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Jacques Hadamard
Jacques Salomon Hadamard
Born	December 8, 1865(1865-12-08) Versailles, France
Died	October 17, 1963 (aged 97) Paris, France
Nationality	French
Field	Mathematician
Institutions	University of Bordeaux Sorbonne Collège de France
Alma mater	École Normale Supérieure
Academic advisor	C. Émile Picard Jules Tannery
Notable students	Maurice René Fréchet Paul Lévy Szolem Mandelbrojt André Weil Xinmou Wu
Known for	Hadamard product Proof of prime number theorem
Notable prizes	Grand Prix des Sciences Mathématiques (1892) Prix Poncelet (1898)

Jacques Salomon Hadamard (December 8, 1865 – October 17, 1963) was a French born mathematician best known for his proof of the prime number theorem in 1896.

Contents 1 Biography 2 On creativity 3 Writings 4 See also 5 External links 6 Further reading 7 References 8 Notes

Hadamard studied at the École Normale Supérieure under the direction of Charles Émile Picard. After the Dreyfus affair, which involved him personally (Dreyfus was his brother-in-law), Hadamard, Jewish himself, became politically active and became a staunch supporter of Jewish causes.^[1]

He introduced the idea of well-posed problem in the theory of partial differential equations. He also gave his name to the Hadamard inequality on volumes, and the Hadamard matrix, on which the Hadamard transform is based. The Hadamard gate in quantum computing uses this matrix.

His students included Maurice Fréchet, Paul Lévy, Szolem Mandelbrojt and André Weil.

In his book Psychology of Invention in the Mathematical Field, Hadamard uses introspection to describe mathematical thought processes. In sharp contrast to authors who identify language and cognition, he describes his own mathematical thinking as largely wordless, often accompanied by mental images that represent the entire solution to a problem. He surveyed 100 of the leading physicists of the day (approximately 1900), asking them how they did their work. Many of the responses mirrored his; some reported seeing mathematical concepts as colors.

Hadamard described the experiences of the mathematicians/theoretical physicists Carl Friedrich Gauss, Hermann von Helmholtz, Henri Poincaré and others as viewing entire solutions with “sudden spontaneousness.”^[2] The same has been reported in literature by many others, such as Denis Brian,^[3] G. H. Hardy,^[4], B. L. van der Waerden,^[5], Harold Ruegg.^[6], Friedrich Kekulé (dreamed of benzene ring) and Tesla.

Hadamard described the process as having four steps of the five-step Graham Wallas creative process model, with the first three also having been put forth by Helmholtz:^[7]

• Preparation
• Incubation
• Illumination
• Verification

Marie-Louise von Franz, a colleague of the eminent psychiatrist Carl Jung, noted that in these unconscious scientific discoveries the “always recurring and important factor … is the simultaneity with which the complete solution is intuitively perceived and which can be checked later by discursive reasoning.” She attributes the solution presented “as an archetypal pattern or image.”^[8] As cited by von Franz,^[9] according to Jung: “Archetypes … manifest themselves only through their ability to organize images and ideas, and this is always an unconscious process which cannot be detected until afterwards.”^[10]

• Jacques Hadamard The Psychology of Invention in the Mathematical Field (Dover, 1954) ISBN 0-486-20107-4 ( Princeton University Press, 1945)
• Jacques Hadamard The Mathematician's Mind: The Psychology of Invention in the Mathematical Field (Princeton, 1996) ISBN 0-691-02931-8

• Cartan–Hadamard theorem
• Cauchy-Hadamard theorem
• Hadamard product
• Hadamard's dynamical system
• Hadamard three-circle theorem
• Hadamard manifold
• Hadamard space
• Ostrowski-Hadamard gap theorem

• O'Connor, John J; Edmund F. Robertson "Jacques Hadamard". MacTutor History of Mathematics archive.  
• Jacques Hadamard at the Mathematics Genealogy Project

• Life and Work of Jacques Hadamard, Vladimir Maz'ya[1] & T. O. Shaposhnikova, American Mathematical Society, February 1998, hardcover, 574 pages, ISBN 0-8218-0841-9

• Denis Brian Einstein: A Life (John Wiley and Sons, 1996) ISBN 0-471-11459-6
• Jacques Hadamard The Psychology of Invention in the Mathematical Field (Dover, 1954) ISBN 0-486-20107-4
• C. G. Jung The Collected Works of C. G. Jung. Volume 8. The Structure and Dynamics of the Psyche. (Princeton, 1981) ISBN 0-691-09774-7
• Robert Kanigel The Man Who Knew Infinity: A Life of the Genius Ramanujan (Washington Square Press, 1992) ISBN 0-671-75061-5
• Marie-Louise von Franz, Psyche and Matter (Shambhala, 1992) ISBN 0-87773-902-1

1. ^ The Psychology of Invention in the Mathematical Field
2. ^ Hadamard, 1954, pp. 13-16.
3. ^ Einstein, after years of fruitless calculations, suddenly had the solution of the general theory of relativity revealed in a dream “like a giant die making an indelible impress, a huge map of the universe outlined itself in one clear vision.” See Brian, 1996, p. 159.
4. ^ G. H. Hardy cited how the mathematician Srinivasa Ramanujan had “moments of sudden illumination.” See Kanigel, 1992, pp. 285-286.
5. ^ von Franz, 1992, p. 297 and 314. Cited work: B. L. van der Waerden, Einfall und Überlegung: Drei kleine Beiträge zur Psychologie des mathematischen Denkens (Gasel & Stuttgart, 1954).
6. ^ von Franz, 1992, p. 297 and 314. Cited work: Harold Ruegg, Imagination: An Inquiry into the Sources and Conditions That Stimulate Creativity (New York: Harper, 1954)
7. ^ Hadamard, 1954, p. 56.
8. ^ von Franz, 1992, pp. 297-298.
9. ^ von Franz, 1992 297-298 and 314.
10. ^ Jung, 1981, paragraph 440, p. 231.

Persondata
NAME	Hadamard, Jacques
ALTERNATIVE NAMES
SHORT DESCRIPTION	French Mathematician
DATE OF BIRTH	December 8, 1865
PLACE OF BIRTH	Versailles, France
DATE OF DEATH	October 17, 1963
PLACE OF DEATH	Paris, France