We investigate the meaning of the wave function by analyzing the mass and charge density distributions of a quantum system. According to protective measurement, a charged quantum system has effective mass and charge density distributing in space, proportional to the square of the absolute value of its wave function. In a realistic interpretation, the wave function of a quantum system can be taken as a description of either a physical field or the ergodic motion of a particle. The (...) essential difference between a field and the ergodic motion of a particle lies in the property of simultaneity; a field exists throughout space simultaneously, whereas the ergodic motion of a particle exists throughout space in a time-divided way. If the wave function is a physical field, then the mass and charge density will be distributed in space simultaneously for a charged quantum system, and thus there will exist gravitational and electrostatic self-interactions of its wave function. This not only violates the superposition principle of quantum mechanics but also contradicts experimental observations. Thus the wave function cannot be a description of a physical field but a description of the ergodic motion of a particle. For the later there is only a localized particle with mass and charge at every instant, and thus there will not exist any self-interaction for the wave function. Which kind of ergodic motion of particles then? It is argued that the classical ergodic models, which assume continuous motion of particles, cannot be consistent with quantum mechanics. Based on the negative result, we suggest that the wave function is a description of the quantum motion of particles, which is random and discontinuous in nature. On this interpretation, the square of the absolute value of the wave function not only gives the probability of the particle being found in certain locations, but also gives the probability of the particle being there. We show that this new interpretation of the wave function provides a natural realistic alternative to the orthodox interpretation, and its implications for other realistic interpretations of quantum mechanics are also briefly discussed. (shrink)

This article analyzes the implications of protective measurement for the meaning of the wave function. According to protective measurement, a charged quantum system has mass and charge density proportional to the modulus square of its wave function. It is shown that the mass and charge density is not real but effective, formed by the ergodic motion of a localized particle with the total mass and charge of the system. Moreover, it is argued that the ergodic motion (...) is not continuous but discontinuous and random. This result suggests a new interpretation of the wave function, according to which the wave function is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations. It is shown that the suggested interpretation of the wave function disfavors the de Broglie-Bohm theory and the many-worlds interpretation but favors the dynamical collapse theories, and the random discontinuous motion of particles may provide an appropriate random source to collapse the wave function. (shrink)

James A. Diefenbeck, Wayward Reflections on the History ofPhilosophyThomas R. Flynn Sartre, Foucault and Historical Reason. Volume 1:Toward an Existential Theory of HistoryMark Golden and Peter Toohey Inventing Ancient Culture:Historicism, Periodization and the Ancient WorldZenonas Norkus Istorika: Istorinis IvadasEverett Zimmerman The Boundaries of Fiction: History and theEighteenth‐Century British Novel.

We investigate the validity of the field explanation of the wave function by analyzing the mass and charge density distributions of a quantum system. It is argued that a charged quantum system has effective mass and charge density distributing in space, proportional to the square of the absolute value of its wave function. This is also a consequence of protective measurement. If the wave function is a physical field, then the mass and charge density will be distributed in space simultaneously (...) for a charged quantum system, and thus there will exist a remarkable electrostatic self-interaction of its wave function, though the gravitational self-interaction is too weak to be detected presently. This not only violates the superposition principle of quantum mechanics but also contradicts experimental observations. Thus we conclude that the wave function cannot be a description of a physical field. In the second part of this paper, we further analyze the implications of these results for the main realistic interpretations of quantum mechanics, especially for de Broglie-Bohm theory. It has been argued that de Broglie-Bohm theory gives the same predictions as quantum mechanics by means of quantum equilibrium hypothesis. However, this equivalence is based on the premise that the wave function, regarded as a Ψ-field, has no mass and charge density distributions, which turns out to be wrong according to the above results. For a charged quantum system, both Ψ-field and Bohmian particle have charge density distribution. This then results in the existence of an electrostatic self-interaction of the field and an electromagnetic interaction between the field and Bohmian particle, which contradicts both the predictions of quantum mechanics and experimental observations. Therefore, de Broglie-Bohm theory as a realistic interpretation of quantum mechanics is probably wrong. Lastly, we suggest that the wave function is a description of some sort of ergodic motion (e.g. random discontinuous motion) of particles, and we also briefly analyze the implications of this suggestion for other realistic interpretations of quantum mechanics including many-worlds interpretation and dynamical collapse theories. (shrink)

We investigate the implications of protective measurement for de Broglie-Bohm theory, mainly focusing on the interpretation of the wave function. It has been argued that the de Broglie-Bohm theory gives the same predictions as quantum mechanics by means of quantum equilibrium hypothesis. However, this equivalence is based on the premise that the wave function, regarded as a Ψ-field, has no mass and charge density distributions. But this premise turns out to be wrong according to protective measurement; a charged quantum system (...) has effective mass and charge density distributing in space, proportional to the square of the absolute value of its wave function. Then in the de Broglie-Bohm theory both Ψ-field and Bohmian particle will have charge density distribution for a charged quantum system. This will result in the existence of an electrostatic self-interaction of the field and an electromagnetic interaction between the field and Bohmian particle, which not only violates the superposition principle of quantum mechanics but also contradicts experimental observations. Therefore, the de Broglie-Bohm theory as a realistic interpretation of quantum mechanics is problematic according to protective measurement. Lastly, we briefly discuss the possibility that the wave function is not a physical field but a description of some sort of ergodic motion (e.g. random discontinuous motion) of particles. (shrink)

