Le plus court chemin entre deux vérités dans
le domaine réel passe par le domaine complexe.
The quote was used in this form (in French) as a epigraph in
"Unitary representations and Osterwalder-Schrader duality"
by Palle E. T. Jorgensen and Gestur Ólafsson, in
The Mathematical Legacy of Harish-Chandra: A Celebration of
Representation Theory and Harmonic Analysis (R. Doran and V. Varadarajan,
eds.), Proc. Sympos. Pure Math., vol. 68, American Mathematical Society,
Providence, R.I., 2000, pp. 333--401.
No source was cited.
It appears to be a free translation of an epigraph found (in English, again without citing a source)
in an article on Jacques Hadamard by Jean-Pierre Kahane in
The Mathematical Intelligencer, volume 13, number 1, winter 1991, page 26:
"The shortest path between two truths in the real domain passes through the complex domain."
A longer and more nuanced formulation appears (in English)
in Hadamard's An Essay on the Psychology of Invention in the
Mathematical Field (Princeton U. Press, 1945; Dover, 1954; Princeton U. Press, as The Mathematician's Mind, 1996),
page 123: "It has been written that the shortest and best way between two truths of the real domain often passes
through the imaginary one."
Note the use of "way" rather than "path", "of" rather than "in", "imaginary" rather than "complex",
the added characterization "and best", the qualification "often", and the mysterious introduction "It has been written...".
This work of Jacques Hadamard's was translated into French
by his daughter Jacqueline Hadamard, revised and augmented by the author, and published as
Essai sur la psychologie de l'invention dans le domaine mathématique
by A. Blanchard, Paris, 1959.
The wording in the French edition (page 114) is
"On a pu écrire depuis que la voie la plus courte et la meilleure entre deux vérités
du domaine réel passe souvent par le domaine imaginaire."
We have not found where this "has been written" (except by Hadamard in the work cited). Send clues to email@example.com.
[30 Jan 2006:]
Where this had "been written" (thanks to A. I. Shtern for passing this information on to us in response to the appeal for clues posted here)
was in Paul Painlevé's Analyse des travaux scientifiques (Gauthier-Villars, 1900;
reprinted in Librairie Scientifique et Technique, Albert Blanchard, Paris, 1967, pp. 1-2; reproduced in Oeuvres de Paul Painlevé, Éditions du CNRS, Paris, 1972-1975,
vol. 1, pp. 72-73):
"Le développement naturel de cette étude conduisit bientôt les géomètres à embrasser dans leurs recherches les valeurs imaginaires de la variable aussi bien que les valeurs réelles. La théorie de la série de Taylor, celle des fonctions elliptiques, la vaste doctrine de Cauchy firent éclater la fécondité de cette généralisation. Il apparut que, entre deux vérités du domaine réel, le chemin le plus facile et le plus court passe bien souvent par le domaine complexe."
[The natural development of this work soon led the geometers in their studies
to embrace imaginary as well as real values of the variable.
The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis,
caused a burst of productivity derived from this generalization.
It came to appear that, between two truths of the real domain,
the easiest and shortest path quite often passes through the complex domain.]
To a mathematical analyst, the "complex" domain, where
"real" and "imaginary" numbers are combined in a single field, is the fruitful conception in
the particular studies alluded to in this quote from Painlevé.
Already in Painlevé's time, in the late nineteenth and early twentieth centuries,
"imaginary" numbers were firmly established as a part of mathematical reality.
By adjusting the phrase to refer directly to the contrast of "imaginary" to "real",
Hadamard brought the image before a different audience, who have remembered him for it.
This "très jolie phrase" (as
describes it) resonates, according to the common non-mathematical senses of these two words
"real" and "imaginary", with whatever it is or may be that our minds are now
reaching to encompass.
Quotes selected by Palle Jorgensen