The Early Development of Mathematical Probability Glenn Shafer
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Creating Modern Probability. Its Mathematics, Physics and Philosophy in Historical Perspective
HM 23 REVIEWS 203 The reviewer hopes this book will be widely read and enjoyed, and that it will be followed by other volumes telling even more of the fascinating story of Soviet mathematics. It should also be followed in a few years by an update, so that we can know if this great accumulation of talent will have survived the economic and political crisis that is just now robbing it of many of its most brilliant stars (see the article, ``To guard the future of Soviet mathematics,'' by A. M. Vershik, O. Ya. Viro, and L. A. Bokut' in Vol. 14 (1992) of The Mathematical Intelligencer). Creating Modern Probability. Its Mathematics, Physics and Philosophy in Historical Perspective. By Jan von Plato. Cambridge/New York/Melbourne (Cambridge Univ. Press). 1994. 323 pp. View metadata, citation and similar papers at core.ac.uk brought to you by CORE Reviewed by THOMAS HOCHKIRCHEN* provided by Elsevier - Publisher Connector Fachbereich Mathematik, Bergische UniversitaÈt Wuppertal, 42097 Wuppertal, Germany Aside from the role probabilistic concepts play in modern science, the history of the axiomatic foundation of probability theory is interesting from at least two more points of view. Probability as it is understood nowadays, probability in the sense of Kolmogorov (see [3]), is not easy to grasp, since the de®nition of probability as a normalized measure on a s-algebra of ``events'' is not a very obvious one. Furthermore, the discussion of different concepts of probability might help in under- standing the philosophy and role of ``applied mathematics.'' So the exploration of the creation of axiomatic probability should be interesting not only for historians of science but also for people concerned with didactics of mathematics and for those concerned with philosophical questions. -
There Is No Pure Empirical Reasoning
There Is No Pure Empirical Reasoning 1. Empiricism and the Question of Empirical Reasons Empiricism may be defined as the view there is no a priori justification for any synthetic claim. Critics object that empiricism cannot account for all the kinds of knowledge we seem to possess, such as moral knowledge, metaphysical knowledge, mathematical knowledge, and modal knowledge.1 In some cases, empiricists try to account for these types of knowledge; in other cases, they shrug off the objections, happily concluding, for example, that there is no moral knowledge, or that there is no metaphysical knowledge.2 But empiricism cannot shrug off just any type of knowledge; to be minimally plausible, empiricism must, for example, at least be able to account for paradigm instances of empirical knowledge, including especially scientific knowledge. Empirical knowledge can be divided into three categories: (a) knowledge by direct observation; (b) knowledge that is deductively inferred from observations; and (c) knowledge that is non-deductively inferred from observations, including knowledge arrived at by induction and inference to the best explanation. Category (c) includes all scientific knowledge. This category is of particular import to empiricists, many of whom take scientific knowledge as a sort of paradigm for knowledge in general; indeed, this forms a central source of motivation for empiricism.3 Thus, if there is any kind of knowledge that empiricists need to be able to account for, it is knowledge of type (c). I use the term “empirical reasoning” to refer to the reasoning involved in acquiring this type of knowledge – that is, to any instance of reasoning in which (i) the premises are justified directly by observation, (ii) the reasoning is non- deductive, and (iii) the reasoning provides adequate justification for the conclusion. -
The Interpretation of Probability: Still an Open Issue? 1
philosophies Article The Interpretation of Probability: Still an Open Issue? 1 Maria Carla Galavotti Department of Philosophy and Communication, University of Bologna, Via Zamboni 38, 40126 Bologna, Italy; [email protected] Received: 19 July 2017; Accepted: 19 August 2017; Published: 29 August 2017 Abstract: Probability as understood today, namely as a quantitative notion expressible by means of a function ranging in the interval between 0–1, took shape in the mid-17th century, and presents both a mathematical and a philosophical aspect. Of these two sides, the second is by far the most controversial, and fuels a heated debate, still ongoing. After a short historical sketch of the birth and developments of probability, its major interpretations are outlined, by referring to the work of their most prominent representatives. The final section addresses the question of whether any of such interpretations can presently be considered predominant, which is answered in the negative. Keywords: probability; classical theory; frequentism; logicism; subjectivism; propensity 1. A Long Story Made Short Probability, taken as a quantitative notion whose value ranges in the interval between 0 and 1, emerged around the middle of the 17th century thanks to the work of two leading French mathematicians: Blaise Pascal and Pierre Fermat. According to a well-known anecdote: “a problem about games of chance proposed to an austere Jansenist by a man of the world was the origin of the calculus of probabilities”2. The ‘man of the world’ was the French gentleman Chevalier de Méré, a conspicuous figure at the court of Louis XIV, who asked Pascal—the ‘austere Jansenist’—the solution to some questions regarding gambling, such as how many dice tosses are needed to have a fair chance to obtain a double-six, or how the players should divide the stakes if a game is interrupted. -
