A probabilistic “New Principle” of the 19th century

Oxford Handbook of Probability and Philosophy, 2016

It has been common to begin histories of probability with the calculations of Fermat and Pascal on games of chance in 1654. Those calculations were the first substantial mathematical results on stochastic phenomena, which, according to a frequentist philosophy of probability, are the true subject-matter of the field. But there is great philosophical interest in studying earlier ideas. That is because we see a struggle to understand a range of basic concepts about probability and uncertain evidence, before the straightjacket of a single formalization was imposed. There is still much to learn from seeing how the problems now studied with the aid of probability theory were dealt with in “bare hands” fashion before that formalization was available.

2009

This short note reviews and details various senses of the word "martingale," with their respective etymologies, in mathematics, gambling, technology, and vernacular language.

2009

This paper examines—by means of the example of the St. Petersburg paradox—how Borel exposited the science of his day. The first part sketches the singular place of popularization in Borel’s work. The two parts that follow give a chronological presentation of Borel’s contributions to the St. Petersburg paradox, contributions that evolved over a period of more than fifty years. These show how Borel attacked the problem by positioning it in a long—and scientifically very rich— meditation on the paradox of martingales, those systems of play that purport to make a gambler tossing a coin rich. Borel gave an original solution to this problem, anticipating the fundamental equality of the nascent mathematical theory of martingales. The paradoxical role played by Félix Le Dantec in the development of Borel’s thought on these themes is highlighted. An appendix recasts Borel’s martingales in modern terms. Bernard Bru, Université René-Descartes, Laboratoire MAP5, 45 rue des Saints-Pères, 75270 P...

Historia Mathematica, 2015

This paper provides a critical discussion of historical and theoretical meaningfulness of the distinction between 'objective' and 'subjective' probability, as it supposedly emerged around 1840, by examining whether and how it appeared in the work of the mid-nineteenth-century British revisionist probabilists. A detailed analysis of the contributions of Augustus De Morgan, John Stuart Mill, George Boole, Robert Leslie Ellis and John Venn to probability is put forward in order to show that in so far as the terms did not appear as contradictories it is not possible to understand or compare these contributions with reference to the modern binary of 'objective' and 'subjective'.

Historia Mathematica, 1980

2009

This paper examines-by means of the example of the St. Peters-burg paradox-how Borel exposited the science of his day. The first part sketches the singular place of popularization in Borel's work. The two parts that follow give a chronological presentation of Borel's contributions to the St. Petersburg paradox, contributions that evolved over a period of more than fifty years. These show how Borel attacked the problem by positioning it in a long-and scientifically very rich-meditation on the paradox of martingales, those systems of play that purport to make a gambler tossing a coin rich. Borel gave an original solution to this problem, anticipating the fundamental equality of the nascent mathematical theory of martingales. The paradoxical role played by Félix Le Dantec in the development of Borel's thought on these themes is highlighted. An appendix recasts Borel's martingales in modern terms. i Bernard Bru, Université René-Descartes, Laboratoire MAP5, 45 rue des Sai...

BSHM Bulletin: Journal of the British Society for the History of Mathematics, 2018

The paper discusses the background to and provides a transcription of a letter from Robert Leslie Ellis (1817-59) to William Walton (1813Walton ( -1901) ) of 1849 on probability theory. A lthough today still a largely forgotten figure, the polymath Robert Leslie Ellis (1817-59) was an important member of the Cambridge mathematical community in the mid-nineteenth century. 1 As a pupil of William Hopkins, student of George Peacock, First Wrangler in 1840, editor of the Cambridge Mathematical Journal and frequent contributor to the important British mathematical journals of the time, Ellis was much admired by his Victorian contemporaries. George Boole, in an 1857 prize-essay about applications of probability theory, wrote that '[t]here' is no living mathematician for whose intellectual character I entertain a more sincere respect than I do for that of Mr. Ellis' (Boole 1857 [1952], 350). A decade later, Francis Galton expressed a similar sentiment when describing Ellis as one 'whose name is familiar to generations of Cambridge men as a prodigy of universal genius' (Galton 1869 [1892], 18). During his tragically short life, Ellis wrote five papers and one note on probability theory (1844a [1863], 1844b [1863], 1844c, 1850a, 1850b, 1854 [1863]). 2 As the Reverend Harvey Goodwin noted in his biographical memoir accompanying The Mathematical and Other Writings of Robert Leslie Ellis, Ellis' papers on probability represented 'as well as possible his special taste [...] with regard to mathematics' (Goodwin 1863, xxix). Given his 'speculative mind', Ellis, who always talked about probability 'with great pleasure' and as a subject in which he was 'thoroughly at home', was naturally drawn to questions of foundations (Goodwin 1863, xxix, xxxv). Ellis' contributions to mathematical probability theory, written in the 1840s to 1850s, roughly consisted of two parts: firstly, an early sketch of a frequentist theory of probability and, secondly, a simplification and extension to any number of unknowns of Pierre-Simon Laplace's demonstration of the method of least squares. 3

