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\noindent{\bf Marquis de CONDORCET}\\
b. 17 September 1743 - d. 29 March 1794\\
\noindent{\bf Summary.} Condorcet applied mathematics in precocious and philosophically
argued fashion to humanistic and social problems. His probabilistic work is full
of interesting ideas on mathematical expectation, inverse probability and taking
into account evolution over time, amongst others.
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Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet was born
in Ribemont, Picardy, in the north of France. His father, a
military man from a Dauphin\'e family, died when Condorcet was only
a few weeks old. The boy was raised by his mother's family in Picardy in a
bourgeois legal setting accustomed to
economic and political reponsibilities. Following his studies with the
Jesuits at Rheims, then at the Coll\`ege de Navarre in Paris, he
devoted himself to pure mathematics, with the direct goal of obtaining
general results in the integral calculus.
His early work, begun towards the end of the the 1750's in collaboration with
his friend and first teacher, the Abb\'e Girault de Keroudou, were both
praised and criticized by Fontaine and D'Alembert (q.v.) who
found fault with the often confused and excessively general style.
Nevertheless, his
research on the integral calculus resulted in his election to
the Academy of Sciences at the age of 26. It culminated in the
early 1780's in a treatise (regrettably unpublished) containing in
particular a theorem on the integration of differential equations
in terms of a finite number of explicit functions, 40 years before Liouville.
Starting in 1767-1770 Condorcet wrote several papers on law,
political arithmetic and the calculus of probabilities; but these
were dated and published only in 1994. After seriously
considering d'Alembert's
doubts on the foundations and relevance of the calculus
of probabilities, and stimulated by Beccaria, the young mathematician
discovered the likelihood principle before Laplace. This principle in its
so called Bayesian interpretation allows us to go
from effects to causes within a probabilistic framework. He
did the same with what is now referred to as the rule of sucession of
Bayes-Laplace, in which if an event occurs $m$ times in $n$ trials,
its probability may be estimated as $(m + 1)/(m+n+2)$. Note that
Bayes' (q.v.) works became known on the Continent only towards 1780.
Condorcet's first researches, concerned with regular arrangements and the
theory of mathematical expectation, were thus already
noteworthy for his
wish to make the calculus of probabilities useful in the political
and moral sciences.
Following his active participation in Turgot's ministry
(1774-1776), Condorcet, already an adjunct secretary of the Academy
of Sciences, took up the position of permanent secretary until the
most violent episodes of the French Revolution. He continued his
research both in pure mathematics and in the calculus of probabilities.
It was mostly after 1783 that he developed in detail and published
his mature work on probabilities, including
their ``inverse" problems, as they are now referred to in
mathematical statistics and the
philosophical and practical conditions of their use. His {\it Essai} of
1785 contained a theory on the motive for belief, and the famous
paradox of votes, but mainly an attempt to prove by an example
(that on judgments) ``that the truths of the moral and polical
sciences are subject to the same certainty as those which form the
system of physical knowledge". This is true, he believed,
so long as one introduced a
quantitative basis for the different types of possible errors. In
particular, the simultaneous evaluation of the probabilities of
acquitting a guilty person and condemning an innocent one was at the
basis of later works of Laplace. J. Neyman (q.v.) found inspiration
for his theory of statistical tests with
errors of the first and second kinds in the latter.
At the same time, Condorcet published six papers on the calculus of
probabilities and articles in the {\it Encyclop\'edie
m\'ethodique}. These contained remarkable innovations: a theory of
mathematical expectation with a solution
of the St. Petersburg problem for a finite horizon, a theory of
complexity of random
sequences in regard to regular arrangements, a model for probabilistic
dependence, which is none other than what are now called
``Markov chains" and even ``semi-Markov processes", and solutions to
the problem of statistical estimation in the case of
time-dependent probabilities of events. One might say that this
foreshadows, perhaps clumsily and not very practicably, the concept of
time series. He also produced a definition of probabilities
starting from classes of events, and a theory of individual economic
choice in a setting of universal risk and competition.
Unfortunately, he was too daring in his writing, which suggested research
programmes rather than concrete theorems; and the exposition of his ideas was
so unclear and impractical that his original contributions were not
understood in his lifetime or even in the two following centuries.
Strongly involved in the encyclopedic movement and a friend of
D'Alembert, Turgot and Voltaire, Condorcet was the last of the
``encyclop\'edistes" and the only one who lived through the French
Revolution. He committed himself deeply to it, developing and
illustrating his scientific vision of politics, while having little
inclination to a romantic view of populist participation. This enabled him
to work out some
very fertile ideas on education, women, slavery,and the
rights of man, but he often had little influence on current events.
However exchange of ideas of political kind influenced developments in the
U.S. through Thomas Jefferson and James Madison.
During the Terror, Condorcet went into hiding, writing his famous
{\it Esquisse d'un Tableau historique des progr\`es de l'esprit humain}.
He was arrested on 27 March 1794, and was found dead in the
prison of Bourg-Egalit\'e (Bourg-la-Reine) two days later. It is not
known whether he committed suicide or died of apoplexy.
Much esteemed during his lifetime, Condorcet was considered a
mediocre mathematician during the next one and a half centuries. It
is only gradually, since 1950, thanks to G. Th. Guilbaud and D.
Black that his scientific work has been reappraised, first in
connection with the summing of preferences related to Arrow's
theorem, and later perceived as that of a ``mathematician-philosopher" who
critically studied the conditions necessary to found the human and
social sciences. His mathematical works, were in fact
only rediscovered in the 1980's.
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\begin{thebibliography}{3}
\bibitem{1} Brian, E. (1994). {\it La mesure de l'Etat}, Albin Michel,
Paris.
\bibitem{2} Condorcet, Le Marquis de (1785). {\it Essai sur l'application de
l'analyse
\`a la probabilit\'e des d\'ecisions rendues \`a la pluralit\'e des
voix}. Imprimerie Royale, Paris. [Reprinted by Chelsea, New York, 1972].
\bibitem{3} Condorcet (1994). {\it Arithm\'etique politique.
Textes rares ou in\'edits}. INED, Paris. [ Edited with commentary by
B. Bru and P. Cr\'epel.]
\bibitem{4} Granger, G.G. (1956). {\it La math\'ematique sociale du
marquis de Condorcet}, PUF, Paris.
\bibitem{5} Rashed, R.R. (1974). {\it Condorcet, math\'ematique et
soci\'et\'e}, Hermann, Paris.
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\hfill{P. Cr\'epel}
\end{thebibliography}
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