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Edith Dudley Sylla - One of the best experts on this subject based on the ideXlab platform.

  • Tercentenary of Ars Conjectandi (1713) Jacob Bernoulli and the Founding of Mathematical Probability
    2015
    Co-Authors: Edith Dudley Sylla
    Abstract:

    Jacob Bernoulli worked for many years on the manuscript of his book Ars Conjectandi, but it was incomplete when he died in 1705 at age 50. Only in 1713 was it published as he had left it. By then Pierre Rémond de Montmort had published his Essay d’analyse sur les jeux de hazard (1708), Jacob’s nephew, Nicholas Bernoulli, had written a master’s thesis on the use of the art of conjecture in law (1709), and Abraham De Moivre had published “De Mensura Sortis, seu de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus ” (1712). Nevertheless, Ars Conjectandi deserves to be considered the founding document of Mathematical Probability, for reasons explained in this paper. By the “art of conjecturing ” Bernoulli meant an approach by which one could choose more appropriate, safer, more carefully considered, and, in a word, more probable actions in matters in which complete certainty is impossible. He believed that his proof of a new fundamental theorem – later called the weak law of large numbers – showed that the mathematics of games of chance could be extended to a wide range civil, moral, and economic problems. Gottfried Wilhelm Leibniz boasted that Bernoulli had taken up the mathematics of Probability at his urging. Abraham De Moivre pursued the project that Bernoulli had begun, at the same time shifting the central meaning of Probability to relative frequency. Key words: conjecture, Abraham De Moivre, G. W. Leibniz, weak law of large numbers. The origin of the frequentist theory of Probability goes back to the question of whether one can compute the long-range frequency of some event E from known frequencies of some related events A, B,.... C. With an unavoidable degree of oversimplification, one might say that the theory of Probability started in 1713, with the publication of the book Ars Conjectandi, by Jacob Bernoulli. Jerzy Nyman (1976, 152) 1. Introduction an

  • Tercentenary of Ars Conjectandi (1713) Jacob Bernoulli and the Founding of Mathematical Probability
    International Statistical Review, 2014
    Co-Authors: Edith Dudley Sylla
    Abstract:

    type="main" xml:id="insr12050-abs-0001"> The Tercentenary of the publication of Jacob Bernoulli's Ars Conjectandi (The Art of Conjecturing) provides an opportunity to look at the origins of Mathematical Probability from Jacob Bernoulli's point of view. Bernoulli gave a Mathematically rigorous proof of what has come to be called the weak law of large numbers, relevant to discovering ratios of unknown factors through sampling. The Art of Conjecturing was a bridge between the mathematics of expectation in games of chance as found in Huygens's On Reckoning in Games of Chance and Mathematical Probability as found in Abraham De Moivre's The Doctrine of Chances. This paper looks at the conceptual context as well as the mathematics of Bernoulli's book.

  • Mendelssohn, Wolff, and Bernoulli on Probability
    Moses Mendelssohn's Metaphysics and Aesthetics, 2011
    Co-Authors: Edith Dudley Sylla
    Abstract:

    In his “On Probability,” Moses Mendelssohn proposed to use a Mathematical ­formulation of the definition of Probability, found in the work of Christian Wolff to support the validity of induction (against Hume) and the view that all our actions, even including those supposed to be the result of free will, are predetermined (in agreement with Leibniz). Mendelssohn went into few of the details of Mathematical Probability as they had been developed by mathematicians like Jacob Bernoulli in the century before he wrote.

Lorraine Daston - One of the best experts on this subject based on the ideXlab platform.

Timothy C. Johnson - One of the best experts on this subject based on the ideXlab platform.

  • Reciprocity as a Foundation of Financial Economics
    Journal of Business Ethics, 2015
    Co-Authors: Timothy C. Johnson
    Abstract:

    This paper argues that the subsistence of the fundamental theorem of contemporary financial mathematics is the ethical concept ‘reciprocity’. The argument is based on identifying an equivalence between the contemporary, and ostensibly ‘value neutral’, Fundamental Theory of Asset Pricing with theories of Mathematical Probability that emerged in the seventeenth century in the context of the ethical assessment of commercial contracts in a framework of Aristotelian ethics. This observation, the main claim of the paper, is justified on the basis of results from the Ultimatum Game and is analysed within a framework of Pragmatic philosophy. The analysis leads to the explanatory hypothesis that markets are centres of communicative action with reciprocity as a rule of discourse. The purpose of the paper is to reorientate financial economics to emphasise the objectives of cooperation and social cohesion and to this end, we offer specific policy advice.

