JACQUES HADAMARD

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

Contents

Biography

On creativity

Writings

See also

External links

Further reading

References

Notes

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.

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.”^[1] The same has been reported in literature by many others, such as Denis Brian,^[2] G. H. Hardy,^[3], B. L. van der Waerden,^[4], Harold Ruegg.^[5], 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:^[6]

★ 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.”^[7] As cited by von Franz,^[8] 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.”^[9]

Writings

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

See also

★ Cartan–Hadamard theorem

★ Hadamard product

★ Hadamard's dynamical system

★ Hadamard three-circle theorem

★ Hadamard manifold

★ Hadamard space

★ Ostrowski-Hadamard gap theorem

External links

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★

Further reading

★ ''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

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

Notes

1. Hadamard, 1954, pp. 13-16.
2. 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.
3. G. H. Hardy cited how the mathematician Srinivasa Ramanujan had “moments of sudden illumination.” See Kanigel, 1992, pp. 285-286.
4. 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).
5. 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)
6. Hadamard, 1954, p. 56.
7. von Franz, 1992, pp. 297-298.
8. von Franz, 1992 297-298 and 314.
9. Jung, 1981, paragraph 440, p. 231.