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Theory Of Probability. A Historical Essay

Figures from the History of Probability and Statistics

John Aldrich, University of Southampton, Southampton, UK.(home)

2005. Latest changes October 2012

Notes on the work of

A further 200+ individuals are mentioned below. Use Search on your browser to find the person you are interested in. It is also worth searching for the ‘principals’ for they can pop up anywhere.

The entries are arranged chronologically, so the document can be read as a story.These are the date markers

with people placed according to date of their first impact. Do not take the placings too seriously and remember that a career may last more than 50 years! At each marker there are notes on developments in the following period. There is more about and about economics than there should be but I know more about them.

For further on-line information there are links to

·         Earliest Uses (Words and Symbols) for details (particularly detailed references) on the topics to which the individuals contributed. (The Words site is organised by letter of the alphabet. See here for a list of entries)

·         MacTutor for fuller biographical information on the ‘principals’ (all but three) and on a very large ‘supporting’ cast. The MacTutor biographies also cover the work the individuals did outside probability and statistics. The MacTutor and References links are to these pages. There is an index to the Statistics articles on the site.

·         ASA Statisticians in History for biographies of mainly recent, mainly US statisticians.

·         Life and Work of Statisticians (part of the Materials for the History of Statistics site) for further links, particularly to original sources.

·         Oscar Sheynin’s Theory of Probability: A Historical EssayAn account of developments to the beginning of the twentieth century, particularly useful for its coverage of Continental work on statistics.

·         Isaac Todhunter’s classic from 1865 A History of the Mathematical Theory of Probability : from the Time of Pascal to that of Laplace for detailed commentaries on the contributions from 1650-1800. The coverage is extraordinary and the entries are still interesting—even their humourlessness has a certain charm.

·         The Mathematics Genealogy Project,abbreviatedMGP, which is useful for tracking modern scholars. The PhD degree is a relatively recent development and in the a very recent one. See my The Mathematics PhD in the UK.

·         Wikipedia for additional biographies. This is an uneven site but it has some useful articles.

The entries contain references to the following histories and books of lives. See below for more literature.

·         Ian Hacking The Emergence of Probability, , Press 1975. (contents)

·         Stephen M Stigler The History of Statistics: The Measurement of Uncertainty before 1900, , : Press 1986. (contents + bibliography)

·         Anders Hald A History of Probability and Statistics and their applications before 1750, : Wiley 1990. (contents)

·         Anders Hald A History of Mathematical Statistics from 1750 to 1930, : Wiley 1998. (contents + bibliography)

·         Jan von Plato Creating Modern Probability,: Cambridge University Press, 1994. (contents)

·         Leading Personalities in Statistical Sciences from the Seventeenth Century to the Present, (ed. N. L. Johnson and S. Kotz) 1997. : Wiley. Contains around 110 biographies and based on entries in Encyclopedia of Statistical Science (ed. N. L. Johnson and S. Kotz.) AbbreviatedLP.

·         Statisticians of the Centuries (ed. C. C. Heyde and E. Seneta) 2001. : Springer. Contains 105 biographies. The coverage is restricted to individuals born before 1900. AbbreviatedSC.

·         Encyclopedia of Social Measurement (ed. K. Kempf-Leonard) 2004. : Elsevier (description). Contains numerous biographies. AbbreviatedESM.

On the web (see also online biblios and texts below)

·         Stochastikon Encyclopedia. Articles in English and German.

·         Portraits of Statisticians on the Materials for the History of Statistics site.

·         Sources in the History of Probability and Statistics by Richard J. Pulskamp.

·         Tales of Statisticians. Vignettes by E. Bruce Brooks.

·         History of Statistics and Probability 18 short biographies from .

·         Glimpses of the Prehistory of Econometrics. Montage by Jan Kiviet.

·         Probability and Statistics Ideas in the Classroom: Lesson from History.  Comments on the uses of history by D. Bellhouse.

·         The History of Statistics in the Classroom.Thumbnail sketches of Gauss, Laplace and Fisher by H. A. David.

·         Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization. Encyclopaedic coverage by M. Friendly & D.J. Denis.

·         Actuarial History. A very comprehensive collection of links, created by Henk Wolthuis,not only to actuarial science and demography, but to statistics as well.

