Jacques Salomon Hadamard

Born: 8 Dec 1865 in Versailles, France
Died: 17 Oct 1963 in Paris, France

Jacques Hadamard's father, Amédée Hadamard, married Claire Marie Jeanne Picard on 6 June 1864. Amédée Hadamard, who was of a Jewish background, was a teacher who taught several subjects such as classics, grammar, history and geography while Jacques' mother taught piano giving private lessons in their home. At the time that Jacques was born Amédée was teaching at the Lycée Impérial in Versailles but the family moved to Paris when Jacques was three years old when his father took up a position at the Lycée Charlemagne.

This was an unfortunate time for a child to be growing up in Paris. The Franco-Prussian War which began on 19 July 1870 went badly for France and on 19 September 1870 the Prussians began a siege of Paris. This was a desperate time for the inhabitants of the town who killed their horses, cats and dogs for food. Hadamard's family, like many others, ate elephant meat to survive. Paris surrendered on 28 January 1871 and the Treaty of Frankfurt, signed on 10 May 1871, was a humiliation for France. Between the surrender and the signing of the treaty there was essentially a civil war in Paris and the Hadamards' house was burnt down.

The war was not the only cause of sadness for the Hadamards. Jacques' young sister Jeanne died in 1870 before the siege of Paris and another sister Suzanne, who was born in 1871, died in 1874.

Jacques began his schooling at the Lycée Charlemagne where his father taught. In his first few years at school he was good at all subjects except mathematics. He excelled in particular in Greek and Latin. He wrote in 1936:-

... in arithmetic, until the fifth grade, I was last or nearly last.

He was not accurate in this statement for although at first it is true he was weak in arithmetic, by the fifth class he was placed second in his class at the Lycée. By this time (1875) he was winning prizes in many subjects in the Concours Général, the national competition for school pupils. It was a good mathematics teacher who turned him towards mathematics and science, when he was in this fifth class.

In 1875 Hadamard's father, having acquired a poor reputation as a teacher, was transferred to the Lycée Louis-le-Grand and Jacques attended this school from 1876. In 1882 he graduated Bachelier ès lettres et ès sciences then, in the following year, he received his Baccalauréat ès sciences. He was awarded first prize in algebra and first prize in mechanics in the Concours Général of 1883.

In 1884 Hadamard took the entrance examinations for École Polytechnique and École Normale Supérieure; he was placed first in both examinations. He chose the École Normale Supérieure, where he soon made friends with his fellow students including Duhem and Painlevé. Among his teachers were Jules Tannery, Hermite, Darboux, Appell, Goursat and Emile Picard. Already at this stage he began to undertake research, investigating the problem of finding an estimate for the determinant generated by coefficients of a power series. He graduated from the École Normale Supérieure on 30 October 1888.

While undertaking research for his doctorate he worked as a school teacher. At first he was attached to the Lycée de Caen but without teaching duties. From June 1889 he taught at the Lycée Saint-Louis and then from September 1890 at the Lycée Buffon where he taught for three years. Although his research went extremely well, his teaching was less appreciated, probably because he demanded more of his pupils than their abilities allowed. His one great success was teaching Fréchet, and the two corresponded over a period of about nine years.

Hadamard obtained his doctorate in 1892 for a thesis on functions defined by Taylor series. This work on functions of a complex variable was one of the first to examine the general theory of analytic functions, in particular his thesis contained the first general work on singularities.

In the same year Hadamard received the Grand Prix des Sciences Mathématiques for his paper Determination of the number of primes less than a given number. The topic proposed for the prize, concerning filling gaps in Riemann's work on zeta functions, had been put forward by Hermite with his friend Stieltjes in mind. Stieltjes had claimed in 1885 to have proved the Riemann hypothesis but had never published his "proof" and, after the prize topic was announced in 1890, Stieltjes discovered a gap in his "proof" which he was unable to fill. He never submitted an entry for the prize but Hadamard, between the time his thesis was submitted and his oral examination, realised that his results could be applied to zeta functions. His paper on entire functions and zeta functions was awarded first prize.

The year 1892 was significant for Hadamard in addition to the academic achievements described above. In June of that year he married Louise-Anna Trénel who was, like Hadamard, of a Jewish background. They had known each other from childhood and shared a love of music. They moved to Bordeaux the following year when Hadamard was appointed as a lecturer at the University. If he had failed to come up to scratch as a teacher at the Lycée Buffon, this was far from the case now, for he impressed everyone with both his research and teaching skills. On 1 February 1896 he was appointed as Professor of Astronomy and Rational Mechanics at Bordeaux.