  • Reciprocity as the foundation of Financial Economics
    arXiv: General Finance, 2013
    Co-Authors: Timothy C. Johnson
    Abstract:

    This paper argues that the fundamental principle of contemporary financial economics is balanced reciprocity, not the principle of utility maximisation that is important in economics more generally. The argument is developed by analysing the Mathematical Fundamental Theory of Asset Pricing with reference to the emergence of Mathematical Probability in the seventeenth century in the context of the ethical assessment of commercial contracts. This analysis is undertaken within a framework of Pragmatic philosophy and Virtue Ethics. The purpose of the paper is to mitigate future financial crises by reorienting financial economics to emphasise the objectives of market stability and social cohesion rather than individual utility maximisation.

Sandy L. Zabell - One of the best experts on this subject based on the ideXlab platform.

  • The Subjective and the Objective
    Philosophy of Statistics, 2011
    Co-Authors: Sandy L. Zabell
    Abstract:

    Publisher Summary This chapter discusses the origins of the subjective-objective distinction in Probability, and the reasons why the resulting debate over the nature of Probability later took the form it did in the twentieth century. It argues that the explicit distinction between subjective and objective arose when it did because of a tension between two distinct scientific currents. On the one hand, the work of James and Daniel Bernoulli, Condorcet, Laplace, Poisson, and Quetelet dramatically extended the applications of Mathematical Probability to areas of pressing social concern, and thus for the first time made the nature and scope of Probability an issue of interest and importance. Kant had attempted to steer a middle course between the extreme rationalism of Leibniz, and the extreme skepticism of Hume; one claimed in effect that all knowledge is objective, the other that all knowledge is subjective.

  • carnap and the logic of inductive inference
    Handbook of the History of Logic, 2011
    Co-Authors: Sandy L. Zabell
    Abstract:

    Publisher Summary This chapter discusses Carnap's work on Probability and induction, using the notation and terminology of modern Mathematical Probability, from the perspective of the modern Bayesian or subjective school of Probability. Carnap used logical Probability as a tool in understanding the quantitative confirmation of a hypothesis based on evidence and in rational decision making. The resulting analysis of induction involved a two step process. The first step included a broad class of possible confirmation functions, commonly called the “regular c-functions,” along with a unique function in that class (early Carnap) or a parametric family (later Carnap) of specific confirmation functions. The first step in the process essentially placed Carnap in substantial agreement with subjectivists. The second step, however, limits the class of probabilities that distinguishes Carnap from the subjectivist brethren. Carnap largely shaped the way current philosophy views the nature and role of Probability, in particular its widespread acceptance of the Bayesian paradigm.

  • ALAN TURING AND THE CENTRAL LIMIT THEOREM
    American Mathematical Monthly, 1995
    Co-Authors: Sandy L. Zabell
    Abstract:

    Because the English mathematician Alan Mathison Turing (1912–1954) is remembered today primarily for his work in Mathematical logic (Turing machines and the “Entscheidungsproblem”), machine computation, and artificial intelligence (the “Turing test”), his name is not usually thought of in connection with either Probability or statistics. One of the basic tools in both of these subjects is the use of the normal or Gaussian distribution as an approximation, one basic result being the Lindeberg-Feller central limit theorem taught in first-year graduate courses in Mathematical Probability. No-one associates Turing with the central limit theorem, but in 1934 Turing, while still an undergraduate, rediscovered a version of Lindeberg's 1922 theorem and much of the Feller-Levy converse to it (then unpublished). This paper discusses Turing's connection with the central limit theorem and its surprising aftermath: his use of statistical methods during World War II to break key German military codes. 1 Introduction Turing went up to Cambridge as an undergraduate in the Fall Term of 1931, having gained a scholarship to King's College. (Ironically, King's was his second choice; he had failed to gain a scholarship to Trinity.) Two years later, during the course of his studies, Turing attended a series of lectures on the Methodology of Science, given in the autumn of 1933 by the distinguished astrophysicist Sir Arthur Stanley Eddington. One topic Eddington discussed was the tendency of experimental measurements subject to errors of observation to often have an approximately normal or Gaussian distribution.

Alexander Shen - One of the best experts on this subject based on the ideXlab platform.