The Open Handbook of Formal Epistemology
THEOPENHANDBOOKOFFORMALEPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. THEOPENHANDBOOKOFFORMAL EPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. Published open access by PhilPapers, 2019 All entries copyright © their respective authors and licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. LISTOFCONTRIBUTORS R. A. Briggs Stanford University Michael Caie University of Toronto Kenny Easwaran Texas A&M University Konstantin Genin University of Toronto Franz Huber University of Toronto Jason Konek University of Bristol Hanti Lin University of California, Davis Anna Mahtani London School of Economics Johanna Thoma London School of Economics Michael G. Titelbaum University of Wisconsin, Madison Sylvia Wenmackers Katholieke Universiteit Leuven iii For our teachers Overall, and ultimately, mathematical methods are necessary for philosophical progress. — Hannes Leitgeb There is no mathematical substitute for philosophy. — Saul Kripke PREFACE In formal epistemology, we use mathematical methods to explore the questions of epistemology and rational choice. What can we know? What should we believe and how strongly? How should we act based on our beliefs and values? We begin by modelling phenomena like knowledge, belief, and desire using mathematical machinery, just as a biologist might model the fluc- tuations of a pair of competing populations, or a physicist might model the turbulence of a fluid passing through a small aperture. Then, we ex- plore, discover, and justify the laws governing those phenomena, using the precision that mathematical machinery affords. For example, we might represent a person by the strengths of their beliefs, and we might measure these using real numbers, which we call credences. Having done this, we might ask what the norms are that govern that person when we represent them in that way. -
The Bayesian Approach to the Philosophy of Science
The Bayesian Approach to the Philosophy of Science Michael Strevens For the Macmillan Encyclopedia of Philosophy, second edition The posthumous publication, in 1763, of Thomas Bayes’ “Essay Towards Solving a Problem in the Doctrine of Chances” inaugurated a revolution in the understanding of the confirmation of scientific hypotheses—two hun- dred years later. Such a long period of neglect, followed by such a sweeping revival, ensured that it was the inhabitants of the latter half of the twentieth century above all who determined what it was to take a “Bayesian approach” to scientific reasoning. Like most confirmation theorists, Bayesians alternate between a descrip- tive and a prescriptive tone in their teachings: they aim both to describe how scientific evidence is assessed, and to prescribe how it ought to be assessed. This double message will be made explicit at some points, but passed over quietly elsewhere. Subjective Probability The first of the three fundamental tenets of Bayesianism is that the scientist’s epistemic attitude to any scientifically significant proposition is, or ought to be, exhausted by the subjective probability the scientist assigns to the propo- sition. A subjective probability is a number between zero and one that re- 1 flects in some sense the scientist’s confidence that the proposition is true. (Subjective probabilities are sometimes called degrees of belief or credences.) A scientist’s subjective probability for a proposition is, then, more a psy- chological fact about the scientist than an observer-independent fact about the proposition. Very roughly, it is not a matter of how likely the truth of the proposition actually is, but about how likely the scientist thinks it to be. -
1 Stochastic Processes and Their Classification
1 1 STOCHASTIC PROCESSES AND THEIR CLASSIFICATION 1.1 DEFINITION AND EXAMPLES Definition 1. Stochastic process or random process is a collection of random variables ordered by an index set. ☛ Example 1. Random variables X0;X1;X2;::: form a stochastic process ordered by the discrete index set f0; 1; 2;::: g: Notation: fXn : n = 0; 1; 2;::: g: ☛ Example 2. Stochastic process fYt : t ¸ 0g: with continuous index set ft : t ¸ 0g: The indices n and t are often referred to as "time", so that Xn is a descrete-time process and Yt is a continuous-time process. Convention: the index set of a stochastic process is always infinite. The range (possible values) of the random variables in a stochastic process is called the state space of the process. We consider both discrete-state and continuous-state processes. Further examples: ☛ Example 3. fXn : n = 0; 1; 2;::: g; where the state space of Xn is f0; 1; 2; 3; 4g representing which of four types of transactions a person submits to an on-line data- base service, and time n corresponds to the number of transactions submitted. ☛ Example 4. fXn : n = 0; 1; 2;::: g; where the state space of Xn is f1; 2g re- presenting whether an electronic component is acceptable or defective, and time n corresponds to the number of components produced. ☛ Example 5. fYt : t ¸ 0g; where the state space of Yt is f0; 1; 2;::: g representing the number of accidents that have occurred at an intersection, and time t corresponds to weeks. ☛ Example 6. fYt : t ¸ 0g; where the state space of Yt is f0; 1; 2; : : : ; sg representing the number of copies of a software product in inventory, and time t corresponds to days. -