The four years which he spent in Bordeaux were not only busy ones for his family life, with two sons Pierre and Étienne being born during this time, but they were also extremely productive ones for Hadamard's research. He published 29 papers during these four years, but they are remarkable more for their depth and the range of the topics which they covered rather than their number. Perhaps his most important result proved during this time was the prime number theorem which he proved in 1896. This states that:-

The number of primes lte n tends to infinity as n/ln n.

This theorem was conjectured in the 18th century, but it was not proved until 1896, when Hadamard and (independently) Charles de la Vallée Poussin, used complex analysis. The proof had been outlined by Riemann in 1851, but the necessary tools had not been developed at that time. This problem was one of the major motivations for the development of complex analysis from 1851 to 1896 when Riemann's outlined proof was finally completed.

Solving this important open problem was not Hadamard's only remarkable contribution of 1896. In the same year he published a paper on properties of dynamic trajectories which won the Bordin Prize of the Academy of Sciences. The topic proposed for the prize had been one on geodesics and Hadamard's work in studying the trajectories of point masses on a surface led to certain non-linear differential equations whose solution also gave properties of geodesics. His work was therefore a major contribution to both geometry and to dynamics.

Another result which Hadamard published during his time in Bordeaux was his famous determinant inequality of 1893. Matrices whose determinants satisfied equality in the relation are today called Hadamard matrices and are important in the theory of integral equations, coding theory and other areas.

Hadamard first became involved in politics during his time in Bordeaux. Alfred Dreyfus, a relation of Hadamard's wife, came from Alsace. Born into a Jewish family, Dreyfus embarked on a military career. In 1894, when he was in the War Ministry, he was accused of selling military secrets to the Germans and he was sentenced to life imprisonment. Although his trial had been highly irregular, the anti-Semitic views of many people made the verdict popular. Forged documents and cover-ups soon showed that the legal process had been suspect. At first Hadamard, like many people, assumed that Dreyfus was guilty. However after moving to Paris in 1897 he began to discover how evidence against Dreyfus had been forged. He became a leading member of those trying to correct the injustice. Painlevé described a conversation he had with Hadamard on the matter in 1897 (see for example [6]):-

For almost an hour, [Hadamard] tried to convince me of Dreyfus's innocence, and at the end, faced with my disbelief, he did his best to make me understand the intrinsic value of his arguments and his complete lack of passion and sentimentality ... he based his belief in his innocence on the facts.

In 1898 the novelist Emile Zola wrote an open letter accusing the army of covering up its mistaken conviction of Dreyfus. The case split France into two opposing camps leading to issues far beyond the guilt or innocence of Dreyfus. There were demands to bring Zola to justice, anti-Semitic riots broke out, and there was a petition demanding that Dreyfus be retried. Zola was sentenced to a year in prison and fined 3,000 francs. By 1899 there had been a confession to the forgeries, followed by a suicide, and Dreyfus was retried, again found guilty, but pardoned. Hadamard took an active part in clearing Dreyfus's name which finally happened on 22 July 1906, when Dreyfus was reinstated and decorated with the Legion of Honour. Laurent Schwartz wrote (see for example [27]):-

It is the Dreyfus Affair which was in the sense of defence of justice the great affair of [Hadamard's] life. From the moment when he understood the enormity of the injustice perpetrated against a man in the name of reason of state, and the consequences which anti-Semitism could have, he devoted himself passionately to the review of the trial. This affair marked his life.

Long before the Dreyfus Affair had ended Hadamard had, as we have indicated, moved from Bordeaux to Paris. In 1897 he resigned his chair in Bordeaux to take up lesser posts, one in the Faculty of Science of the Sorbonne and one at the Collège de France. Soon after arriving in Paris in October 1897, he published the first volume of Leçons de Géométrie Elémentaire . This volume on two dimensional geometry appeared in 1898, and was followed by a second volume on three dimensional geometry in 1901. This work had been requested by Darboux and it went on to have a major influence on the teaching of mathematics in France.

Hadamard received the Prix Poncelet in 1898 for his research achievements over the preceding ten years. His research turned more towards mathematical physics from the time he took up the posts in Paris, yet he always argued strongly that he was a mathematician, not a physicist. He believed that it was the rigour of his work which made it mathematics. In particular he worked on the partial differential equations of mathematical physics producing results of outstanding importance. His famous 1898 work on geodesics on surfaces of negative curvature laid the foundations of symbolic dynamics. Among the topics he considered were elasticity, geometrical optics, hydrodynamics and boundary value problems. He introduced the concept of a well-posed initial value and boundary value problem.