To help place individuals I have used modern terms for occupation (e.g. physicist or statistician). For the earlier figures these terms are anachronistic but, I hope, not too misleading. I have not given nationality as people move and states come and go. MacTutor has plenty of geographical information.

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1650-1700The origins of probability and statistics are usually found in this period, in the mathematical treatment of games of chance and in the systematic study of mortality data. This was the age of the Scientific Revolution and the biggest names, Galileo (Materials and Todhunter ch.I (4-6).) and Newton (LP) gave some thought to probability without apparently influencing its development. For an introduction to the Scientific Revolution, see Westfall’s Scientific Revolution (1986).

·    There were earlier contributions to probability, e.g. Cardano (1501-76) gave some ‘probabilities’ associated with dice throwing, but a critical mass of researchers (and results) was only achieved following discussions between Pascal and Fermatand the publication of the first book by Huygens.HackingChapters 1-5 discusses thinking before Pascal. James Franklin’s The Science of Conjecture: Evidence and Probability Before Pascal (2001) examines this earlier work in depth. A recent issue of the JEHPSis devoted to Medieval probabilities.

·    Statisticsin the form of population statistics was created by Graunt. Graunt’s friend William Petty gave the name Political Arithmetic to the quantitative study of demography and economics. Gregory King was an important figure in the next generation. However the economic line fizzled out. Adam Smith, the most influential C18 British economist, wrote,“I have no great faith in political arithmetic...” Wealth of Nations (1776) B.IV, Ch.5, Of Bounties.

·    A form of life insurance mathematics was developed from Graunt’s work on the life table by the mathematicians Halley, Hudde and de Witt. Many later ‘probabilists’ wrote on actuarial matters, including de Moivre,Simpson, Price,De Morgan,Gram, Thiele, Cantelli, Cramér and de Finetti. In the C20 the Skandinavisk aktuarietidskrift and the Giornale dell'Istituto Italiano degli Attuari were important journals for theoretical statistics and probability. Actuarial questions and friendship with the actuary G. J. Lidstone stimulated the Edinburgh mathematicians, E. T. Whittaker and A. C. Aitken (MGPP), to contribute to statistics and numerical analysis. The C17 work is discussed by Hacking (1975): Chapter 13, Annuities. See also Chris Lewin’s The Creation of Actuarial Science and the other historical links on the International Actuarial Links page. There are historical articles in the Encyclopedia of Actuarial Science. Classics are reprinted in History of Actuarial Science. There is a nice review of the early literature in the catalogue of the Equitable Life Archive.

·    New institutions, rather than the traditional universities, underpinned these developments. In and private discussion groups, like that of Mersenne, were forerunners of the Académie des Sciences and the Royal Society of London (archives). The latter’s Philosophical Transactions (Gallica) published many important contributions to probability and statistics, including papers by Halley, de Moivre, Bayes, Pearson, Fisher, Jeffreys and Neyman. The Berlin and St. Petersburg academies were formed a bit later. The Royal Society was a forerunner of the modern scientific society, while the continental academies were more like research institutes.

Life & Work has links to the writings of many of these people. For the period generally see Todhunter ch. I-VI (pp. 1-55) and Hald (1990, ch. 1-12).

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Blaise Pascal(1623-1662) Mathematician and philosopher. MacTutorReferencesSC, LP. Pascal was educated at home by his father, himself a considerable mathematician. The origins of probability are usually found in the correspondencebetween Pascal and Fermat where they treated several problems associated with games of chance. The letters were not published but their contents were known in Parisian scientific circles. Pascal’s only probability publication was the posthumously published Traité du triangle arithmétique (1654, published in 1665 and so after Huygens’s work); this treated Pascal’s triangle with probability applications. Pascal introduced the concept of expectation and discussed the problem of gambler’s ruin. Pascal’s wager, is now often read as a pioneering analysis of decision-making under uncertainty although it appeared, not in his mathematical writings, but in the Pensées, his reflections on religion. The last chapter of the Port-Royal Logicpp. 365ff by Pascal’s friends Arnauld and Nicole has a brief treatment of the use of probability in decision making, with an allusion to the wager. See Ben Rogers Pascal's Life & Times,Life & Work,A.W.F. Edwards on the triangle and Todhunter ch.II (pp. 7-21). See also Hald 1990, chapter 5, The Foundations of Probability Theory by Pascal and Fermat in 1654 and Hacking 1975, chapter 7, The (1654) andchapter 8, The Great Decision (1658?).