1 History of Probability (Part 2)
History of Probability (Part 2) - 17 th Century France The Problem of Points: Pascal, Fermat, and Huygens Every history of probability emphasizes the Figure 1 correspondence between two 17 th century French scholars, Blaise Pascal and Pierre de Fermat . In 1654 they exchanged letters where they discussed how to solve a particular gambling problem, now referred to as The Problem of Points (also called the problem of division of the stakes) . Simply stated, the problem is how to split the pot if the game is interrupted before someone has won. Pascal and Fermat’s work on this problem led to the development of formal rules for probability calculations. Blaise Pascal 1623 -1662 Pierre de Fermat 1601 -1665 The problem of points concerns a game of chance with two competing players who have equal chances of winning each round. [Imagine that one round is a toss of a coin. Player A has heads, B has tails.] The winner is the first one who wins a certain number of pre-agreed upon rounds. [Suppose, for example, they agree that the first person to win 3 tosses wins the game.] Whoever wins the game gets the whole pot. [Say, each player puts in $6, so the total pot is $12 . If A gets three heads before B gets three tails, A wins $12.] Now suppose that the game is interrupted before either player has won. How does one then divide the pot fairly? [Suppose they have to stop early and at that point A has won two tosses and B has won one.] Should each get $6, or should A get more since A was ahead when they stopped? What is fair? Historically, it is important to note that all of these gambling problems were framed in terms of odds for winning a game and not in terms of probabilities of individual events. -
Most Honourable Remembrance. the Life and Work of Thomas Bayes
MOST HONOURABLE REMEMBRANCE. THE LIFE AND WORK OF THOMAS BAYES Andrew I. Dale Springer-Verlag, New York, 2003 668 pages with 29 illustrations The author of this book is professor of the Department of Mathematical Statistics at the University of Natal in South Africa. Andrew I. Dale is a world known expert in History of Mathematics, and he is also the author of the book “A History of Inverse Probability: From Thomas Bayes to Karl Pearson” (Springer-Verlag, 2nd. Ed., 1999). The book is very erudite and reflects the wide experience and knowledge of the author not only concerning the history of science but the social history of Europe during XVIII century. The book is appropriate for statisticians and mathematicians, and also for students with interest in the history of science. Chapters 4 to 8 contain the texts of the main works of Thomas Bayes. Each of these chapters has an introduction, the corresponding tract and commentaries. The main works of Thomas Bayes are the following: Chapter 4: Divine Benevolence, or an attempt to prove that the principal end of the Divine Providence and Government is the happiness of his creatures. Being an answer to a pamphlet, entitled, “Divine Rectitude; or, An Inquiry concerning the Moral Perfections of the Deity”. With a refutation of the notions therein advanced concerning Beauty and Order, the reason of punishment, and the necessity of a state of trial antecedent to perfect Happiness. This is a work of a theological kind published in 1731. The approaches developed in it are not far away from the rationalist thinking. -
Maty's Biography of Abraham De Moivre, Translated
Statistical Science 2007, Vol. 22, No. 1, 109–136 DOI: 10.1214/088342306000000268 c Institute of Mathematical Statistics, 2007 Maty’s Biography of Abraham De Moivre, Translated, Annotated and Augmented David R. Bellhouse and Christian Genest Abstract. November 27, 2004, marked the 250th anniversary of the death of Abraham De Moivre, best known in statistical circles for his famous large-sample approximation to the binomial distribution, whose generalization is now referred to as the Central Limit Theorem. De Moivre was one of the great pioneers of classical probability the- ory. He also made seminal contributions in analytic geometry, complex analysis and the theory of annuities. The first biography of De Moivre, on which almost all subsequent ones have since relied, was written in French by Matthew Maty. It was published in 1755 in the Journal britannique. The authors provide here, for the first time, a complete translation into English of Maty’s biography of De Moivre. New mate- rial, much of it taken from modern sources, is given in footnotes, along with numerous annotations designed to provide additional clarity to Maty’s biography for contemporary readers. INTRODUCTION ´emigr´es that both of them are known to have fre- Matthew Maty (1718–1776) was born of Huguenot quented. In the weeks prior to De Moivre’s death, parentage in the city of Utrecht, in Holland. He stud- Maty began to interview him in order to write his ied medicine and philosophy at the University of biography. De Moivre died shortly after giving his Leiden before immigrating to England in 1740. Af- reminiscences up to the late 1680s and Maty com- ter a decade in London, he edited for six years the pleted the task using only his own knowledge of the Journal britannique, a French-language publication man and De Moivre’s published work. -
