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\noindent{\bf D'ALEMBERT}\\
b. 17 November 1717 - d. 29 October 1783
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\noindent{\bf Summary}. D'Alembert's doubts concerning the calculus of probabilities
are more pertinent than has long been believed, and were the starting point for
the work of Condorcet and Laplace.
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D'Alembert was the illegitimate son of
the officer Destouches-Canon, and Madame de Tencin. He was
abandoned on the steps of the church of Saint-Jean-le-Rond in Paris, and put
in the care of a glazier's wife. Initially called Jean Le Rond, he
later gave himself the name of
J.B. Daremberg and then Dalembert or D'Alembert. In fact his father
did not totally abandon him, since he provided for his
education. The young man was a pupil at the Coll\`ege des
Quatre-Nations, where he studied law and medicine, but he rapidly
began to concentrate on mathematics. He became a member of the
French Royal Academy of Sciences in 1741, as ``adjunct astronomer"
and soon became known through his {\it Trait\'e de dynamique}
(1743) which based mechanics on a new principle. In the few years
until the beginning of the 1750's he published some papers of
remarkable originality on fluids, vibrating strings, the precession
of equinoxes, and lunar theory. We owe to the young D'Alembert
the first equations with partial derivatives, the fundamental
theorem of algebra, an essential contribution to the equations for
perfect fluids, and the statement of a paradox in hydrodynamics.
The style of the young author exhibited a number of characteristics
which remained with him for life: the desire to construct theories
starting only from a minimum of principles, a demanding nature
which admitted nothing without an investigation of principles, a
critical study of the link between mathematics and the conditions
for its application, and a method of mathematical exposition which
was often difficult to follow.
D'Alembert was also given to polemics, inflexible in his
priorities and relentless against his adversaries; among these were
Daniel Bernoulli (q.v.), Clairaut and Euler, who gave as good as they
got.
In 1751 the first volume of the {\it l'Encyclop\'edie} appeared,
with the preface ``Discours pr\'eliminaire" written by D'Alembert.
He had been made responsible for the mathematical section of this
enterprise, and it was in this capacity that he wrote his first
contributions to the calculus of probabilities. Following some
indirectly critical articles on this topic a scandal broke out about
the article on heads or tails, (croix ou pile) in volume IV of the
{\it Encyclop\'edie}. D'Alembert contested principles accepted by
everyone: he cast doubt, for example, on the fact that the
probability of obtaining heads in two trials was 3/4, and suggested
that it might be 2/3, arguing that there were in fact only three
possible events (the coin is not thrown again if heads occurs at the first
trial), namely heads, tails followed by heads, and tails
followed by tails, the first two being favourable, while the third
was not. D'Alembert was thus provocatively initiating a prickly
debate on the question of equally likely events; this debate became
even more prickly when the calculus of probabilities was applied to
events in real life rather than simple games of chance. In his
contribution to the famous St. Petersburg problem, he disputed not
only Daniel Bernoulli's solution, in which the fate of a player is
evaluated by a criterion other than the mathematical expectation,
but also other solutions involving limiting the number of trials in
the game, or neglecting smaller probabilities, etc. In other words,
D'Alembert doubted not so much the abstract rules of probability,
but rather the relevance of these rules to moral, that is human and
social, behaviour. He would not tolerate the separation of the
calculus of probabilities from its use in concrete examples. He
refused to assume without severe appraisal all the simplifying
hypotheses which one usually makes without much thought: the assumption that
events are equally likely, their independence, the additivity of
probabilities,
and the linearity of
expectation. He also found it difficult to bear not being able to take the
effect of time
into account in his calculations. In the calculus of probabilities, as also in
mechanics and in the rest of science, he contested the principle of
proportionality of causes to effects.
D'Alembert developed his doubts and objections in various works
until his death, for example in his {\it Opuscules
math\'ematiques}, and in books accessible to a larger public like
his {\it M\'elanges de philosophie}. An ideal case for the
concrete discussion of the principles of the calculus of
probabilities was provided by the controversy on inoculation which
raged in the 1750's and 1760's. A supporter of inoculation, he did
not accept the fact that one should measure its advantage simply by
evaluating the increase in life expectancy. He was not happy with
crude summary measures; he queried those indicators which we call the mean and
the median, indirectly requiring a study of the variability. He
wanted made explicit the manner of dealing, within the same
calculation, with events separated by time; and insisted that one
should distinguish the viewpoint of
the individual from that of the State. In all these disputes
Daniel Bernoulli was D'Alembert's main adversary. Bernoulli
was a very skilled analyst, inclined to choose effective simplifications;
he constructed striking and
seductive theories, but refused to be dragged into discussions on the
principles involved.
