Bertrand Paradox
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Bertrand’s Paradox and the Principle of Indifference
Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. Bertrand's Paradox and the Principle of Indifference will appeal to scholars and advanced students working in the philosophy of mathematics, epistemology, philosophy of science, probability theory and mathematical physics.
An Introduction to Geometrical Probability
A useful guide for researchers and professionals, graduate and senior undergraduate students, this book provides an in-depth look at applied and geometrical probability with an emphasis on statistical distributions. A meticulous treatment of geometrical probability, kept at a level to appeal to a wider audience including applied researchers who will find the book to be both functional and practical with the large number of problems chosen from different disciplines A few topics such as packing and covering problems that have a vast literature are introduced here at a peripheral level for the purpose of familiarizing readers who are new to the area of research.
Linguistic Copenhagen Interpretation of Quantum Theory
Author: Shiro ISHIKAWA
language: en
Publisher: Shiho-Shuppan Publisher
Release Date: 2023-11-15
Various interpretations of quantum mechanics have been proposed such as the Copenhagen interpretation and the many-worlds interpretation. The linguistic Copenhagen interpretation in this book is a kind of the Copenhagen interpretation derived from von Neumann's formulation of quantum mechanics on Hilbert spaces. Von Neumann had the widest coverage of any mathematician of his time, integrating pure and applied sciences and making major contributions to many fields, including mathematics, physics, economics, computing, and statistics. He was not a genius who specialised only in mathematics and physics, but an all-round genius. From this fact we are tempted to expect the following. (#1) the quantum theory generated from the linguistic Copenhagen interpretation called Quantum Language (QL) is a very large theory that includes not only quantum mechanics of physics but also classical statistics. More generally we may say (#2) QL is the scientific realisation of the dualistic idealism of philosophy. In this book I devote myself to proving (#1). QL consists of two axioms (measurement and causality) and the linguistic Copenhagen interpretation. I first prove von Neumann-Lüders projection postulate in QL. This is a solution in QL, and it is undecided whether it is a physical solution, but the theorem allows QL to be discussed without being interfered by various paradoxes (e.g., Schrödinger's cat, etc.). Also, recall that there are no axioms in statistics. This means that we do not yet have 'theoretical statistics'. However, if we consider that QL for classical systems = theoretical statistics, we can then introduce an elegant understanding into statistics. In most books of statistics, Fisher's maximum likelihood method is not given a due treatment. From the quantum linguistic point of view, the most basic arguments are Fisher's maximum likelihood method and regression analysis. They are strongly linked to the measurement and causality axioms, respectively. As modern statistics continues to develop rapidly in the direction of application now, it is essential to take an overview of statistics as a whole under an umbrella of theoretical statistics. For (#2), refer to my previous book (i.e., History of Western Philosophy from a perspective of quantum theory- Introduction to theory of everyday science– Shiho-Shuppan Publisher, 425 p. (2023)). Throughout this book as wall as the one above, I assert that von Neumann's formulation of quantum mechanics should not be confined in physics, but should be regarded as a fundamental theory of science.