Archive for History of Exact Sciences, 1976

During the Synod of Constance (Konstanz) (1414 -1418) the controversial topic of probabilism was publicly discussed probably for the first time in history. But only one hundred years later the Spanish Dominican Bartholomé de Medina (1527 -1581) led the way to probabilism as an acknowledged principle in the moral teaching of the Catholic Church. Soon afterwards probabilism was taken up by the Jesuits and subsequently further developed to a formal theory. In 1662 the book La Logique ou l'Art de Penser (in Latin Ars cogitandi) was published anonymously in Paris. In this influential book the concept of probability is used in the context of chance. Since then, numerous scientists have given various interpretations of the notion of probability, which invariably proved to be ineligible. In this paper the historical development of the notion probability is sketched and, finally, based on Jakob Bernoulli's ideas, a sustainable solution is presented.

Advances in mathematics education, 2014

This chapter presents a twenty first century historical and philosophical perspective on probability, related to the teaching of probability. It is important to remember the historical development as it provides pointers to be taken into account in developing a modern curriculum in teaching probability at all levels. We include some elements relating to continuous as well as discrete distributions. Starting with initial ideas of chance two millennia ago, we move on to the correspondence of Pascal and Fermat, and insurance against risk. Two centuries of debate and discussion led to the key fundamental ideas; the twentieth century saw the climax of the axiomatic approach from Kolmogorov. Philosophical difficulties have been prevalent in probability since its inception, especially since the idea requires modelling-probability is not an inherent property of an event, but is based on the underlying model chosen. Hence the arguments about the philosophical basis of probability have still not been fully resolved. The three main theories (APT, FQT, and SJT) are described, relating to the symmetric, frequentist, and subjectivist approaches. These philosophical ideas are key to developing teaching content and methodology. Probabilistic concepts are closer to a consistent way of thinking about the world rather than describing the world in a consistent manner, which seems paradoxical, and can only be resolved by a careful analysis.

1996

Richard Jeffrey has labelled his philosophy of probability "radical probabilism" and qualified this position as "Bayesian", "nonfoundational" and "anti-rationalist". This paper explores the roots of radical probabilism, to be traced back to the work of Frank P. Ramsey and Bruno de Finetti.

Early Science and Medicine, 2022

Rudolf Schuessler has claimed that 16th-century thinkers developed a concept of equal probability that was virtually absent before 1500 and that may have facilitated the birth of mathematical probability shortly after 1650. I show that this concept was in fact generally available to medieval thinkers. My note should be read in conjunction with Schuessler's generous response in the same issue.

Kolmogorov’s Heritage in Mathematics

©Scholedge International Journal of Management & Development, 2016

Jerzy Neyman analyzed an imaginary, non existent, Urn ball problem that he thought was taken from J M Keynes's A Treatise on Probability in his Lectures and Conferences on Mathematical Statistics and Probability (1952). Neyman apparently never read the book for himself. He apparently relied on some, other, unknown source to provide him with the problem that he thought came from J M Keynes's A Treatise on Probability. The problem can be analyzed based on an "as if" approach to discover if Neyman realized that Keynes's Principle of Indifference is a substantially different technique from Laplace's concoction that, when applied, will lead to substantially different answers from those obtained by the use of Laplace's Principle of Non-Sufficient Knowledge.