International Symposium
Bruno de Finetti Centenary Conference
Rome, November 15-17, 2006
November 15, 2006 (from 12 a.m. to 6 p.m.)
Library of Department of
Mathematics "G. Castelnuovo"
Università di Roma La Sapienza
piazzale Aldo Moro 5 - Roma
Exhibition of Historical books in Probability
Probability from Cardano to de Finetti |
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Ma sopra tutte le invenzioni stupende, qual eminenza di mente fu quella di colui che s'immaginò di trovar modo di comunicare i suoi piú reconditi pensieri a qualsivoglia altra persona, benché distante per lunghissimo intervallo di luogo e di tempo? parlare con quelli che son nell'Indie, parlare a quelli che non sono ancora nati né saranno se non di qua a mille e dieci mila anni? e con qual facilità? con i vari accozzamenti di venti caratteruzzi sopra una carta.
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The early
period of Probability
This
section of the exhibition contains some of
the most important contributions to probability from the very beginning
to the
19th century. The main source on
the development of probability in this period is Todhunter's fundamental
book A History of the Mathematical Theory
of Probability , 1865.
1. The very beginning - XVII
century
Girolamo Cardano
(1501-1576), Liber de
Ludo Aleae, in Opera Omnia, 1663
Galileo
Galilei (1564-1642), Discorso
sopra il gioco dei dadi, in
Opere, 1744
Blaise Pascal (1623-1662),
Lettres
à M. Fermat, in Oeuvres, 1779
Christiaan Huygens (1629-1695), De Ratiociniis in
Ludo Aleae, in Opera varia, 1724
The exhibition starts with Liber de Ludo Aleae (The book on Games of
chance) by Girolamo Cardano (1501-1576). The book was published in 1663, long
after the author's death, in the first volume of the Opera Omnia (Opera
moralia, pages 262-276).
“Cardano's treatise occupies fifteen
folio pages, each containing two columns. It may be considered as a
gambler's
manual. It contains much miscellaneous matter connected with gambling,
such as
description of games and an account of the precautions necessary to be
employed
in order to guard against adversaries disposed to cheat: the discussion
relating to chances forms but a small portion of the treatise. In
Chapter
XIII Cardano shows the numbers of cases which are favourable for each
throw
that can be made with two dice (the quantity of interest is the sum) . He
also treat correctly the case of three dice. Furthermore Cardano treats
the
"Ludo fritilli, (a not completely clear variant of the previous
case) giving
a corresponding list of the number of favourable cases.”
(taken freely from Todhunter's fundamental
book A History of … Probability ,1865)
The second author presented is Galileo Galilei (1564-1642) with Considerazioni
sopra il Giuoco dei Dadi (Considerations on the game
of dice). The
date of this piece is unknown; it first appeared in the
edition of Galileo’s work published at
“Galileo wrote it
answering the following question at the request of a friend: with three
dice
the number 9 and 10 can each be produced by six different
combinations,
and yet experience shows that the number 10 is oftener thrown than the
number
9. Galileo makes a careful analysis of all the cases which can occur,
and he
shows that out of the 216 possible cases 27 are favourable to the
appearance of
the number 10, and 25 are favourable to the appearance of number 9.”
(from Todhunter's fundamental
book A History of…Probability ,1865)
The
next author is Blaise Pascal (1623-1662). The letter to Fermat
presented in the exhibition was written on July 29, 1654. In the same
year,
1654, the Traité du triangle arithmetique ("Properties of the
Figurate Numbers"), had been printed, but it was not published until
1665,
after the author’s death [a table of the arithmetical triangle, from
the
Ouvres, 1779, is shown in the exhibition].
The year 1654 is usually considered as the birth’s year of the
Theory of
Probability.
“It appears that the Chevalier de
Méré proposed
certain questions to Pascal; and Pascal corresponded with Fermat on the
subject
of these questions. […] Three letters of Pascal to Fermat on this
subject,
which were all written in 1654, were published in the Varia Opera D.
Petri de
Fermat, 1679. […] Pascal’s first letter indicates that some previous
correspondence had occurred which we do not posses; the letter is dated
July
29, 1654. In
this letter Pascal discusses the problem to which […] he attached the
greatest importance. It is called in English the Problem of Points, and
it is
thus enunciated: two players want each a given number of points in
order to
win; if they separate without playing out the game, how should the
stakes be
divided between them?”
The problem proposed
by de Méré to Pascal was not new: it had been posed by Luca Pacioli
(1445-1515), but his solution was not satisfactory. After giving the
now famous
solution to this problem, Pascal treats briefly another problem
proposed by de
Méré, and which is now known after his name. In modern terms we can
rephrase
the problem as follows: is the
probability of getting at least one six by throwing four times a die
equal to
the probability of getting at least a double six by throwing
twenty-four times
a couple of dice?
The solutions given by
Pascal to these problems are based on combinatorial calculus, the
subject of
his famous Traité du triangle
arithmétique.
