Various processes are often classified as both deterministic and random or chaotic. The main difficulty in analysing the randomness of such processes is the apparent tension between the notions of randomness and determinism: what type of randomness could exist in a deterministic process? Ergodic theory seems to offer a particularly promising theoretical tool for tackling this problem by positing a hierarchy, the so-called ‘ergodic hierarchy’, which is commonly assumed to provide a hierarchy of increasing degrees of randomness. However, that notion (...) of randomness requires clarification. The mathematical definition of EH does not make explicit appeal to randomness; nor does the usual way of presenting EH involve a specification of the notion of randomness that is supposed to underlie the hierarchy. In this paper we argue that EH is best understood as a hierarchy of random behaviour if randomness is explicated in terms of unpredictability. We then show that, contrary to common wisdom, EH is useful in characterising the behaviour of Hamiltonian dynamical systems. (shrink)

The so-called ergodic hierarchy (EH) is a central part of ergodic theory. It is a hierarchy of properties that dynamical systems can possess. Its five levels are egrodicity, weak mixing, strong mixing, Kolomogorov, and Bernoulli. Although EH is a mathematical theory, its concepts have been widely used in the foundations of statistical physics, accounts of randomness, and discussions about the nature of chaos. We introduce EH and discuss how its applications in these fields.

Single-case and long-run propensity theories are among the main objective interpretations of probability. There have been various objections to these theories, e.g. that it is difficult to explain why propensities should satisfy the probability axioms and, worse, that propensities are at odds with these axioms, that the explication of propensities is circular and accordingly not informative, and that single-case propensities are metaphysical and accordingly non-scientific. We consider various propensity theories of probability and their prospects in light of these objections. We (...) argue that while propensity theories face challenges, these challenges do not undermine their validity as prospective interpretations of probability in science. (shrink)

In this paper and its sequel, I consider the significance of Jarrett’s and Shimony’s analyses of the so-called factorisability condition for clarifying the nature of quantum non-locality. In this paper, I focus on four types of non-locality: superluminal signalling, action-at-a-distance, non-separability and holism. In the second paper, I consider a fifth type of non-locality: superluminal causation according to ‘logically weak’ concepts of causation, where causal dependence requires neither action nor signalling. In this connection, I pay special attention to the difficulties (...) that superluminal causation raises in relativistic space–time. I conclude by evaluating the relevance of Jarrett’s and Shimony’s analyses for clarifying the question of the compatibility of quantum non-locality with relativity theory. My main conclusions are, first: these analyses are significant for clarifying the questions of superluminal signalling in quantum phenomena and for the compatibility of these phenomena with relativity. But, second, by contrast: these analyses are not very significant for the study of action-at-a distance, superluminal causation, non-separability and holism in quantum phenomena. (shrink)

De Finetti is one of the founding fathers of the subjective school of probability. He held that probabilities are subjective, coherent degrees of expectation, and he argued that none of the objective interpretations of probability make sense. While his theory has been influential in science and philosophy, it has encountered various objections. I argue that these objections overlook central aspects of de Finetti’s philosophy of probability and are largely unfounded. I propose a new interpretation of de Finetti’s theory that highlights (...) these aspects and explains how they are an integral part of de Finetti’s instrumentalist philosophy of probability. I conclude by drawing an analogy between misconceptions about de Finetti’s philosophy of probability and common misconceptions about instrumentalism. (shrink)

De Finetti is one of the founding fathers of the subjective school of probability. He held that probabilities are subjective, coherent degrees of expectation, and he argued that none of the objective interpretations of probability make sense. While his theory has been influential in science and philosophy, it has encountered various objections. I argue that these objections overlook central aspects of de Finetti’s philosophy of probability and are largely unfounded. I propose a new interpretation of de Finetti’s theory that highlights (...) these aspects and explains how they are an integral part of de Finetti’s instrumentalist philosophy of probability. I conclude by drawing an analogy between misconceptions about de Finetti’s philosophy of probability and common misconceptions about instrumentalism. (shrink)

