy of Sciences on April 3, 1922.

Personality

Baire was a quiet young man who kept to himself and was profoundly introspective.

Physical Characteristics: From time to time, Baire's health would prevent him from working or studying. Consequently, the bad spells became more frequent, immobilizing him for long periods of time. Over time, he had developed a kind of psychological disorder which made him unable to undertake work which required long periods of concentration. At times this would make his ability to research mathematics impossible. Between 1909 and 1914 this problem continually plagued him and his teaching duties became more and more difficult.

Connections

associate:

Henri Léon Lebesgue

Baire’s work, held in high esteem by Henri Lebesgue, exerted considerable influence in France and abroad while its author found himself incapable of continuing or finishing the task he had set himself.

References

• The Calculus Gallery: Masterpieces from Newton to Lebesgue More than three centuries after its creation, the calculus remains a dazzling intellectual achievement and the gateway into higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century and continuing to Henri Lebesgue at the dawn of the twentieth - mathematicians whose achievements are comparable
  2008
• Encyclopaedia Britannica
  1990
• Dictionary of Scientific Biography
  1970