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Montmort (Essay d'analysis sur les jeux de
hazard, 1708) was the first to attach Pascal's name to the
triangle. Behind the Traité were many related investigations
spanning many centuries and many countries and the triangle has had
several names, e.g. in (from Earliest Known Uses of Some of the Words of Mathematics) |
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« Elle doit la naissance à deux Géomètres français du
dix-septième siècle, si féconde en grands hommes et en grandes
découvertes, et peut-être de tous les siècles celui qui fait le plus
d’honneur à l’esprit humain. Pascal et Fermat se proposèrent et
résolurent quelques problèmes sur les probabilités. » Laplace, page 3 of
Théorie Analytique des Probabilités (1812) |
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« Une problème relatif aux jeux
de hasard, propose a un austère janséniste par un homme de monde
(Chevalier de Méré a été l’origine du calcul des probabilités. » Poisson, Recherches.. page 1 |
Pascal did not
published anything on Probability during his life. The first to publish
a
treatise (in 12 pages) on this subject, in 1657, was Christiaan Huygens (1629-1695).
Huygens wrote the
treatise De
Ratiocinis ... in his mother language, and his instructor in
Mathematics,
van Schooten, translated it into Latin.
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The term Expectatio
appears in the Latin translation by van Schooten and expectation
appears in the English translation of the Latin,
while Huygens' Dutch text has no term that translates as "expectation;"
it speaks rather of the "value" of a game. (from Earliest Known Uses of Some of the Words of Mathematics) |
Huygens’ treatise
contains fourteen propositions and ends with five problems [a copy of
the
problems taken from the Opera varia, 1724 is shown in the exhibition] .
The
first proposition asserts "If I expect a or b, and have
an
equal chance of gaining either of them, my Expectation is worth (a
+ b)/2."
“The third proposition asserts that if a
player has p chances of gaining a and q chances of gaining a, his
expectation
is (pa+qb)/(p+q). […] Propositions
IV—VII discuss the Problem of Points. His method is similar to
Pascal’s. […]
The fourteen proposition consists in the following problem. A and B
play with
two dice on the condition that A is to have the stake if he throws six
before B
throws seven, and that B is to have the stake if he throws seven before
A
throws six; A is to begin, and they are to throw alternately; compare
the
chances of A and B.” The solution given by Huygens is based on the
third
proposition, and therefore Huygens has not to show that the game will
eventually end.
The
treatise ends with “the following five problems:
(1) A and B play with two dice on
this
condition, that A gains if he throws six, and B gains if he throws
seven. A
first has one throw, then B has two throws, then A has two throws, and
so on
until one or the other gains. Show that A’s chance is to B as 10355 to
12276.
(2) Three players A, B, C take
twelve balls,
eight of which are black and four white. They play on the following
condition;
they have to draw blindfold, and the first who draws a white ball wins.
A is to
have the first turn, B the next, C the next, then A again, and so on.
Determine
the chance of the players.
(3) There are forty cards forming
four sets
each of ten cards; A plays with B and undertakes in drawing four cards
to
obtain one of each set. Show that A’s chance is to B’s as 1000 to 8139.
(4) Twelve balls are taken, eight
of which are
black and four are white. A plays with B and undertakes in drawing
seven balls
blindfold to obtain three white balls. Compare the chance of A and B.
(5) A and B take each twelve
counters and play
with three dice on this condition, that if eleven is thrown A gives a
counter
to B; and if fourteen is thrown B gives a counter to A; and he wins the
game
who first obtains all the counters. Show that A’s chance is to B’s as
244140625
is to 282429536481.”
(from Todhunter's fundamental
book A History of … Probability ,1865)
2. The first books and the first
asymptotic results - XVIII century
Jakob Bernoulli (1654-1705), Ars Conjectandi, published posthumously, 1713
Pierre Rémond de Montmort (1678-1719), Essay d'analyse sur les
jeux de hazard,
II edition, 1713
Abraham de Moivre
(1667-1754) , The Doctrine of Chance, 1718
Gottfried Wilhelm Leibniz
(1646-1716), Lettres a M. Montmort, in Opera omnia, 1768
Johann Bernoulli (1667-1748), a letter to Leibniz
on De Moivre and Montmort,
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in Virorum celeberr. Got. Gul. Leibnitii et Johan. Bernoullii Commercium philosophicum et mathematicum, 1745 |
“The treatise by Huygens
continued to form the
best account of the subject until it was superseded by the more
elaborated
works of J. Bernoulli, Montmort, and de
Moivre.”
(from
Todhunter's
fundamental
book A History of … Probability ,1865)
The works of Jakob
Bernoulli, Montmort, and de Moivre are
to be considered the first books on Probability, and all of them
appeared at
the beginning of the 18th century, a little more than half a
century
after Huygens’ treatise. All the five problems proposed by Huygens were
considered by the above authors.