D'Alembert is not known for any theorems in the calculus of
probabilities, nor what in the twentieth century is referred to as
mathematical statistics. The essence of his contributions consists
of criticisms, but it would be wrong to believe that he
played a purely destructive role. On the one hand, his
articles and papers contain propositions, such as, for example, a
mathematical theory of inoculation (11th paper in the {\it Opuscules}), and a
comparative criterion for distribution functions (23rd paper).
On the other hand, D'Alembert's doubts stimulated the researches of
young French mathematicians in the second half of the 18th century,
in particular Condorcet (q.v) and Laplace (q.v.). This is clear from their
writings: each in his own way, took some
difficult questions posed by D'Alembert as a starting point to
construct their own theories. Laplace's famous ``M\'emoire sur les
probabilit\'es des causes par les
\'ev\'enements", published in 1774, which may in some sense be
thought of as the starting point of modern mathematical statistics,
is the most obvious example.
D'Alembert's ideas were partially eclipsed in the general area of
mathematics by Euler and Lagrange,`` geometers" who were perhaps more
able, but also gifted with a far more readable style, which
appeared less archaic to the mathematicians of the 19th and 20th
centuries. In the area of the calculus of probabilities, D'Alembert was
considered very eccentric. Lacroix,
who was among his more polite critics wrote in his treatise on
probabilities (1816) ``it can happen to the most eminent that they lose
their way even in a very simple subject". However, this long-unquestioned
judgement has been revised in the decade 1970-80; one
may note today that D'Alembert had pinpointed some of the very real
difficulties in the calculus of probabilities, particularly in its
relation to reality. Looking back, one can see that he did not at
the time have the methods to overcome a number of these problems.
D'Alembert took part in many other activities. A close acquaintance
of Voltaire and Turgot, like Pascal (q.v.) before him an enemy of the
Jesuits, he was one of the
better known personalities in the {\it philosophes} movement.
D'Alembert was first a member, and later permanent secretary (1772)
of the Acad\'emie fran\c caise, and a friend of Frederick
II of Prussia. In this role he was a prime mover in the
foundation of the Academy of Berlin and until his death in 1783,
played a very important role in the organization of intellectual
life in Europe.
His {\it Oeuvres Compl\`etes} are currently in the
course of publication. His name is known even to students of
elementary mathematics through
``D'Alembert's test" for convergence.
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\begin{thebibliography}{3}
\bibitem{1} D'Alembert (1754). Croix ou pile. {\it Encyclop\'edie},
Vol.IV. Briasson, Paris. pp512-513.
\bibitem{2} D'Alembert (1761-1780). {\it Opuscules math\'ematiques},
8 volumes, Briasson/Jombert, Paris. [ In particular m\'emoires Nos.: 10, 11,
23, 27,
36, 52.]
\bibitem{3} Brian, E.(1996). L'Objet du doute: les articles de D'Alembert sur
l'analyse des
hasards dans les quatre premiers tomes de l'Encyclop\'edie.{\it Recherches sur
Diderot et sur l'Encyclop\'edie,} No. 21, 163-178.
\bibitem{4} Daston, L. (1988). {\it Classical Probability in the
Enlightenment}, Princeton University Press, Princeton.
\bibitem{5} Hankins, T.L. (1970). {\it Jean d'Alembert: Science and
the Enlightenment}, Clarendon Press, Oxford.
\bibitem{6} Paty, M. (1988). D'Alembert et les probabilit\'es. In
R. Rashed (Ed.), {\it Sciences \'a l'\'epoque de la R\'evolution
fran\c{c}aise. Recherches historiques}, A. Blanchard, Paris, pp.
203-265.
\bibitem{7} Pearson, K. (1978). {\it The History of Statistics in the 17th
and 18th Centuries,} Griffin, London, pp. 506-573.
\bibitem{8} {\it Dix-huiti\'eme si\'ecle} (1984). Special Issue on
"D'Alembert", No.16, PUF, Paris.
\bibitem{9} {\it Jean D'Alembert, savant et philosophe, portrait
\`a plusieurs voix}, Ed. des Archives contemporaines, Paris, 1989.
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\hfill{P. Cr\'epel}
\end{thebibliography}
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