This article re-examines Schrödinger’s charge density hypothesis, according to which the charge of an electron is distributed in the whole space, and the charge density in each position is proportional to the modulus squared of the wave function of the electron there. It is shown that the charge distribution of a quantum system can be measured by protective measurements as expectation values of certain observables, and the results as predicted by quantum mechanics confirm Schrödinger’s original hypothesis. Moreover, the physical origin (...) of the charge distribution is also investigated. It is argued that the charge distribution of a quantum system is effective, that is, it is formed by the ergodic motion of a localized particle with the charge of the system. (shrink)

It is argued that in Bohmian mechanics the effective wave function of a subsystem of the universe does not encode the influences of other particles on the subsystem. This suggests that the ontology of Bohmian mechanics does not consist only in Bohmian particles and their positions. It is nonetheless pointed out that since the wave function in configuration space may represent the state of ergodic motion of non-Bohmian particles in three-dimensional space, the ontology of Bohmian mechanics may still (...) consist only in particles. (shrink)

All physical processes are deterministic de iure. Physicists speak about different types of determinism of physical processes, depending on the degree with which their course can be anticipated. Usually, the course of ergodic processes can be predicted with less certainty than the non-ergodic ones, the latter being integrable. Recent measurements of motions of single particles in composite systems, especially in living biological cells, show that such motions are, in most cases, breaking the Boltzmann’s ergodic hypothesis. On the (...) other hand, their trajectories are random, i.e., one cannot know a priori where the particle will be even in near future. This leads to conclusion that many existing in nature processes are nonergodic but not integrable, therefore predictable only in the mean, representing still other type of determinism. (shrink)

For the ancient Greeks, the world was both Eros, the god of chaos and creativity, and Logos, the regularity of the heavens as law. From chaos the world came forth. The world was home to ultimate creativity. Two thousand years later Kepler, Galileo, and then mighty Newton created deterministic classical physics in which all that happens in the universe is determined by the laws of motion, initial and boundary conditions. The Theistic God who worked miracles became the Deistic God (...) who set up the universe and let Newton’s laws take over. Eros, raw creativity, is dead, all is Logos.Quantum mechanics replaced the determinism of classical physics with fundamental indeterminism. This was a major crisis in physics, but remained entirely within the Newtonian paradigm of laws, here the Schrödinger equation, initial and boundary conditions. The Schrödinger equation entails the deterministic propagation of a probability distribution. The probabilities concern the indeterminate outcomes of quantum measurement events.The central issue of this article is to show that no laws at all determine or entail the becoming of our biosphere or any other among the 1022 solar systems in the universe. This claim is radical. If no laws determine or entail the becoming of biospheres, and biospheres are part of the universe, the Pythagorean dream that All is Number, All is Logos, is dead. There is no Final Theory that entails all that become in the universe. Eros is again, and always was, rambunctiously alive in the raw creativity of the becoming of life anywhere in the universe. The becoming of life is based on physics, on Logos, but beyond it, emerging from Eros.The reasons Eros is at play in the evolution of biospheres are fourfold.First, the universe will not make all possible complex things. That is, the universe is vastly non-ergodic. Yet complex things such as the human heart exist.Second, the reason human hearts exist is that organisms are Kantian Wholes in which the parts exist for and by means of the whole. Hearts exist because they fulfill the function of pumping the blood that keeps the whole organism alive. Such organisms propagate progeny that carry with them the hearts that keep them alive.Third, adaptations like hearts, the flagellar motor, the loop of Henle in kidneys that concentrates urine, and flight feathers, are tinkered-together contraptions stumbled upon in evolution. Parts and processes that arise in evolution are jury-rigged for unprestatable new functions that help keep the entire Kantian Whole organism alive and propagating in the evolving biosphere. There is no deductive theory of jury rigging. We cannot deduce the emergence of such novel functions.Fourth, the functions of parts of organisms emerge in unprestatable ways and form the unprestatable and ever-changing phase space of evolution. Since we cannot prestate the ever-changing phase space of evolution, we can write no laws of motion in differential equation form, so cannot integrate the equations we do not have. Thus, no laws entail the becoming of any biosphere.Evolving biospheres are everywhere creative. This is Eros, the chaos from which the world emerges. This profound creative emergence is the stuff of story, of narrative. Evolving biospheres always were and always will be both Eros and Logos, the stuff of story and the stuff of law. We always live in a world of Art and Science. (shrink)

