Weil on Analogy in Mathematics

Notes by Steven H. Cullinane, March 5, 2004

From John Baez,
http://math.ucr.edu/home/baez/week198.html:

"Analogies are incredibly important in mathematics.
Some can be made completely precise and their content fully
captured by a theorem, but the 'deep' ones, the truly fruitful ones,
are precisely those that resist complete encapsulation and
only yield their secrets a bit at a time.  Corfield* quotes Andre Weil, who describes the phenomenon as only a Frenchman could -
even in translation, this sounds like something straight out of Proust:

     As every mathematician knows,
     nothing is more fruitful than these obscure analogies,
     these indistinct reflections of one theory into another,
     these furtive caresses,
     these inexplicable disagreements;
     also nothing gives the researcher greater pleasure.

I actually doubt that every mathematician gets so turned on by
analogies, but many of the 'architects' of mathematics do, and
Weil was one." 

* David Corfield, Towards a Philosophy of Real Mathematics, Cambridge U. Press, Cambridge, 2003.

From an American Scientist book review:

"Nothing is more fruitful—all mathematicians know it—
than those obscure analogies,
those disturbing reflections of one theory on another;
those furtive caresses,
those inexplicable discords;
nothing also gives more pleasure to the researcher.
The day comes when this illusion dissolves:
the presentiment turns into certainty;
the yoked theories reveal their common source
before disappearing.
As the Gita teaches, one achieves
knowledge and indifference at the same time."

From a French Web site:

«Rien  n’est  plus  fécond,  tous  les  mathématiciens  le  savent,
que  ces obscures analogies,
ces troubles reflets d’une théorie à une autre,
ces furtives caresses,
ces brouilleries inexplicables:
rien aussi ne donne plus de plaisir au chercheur.
Un  jour  vient  où  l’illusion  se  dissipe;
le pressentiment se change en certitude [ ... ].
Heureusement pour les chercheurs,
à mesure que les brouillards se dissipent sur un point,
c’est pour se reformer sur un autre.»

André Weil, «De la métaphysique aux mathématiques»,
Œuvres, t. II, p. 408.

Page created March 5, 2004