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Hadamard, Jacques

Category: Scientist

 

Jacques Salomon Hadamard (8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.

The son of a teacher, Amédée Hadamard, of Jewish descent, Jacques Hadamard married Louise-Anna Trénel, also of Jewish descent, with whom he had three sons and two daughters.  Although he was not religious, he was spiritually oriented having a fascination with the source of inspiration.

He was a bright little boy.  He was placed first in both entrance examinations for the schools he attended and he obtained his doctorate in 1892.  In the same year, he was awarded the Grand Prix des Sciences Mathématiques for his essay on the Riemann zeta function.

 

In 1893, he proved his celebrated inequality on determinants, which led to the discovery of Hadamard matrices when equality holds.

In 1896, he proved the prime number theorem, using complex function theory.

He was awarded the Bordin Prize of the French Academy of Sciences for his work on geodesics in the differential geometry of surfaces and dynamical systems. He also studied geodesics on surfaces of negative curvature. For his cumulative work, he was awarded the Prix Poncelet in 1898.  Hadamard was elected to the French Academy of Sciences in 1916, in succession to Poincaré, whose complete works he helped edit.

 

In the 1920s, Hadamard concentrated his interests on the problems of mathematical physics, in particular partial differential equations, the calculus of variations and the foundations of functional analysis.

He introduced the idea of well-posed problem and the method of descent in the theory of partial differential equations. Later in his life he wrote on probability theory and mathematical education.

Hadamard stayed in France at the beginning of the Second World War and escaped to southern France in 1940. The Vichy government permitted him to leave for the United States in 1941 and he obtained a visiting position at Columbia University in New York. He moved to London in 1944 and returned to France when the war ended in 1945.

 

He died in Paris in 1963, aged ninety-seven.

Why is he on the site?  In his book Psychology of Invention in the Mathematical Field, Hadamard not only describes his own mathematical thinking as largely wordless, often accompanied by mental images that represent the entire solution to a problem – inner speech – but also surveyed 100 of the leading physicists of the day (approximately 1900), asking them how they did their work, discovering in the process, they used much the same approach. 

And it his explanation as to how he obtained inspiration, as well as his research on the experiences of the mathematicians/theoretical physicists Carl Friedrich Gauss, Hermann von Helmholtz, Henri Poincaré and others that has provided the site with a wealth of proof that much wisdom in mathematics and physics is discovery of an existing spiritual truth, not an invention. 

An Essay on the Psychology of Invention in the Mathematical Field – Jacques Hadamard
One cannot say that various kinds of invention proceed in exactly the same way …. There is, between the artistic domain and the scientific one, the difference that art enjoys a greater freedom, since the artist is governed only by his own fantasy [imagination], so that works of art are true inventions………
The scientist behaves quite otherwise and his work properly concerns discoveries.  As my master Hermite told me ‘We are rather servants than Masters in mathematics”.  Although the truth is not yet known to us, it pre-exists and inescapably imposes on us the path we must follow under penalty of going astray

 

Invention may spring from the discovery, but as Hadamard says, most mathematicians, chemists and physicists are simply like Columbus discovering America, the answer was always ‘there’ it just had to be ‘visited’ to be found. 

In his case and the cases of the 100 or so people he interviewed this visiting was via dreams and other forms of spiritual experience – often the solution was derived through Inner speech – the viewing of entire solutions with "sudden spontaneousness".

 In the observations from his book that follow, I have grouped all those which are his inspiration under the name 'Hadamard, Jacques'.  All those of other people are grouped under the heading 'Jacques Hadamard' with a reference to the person concerned in the heading.  The exception is where I have an entry for that person on the site as a source.  In that case the observation can be found under their name, but the description within the observation indicates that it was Hadamard who collected the observation.

 

Observations

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