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It may be
interesting to note that “Jakob Bernouilli solves the second
problem of Huygens’ text [(2) Three
players A, B, C take twelve balls, eight of which are black and four
white. They play on the following condition; they have to draw
blindfold, and the first who draws a white ball wins. A is to have the
first turn, B the next, C the next, then A again, and so on. Determine
the chance of the players ] on three suppositions as to the meaning; first
he suppose that each ball is replaced after it is drawn; secondly he
suppose that there is only one set of twelve balls, and that the balls
are not replaced after being drawn; thirdly he supposes that each
player has his own set of twelve balls, and that the balls are not
replaced after being drawn. […] In
solving the fourth problem of Huygens’ text [
(4) Twelve balls are
taken, eight of which are black and four are white. A plays with B and
undertakes in drawing seven balls blindfold to obtain three white
balls. Compare the chance of A and B. ] de Moivre took the meaning to be that A is to
draw three white balls at least. Montmort had taken the meaning to be
that A is to draw exactly
three white balls. Johann Bernouilli in his letter to Montmort
(published in the second edition of Montmort’s book) took the meaning
to be that A is to draw three white balls at least. Jakob Bernouilli had considered both the meanings.” (from Todhunter's
fundamental book A History of … Probability ,1865) |
The most
important book of the exhibition is the Ars Conjectandi, by Jakob Bernoulli (1654-1705). Jakob
Bernoulli died in 1705 before finishing
his opera. The fourth part of the opera proposed to apply the Theory of
Probability to questions of interest in morals and economic sciences,
but was
left unfinished. The publisher had asked to finish it first to
the Jakob
Bernoulli's brother, Johann Bernoulli, and then to Nicolas Bernoulli, a
nephew
of Jakob and Johann. Nicolas did not consider himself adequate to the
task, and
the opera was published as the author had left if, eight years after
the
author's death, in 1713.
“The Ars
Conjectandi is divided into four parts. The first part consists of a
reprint of the
treatise of Huygens' work, completed with some annotations by Jakob
Bernoulli (the
first page of this part is shown in this exhibition). The
second part is devoted to the theory
of permutations and combinations. The third part consists of the
solution of
various problems relating to the games of chance. The fourth part, as
already
recalled, was unfinished; the most remarkable subject contained in it
is the
enunciation and investigation of we now call Bernoulli’s Theorem. It is
introduced in terms which show a high opinion of its importance:”
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« Hoc igitur est illud Problema, quod
evulgandum hoc loco proposui, postquam jam per vicennium pressi, et
cujus tum novitas, tum summa utilitas cum pari conjucta difficultate
omnibus reliquis hujus doctrinae capitibus pondus et pretium
superaddere potest. » Bernoulli Ars
conjectandi, page 227 |
(taken from Todhunter's
fundamental book A History of …Probability ,1865)
Abraham de Moivre (1667-1754) published The Doctrine of Chance: A method of
calculating the
probabilities of events in play in 1718, although a shorter version
in
Latin, De mensura sortis, had been
presented to the Royal Society and published in the Philosophical
Transactions in 1711. Francis Robartes, who later became the Earl
of
Radnor, suggested to de Moivre that he present a broader picture of the
principles of probability theory than those which had been presented by
Montmort
in Essay d'analyse sur les jeux de hazard (1708). Clearly this
work by Montmort
and that by Huygens which de Moivre had read, contained the problems
which de
Moivre attacked in his work and this led Montmort to enter into a
dispute with
de Moivre concerning originality and priority (the argument with
Montmort
appears to have been settled amicably).
The Doctrine of Chance appeared in expanded editions in 1718, 1738 and
1756. The first edition
was dedicated to
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« Two Events are independent, when they have no
connexion one with the other, and that the happening of one neither
forwards nor obstructs the happening of the other. Two events are
dependent, when they are so connected together as that the Probability
of either's happening is alter'd by the happening of the other. » The Doctrine
of Chance (1738 edition) |
Among other problems,
the "gamblers' ruin" problem appears as Problem LXV in the 1756
edition. In A history of the mathematical theory of probability
(1865), Todhunter
says that probability:
“... owes more to [de Moivre] than any other
mathematician,
with the single exception of
The
1756 edition of The Doctrine
of Chance contained what is probably de Moivre's most significant
contribution to this area, namely the approximation to the binomial
distribution by the normal distribution in the case of a large number
of
trials. De Moivre first published this result in a Latin pamphlet dated
In Miscellanea
Analytica (1730) appears Stirling's formula [ n!
» c nn+½ e-n ] (wrongly attributed to Stirling) which de Moivre used in 1733 to derive the
normal curve as an
approximation to the binomial. In the second edition of the book in
1738 de
Moivre gives credit to
« I desisted in proceeding
farther till my worthy and learned friend Mr James Stirling, who had applied after me to that inquiry, [discovered
the value of the c=(2π)½]. »
(taken from
the biography of de Moivre -The Mac
Tutor History of Mathematics archive)
Pierre Rémond de Montmort (1678-1719)
published the Essay d'analyse sur les jeux de hazard
for the first time in 1708,
three years after Bernoulli’s death, and about half a century after
Huygens’ De Ratiociniis … . “Montmort
divides his work into four parts:
A peculiarity of this
book is the following: “The name of
Montmort does not appear in the title page or in the work, except once
on page
338, where it is used with respect to a place.”
Montmort explains in
the preface that “his studies has arisen
from the request of some friends that he would determine the advantage
of the
banker at the game of Pharaon; he had
been led on to compose a work which might compensate for the loss of
Jakob Bernoulli’s
work”, due to his premature death. “Montmort
refers briefly to his predecessors, Huygens, Pascal, and Fermat. He
says that
his work is intended principally for mathematicians, and that he fully
explains
the various games which he discusses because, «pour l’ordinaire les
Sçavans ne
sont pas Joueurs ».”
