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Metaphysicians speak of laws of nature in terms of necessity and universality; scientists, in terms of symmetry and invariance. In this book van Fraassen argues that no metaphysical account of laws can succeed. He analyzes and rejects the arguments that there are laws of nature, or that we must believe there are, and argues that we should disregard the idea of law as an adequate clue to science. After exploring what this means for general epistemology, the author develops the empiricist (...) |
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The principle of indifference is supposed to suffice for the rational assignation of probabilities to possibilities. Bertrand advances a probability problem, now known as his paradox, to which the principle is supposed to apply; yet, just because the problem is ill‐posed in a technical sense, applying it leads to a contradiction. Examining an ambiguity in the notion of an ill‐posed problem shows that there are precisely two strategies for resolving the paradox: the distinction strategy and the well‐posing strategy. The main (...) |
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Jaynes contends that in many statistical problems a seemingly indeterminate probability distribution is made unique by the transformation group of necessarily implied invariance properties, thereby justifying the principle of indifference. To illustrate and substantiate his claims he considers Bertrand's Paradox. These assertions are here refuted and the traditional attitude is vindicated. |
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water paradox has long served as an argument against the Principle of Indifference. A solution to the paradox is proposed, with a view toward resolving general difficulties in applying the principle. |
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The Twentieth Century has seen a dramatic rise in the use of probability and statistics in almost all fields of research. This has stimulated many new philosophical ideas on probability. _Philosophical Theories of Probability_ is the first book to present a clear, comprehensive and systematic account of these various theories and to explain how they relate to one another. Gillies also offers a distinctive version of the propensity theory of probability, and the intersubjective interpretation, which develops the subjective theory. |
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Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...) |
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The epistemic state of complete ignorance is not a probability distribution. In it, we assign the same, unique, ignorance degree of belief to any contingent outcome and each of its contingent, disjunctive parts. That this is the appropriate way to represent complete ignorance is established by two instruments, each individually strong enough to identify this state. They are the principle of indifference (PI) and the notion that ignorance is invariant under certain redescriptions of the outcome space, here developed into the (...) |
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This book: * assumes no mathematical background and keeps the technicalities to a minimum * explains the most important applications of probability theory to ... |
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_Probability: A Philosophical Introduction_ introduces and explains the principal concepts and applications of probability. It is intended for philosophers and others who want to understand probability as we all apply it in our working and everyday lives. The book is not a course in mathematical probability, of which it uses only the simplest results, and avoids all needless technicality. The role of probability in modern theories of knowledge, inference, induction, causation, laws of nature, action and decision-making makes an understanding of (...) |
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Bertrand's random-chord paradox purports to illustrate the inconsistency of the principle of indifference when applied to problems in which the number of possible cases is infinite. This paper shows that Bertrand's original problem is vaguely posed, but demonstrates that clearly stated variations lead to different, but theoretically and empirically self-consistent solutions. The resolution of the paradox lies in appreciating how different geometric entities, represented by uniformly distributed random variables, give rise to respectively different nonuniform distributions of random chords, and hence (...) |
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In the philosophy of probability there are two central questions we are concerned with. The first is: what is the correct formal theory of probability? Orthodoxy has it that Kolmogorov’s axioms are the correct axioms of probability. However, we shall see that there are good reasons to consider alternative axiom systems. The second central question is: what do probability statements mean? Are probabilities “out there”, in the world as frequencies, propensities, or some other objective feature of reality, or are probabilities (...) |
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This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and (...) |
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Many statistical problems, including some of the most important for physical applications, have long been regarded as underdetermined from the standpoint of a strict frequency definition of probability; yet they may appear wellposed or even overdetermined by the principles of maximum entropy and transformation groups. Furthermore, the distributions found by these methods turn out to have a definite frequency correspondence; the distribution obtained by invariance under a transformation group is by far the most likely to be observed experimentally, in the (...) |
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Scientific reasoning is—and ought to be—conducted in accordance with the axioms of probability. This Bayesian view—so called because of the central role it accords to a theorem first proved by Thomas Bayes in the late eighteenth ... |
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The classical interpretation of probability together with the principle of indifference is formulated in terms of probability measure spaces in which the probability is given by the Haar measure. A notion called labelling invariance is defined in the category of Haar probability spaces; it is shown that labelling invariance is violated, and Bertrand’s paradox is interpreted as the proof of violation of labelling invariance. It is shown that Bangu’s attempt to block the emergence of Bertrand’s paradox by requiring the re-labelling (...) |
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E. T. Jaynes' resolution of Bertrand's paradox in terms of invariance principles is criticized. An experimental setup is considered which generates general solutions to Bertrand's problem by rotating a line around a point a distancer+d from a circle of radiusr. The general solution obtained is neither translationally nor scale invariant, but depends on the value ofr/d. Only in the limitr/d » 0, when the line is just translating across the circle, is the distribution translationally invariant and scale invariant. In this (...) |
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This is the only book to chart the history and development of modern probability theory. It shows how in the first thirty years of this century probability theory became a mathematical science. The author also traces the development of probabilistic concepts and theories in statistical and quantum physics. There are chapters dealing with chance phenomena, as well as the main mathematical theories of today, together with their foundational and philosophical problems. Among the theorists whose work is treated at some length (...) |
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This paper justifies the inference of probabilities from symmetries. I supply some examples of important and correct inferences of this variety. Two explanations of such inferences -- an explanation based on the Principle of Indifference and a proposal due to Poincaré and Reichenbach -- are considered and rejected. I conclude with my own account, in which the inferences in question are shown to be warranted a posteriori, provided that they are based on symmetries in the mechanisms of chance setups. |
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Bertand’s paradox is a fundamental problem in probability that casts doubt on the applicability of the indifference principle by showing that it may yield contradictory results, depending on the meaning assigned to “randomness”. Jaynes claimed that symmetry requirements solve the paradox by selecting a unique solution to the problem. I show that this is not the case and that every variant obtained from the principle of indifference can also be obtained from Jaynes’ principle of transformation groups. This is because the (...) |
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The Principle of Indifference, which dictates that we ought to assign two outcomes equal probability in the absence of known reasons to do otherwise, is vulnerable to well-known objections. Nevertheless, the appeal of the principle, and of symmetry-based assignments of equal probability, persists. We show that, relative to a given class of symmetries satisfying certain properties, we are justified in calling certain outcomes equally probable, and more generally, in defining what we call relative probabilities. Relative probabilities are useful in providing (...) |
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This is the only book to chart the history and development of modern probability theory. It shows how in the first thirty years of this century probability theory became a mathematical science. The author also traces the development of probabilistic concepts and theories in statistical and quantum physics. There are chapters dealing with chance phenomena, as well as the main mathematical theories of today, together with their foundational and philosophical problems. Among the theorists whose work is treated at some length (...) |
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Bertrand's Paradox Revisited: Why Bertrand's 'Solutions' Are All Inapplicable. |
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In probability textbooks, it is widely claimed that zero probability does not mean impossibility. But what stands behind this claim? In this paper I offer an explanation to this claim based on Kolmogorov's formalism. As such, this explanation is relevant to all interpretations of Kolmogorov's probability theory. I start by clarifying that this claim refers only to nonempty events, since empty events are always considered as impossible. Then, I offer the following three reasons for the claim that nonempty events with (...) |
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