Jacques Hadamard

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

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

[edit] Biography

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), he became politically active and became a staunch supporter of Jewish causes.

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.

[edit] On Creativity

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.”<ref> Hadamard, 1954, pp. 13-16.</ref> The same has been reported in literature by many others, such as Denis Brian,<ref> 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.</ref> G. H. Hardy,<ref>G. H. Hardy cited how the mathematician Srinivasa Ramanujan had “moments of sudden illumination.” See Kanigel, 1992, pp. 285-286.</ref>, B. L. van der Waerden,<ref>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).</ref> and Harold Ruegg.<ref>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).</ref>

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:<ref> Hadamard, 1954, p. 56.</ref>

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.”<ref> von Franz, 1992, pp. 297-298.</ref> As cited by von Franz,<ref>von Franz, 1992 297-298 and 314.</ref> 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.”<ref>Jung, 1981, paragraph 440, p. 231.</ref>

[edit] Writings

Jacques Hadamard The Psychology of Invention in the Mathematical Field (Dover, 1954) ISBN 0-486-20107-4 [Copyright 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]

[edit] See also

[edit] External links

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

[edit] Further reading

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

[edit] References

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