Maurice Fréchet | Biography

Maurice Fréchet was one of the creators of the 20th-century discipline of topology, which deals with the properties of geometric configurations; his research added a new degree of abstractness to the mathematical advances of the previous generation. He profited from a rich mathematical environment during his studies and in turn passed on a wealth of ideas to his students over a long career. Some of the mathematicians who had learned their skills before Fréchet's work appeared could not help wondering whether there were advantages to the new degree of generality in his work. The answer lay in the fruitfulness of the methods of Fréchet for addressing problems whose solution included concrete problems of long standing.

René Maurice Fréchet was born on the 10th of September in 1878 in Maligny, a small town in provincial France, where his father Jacques directed an orphanage. Soon after Fréchet's birth the family moved to Paris, much to his advantage in terms of mathematical environment. His mother Zoé was responsible for a boardinghouse for foreigners, which early put Fréchet in contact with a cosmopolitan community. This may be reflected in his subsequent hospitality toward students and collaborators from all over the world. At his lycée (high school), he was singularly fortunate in learning mathematics from Jacques Hadamard , already a mathematician of distinction who would shortly thereafter provide a proof of a central result of number theory called the prime number theorem.

Fréchet's talents blossomed under Hadamard's encouragement and he was well prepared to enter the École Normale Supérieure, the great French scientific university, in 1900. Hadamard was not the only mathematical influence on the young Fréchet. After graduating from the École Normale, he began to work with mathematician Émile Borel, who was only seven years older than Fréchet but who had started his career so early that he may have seemed to belong to an earlier generation. Fréchet collaborated with Borel on the publication of a series of the latter's lectures and continued to be involved with the publishing of the so-called Borel collection for Gauthier-Villars. Even though Borel's role in the collection was primarily an editor's, he also wrote all the volumes to begin with, the first exception being one written by Fréchet. In turn, Fréchet undertook the editing of a series on general analysis published by Hermann (the other great mathematical publisher in Paris) and undertook the writing of several of the volumes as well.

Fréchet wrote his thesis under Hadamard, who had returned to Paris, and then followed Hadamard in teaching at the level of the lycée for a few years. His marriage in 1908 to Suzanne Carrive produced four children, whom he supported with professorships outside of Paris until 1928. He was officially connected with the University at Poitiers from 1910 to 1918, but World War I took him out of mathematics and into the less familiar surroundings of working as an interpreter with the British army, where his early exposure to different languages was of help. After his return from military service, he was head of the Institute of Mathematics at the University of Strasbourg, still a provincial appointment. It was not until 1928 that he was called to the University of Paris.

Proposes Revolutionary Variations of Topology Theory

One of the reasons for the delay in the recognition of Fréchet's work by the French academic establishment was its revolutionary character. The notions of set theory as introduced by the German mathematician Georg Cantor in the previous century were slowly winning converts, although there were differences of opinion about which axioms ought to be accepted. What Fréchet did in his thesis and in the most influential of his subsequent work was to bring the ideas of general set theory to bear on questions of the new discipline of topology, the generalization of geometry that had been given a good deal of prominence in French mathematics by the work of Jules Henri Poincaré . The questions that Poincaré had raised were new, but they were in the context of classical mathematics, centered on space with standard Cartesian coordinates (those points commonly expressed as located along x, y, and z axes), although perhaps in more than three dimensions.

This much of a revolution the mathematical community had come to accept, but Fréchet's thesis pushed the level of abstractness to new heights. Rather than looking just at sets of points in Cartesian space, he was prepared to handle sets of points in arbitrary spaces--so-called abstract spaces. The important tool that he used to handle such sets was a distance function. The ordinary distance function for sets of points with Cartesian coordinates (x, y) comes from the Pythagorean theorem and involves taking the square root of the sums of the squares of the differences in each coordinate. Since in abstract spaces there weren't necessarily any coordinates to assign to points, the distance function had to be more general and governed by some of the principles that applied to the Cartesian version.

The advantage of the new approach of Fréchet was that complicated algebraic expressions could be replaced by general considerations about distance. Spaces with a distance function were called metric spaces and proved to be the setting for expressing many of the results hitherto considered limited to spaces with real numbers as coordinates. Having once introduced these ideas into topology, Fréchet proceeded to look at calculus in metric spaces, an area that became known as functional analysis. Again, the basis for progress on long-standing problems was the avoidance of the complicated calculations that had bedeviled earlier work and the application of general notions from topology instead. Fréchet extended the notions of derivatives and integrals from standard calculus so that they could be used in the setting of a metric space; in addition, he introduced new types of functions called functionals, which took real numbers as values but could operate on the points of abstract spaces. Much of his work from his thesis onward was summarized in Les espaces abstraits, published in 1926.

Fréchet taught at the University of Paris until 1949, and a good deal of his time there was spent on questions of probability. Just as general questions about calculus could be asked in the setting of abstract metric spaces, so the techniques of probability could be moved there as well. The application of probability to continuous quantities, as opposed to discrete quantities that took only a finite number of values, had always been dependent on calculus, and Fréchet's results showed that the extension to the abstract setting could be fruitful as well. As with functional analysis in general, the more one could move away from messy computations, the more one could hope that the idea behind a proof could be visible.

Another possible reason for Fréchet's move into probability was the hope that a more concrete area would make the techniques of abstract spaces more palatable to the part of the mathematical community uneasy about getting too far from applications. If so, the efforts proved largely unavailing, at least in France, although the level of abstractness introduced by Fréchet was one of the inspirations for the Polish mathematical school between World Wars I and II. It is perhaps indicative of the relative opinions of his work that Fréchet was elected to the Polish Academy of Sciences in 1929 but not to the French Académie Royale des Sciences until 1956. He was recognized as a member of the Legion of Honor, and some accumulation of praise could hardly be avoided as he lived into his nineties. He died in Paris on the 4th of June in 1973, having earned belated recognition of his role in bringing mathematics into the twentieth century on the wings of abstractness.