During Hadamard's first five years in Paris another three children were born, first another son Mathieu and then two daughters Cécile and Jacqueline. He continued to receive prizes for his research and he was further honoured in 1906 with election as President of the French Mathematical Society. In 1909 he was appointed to the chair of mechanics at the Collège de France. In the following year he published Leçons sur le calcul des variations which helped lay the foundations of functional analysis (he introduced the word functional). Then in 1912 he was appointed as professor of analysis at the École Polytechnique where he succeed Jordan.

Poincaré had strongly supported Hadamard for this chair but, within a few months, he died at the tragically young age of 58. Hadamard then undertook the hugely difficult task of surveying Poincaré's work and by the end of the summer of 1912 he had produced two major articles. As Paul Lévy wrote:-

One had to be Hadamard to dare to undertake the exposition of all of [Poincaré's] immense work which dealt with so many different areas, and to finish it in one summer.

Near the end of 1912 Hadamard was elected to the Academy of Sciences to succeed Poincaré. He wrote later that his many years of "pure joy", beginning from the time of his marriage, came to an end in 1916. It was World War I which led to a great tragedy for Hadamard with his two older sons being killed in action. Both were killed at Verdun and Hadamard was lecturing in Rome when Pierre was killed. He left before receiving the news which he did not discover until arriving back in Paris despite the best efforts of Fano, Volterra's wife and others to get news to him. Étienne was killed near Verdun about two months later.

Hadamard knew only one way to push the pain of these tragedies away enough to allow him to keep going, and that was to throw himself even more vigorously into mathematics. He was appointed to Appell's chair of analysis at the École Centrale in 1920 but retained his positions in the École Polytechnique and the Collège de France. During the years between his appointment and 1933 he travelled widely visiting the United States twice, Spain, Czechoslovakia, Italy, Switzerland, Brazil, Argentina, and Egypt.

He continued to produce books and papers of the highest quality, publishing perhaps his most famous text Lectures on Cauchy's problem in linear partial differential equations in 1922. The book was based on a lecture course he had given at Yale University in the United States. He also took up new topics, writing several papers on probability theory, in particular on Markov chains. He also published many articles on mathematical education and education in general.

Between the wars Hadamard's politics moved towards the left, mainly in response to the Nazis rise to power in 1933. After the start of World War II, when France fell in 1940, Hadamard and his family escaped to the United States where he was appointed to a visiting position at Columbia University. However, he failed to find a permanent post in America and in 1944 received the terrible news that his third son Mathieu had been killed in the war. Hadamard left the United States soon after and spent a year in England before returning to Paris as soon as was possible after the end of the war.

After the War he became an active peace campaigner and it required the strong support of mathematicians in the USA to allow him to enter the country for the International Congress in Cambridge, Massachusetts in 1950. He was made honorary president of the Congress.

One further tragedy was to hit Hadamard before his death. In 1962, when he was 96 years old, his grandson Étienne was killed in a mountaineering accident. This seemed to finally kill Hadamard's spirit and he did not leave his house after this, almost waiting to die.

The following tribute is paid in [3]:-

Hadamard was the doyen not only of the Academy of Sciences, to which he was elected in December 1912, in the seat left vacant by the death of Henri Poincaré, but of the entire Institut de France. In December last year, he was formally presented with a gold medal specially struck to commemorate the fiftieth anniversary of his election to the academy, and tributes were paid to him by scientists all over the world.

There is no way that an article of this length can even indicate the range of Hadamard's mathematical contributions. As well as about 300 scientific papers and books Hadamard also wrote books for a wider audience. His book The psychology of invention in the mathematical field (1945) is a wonderful work about mathematics. We should also, however, indicate Hadamard's style of teaching. At the conference to celebrate the centenary of his birth, one of his students said he had been taught by:-

... a teacher who was active, alive, whose reasoning combined exactness and dynamism. Thus the lecture became a struggle and an adventure. Without rigour suffering, the importance of intuition was restored to us, and the better students were delighted.

Laurent Schwartz spoke about Hadamard at this ceremony held to celebrate the centenary of Hadamard's birth:-

I believe that he had a fantastic influence on his time, and that all living analysts were shaped by him, directly or indirectly.

Article by: J J O'Connor and E F Robertson

October 2003

MacTutor History of Mathematics