“After the preface follows an
Advertissement, where he says that two
small treatises on the subject had appeared since his first edition;
namely a
thesis by Nicolas Bernoulli De arte Conjectandi in Jure, and a memoir
by de
Moivre, De mensura sortis .” The latter had been published in 1711,
and “Montmort seems to have been displeased with
the terms in which reference was made to
him by de Moivre”.
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«M. de Montmort fut vivement piqué
de cet ouvrage, qui lui parut avoir été entiérement fait sur le sien,
et d’apres le sien. » (Fontenelle, Eloge de M. de Montmort, 1719) |
Montmort died of small
pox at Paris in 1719, one year after the publication of the Doctrine
of Chance, by de Moivre. He was
31 years old.
(taken freely from Todhunter's fundamental
book A History of …Probability ,1865)
Gottfried Wilhelm Leibniz (1646-1716) did not contribute much to
probability, nevertheless he was very
interested in the subject. Among others, Leibniz corresponded with
Montmort,
with Jakob Bernoulli and his brother Johann Bernoulli. In the
exhibition one
can find a letter from Leibniz to Montmort, in which he records a very
favourable opinion of
the Essay d'analyse . He
says however
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« … J’aurois souhaité les loix
des Jeux un peu mieux decrites, et les termes expliqués en faveur des
étranges et de la postérité. » Leibniz, 1708
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In the exhibition one
can also find a letter from Johann Bernoulli (1667-1748) to Leibniz:
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« … Dominus Moyavraeus, insignis certe Geometra, qui
haud dubie adhuc haeret Londini,
luctans, ut audio, cum fame et miseri, quas ut depellant, victum
quotidianum ex informationibus
adolescentum petere cogitur. O duram sortem hominis! Et parum aptam ad
excitanda ingenia nobilia; quis non tandem succumberet sub tam iniquae fortunae vexationibus? vel quodnam
ingenium etiam fervidissimum non algeat
tandem? Miror
certe Moyvraeum tantis angustistiis pressum ea tamen adhuc praestare, quae praestat. » Johann Bernoulli, 26 April 1710 |
We end this part of this brief
note by recalling that in the preface
Montmort explains why he did not devote “a
part to the application of his subject to political, economical, and
moral
questions, in conformity with the known design of Jakob Bernoulli; […]
and he
thus states the conditions under which we may attempt the applications
with
advantage:”
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« 1° borner la question que
l’on se propose une petit nombre de suppositions, établies sur des
faits certains ; 2° faire abstraction de toutes les circonstances
ausquelles la liberté de l’homme, cet écueil perpétuel de nos
connaissances, pourroit avoir quelque part. » Montmort,
Essai d’Analyse … |
The unfinished design of Jakob Bernoulli to
apply
probability to political, economical, and moral questions started to be
implemented mainly after 1763, when Richard Price published
posthumously the
single paper by Thomas Bayes (1702-1761): An Essay towards solving a Problem in the
Doctrine of Chances
(Philosophical Transactions of the
Royal Society of London 53
(1763), 370-418). This work
may be considered as a starting
point for the application of probability to this kind of questions.
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The Essay considers the
problem: "Given the number of times in which an unknown event
has happened and failed: Required the chance of the probability
of its happening in a single trial lies somewhere between any two
degrees of probability that can be named." Or, in modern terms: given
the outcomes of a number of Bernoulli trials, find the posterior
distribution of the probability of a success. For the prior Bayes took
a uniform distribution on the unit interval. […] Inverse Probability was the term used in the 19th
and early 20th centuries for the probability found when
reasoning from effects to causes, direct probabilities being
used when reasoning from causes to effects. […] Today the terms Bayes's
Formula, Rule and Theorem are associated with a basic
theorem on conditional probability. "La règle de Bayes" appears with this meaning
in 1843 in A. A. Cournot's Exposition de la Théorie des Chances et des
Probabilités
(pp. 158-9). Cournot
says the rule is "attributed to Bayes." It is not in the Essay
but comes from (from Earliest Known Uses of Some of the Words of Mathematics) |
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In the exhibition, in a
small separate section devoted to a few modern translations of antique
books, one can find a German translation of Bayes contribution
published in the interesting Ostwald’s collection Klassiker der exakten
Wissenschaften (in an edition of 1908). |
3. Probability as a degree of
belief and applications - XVIII and XIX century
Gottfried Wilhelm Leibniz (1646-1716), Nouveaux Essaies sur l'entendement humain,
in Oeuvres
philosophiques latines et françoise, 1765
Marie Jean Antoine de Condorcet (1743-1794), Essai sur l'Application de
l'Analyse à la
Probabilité des Décisions rendues à la pluralité des voix., 1785
Pierre Simon de Laplace (1749–1827), Théorie Analiytique des
Probabilités, III edition, 1820, containing the Essai
philosophique sur les probabilités as an Introduction.
