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\noindent{\bf \'Emile BOREL}\\
b. 7 January 1871 - d. 3 February 1956
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\noindent{\bf Summary}. After his celebrated contributions to pure mathematics, Borel
militated in support of the calculus of probabilities all his life. He gave
impetus to work on almost sure convergence.
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\'Emile Borel was born in Saint-Affrique, in the department of the
Aveyron (southern Massif Central), France, and died in Paris. He began his
studies at the private school run by his father, the pastor of
Saint-Affrique, and continued them at Montauban and then Paris, where
he made a great impression with his scholastic successes.
At the age of 18, he won the first prize for mathematics in the
`Concours general' (competitive examination) and was selected as the
top candidate at both the \'Ecole Polytechnique and the \'Ecole Normale
Sup\'erieure, like Darboux, Picard and Hadamard before him. On the
advice of Darboux, he chose to attend the second. This was considered
an ideal career path in the closed world of French Universities at the
end of the 19th century.
He attended Hermite's and Poincar\'e's courses at the Sorbonne; both
were to have a profound influence on his scientific life. It was due to
Hermite that he was initiated ``into the redoubtable and sacred mysteries
of the divinity of number''; this grew into a deep love of mathematics,
a youthful love which would never leave him. From Poincar\'e he
absorbed an openness of mind, a feeling for the unity and universality
of scientific culture at the highest level, the ``objective value of
Science'', even though Borel soon found himself opposed to
Poincar\'e's fundamental scepticism. Poincar\'e, knowing everything
about everything, knew better then most ``that we know nothing''.
Like many young mathematicians of his generation, Borel was deeply
influenced by both Cantorian ``romanticism'' and Weierstrassian ``discipline''.
In 1894 he defended a dazzling thesis in which he used the Cantorian
theory of (linear) sets which are now called after him, to break the confines
into which Weierstrass had enclosed analytic function theory, and showed the
potential richness of the concept of a set of points of measure zero
in the theory of functions. Borelian theory of functions, which he taught
the young `Normaliens' at the end of the 19th century, including Baire
and Lebesgue, was soon to infiltrate all real analysis.
In 1893, even before defending his thesis, he was appointed ma\^itre de
conf\'erences (lecturer) in the Faculty of Sciences at Lille, and in
1897 at the \'Ecole Normale Sup\'erieure, and then the University of
Paris in 1904. In 1909 he became a full professor appointed to a Chair
in the Theory of Functions specially created for him. At the same time,
he held the post of Assistant Director of the \'Ecole Normale
Sup\'erieure between 1910 and 1920. He remained a professor in the
Faculty of Sciences at the University of Paris until his retirement in
1941.
In April 1906, in {\it La Revue du Mois}, the monthly periodical which
he had just founded, Borel defined in a specific manner the new
scientific programme which he intended to follow thereafter, namely
``The practical worth of the calculus of probabilities''. Borel at
once disposed of one of the principal interpretations of the calculus
of probabilities, which the 19th century had accepted as a last resort
. This was the double meaning (double nature or double
origin) of probability, in that it could be ``subjective" or
``objective". Borel wrote ``There is no difference in nature
between objective and subjective probability, only a difference of degree.
A result in the calculus of probabilities deserves to be called
objective, when the probability is sufficiently large to be
practically equivalent to certainty. It matters little whether one is
predicting future events or reviewing past events; one may equally aver that a
probabilistic law will be, or has been, confirmed."
Borel's eventual position on the double nature of probability evolved
very slightly after the appearance of Keynes' (q.v.) (1921) treatise on
probabilities, in which Keynes adapted a subjectivist-logical
position. Borel distanced himself from it, judging that ``if Keynes is
correct from the theoretical point of view, Poincar\'e's opinion is
practically justified and exact.'' Borel's thinking was no doubt
fairly close to this austere dualism: theoretical
subjectivity-practical objectivity.
Simultaneously Borel published, one after the other, two mathematical
papers on the calculus of probabilities. The first, in 1905, explained
how Borel measure allowed one to extend and make more precise the
calculation of ``geometric probabilities''. The second was more
ambitious: it attempted to show that the calculus of probabilities,
while certainly playing an interesting role in human affairs, could
also explain the paradoxes of statistical physics. Thus, even adopting
Poincar\'e's viewpoint of science pursued for its own sake, the
calculus of probabilities could shed light on some of its darkest
shadows, namely those surrounding the kinetic theory of gases.
The only possibility open to a scholar was to reason probabilistically
and to seek, as in a game of heads or tails, those motions among all
the possible ones which were the most probable, particularly as these
rapidly become the only ones physically possible. As is well known,
Borel put forward the parable of the monkey typists to give an idea of
the extreme improbability of a small deviation from the most probable
configurations. It is easier for an army of monkeys typing on the
keyboards of typewriters, to compose all the books in all the
libraries of the world, than for a mixture of two initially separate
gases to return, by the random collision of their molecules, to their
initial states.
The {\it Revue du Mois} having proclaimed (the first in the world to do
so) both the practical and scientific values of the calculus of
probabilities, there remained for its director only to pursue the good
fight wherever he could. Borel did so initially at the Sorbonne where
he gave his first course on the calculus of probabilities in
1908--1909, the same year that he had been appointed as full professor
to the Chair of the Theory of Functions. In the same year 1909, Borel
took an active part in the Congress of the International Statistical
Institute (ISI) held in Paris, campaigning to regain a place for the
application of the calculus of probabilities to statistics. Until
then, the ISI had concentrated on official statistics.
In the same year, 1909, Borel published a classical work in
the mathematical theory of probability, on ``denumerable
probabilities". The so-called Borel-Cantelli Lemma originates in this work.