Probability and Logic
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Applied Logic 1 (2003) 151–165 www.elsevier.com/locate/jal Probability and logic Colin Howson London School of Economics, Houghton Street, London WC2A 2AE, UK Abstract The paper is an attempt to show that the formalism of subjective probability has a logical in- terpretation of the sort proposed by Frank Ramsey: as a complete set of constraints for consistent distributions of partial belief. Though Ramsey proposed this view, he did not actually establish it in a way that showed an authentically logical character for the probability axioms (he started the current fashion for generating probabilities from suitably constrained preferences over uncertain options). Other people have also sought to provide the probability calculus with a logical character, though also unsuccessfully. The present paper gives a completeness and soundness theorem supporting a logical interpretation: the syntax is the probability axioms, and the semantics is that of fairness (for bets). 2003 Elsevier B.V. All rights reserved. Keywords: Probability; Logic; Fair betting quotients; Soundness; Completeness 1. Anticipations The connection between deductive logic and probability, at any rate epistemic probabil- ity, has been the subject of exploration and controversy for a long time: over three centuries, in fact. Both disciplines specify rules of valid non-domain-specific reasoning, and it would seem therefore a reasonable question why one should be distinguished as logic and the other not. I will argue that there is no good reason for this difference, and that both are indeed equally logic. -
Pascal's and Huygens's Game-Theoretic Foundations For
Pascal's and Huygens's game-theoretic foundations for probability Glenn Shafer Rutgers University [email protected] The Game-Theoretic Probability and Finance Project Working Paper #53 First posted December 28, 2018. Last revised December 28, 2018. Project web site: http://www.probabilityandfinance.com Abstract Blaise Pascal and Christiaan Huygens developed game-theoretic foundations for the calculus of chances | foundations that replaced appeals to frequency with arguments based on a game's temporal structure. Pascal argued for equal division when chances are equal. Huygens extended the argument by considering strategies for a player who can make any bet with any opponent so long as its terms are equal. These game-theoretic foundations were disregarded by Pascal's and Huy- gens's 18th century successors, who found the already established foundation of equally frequent cases more conceptually relevant and mathematically fruit- ful. But the game-theoretic foundations can be developed in ways that merit attention in the 21st century. 1 The calculus of chances before Pascal and Fermat 1 1.1 Counting chances . .2 1.2 Fixing stakes and bets . .3 2 The division problem 5 2.1 Pascal's solution of the division problem . .6 2.2 Published antecedents . .7 2.3 Unpublished antecedents . .8 3 Pascal's game-theoretic foundation 9 3.1 Enter the Chevalier de M´er´e. .9 3.2 Carrying its demonstration in itself . 11 4 Huygens's game-theoretic foundation 12 4.1 What did Huygens learn in Paris? . 13 4.2 Only games of pure chance? . 15 4.3 Using algebra . 16 5 Back to frequency 18 5.1 Montmort . -
From Abraham De Moivre to Johann Carl Friedrich Gauss
International Journal of Engineering Science Invention (IJESI) ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726 www.ijesi.org ||Volume 7 Issue 6 Ver V || June 2018 || PP 28-34 A Brief Historical Overview Of the Gaussian Curve: From Abraham De Moivre to Johann Carl Friedrich Gauss Edel Alexandre Silva Pontes1 1Department of Mathematics, Federal Institute of Alagoas, Brazil Abstract : If there were only one law of probability to be known, this would be the Gaussian distribution. Faced with this uneasiness, this article intends to discuss about this distribution associated with its graph called the Gaussian curve. Due to the scarcity of texts in the area and the great demand of students and researchers for more information about this distribution, this article aimed to present a material on the history of the Gaussian curve and its relations. In the eighteenth and nineteenth centuries, there were several mathematicians who developed research on the curve, including Abraham de Moivre, Pierre Simon Laplace, Adrien-Marie Legendre, Francis Galton and Johann Carl Friedrich Gauss. Some researchers refer to the Gaussian curve as the "curve of nature itself" because of its versatility and inherent nature in almost everything we find. Virtually all probability distributions were somehow part or originated from the Gaussian distribution. We believe that the work described, the study of the Gaussian curve, its history and applications, is a valuable contribution to the students and researchers of the different areas of science, due to the lack of more detailed research on the subject. Keywords - History of Mathematics, Distribution of Probabilities, Gaussian Curve. ----------------------------------------------------------------------------------------------------------------------------- --------- Date of Submission: 09-06-2018 Date of acceptance: 25-06-2018 ----------------------------------------------------------------------------------------------------------------------------- ---------- I.