Siméon Denis Poisson (1781-1840),
Recherches
sur la probabilité des jugements en matière criminelle et en matière
civile, 1837
As
already recalled, G. W. Leibniz was very
interested in
the subject, and especially in its applications. This
is clearly explained in many of his
contributions. In the exhibition it is shown the Nouveaux
Essais sur
l'entendement humain ("New Essays on Human
Understanding")
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Nouveaux Essais sur
l'entendement humain was a chapter-by-chapter rebuttal by
Gottfried Leibniz of John Locke's major work, An Essay Concerning Human Understanding. It is one of only two
full-length works by Leibniz (the other being the Theodicy). Like many
philosophical works of the time, it is written in dialogue form. It was
originally written in 1704, but Leibniz refrained from publishing it at
that time in deference to Locke's death that same year, and it did not
see print until 1764, at which time much of Leibniz's philosophy was
out of favour. (from Wikipedia Nouveaux
Essais sur l'entendement
humain) |
In the exhibition it
is shown the beginning of Chapter XVI-Livre IV. The external margins
contain
the following indications
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« De differens degrés de
probabilité en Jurisprudence, en Medicine , dans les
Mathematiques » [sic] Leibniz , Nouveaux Essais
sur l'entendement humain,
1704 (1765) |
It may be interesting
to observe that in Nouveaux Essais ...,
the title of Livre IV is De la Connaissance, and that (in this
Livre) the titles of Chapters XIV, XV, and XVI are Du
Jugement , De
la Probabilité,
and De degrés
d’Assentiment,
respectively.
The book
Essai
sur l'Application de l'Analyse à la Probabilité des Décisions ..., by Marie Jean
Antoine de Condorcet
(1743-1794) was published in 1785, a few years before the French
Revolution, and less than ten years afterwards Condorcet perished a
victim of
the Revolution. The book is divided into
five parts; the first and second part are devoted to examine the
probability
that a “correct” decision is taken under different conditions
(Hypotheses). The
conditions concern the number of voters, the majority in order to take
a
decision, and the probability to take the correct decision (vérite)
or the wrong one (erreur).
The third part is devoted to the inverse problem. As an
example we quote the first of the thirteen preliminary problems in this
part:
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Soient deux évènements seul
possibles A et N, dont on ignore la probabilité, et qu’on sache
seulement que A est arrivé m fois, et N, n fois. On suppose que l’un
des deux évènements arrivés, et on demande la probabilité que c’est
l’évènement A, ou que c’est l’évènement N, dans l’hypothèse que la
probabilité de chacun de deux évènements est constamment la même. Condorcet, Essai sur l'Application de l'Analyse à la Probabilité des
Décisions..,1785 |
With respect to these
preliminary problems Condorcet makes the following historical remark
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L’idée de chercher la probabilité
des évènements futurs d’après des évènements passés, paroît s’être
présentée a Jaques Bernoulli e à Moivre, mais ils n’ont donné dans
leurs ouvrage aucune méthode pour y parvenir. M.rs Bayes et Price en ont donné une
dans les Transaction philosophiques, années 1764 et 1765, et M. de la
Place est le premier qui ait traité cette question d’une manière
analytique. Condorcet, Essai sur l'Application de l'Analyse à la Probabilité des
Décisions…,1785 page LXXXIII |
The fourth part is
devoted to examine the modifications which the results of the preceding
part of
his book require, before they can be applied to practice. For instance
we
cannot in practice assume that all voters are of equally skill and
honesty. […]
Condorcet asserts that the fifth part is devoted to some
examples. But it would be rather more correct to
describe this part as furnishing some additions to the preceding
investigations
than as giving examples of them.
(from Todhunter's fundamental
book A History of …Probability ,1865)
Pierre Simon de Laplace (1749–1827) published the Théorie Analiytique des Probabilités in 1812. The
third edition, 1820, (shown in the exhibition) contains the Essai philosophique sur les
probabilités as an Introduction. His main
contributions concern generating functions, the normal approximation,
and the
development of the now called Bayes’ rule.
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The first edition of Later editions of the Théorie Analytique
des Probabilités also contains supplements which consider
applications of probability to: errors in observations; the
determination of the masses of Jupiter, Saturn and Uranus;
triangulation methods in surveying; and problems of geodesy in
particular the determination of the meridian of (from
Laplace’s
biography -The Mac Tutor History of Mathematics archive) |
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In the 1730s De Moivre had found
the normal approximation to the binomial. Then in the early 19th
century Laplace, Poisson, Cauchy and others worked on normal
approximations in connection with the theory of least squares. (from
Earliest Known Uses of Some of the Words of Mathematics) |
In Recherches sur la probabilité
des jugements en matière criminelle et en matière civile, 1837, Siméon Denis Poisson
(1781-1840) generalized Bernoulli’s Theorem (to the case of
trials with
unequal probabilities) and introduced the expression "law of large
numbers" .
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In Recherches sur la probabilité des
jugements en matière criminelle et matière civile, an important
work on probability published in 1837, the Poisson distribution first
appears. The
Poisson distribution describes the probability that a random event will
occur in a time or space interval under the conditions that the
probability of the event occurring is very small, but the number of
trials is very large so that the event actually occurs a few times. He
also introduced the expression "law of large numbers". Although we now
rate this work as of great importance, it found little favour at the
time, the exception being in (from
Poisson’s
biography -The Mac Tutor History of Mathematics archive) |
A new era began with
Pafnuty
Lvovich Chebyshev
(1821-1894). The collection of his works Oeuvres , 1899-1907,
[shown in the exhibition] was edited by his two students M. A. Markov and N.