In a preceding paper, I studied the significance of Jarrett's and Shimony's analyses of 'factorisability' into 'parameter independence' and 'outcome independence' for clarifying the nature of non-locality in quantum phenomena. I focused on four types of non-locality; superluminal signalling, action-at-a-distance, non-separability and holism. In this paper, I consider a fifth type of non-locality: superluminal causation according to 'logically weak' concepts of causation, where causal dependence requires neither action nor signalling. I conclude by considering the compatibility of non-factorisable theories with relativity (...) theory. In this connection, I pay special attention to the difficulties that superluminal causation raises in relativistic spacetime. My main findings in this paper are: first, parameter-dependent and outcome-dependent theories both involve superluminal causal connections between outcomes and between settings and outcomes. Second, while relativistic deterministic parameter-dependent theories seem impossible on pain of causal paradoxes, relativistic indeterministic parameter-dependent theories are not subjected to the same challenge. Third, current relativistic non-factorisable theories seem to have some rather unattractive characteristics. (shrink)

In this paper and its sequel, I consider the significance of Jarrett’s and Shimony’s analyses of the so-called factorisability condition for clarifying the nature of quantum non-locality. In this paper, I focus on four types of non-locality: superluminal signalling, action-at-a-distance, non-separability and holism. In the second paper, I consider a fifth type of non-locality: superluminal causation according to ‘logically weak’ concepts of causation, where causal dependence requires neither action nor signalling. In this connection, I pay special attention to the difficulties (...) that superluminal causation raises in relativistic space–time. I conclude by evaluating the relevance of Jarrett’s and Shimony’s analyses for clarifying the question of the compatibility of quantum non-locality with relativity theory. My main conclusions are, first: these analyses are significant for clarifying the questions of superluminal signalling in quantum phenomena and for the compatibility of these phenomena with relativity. But, second, by contrast: these analyses are not very significant for the study of action-at-a distance, superluminal causation, non-separability and holism in quantum phenomena. (shrink)

Modal interpretations are hidden-variable, no-collapse interpretations of quantum mechanics that were designed to solve the measurement problem and reconcile this theory with relativity. Yet, as no-go theorems by Dickson and Clifton, Arntzenius and Myrvold demonstrate, current modal interpretations are incompatible with relativity. In the mainstream modal interpretations, properties of composite systems are generally unrelated to the properties of their subsystems. We propose holistic and relational interpretations of properties to explain this failure of property composition. Based on these interpretations, we consider (...) strategies for circumventing Myrvold's theorem, which are also effective against the other two theorems. (shrink)

Cartwright and Humphreys have suggested theories of probabilistic causation for singular events, which are based on modifications of traditional causal linear modelling. On the basis of her theory, Cartwright offered an allegedly local, and non-factorizable, common-cause model for the EPR experiment. In this paper I consider Cartwright's and Humphreys' theories. I argue that, provided plausible assumptions obtain, local models for EPR in the framework of these theories are committed to Bell inequalities, which are violated by experiment.

Bruno de Finetti is one of the founding fathers of the subjectivist school of probability, where probabilities are interpreted as rational degrees of belief. His work on the relation between the theorems of probability and rationality is among the corner stones of modern subjective probability theory. De Finetti maintained that rationality requires that degrees of belief be coherent, and he argued that the whole of probability theory could be derived from these coherence conditions. De Finetti’s interpretation of probability has been (...) highly influential in science. This paper focuses on the application of this interpretation to quantum mechanics. We argue that de Finetti held that the coherence conditions of degrees of belief in events depend on their verifiability. Accordingly, the standard coherence conditions of degrees of belief that are familiar from the literature on subjective probability only apply to degrees of belief in events which could be jointly verified; and the coherence conditions of degrees of belief in events that cannot be jointly verified are weaker. While the most obvious explanation of de Finetti’s verificationism is the influence of positivism, we argue that it could be motivated by the radical subjectivist and instrumental nature of probability in his interpretation; for as it turns out, in this interpretation it is difficult to make sense of the idea of coherent degrees of belief in, and accordingly probabilities of unverifiable events. We then consider the application of this interpretation to quantum mechanics, concentrating on the Einstein-Podolsky-Rosen experiment and Bell’s theorem. (shrink)

