Jacques Hadamard | Biography

Widely considered the preeminent French mathematician of the 20th century, Jacques Hadamard has made an impact on many fields of mathematics. Although an analyst and a student of theoretical calculus by training, he has influenced topology, number theory, and even psychology. His work on defining functions won him the Grand Prix of the Académie des Sciences early in his career, and his proof of the prime number theorem solidified his importance in the mathematical world. He wrote several textbooks on a variety of mathematical subjects, including one which explained a mathematician's thought processes. Hadamard was first and foremost a teacher, however, and he used his position to help both students and colleagues alike see the connections between seemingly unrelated fields.

Born in Versailles on December 8, 1865, Jacques-Salomon Hadamard was the son of two teachers. His mother, Claude-Marie Picard, taught piano, while his father, Amédée, taught Latin at a prominent Paris high school. In 1884, at the age of eighteen, Hadamard began studying at the École Normale Supérieure. His first teaching job was at a high school in Paris, the Lycée Buffon, in 1890. When he was not teaching, he worked on his doctoral dissertation, and the research he did during this period led to his first breakthrough in mathematics.

Hadamard's dissertation concerned determining the shape of a function and finding certain points on that function where division by zero was involved in the original equation. Such functions had previously been considered undefined and unsolvable, but Hadamard found a way to solve them using a set of fractions known as the Taylor series. Published in 1892, his work was so revolutionary that the French Académie des Sciences immediately awarded him its highest honor, the Grand Prix. This was also the year Hadamard married Louise-Anna Trenel, with whom he would have five children. In 1892 Hadamard also accepted a position as lecturer at the Faculté des Sciences of Bordeaux, where he continued his work. Although his accomplishments in defining functions had been important to the mathematical community at large, for Hadamard it was just another step toward a larger goal. He wanted to find a proof of the prime number theorem. For years, some of the world's best mathematicians had attempted to prove that the total number of primes could be defined and that individual primes could be determined by something other than the endless testing of possible factors. Many had discovered estimates and close guesses, but no one had achieved accurate results.

Hadamard used his work on the Taylor series as a guide, and he established that the number of primes below any given number could be determined by using complex numbers, also known as imaginary numbers. While his theory only works when the numbers used are sufficiently large, mathematicians generally only concern themselves with primes when such large numbers are involved. Later attempts to improve upon or generalize Hadamard's 1896 prime number theorem by such noted mathematicians as S. I. Ramanujan have failed.

Following publication of the proof of the prime number theorem, Hadamard left Bordeaux for a lectureship at the Sorbonne in Paris. A return to the intellectual center of Paris also meant greater involvement with the mathematical community, in which Hadamard had earned a high place. While many mathematicians were content to specialize in a small area of mathematics, Hadamard saw the importance of finding connections between the various fields. He was openly critical of mathematicians who limited their work to their immediate subject. In 1902 he argued, for example, that the definitions Vito Volterra had offered for the calculus terms continuity, derivative, and differential were inadequate because they could not be generalized to other fields, especially the relatively new area of topology. Instead of merely criticizing Volterra, however, Hadamard applied himself to generalizing analysis so it would be more applicable to other fields. His creation and definition of the term functional, first put forth in 1903, is one result of this generalization. Though Hadamard had used standard analysis to come up with functionals, the application of the idea to topology was important to establishing the validity of that field.

Hadamard's work forming connections between topology and analysis was interrupted in 1904 by a debate over mathematical logic which raged through the mathematical community. Ernst Zermelo, a German mathematician, had proposed that given an infinite number of sets, it would be possible to select exactly one, definable item from each set. This proposal was called the axiom of choice. Zermelo argued that it was obvious and thus needed no proof, but many of the most prominent mathematicians of the time, including Émile Borel, Jules Henri Poincaré, and Henri Lebesgue, disputed it. As Morris Kline describes the controversy in Mathematics: The Loss of Certainty: "The nub of the criticism was that, unless a definite law specified which element was chosen from each set, no real choice had been made, so the new set was not really formed." Yet the axiom of choice was necessary to establish sections of abstract algebra, topology, and standard analysis. Hadamard supported Zermelo. He rejected the idea that the item taken from the set could necessarily be defined, yet he felt that any theory which allowed mathematics to progress should be accepted, with or without formal proof.

In 1908, Hadamard spoke at the Fourth International Congress of Mathematicians in Rome, where he met the famous German topologist L. E. J. Brouwer. They began a correspondence relating to the mathematical ideas of their time, and the exchange of letters was crucial to Brouwer. The German mathematician used Hadamard's ideas as a springboard to some of his most important topological discoveries. In 1909, Hadamard left the Sorbonne for a more prestigious appointment as professor at the École Centrale des Arts et Manufactures. He would remain there, teaching concurrently at the Collège de France after 1920, until his retirement at the age of 71.

In 1912, Hadamard's friend and colleague Jules Henri Poincaré died. Poincaré, like Hadamard, had been involved in several different fields of mathematics, and his work had greatly influenced Hadamard's interest in generalization. Saddened by the loss of this great mathematician, Hadamard devoted a great deal of his research time after Poincaré's death to writing biographical works of his friend. Hadamard did his last piece of major research in the field of calculus in 1932, when he addressed a problem posed by the French mathematician Augustin-Louis Cauchy. But even after his retirement in 1937, Hadamard continued to ponder some of the questions that had concerned him throughout his career. The old controversy over the axiom of choice became the basis of a new book on the importance of accepting intuition for the sake of mathematical progress. He published The Psychology of Invention in the Mathematical Field in 1945, at the age of 80, and it was widely considered an innovative attempt at understanding how mathematicians come up with their ideas. Some of the work on this book was done in the United States, where he was a visiting professor at Columbia University in New York in 1941. Unlike many European mathematicians, however, Hadamard did not stay in America. He returned home to France, living out the rest of his life quietly. He died in Paris on October 17, 1963, at the age of 97.