Sonin. The
collection contains some papers devoted to probability and in
particular the two
papers (1867, 1887) where Chebyshev gave the proof of the
inequality which has taken its name.
|
The inequality in the discrete case had been
already proved by L. J. Bienaymé in Considérations à l'appui de la découverte de
Laplace ... Comptes Rendus de l'Académie des Sciences, 37, (1853), 309-324. In the later literature Bienaymé's
contribution was often overlooked, thus A. A. Markov refers to "die
Ungleichheit von Tschebyscheff." in Wahrscheinlichkeitsrechnung (1912, p. 56). (from
Earliest Known Uses of Some of the Words of Mathematics) |
In
the second paper (1887) Chebyshev extends the result to the continuos case
|
… and gives the basis for applying the theory
of probability to statistical data, generalising the central limit
theorem of de Moivre and Laplace. Of this Kolmogorov wrote The principal meaning of Chebyshev's work is
that through it he always aspired to estimate exactly in the form of
inequalities absolutely valid under any number of tests the possible
deviations from limit regularities. Further, Chebyshev was the first to
estimate clearly and make use of such notions as "random quantity" and
its "expectation (mean) value". (from
Chebyshev’s
biography -The Mac Tutor History of Mathematics archive) |
The birth of modern Probability
This
section is divided into two parts.
4a. The first part starts with some
fundamental books on probability, published between the end of the 19th
century
and the beginning of the 20th century, such as
Calcul de Probabilités (1888), by J. Bertrand
Calcul de Probabilités (II edition 1912), by H. Poincaré [the I edition appeared in
1896]
Wahrscheinlichkeitsrechnung
und ihre Anwendung ...
( II edition in two
volumes 1908-10), by E. Czuber (1851-1925) [the I edition in
one volume appeared in 1902]
Wahrscheinlichkeits-Rechnung (1912, German
translation from Russian), by A. Markov
The books by J. Bertrand, H. Poincaré, E.
Czuber, and A. Markov are listed in
Calcolo delle
Probabilità, (I edition 1919 in one volume, II
edition 1925 in two volumes) by G. Castelnuovo (1865-1952),
as
the most important books the author, Guido Castelnuovo, consulted while
writing
his own book on probability. This list appears just after the
preface of
the II edition. In the revised reprint of the II edition of 1933,
Castelnuovo
mentions some other books. Most of the books in Castelnuovo’s list are
presented in this Section.
|
For many years Castelnuovo's book was the
(almost) unique book on probability, written in Italian. The book had a
fundamental role for Bruno de Finetti, who started to study probability
by reading this book and the book by Czuber. [see the text of his last lesson,
Scientia, 1976] [see also D.Cifarelli and E. Regazzini de Finetti's Contribution to
Probability and Statistics (in
JSTOR)] |
Castelnuovo
quotes also the book by L. Bachelier, Calcul des probabilités (1912), but he states that the book
by Bachelier has been written using an approach different from that of
his own
book: "Notevole per elevatezza di mezzi matematici impiegati, ma
redatto secondo un indirizzo diverso da quello del mio Trattato". In
the list of the most important books consulted by Calstelnuovo there is
also
the book Eléments de la théorie des probabilités, by
Émile
Borel. This book does not appear in the exibition, while the
exibition
will show some of the volumes of the collection
Traité du
Calcul des Probabilités et de ses Applications (1925), edited
E. Borel
and in particular
Volume1 I. Principes et
formules classiques du calcul des probabilités, by E.Borel
(1871-1956), Leçons professées à la Faculté
des Sciences de Paris. Redigées par R.Lagrange. 159 p.
Volume3 I. II. Les principes de la Théorie des probabilités, by Maurice Fréchet (1878-1973).
In the revised reprint of 1933, Castelnuovo
also quotes the following books, published after the second
edition of
1925
Calcul de Probabilités(1925), by Paul Lévy (1886-1971)
Wahrscheinlichkeitsrechnung und
ihre Anwendung ...(1931),
by Richard
Von Mises (1883-1953)
|
At that time the collection Traité
du Calcul des Probabilités et de ses Applications, edited
E. Borel, was not yet complete.
Furthermore the second list contained also the books by A.Fischer, The mathematical Theory of Probabilities (1925) and by
J.L.Coolidge, An Introduction to mathematical Probability (1925),
not presented in the exhibition. |
4.b The second part of this section
contains some works, which played a
fundamental
role in the birth of modern probability. In particular we present here
two
papers of the authors that are considered as the fathers of what is
called Brownian
(or Wiener) process, L. Bachelier (1870-1946) and A. Einstein
(1879-1955) [the
paper by Wiener “Differential spaces”(1923) is not presented here].
