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36
The Sources of Kolmogorov’s Grundbegriffe
, 2006
"... Andrei Kolmogorov’s Grundbegriffe Wahrscheinlichkeitsrechnung put probability’s modern mathematical formalism in place. It also provided a philosophy of probability—an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two a ..."
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Andrei Kolmogorov’s Grundbegriffe Wahrscheinlichkeitsrechnung put probability’s modern mathematical formalism in place. It also provided a philosophy of probability—an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two aspects of the Grundbegriffe—the work of the earlier scholars whose ideas Kolmogorov synthesized.
Dynamical bias in the coin toss
, 2004
"... We analyze the natural process of flipping a coin which is caught in the hand. We prove that vigorouslyflipped coins are biased to come up the same way they started. The amount of bias depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measureme ..."
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Cited by 11 (1 self)
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We analyze the natural process of flipping a coin which is caught in the hand. We prove that vigorouslyflipped coins are biased to come up the same way they started. The amount of bias depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measurements of this parameter based on highspeed photography are reported. For natural flips, the chance of coming up as started is about.51.
Time, quantum mechanics, and probability
 Synthese
, 1998
"... ABSTRACT. A variety of ideas arisiüg in decoherence theory, and in the ongoing debate over Everett’s relativestate theory, can be linked to issues in relativity theory and the philosophy of time, speci…cally the relational theory of tense and of identity over time. These have been systematically pr ..."
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ABSTRACT. A variety of ideas arisiüg in decoherence theory, and in the ongoing debate over Everett’s relativestate theory, can be linked to issues in relativity theory and the philosophy of time, speci…cally the relational theory of tense and of identity over time. These have been systematically presented in companion papers (Saunders 1995, 1996a); in what follows we shall consider the same circle of ideas, but speci…cally in relation to the interpretation of probability, and its identi…cation with relations in the Hilbert space norm. The familiar objection that Everett’s approach yields probabilities di¤erent from quantum mechanics is easily dealt with. The more fundamental question is how to interpret these probabilities consistent with the relational theory of change, and the relational theory of identity over time. I shall show that the relational theory needs nothing more than the physical, minimal criterion of identity as de…ned by Everett’s theory, and that this can be transparently interpreted in terms of the ordinary notion of the chance occurrence of an event, as witnessed in the present. It is in this sense that the theory has empirical content. 1
Quantum Probability Theory
, 2006
"... The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this e ..."
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Cited by 10 (3 self)
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The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems
Basic Elements and Problems of Probability Theory
, 1999
"... After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probabil ..."
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Cited by 6 (0 self)
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After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the measuretheoretical codification of stochastic processes genuine chance processes can be defined rigorously as socalled regular processes which do not allow a longterm prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes. Another individual description refers to algorithmic procedures which connect the intrinsic randomness of a finite sequence with the complexity of the shortest program necessary to produce the sequence. Finally, we ask why there can be laws of chance. We argue that random events fulfill the laws of chance if and only if they can be reduced to (possibly hidden) deterministic events. This mathematical result may elucidate the fact that not all nonpredictable events can be grasped by the methods of mathematical probability theory.
Nonstationarity versus scaling in hydrology
 J. Hydrol
"... Abstract The perception of a changing climate, which impacts also hydrological processes, is now generally admitted. However, the way of handling the changing nature of climate in hydrologic practice and especially in hydrological statistics has not become clear so far. The most common modelling app ..."
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Abstract The perception of a changing climate, which impacts also hydrological processes, is now generally admitted. However, the way of handling the changing nature of climate in hydrologic practice and especially in hydrological statistics has not become clear so far. The most common modelling approach is to assume that longterm trends, which have been found to be omnipresent in long hydrological time series, are “deterministic ” components of the time series and the processes represented by the time series are nonstationary. In this paper, it is maintained that this approach is contradictory in its rationale and even in the terminology it uses. As a result, it may imply misleading perception of phenomena and estimate of uncertainty. Besides, it is maintained that a stochastic approach hypothesizing stationarity and simultaneously admitting a scaling behaviour reproduces climatic trends (considering them as largescale fluctuations) in a manner that is logically consistent, easy to apply and
What is a Random Sequence
 The Mathematical Association of America, Monthly
, 2002
"... there laws of randomness? These old and deep philosophical questions still stir controversy today. Some scholars have suggested that our difficulty in dealing with notions of randomness could be gauged by the comparatively late development of probability theory, which had a ..."
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Cited by 4 (1 self)
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there laws of randomness? These old and deep philosophical questions still stir controversy today. Some scholars have suggested that our difficulty in dealing with notions of randomness could be gauged by the comparatively late development of probability theory, which had a
Probability as typicality
, 2006
"... The concept of typicality refers to properties holding for the “vast majority” of cases and is a fundamental idea of the qualitative approach to dynamical problems. We argue that measuretheoretical typicality would be the adequate viewpoint of the role of probability in classical statistical mechan ..."
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Cited by 3 (0 self)
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The concept of typicality refers to properties holding for the “vast majority” of cases and is a fundamental idea of the qualitative approach to dynamical problems. We argue that measuretheoretical typicality would be the adequate viewpoint of the role of probability in classical statistical mechanics, particularly in understanding the micro to macroscopic change of levels of description. Keywords: Statistical mechanics; Typicality; Probability.