Quantum mechanics portrays the universe as involving non-local influences that are difficult to reconcile with relativity theory. By postulating backward causation, retro-causal interpretations of quantum mechanics could circumvent these influences and accordingly reconcile quantum mechanics with relativity. The postulation of backward causation poses various challenges for the retro-causal interpretations of quantum mechanics and for the existing conceptual frameworks for analyzing counterfactual dependence, causation and causal explanation. In this chapter, we analyze the nature of time, causation and explanation in a local, (...) deterministic retro-causal interpretation of quantum mechanics that is inspired by Bohmian mechanics. This interpretation of quantum mechanics offers a deterministic, local ‘hidden-variables’ model of the Einstein-Podolsky-Rosen experiment that poses a new challenge for Reichenbach’s principle of the common cause. In this model, the common cause – the state of particles at the emission from the source – screens off the correlation between its effects – the distant measurement outcomes – but nevertheless fails to explain the correlation between the effects. (shrink)

The correlations between distant systems in typical quantum situations, such as Einstein-Podolosky-Rosen experiments, strongly suggest that the quantum realm involves curious types of non-Iocal influences. In this paper, I study in detail the nature of these non-Iocal influences, as depicted by various quantum theories. I show how different quantum theories realise non-Iocality in different ways, whichreflect different ontological settings.

The correlations between distant systems in typical quantum situations, such as Einstein-Podolosky-Rosen experiments, strongly suggest that the quantum realm involves curious types of non-Iocal influences. In this paper, I study in detail the nature of these non-Iocal influences, as depicted by various quantum theories. I show how different quantum theories realise non-Iocality in different ways, whichreflect different ontological settings.

Recent no go theorems by Dickson and Clifton (1998), Arntzenius (1998) and Myrvold (2002) demonstrate that current modal interpretations are incompatible with relativity. In this paper we propose strategies for how to circumvent these theorems. We further show how these strategies can be developped into new modal interpretations in which the properties of systems are in general either holistic or relational. We explicitly write down an outline of dynamics for these properties which does not pick out a preferred foliation of (...) spacetime. (shrink)

This volume focuses on various questions concerning the interpretation of probability and probabilistic reasoning in biology and physics. It is inspired by the idea that philosophers of biology and philosophers of physics who work on the foundations of their disciplines encounter similar questions and problems concerning the role and application of probability, and that interaction between the two communities will be both interesting and fruitful. In this introduction we present the background to the main questions that the volume focuses on (...) and summarize the highlights of the individual contributions. (shrink)

Robert Larmer and Alvin Plantinga have argued that modern physics is compatible with the idea that the physical universe is open to God’s supernatural action and that such action would not involve any violation of laws of nature. Thus, they have concluded that supernatural miracles are compatible with modern science. I argue that their line of reasoning is based on an incorrect interpretation of conservation laws and that supernatural miracles would involve violations of laws of nature.

Review of: Peter Smith, Explaining chaos. Cambridge: Cambridge univeristy Press, 1998. ISBN 0 521 47747 6.

Two of the main interpretative problems in quantum mechanics are the so-called measurement problem and the question of the compatibility of quantum mechanics with relativity theory. Modal interpretations of quantum mechanics were designed to solve both of these problems. They are no-collapse (typically) indeterministic interpretations of quantum mechanics that supplement the orthodox state description of physical systems by a set of possessed properties that is supposed to be rich enough to account for the classical-like behavior of macroscopic systems, but sufficiently (...) restricted so as to avoid the no-hidden-variables theorems. But, as recent no-go theorems suggest, current modal interpretations are incompatible with relativity. In this paper, we suggest a strategy for circumventing these theorems. We then show how this strategy could naturally be integrated in a relational version of the modal interpretation, where quantum-mechanical states assign relational rather than intrinsic properties. (shrink)

Review of: Peter Smith, Explaining chaos. Cambridge: Cambridge univeristy Press, 1998. ISBN 0 521 47747 6.

Review of: Peter Smith, Explaining chaos. Cambridge: Cambridge univeristy Press, 1998. ISBN 0 521 47747 6.

Review of: Peter Smith, Explaining chaos. Cambridge: Cambridge univeristy Press, 1998. ISBN 0 521 47747 6.