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Christiaan Huygens(1629-94) Mathematician and physicist. MacTutorReferencesSC, LP. As a youthHuygens was expected to become a diplomat but instead he became a gentleman scientist, making important contributions to mathematics, physics and astronomy. He was educated at the of and at the of at . He spent 14 years in at the Académie des Sciences. Huygens wrote the first book on probability, a pamphlet really, Van Rekeningh in Spelen van Geluck, translated into Latin by his teacher van Schooten as De Ratiociniis in Ludo Aleae (1657) and then into English as The Value of all Chances in Games of Fortune etc. Huygens drew on the ideas of Pascal and Fermat, which he had encountered when he visited . Much of the book is devoted to calculating the value or, as it would be called now the expectation, of a game of chance. The problems contained in the book include the gambler’s ruin and Huygens treated the hypergeometric distribution. His book was widely read and the first part ofJakob Bernoulli’s Ars Conjectandi is a commentary on it. See Life & Work and Todhunter ch.III (pp. 22-5). See also Hald (1990): Chapter 6, Huygens and De Ratiociniis in Ludo Aleae, 1657 and Hacking (1975), Chapter 11, Expectation. C.J. (Kees) Verduin is constructing an impressive Christiaan Huygens site. See Peter Doyle Hedging Huygens.

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No authentic portrait of Graunt is known

John Graunt(1620-74) Merchant. Wikipedia. SC, LP, ESM. Graunt is unique among the figures described here in not having had a university education. He published only one work, Observations Made upon the Bills of Mortality (1662). However, through this work and his friendship with William Petty, he became a fellow of the Royal Society of London and his work became known to savants like Halley. The weekly bills of mortality, which had been collected since 1603, were designed to detect the outbreak of plague. Graunt put the data into tables and produced a commentary on them; he does basic calculations. He discusses the reliability of the data. He compares the number of male and female births and deaths. In the course of Chapter XI on estimating the population of London Graunt produces a primitive life table see the JIA articles by Glass, Renn, Benjamin and Seal. The life table became one of the main tools of demography and insurance mathematics. Halley produced a life table using data from Caspar Neumann (SC) in . See Life & Work for writings by Graunt, Petty and Halley. See alsoTodhunter ch. V (pp. 37-43),Hald (1990):Chapter 7, John Graunt and the Observations upon the Bills of Mortality, 1662 and Hacking (1975): Chapter 12, Political Arithmetic 1662.

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1700-50The great leap forward is Hald’s (1990) name for the decade 1708-1718: there were so many important contributions to such a greatly expanded subject. The roots of probability and statistics were quite distinct but by the early C18 it was understood that the subjects were closely related.

·    Jakob Bernoulli’s Ars Conjectandi, like Arnauld’sLogique (1682) pp. 365ff, suggested a conception of probability broader than that associated with games of chance.Bernoulli’s law of large numbers provided a theory to link between probability and data. See Hacking (1975): Chapters 16-7.

·    Montmort’s(SC, LP) Essay d'analyse sur les jeux de hazard (1708) and de Moivre’sDoctrine of Chances (1718) produced many new results on games of chance, greatly extending the work of Pascal and Huygens.

·    Arbuthnot’s (SC, LP) 1710 paper An Argument for Divine Providence, taken from the constant Regularity observed in the Births of both Sexes used a significance test (sign test) to establish that the probability of a male birth is not ½. The calculations were refined by 'sGravesande (LP) and Nicholas Bernoulli (LP).Apart from being an early application of probability to social statistics, Arbuthnot’s paper illustrates the close connection between theology and probability in the literature of the time. The work of John Craig provides another example.

·    Consideration of the valuation of a risky prospect, dramatised by the St. Petersburg paradox(formulatedby Nicholas Bernoulli in1713 and discussed by Gabriel Cramer)led to Daniel Bernoulli’s (1737) theory of moral expectation (or expected utility).

See Life & Work for writings by Montmort, Euler, Lagrange, etc. For the period see Todhunter ch. VII-X(pp. 56- 212)Hald (1990, ch. 1-12), HackingChapter 16-9.

