Emile Borel (1871-1956) was born in Saint-Affrique (South of France), a very
small city near Roquefort (this delicious but harsh cheese the Americans
like so much that they are ready to pay it twice the price), surrounded by
semi-desertic countryside (le Larzac), with very impressive Templar castles
and memories from the Crusades.
As detailed in Pierre Guiraldenq's "Emile Borel" (published just a propos,
the author being himself a Saint-Affrican and organiser of the Colloque),
the mathematician was also a prominent politician, in local and national
affairs, mayor of Saint-Affrique, deputy and a short time Minister of the
French Marine (where he used to answer to whom seemed doubting of the
capacity of a mathematician to occupy such a charge: *as a mathematician, I
know why the inner tube of the Marine's cannons are rayed opposite way to
the Earth Army's ones*).
Important lectures were made, which placed or discussed the centrality of
the position of Borel in the mathematics of the beginning of this century,
as well as in those of to-day. His famous "covering theorem" (known as
Borel's, or Borel-Lebesgue's, or Dirichlet-Heine-Borel's... the list may not
be closed), and his measure theory were meticulously examined in historical
and epistemological points of view (even if the word "epistemological" was
not pronounced: it is not fashionable any more).
A. Micali told us about a memorable visit Borel made to Brazil. In a country
largely superstructured by Auguste Comte's official and hidden cults, one
reproved the study of sciences which contradicted those of the past
century. By the efforts of Brazilian scientists like Manuel A. Costa or
Henrique Morize the positivistic dogma was somewhat shaded, and the country
could open to new ideas. The coup d'envoi for the new science took place in
Northern Brazil in May 1919 for a solar total eclipse during which
Eddington made his famous observations. And in September 1922 Borel could
speak of the conception of a curved universe in the theory of Relativity.
His visit had an influence we can retrace in the works of a mathematician
like Theodoro A. Ramos (1895-1935).
Pierre Dugac was in charge of giving the last echo of the Bourbakian School,
which hatred Borel but waited until very recently (Laurent Schartz 1997'
book of memories) to say it openly. And, as Bourbakian times are now over,
one assists to a return to Borel's "practical" point of view. It is an
undefinite task to trace what is exactly the meaning of this "practical",
but one can easily understand it somehow opposes to the pure metaphysic
(Cantor), "abstract" (Frechet, Hausdorff, Banach) or "structural"
(Bourbaki).
Pierre Crepel treated of probabilities. Borel's essential concept of
"denumerable probabilities" (1909), and his measure theory applied
immediately to probability theory (1905), gave a long-waited understanding of
the "continuous", and were the foundation on which the axiomatic
Kolmogorovian could spread. Borel, by his teaching at the Sorbonne and at
the Institut Poincare (that he created), by his own monographies as well as
by editing other author's in a "Treaty on probabilities and their
applications", animated the French school of probabilities from 1905 until
his death. Insisting on Markov, Crepel situated the important works of Ville
and Paul Levy. He made (on reading notes written by Bernard Bru) a
passionating examination of Poincare's role in Affaire Dreyfus (remember
that Dreyfus was Hadamard's cousin). Poincare was named an expert by the
Court to discuss previous experts' conclusions applying probability theory
to decide wether the "bordereau" was forged by the accused. The way Poincare
destroys, by proper means of probability theory, the others' so-called
"proofs" of Dreyfus's culpability, is a marvelous morceau de bravoure.
Helene Gisbert spoke about Borel's role in the development of functions'
theory. Not forgetting du Bois-Reymond and his deja use of the diagonal
method, she recalled the work of Cantor and how the French were suspicious
about, despite the attempts of Hadamard, Tannery, Painleve, Darboux, Jordan
and Baire-1st-period to introduce the Cantorian set theory into that of
functions. Then Borel came, with his thesis (Sur quelques points de la
theorie des fonctions, June 1894), preceeded by a short announcement at the
French Academy (same title, February 1894), and, above all, his *Lessons sur
la theorie des fonctions* (1898), where he says he wants to "expose some
recent researches of an increasing importance, of which is the set theory",
but announces that "speaking of sets", he has "tried to never loose the
applications' view". Baire-2nd-period and Lebesgue heard the message and
made their well known and crucial contributions. Again here we see Borel
initiating a large process by his own works, his teaching at the College de
France and at the Ecole normale superieure, and his editing of other's works
in his famous "Collection de monographies sur la theorie des fonctions".
At this point the second anti-Borel bullet was shot, by Andre Warusfel,
whose expose was on the different "states" of Borel's covering theorem. He
insisted on saying that if Borel is the genius, Lebesgue is the great
mathematician!
It was let to F. Pham to tell us how Borel was actual. With pedagogy and
humour, Pham treated of divergent series. Poincare's "series asymptotiques"
were too qualitative for Borel, who introduced his "procede de sommation",
which is, through many avatars, very much on use to-day.
venedem@wanadoo.fr
Note: to get more information about some books referred to here, you are
kindly requested to go to my site
http://perso.wanadoo.fr/alta.mathematica/