Furthermore
there is a paper by Percy John
Daniell (1889-1946), which is a
fundamendal link
between the Theory of Probability and Integration.
|
We
quote here only the first one to be published: Théorie
de la spéculation (1900), by L.Bachelier [Annales
Scientifiques de l'École Normale Supérieure Sér. 3, 17 (1900), p. 21-86] |
The rest of this part concerns de Finetti and
Kolmogorov,
and presents the papers on stochastic processes with independent
increments
[now known as Lévy’s processes], that these authors published in The
Rendiconti
of the Accademia delle Scienze in the years 1929, 1930, 1931, and 1932.
[These
papers were presented to the Accademia by Guido Castelnuovo]
|
We
quote here only the paper by Bruno de Finetti (1906-1985) Le funzioni caratteristiche di legge
istantanea. Atti della Reale
Accademia Nazionale dei Lincei. Rendiconti Classe di Scienze Fisiche, Matematiche
e Naturali, (6), 12 (1930),
pp. 278-282 and
the paper by Andrei Nikolaevich Kolmogorov (1903-1987) Sulla forma
generale di un processo stocastico omogeneo (Un problema di Bruno de
Finetti). Atti della Reale
Accademia Nazionale dei Lincei. Rendiconti Classe di Scienze Fisiche, Matematiche
e Naturali, (6), 15 (1932),
pp. 805-808.
|
This
part ends with
Grundbegriffe der
Wahrscheinlichkeitsrechnung (1933),
by A.N. Kolmogorov
and
with
Bruno de Finetti’s
"La prévision: ses lois logiques, ses
sources subjectives," Annales de
l'Institute Henri Poincaré, 7, (1937) 1-68.
These
two works are two cornerstones
in the modern theory of probability: the one by Kolmogorov in the
axiomatization of probability, and the one by de Finetti in the
subjective
interpretation of probability.
|
Interest in EXCHANGEABLE
random variables dates from the 1920s with Bruno de Finetti (1906 - 1985) the most
influential contributor. However "exchangeable" emerged only slowly as
the standard term. De Finetti used "équivalent" in his most widely read
work, "La prévision: ses lois logiques, ses sources
subjectives,"
Annales de l'Institute Henri Poincaré, 7, (1937)
1-68. L. J. Savage The Foundations of Statistics (1954) used
the term "symmetric." Pólya suggested the term "échangeable" and it
appeared in a 1943 book by Fréchét. De Finetti took up the term and
"exchangeable" was adopted in the 1964 English translation of his 1937
work. David (2001) cites M. Loève's Probability Theory (1955)
for the first occurrence of "exchangeable" in the English literature. [This entry was
contributed by John Aldrich, based on J. von Plato Creating Modern
Probability (1994).] COHERENT in subjective probability
theory. The term is
derived from the "cohérence" of B. de Finetti's "La prévision: ses lois logiques, ses sources
subjectives,"
Annales de l'Institute Henri Poincaré, 7, (1937)
1-68. The English
term is found in the mid 1950s, most conspicuously in Abner Shimony
"Coherence and the Axioms of Confirmation," Journal of Symbolic Logic,
20, (1955), 1-28. The term
"consistency" was used in F. P. Ramsey's treatment of subjective
probability, "Truth and Probability" (1926) (published in The
Foundations of Mathematics and other Logical Essays (1931)): the
calculus of probabilities can be "interpreted as a consistent calculus
of partial belief." (Based on a note to the translation of de
Finetti (1937) in H. E. Kyburg Jr. & H. E. Smokler (eds) Studies
in Subjective Probability (1964)) (from
Earliest Known Uses of Some of the Words of Mathematics) |
de
Finetti
and Probability
The first work in this section is
the copy of Introduzione matematica alla statistica metodologica, the manuscript of the Lessons that
Bruno de Finetti gave at the Istituto Centrale di Statistica, when he was 24
years old.
A dedication to G. Castelnuovo can be found in the front page.
Then
one
can find some works on the subjective interpretation of probability, in
chronological order, such as Probabilismo:
Saggio critico sulla teoria
delle probabilità e sul valore della scienza (1931)
and Sul significato
soggettivo della probabilità (1931). This part ends with
Compte rendu critique du colloque de
Genève sur la théorie des probabilités (1939)
Which is presented together with the collection of the
Colloquie consacré a la théorie
des probabilités /
présidé par Maurice Fréchet. (1938)
containing
contributions by
M. Fréchet, W. Heisenberg, G. Pólya, P. Cantelli, I.F. Steffensen, W.
Feller,
P. Lévy, R.Von Mises, A. Wald, H. Cramér, O. Onicescu, E. Hopf ,
E.
Slutsky, S. Bernstein, H. Steinhaus, B. de Finetti, J. Neyman,
V.Glivenko, E. Dodd, G. Jordan, N. Obrechkoff.
|
The collection is devoted to a
conference which was held in Genève in the week 10-15 October 1937,
thanks to an anonymous sponsor. Some of the invited Speakers could not
participate to the conference, nevertheless they contributed with a
paper in this collection.
|
The exhibition
continues with some papers on
insurance problems, such as
Il
problema dei
“pieni”.
Giornale
dell'Istituto Italiano degli Attuari, (11) 1940, pp. l-88
and
Impostazione
individuale e impostazione collettiva, del problema della
riassicurazione.