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Jakob (James) Bernoulli (1654-1705) Mathematician. MacTutorReferencesSC, LP, ESM. Eight members of the Bernoulli family have biographies in MacTutor (family tree) and several wrote on probability. The most important contributors were Jakob, Daniel and Niklaus. Jakob and his younger brother Johann were the first of the mathematicians. Jakob studied philosophy at the but learnt mathematics on his own. Eventually he became professor of mathematics at . The posthumously publishedArs Conjectandi(title page) (1713) was his only probability publication but it was extremely influential.The first part is a commentary on Huygens’sDe Ratiociniis. The work was an important contribution to combinatorics: the term permutationoriginated here. (The combinatorial side of probability remained important and in late C19 Britain probability and combinatorial analysis were taught together in algebra courses.) Bernoulliused the terms a priori and a posteriori to distinguish two ways of deriving probabilities (see posterior probability): deduction a priori (without experience) is possible when there are specially constructed devices, like dice but otherwise it is possible to make a deduction from many observed outcomes of similar events. Bernoulli’stheorem, or thelaw of large numberswas the work’s most spectacular contribution but see also the entries formorally certainandbinomial distributionThe eponymous Bernoullitrials, numbers and random variable all refer to this Bernoulli and his Ars Conjectandi. See Sheynin ch. 3Life & Work (which also has links for the contributions of Niklaus and Jean III) and Todhunter ch.VII (pp. 56-77). See Stigler (1986): Chapter 2, Probabilists and the Measurement of Uncertainty. See also Hald (1990): Chapter 15, James Bernoulli and Ars Conjectandi, 1713;Chapter 16, Bernoulli’s Theorem. Sheynin has recently translated Part IV of the Ars Conjectandi. A new translation of the whole work has appeared: The Art of Conjecturing translated by Edith Dudley Scylla Amazon. The proceedings of a conference on the Ars Conjectandi are online at the JEHPS: part 1andpart 2.

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Abraham de Moivre(1667-1754)MathematicianMacTutorReferencesSC, LP. De Moivre came to from as a refugee aged about 20 and, although he gained recognition as a mathematician and became a fellow of the Royal Society, he never obtained an academic appointment. De Moivre had read Huygens’s book before leaving but his first paper on probability was published in 1711. In 1718 he published The Doctrine of Chances: or, a Method of Calculating the Probability of Events in Play (title page). He published other pieces on probability, putting the results into new editions of the Doctrine; the third appeared in 1733. The book began with an influential definition ofprobability. De Moivre obtained the normalapproximation to the binomial distribution (a forerunner of the central limit theorem) and almost found thePoisson distribution. His technical innovations included the use ofprobability generating functions, which he used to find the distribution of the sum ofuniformvariables. De Moivre also wrote about life insurance mathematics when analysing annuities Seesurvival function. Todhunter ranked de Moivre’s contributions very highly, “it will not be doubted that the Theory of Probability owes more to him than to any other mathematician, with the sole exception of

Title: Theory of Probability. A Historical Essay

Authors:Oscar Sheynin

(Submitted on 27 Feb 2018)

Abstract: This book covers the history of probability up to Kolmogorov with essential additional coverage of statistics up to Fisher. Based on my work of ca. 50 years, it is the only suchlike book. Gorrochurn (2016) is similar but his study of events preceding Laplace is absolutely unsatisfactory. Hald (1990; 1998) are worthy indeed but the Continental direction of statistics (Russian and German statisticians) is omitted, it is impossible to find out what was contained in any particular memoir of Laplace and the explanation does not always explain the path from, say, Poisson to a modern interpretation of his results. Finally, the reader ought to master modern math. statistics. I included many barely known facts and conclusions, e. g., Gauss' justification of least squares (yes!), the merits of Bayes (again, yes!), the unforgivable mistake of Laplace, the work of Chebyshev and his students (merits and failures) etc., etc. The book covers an extremely wide field, and is targeted at the same readers as any other book on history of science. Mathematical treatment is not as difficult as it is for readers of Hald.

Submission history

From: Oscar Sheynin [view email]
[v1] Tue, 27 Feb 2018 15:29:25 GMT (1638kb)

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