In this paper, for the first time, it was stated and
(almost) proved that the sequence of relative frequencies of tails, in an
indefinite sequence (``denumerable'') of games of heads and tails,
converges to the probability of a tail in a single trial, except for a
set of games of probability zero. This statement was to accelerate
appreciably the rather slow development of probability theory, and may without
exaggeration be considered the starting point of modern probability,
giving it a new dimension, embodied in
almost sure convergence, and Borel sets of probability measure zero.
Borel's article was also read by mathematicians,
partly because it contains statements on the theory of numbers which
were of immediate current mathematical interest, particularly the
first among them. This was that the set of real numbers on the unit
interval consists almost surely of normal numbers, the frequencies of
each of the digits in their expression (in any base whatever) being
asymptotically equal with probability 1, if the digits are drawn at
random one after the other. As Borel said, and Lebesgue further explicated to
him, the set of normal numbers has the Borel measure
1. This article contains a second application to the theory of
numbers, an account of the properties of the sequence of digits in the
continued fraction expansion of a number drawn at random from the
unit segment.
Did Borel train a French ``school'' of probabilists? Any reply to this
question must be qualified. As a humanistic scholar, Borel elevated
the calculus of probabilities to the highest level, possibly too high
a level; as a mathematician, he set it too low. Borel thought that
mathematicians should first train themselves in mathematics by
studying the great mathematical theories (for example, the theory of
functions). He thought that the calculus of probabilities was somewhat
remote from the ``great problems'', such as those of Hermite, Darboux
or Poincar\'e. What is more, and rather paradoxically, Borel was not
very interested in the axiomatisation of probabilities.
The First World War oriented Borel's
activities even more towards applications. France was in
danger, and the scientific community mobilised itself to defend and
protect it from ``German barbarism''. Painlev\'e, who had entered the
political arena after the Dreyfus affair, was elected deputy for
Paris in 1910, and was entrusted in 1915 with the direction of the large
ministry of Public Education, Arts and Inventions relevant to National
Defence. Within the latter, he created the Directorate for inventions
in the service of National Defence, and named Borel ``Head of the
Technical Office'' . When Painlev\'e
was Minister for War from March to September 1917, Borel was
Director of the Ministry's Technical Services, and when Painlev\'e
rose to the Presidency of the Council at the most tragic moment of the
war in the autumn of 1917, Borel was the general Secretary of the
government, entrusted with all missions, both possible and impossible,
including those connected with the scientific aspects of the war. He
later became deputy for the department of the Aveyron from 1924 to
1936, mayor of Saint-Affrique from 1929 to 1941 and from 1945 to 1947,
and Minister for the Navy in 1925.
Borel's political activities did not prevent him from devoting some
of his time to new ``flashes'' of inspiration in the theory of
probability, particularly on the theory of games (1920) or queues
(1942). It was also he, together with Lucien March, Director of the
Statistique G\'en\'erale de la France (French Statistical Bureau) and
Fernand Faure, holder of the Chair of Statistics in the Faculty of Law
of the University of Paris (the first such chair in France, founded in
1892), who was responsible for the creation of the Institut de
Statistique de l'Universit\'e de Paris, an institute serving the
four faculties of the University of Paris (Law, Literature, Medicine,
Sciences), which starting in 1922 was to train the competent
statisticians necessary for Borel's France.
At the same time, Borel launched a new series with Gauthier-Villars
(the publishers) of which he was the sole editor. This was the {\it
Trait\'e du Calcul des Probabilit\'es et de ses Applications}, which
consisted of 5 volumes and 18 fascicles published between 1925 and
1939, five of these being written by Borel himself. This was an
important work, which fell fairly rapidly into relative obscurity
after the war. The great treatises on probability published directly
after the war, namely those of Cram\'er (q.v.), Doob, Feller, Lo\`eve,
$\ldots$, resolutely adopted an expository style which set the pattern for
the mathematical works of the time and thereafter:
explicit axioms, precise enunciations, full proofs in a logical
order. Yet one knows very well since Cournot (q.v.), that this is neither the
rational nor the practical order, which was why Borel had rejected it.
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\begin{thebibliography}{3}
\bibitem{1} Borel, \'E. (1972). {\it Oeuvres d'\'Emile Borel}, 4 vol.,
CNRS, Paris.
\bibitem{2} Borel, \'E. (1909). Les probabilit\'es denombrables et
leurs applications arithm\'etiques, {\it Rendiconti del Circolo
Matematico di Palermo},
{\bf 27}, 1055 - 1079.
\bibitem{3} Borel, \'E. (1914). {\it Le Hasard}, Paris, Alcan. [Most
recent edition, Alcan, 1948.]
\bibitem{4} Fr\'echet, M. (1972). La vie et l'oeuvre d'\'Emile Borel,
in: Borel (1972), Vol. 1, pp. 5-116.
\bibitem{5} Knobloch, E. (1987). Emile Borel as a probabilist, in
L. Kr\"uger, L. Daston, M. Heidelberger (eds), {\it The Probabilistic
Revolution}, Vol. 1, pp. 215-233.
\bibitem{6} Lebesgue, H. (1991). Lettres \`a E. Borel, edited by B. Bru
and P. Dugac, {\it Cahiers du s\'eminaire d'histoire des
math\'ematiques}, Vol. 12, Institut Henri Poincar\'e, Paris.
\bibitem{7} Seneta, E. (1992). On the history of the strong law of large numbers
and Boole's inequality. {\it Historia Mathematica}, {\bf 19} 24-39.
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\hfill{Bernard Bru}
\end{thebibliography}
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