Giornale
dell'Istituto Italiano degli Attuari, 13
(1942), pp. 28-53
which have been now
discovered in the international scientific community.
|
In the part
dedicated to insurance we find also a small contribution that de
Finetti gave to the Enciclopedia Treccani. In the Appendix of
Enciclopedia Treccani (1949/1960) he wrote the voice Attuariale,
matematica. He never mentioned it: this small contribution was not
even quoted in the list of his works prepared by himself, and his
daughter Fulvia did not know anything about it. It has been added to
the list of his work only very recently [w.r.t. It was not easy to find out this
contribution in the Enciclopedia, because in the general index was
quoted as Matematica, but at this voice is due to
Tricomi. Maria Teresa Liuzza, of the Istituto Treccani, helped Giovanna
Nappo to find out it. She also kindly provided the
following list concerning quotations of Bruno de Finetti in the
Enciclopedia Treccani. volume XXIV pag. 284, voce NASCITA (tabella relativa ai saggi di
fecondità matrimoniale - formula di Tait), autore Luigi Galvani; volume XXVII pag. 928, voce POPOLAZIONE ( formula descrittiva
dell'evoluzione di una popolazione considerata come funzione del
tempo)autore Ugo Giusti; Appendice II vol. 1 pag. 742, voce CURVE STATISTICHE (Annali di statistica )
- autore Rodolfo Benini; Appendice III vol. 2 - pagg. 485-486 – voce PROBABILITA' (CALCOLO DELLE)- autore
Giuseppe Pompilj; Appendice IV vol. 1 pag. 189, voce ATTUARIALE, MATEMATICA – autore Giuseppe
Ottaviani; Appendice IV vol. 3 pag. 53, voce PROBABILITA' (CALCOLO DELLE), autore
Giorgio dell'Aglio; Appendice V vol. 2 pag. 330, voce FRATTALI (Autosomiglianza), autore Luigi
Accardi; Appendice V vol. 2 pag. 691, voce INFERENZA STATISTICA (Metodi
Bayesiani , probabilità soggettiva), autore Ludovico Picconato; Appendice V vol. 4 pag. 272, voce PROBABILITA' (CALCOLO DELLE), autore
Luigi Accardi; Appendice V vol. 5 pagg. 211-212, voce
STATISTICA, autore Ester Capuzzo. |
The
exhibition contains also the
famous textbook Matematica logico intuitiva
(III edition, 1959), but especially the book Teoria delle
probabilità. Besides
the edition of 1970, the Library has
also two copies of a preliminary version (in two chapters), containing
some
corrections of the author. A few years after the 1970 edition (in
Italian) the
book was translated into English by A. Smith and A. Machì.
The exhibition ends with some books
dedicated to de Finetti (or named after
de Finetti’s Representation Theorem), and a few collections
of his works. The last one (Opere scelte in 2 volumes,
2006) has been published by U.M.I. this
year (one century after his birth) and is presented in the Bruno de
Finetti
Centenary Conference, just before the opening of the exhibition.
|
In the exhibition, in a
small separate section devoted to a few modern translations of antique
books, one can find an English translation
of Cardano’s Liber de Ludo Aleae (in an edition of 1961), a German translation of
Bayes-Price contribution, published in the interesting Ostwald’s
collection Klassiker der exakten
Wissenschaften (in an edition of 1908), a German translation of
and finally the English
version of Kolmogorov book “Grundbegriffe…” |
|
|
Probability from Cardano to de Finetti |
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A
special thank to the staff of the Library: Angelo Bardelloni, Carolina Del Bufalo,
Alessandra Foti, Ninfa Inguì, Chiara Tullio, and
especially to Adele Piccolomo, which
coordinated their (and my) work, besides working hard on the
project; a thank also to Lucilla Vespucci which was
fundamental in the starting steps of the
organization.
Their work
and their suggestions were precious (and essential) for the realization
of the
exhibition.
Giovanna
Nappo
LINKS to web-sites on the
Hystory of Probability and Statistics
·
Figures from the History of
Probability and Statistics
·
A
short history of Probability
·
The Beginnings of Modern Probability Theory (section
4 of A Handbook for the Early History of Modern Science by Ian C.
Johnston)
·
Studies in the History of Statistics
and Probability I and II
(from Materials for the History of Statistics )
·
Life and Work of Statisticians ( from Materials for the History of Statistics )
LINKS to web-sites
containing digital copies of Historical Books on Mathematics
Cornell University Library. Historical Math
Monographs
University of Michigan
Livres Numérisés
Mathématiques
(LiNuM)
LINKS to papers on Bruno
de Finetti (available only from the web-site uniroma1.it)
- D.Cifarelli and E. Regazzini de Finetti's
Contribution to Probability and Statistics (JSTOR)
- Lindley obituary
LINKS to papers of
Bruno de Finetti (some of them are available only from the web-site
uniroma1.it)
- De Finetti, Bruno - Probabilità e induzione - The book consist of the English translation
of some papers by de Finetti, together with the original Italian
versions. It also contains two introductions: R. Jeffrey "de
Finetti radical probabilism", and D. Fürst "de
Finetti and the teaching of Mathematics". (The book is
available on line, thanks to the AMS
Historica Project of the
Alma Mater University of Bologna)