Here is an accurate and readable translation of a seminal article by Henri Poincaré that is a classic in the study of dynamical systems popularly called chaos theory. In an effort to understand the stability of orbits in the solar system, Poincaré applied a Hamiltonian formulation to the equations of planetary motion and studied these differential equations in the limited case of three bodies to arrive at properties of the equations' solutions, such as orbital resonances and horseshoe orbits. Poincaré (...) wrote for professional mathematicians and astronomers interested in celestial mechanics and differential equations. Contemporary historians of math or science and researchers in dynamical systems and planetary motion with an interest in the origin or history of their field will find his work fascinating. (shrink)

Boltzmann’s approach to statistical mechanics is widely believed to be conceptually superior to Gibbs’ formulation. However, the microcanonical distribution often fails to behave as expected: The ergodicity of the motion relative to it can rarely be established for realistic systems; worse, it can often be proved to fail. Also, the approach involves idealizations that have little physical basis. Here we take Khinchin’s advice and propose a de…nition of equilibrium that is more realistic: The de…nition re‡ects the fact that the (...) system is made of a great number of particles, and implies that all measurable macroscopic observables have steady values. (shrink)

The spectral fluctuations of quantum (or wave) systems with a chaotic classical (or ray) limit are mostly universal and faithful to random-matrix theory. Taking up ideas of Pechukas and Yukawa we show that equilibrium statistical mechanics for the fictitious gas of particles associated with the parametric motion of levels yields spectral fluctuations of the random-matrix type. Previously known clues to that goal are an appropriate equilibrium ensemble and a certain ergodicity of level dynamics. We here complete the reasoning by (...) establishing a power law for the ħ dependence of the mean parametric separation of avoided level crossings. Due to that law universal spectral fluctuations emerge as average behavior of a family of quantum dynamics drawn from a control parameter interval which becomes vanishingly small in the classical limit; the family thus corresponds to a single classical system. We also argue that classically integrable dynamics cannot produce universal spectral fluctuations since their level dynamics resembles a nearly ideal Pechukas–Yukawa gas. (shrink)

A formulation of probabilistic causality is given in terms of the theory of abstract dynamical systems. Causal factors are identified as invariants of motion of a system. Repetition of an experiment leads to the notion of stationarity, and causal factors yield a decomposition of the stationary probability law of the experiment into ergodic components. In these, statistical behaviour is uniform. Control of identified causal factors leads to a corresponding statistical law for the events, which is offered as a (...) notion of probabilistic causality. After a suggestion by Feller, randomization is identified as mixing, formulated in above terms. (shrink)

We study a model proposed by Liboff (Found. Phys. Lett. 10:89, 1997) to violate the second law of thermodynamics. Discs are moving without friction in three connected channels inclined by π/3 with respect to each other. Based on the geometry considerations, it was argued that eventually all the discs end up in the middle channel regardless of their initial positions. This would mean a decrease of the entropy of the system and violation of the second law. We argue that no (...) such anomalous behavior occurs in the system. (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 its applications in these fields.

This paper analyzes the ergodic hypothesis in the context of Boltzmann’s late work in statistical mechanics, where Boltzmann lays the foundations for what is today known as the typicality account. I argue that, based on the concepts of stationarity (of the measure) and typicality (of the equilibrium state), the ergodic hypothesis, as an idealization, is a consequence rather than an assumption of Boltzmann’s account. More precisely, it can be shown that every system with a stationary measure and an (...) equilibrium state (be it a typical state with respect to the phase space or the time average) behaves essentially as if it were ergodic. I claim that Boltzmann was aware of this fact as it grounds both his notion of equilibrium, relating it to the thermodynamic notion of equilibrium, and his estimate of the fluctuation rates. (shrink)

We argue that, contrary to some analyses in the philosophy of science literature, ergodic theory falls short in explaining the success of classical equilibrium statistical mechanics. Our claim is based on the observations that dynamical systems for which statistical mechanics works are most likely not ergodic, and that ergodicity is both too strong and too weak a condition for the required explanation: one needs only ergodic-like behaviour for the finite set of observables that matter, but the behaviour (...) must ensure that the approach to equilibrium for these observables is on the appropriate time-scale. (shrink)

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)

Why does classical equilibrium statistical mechanics work? Malament and Zabell (1980) noticed that, for ergodic dynamical systems, the unique absolutely continuous invariant probability measure is the microcanonical. Earman and Rédei (1996) replied that systems of interest are very probably not ergodic, so that absolutely continuous invariant probability measures very distant from the microcanonical exist. In response I define the generalized properties of epsilon-ergodicity and epsilon-continuity, I review computational evidence indicating that systems of interest are epsilon-ergodic, I adapt (...) Malament and Zabell’s defense of absolute continuity to support epsilon-continuity, and I prove that, for epsilon-ergodic systems, every epsilon-continuous invariant probability measure is very close to the microcanonical. (shrink)

The traditional use of ergodic theory in the foundations of equilibrium statistical mechanics is that it provides a link between thermodynamic observables and microcanonical probabilities. First of all, the ergodic theorem demonstrates the equality of microcanonical phase averages and infinite time averages (albeit for a special class of systems, and up to a measure zero set of exceptions). Secondly, one argues that actual measurements of thermodynamic quantities yield time averaged quantities, since measurements take a long time. The combination (...) of these two points is held to be an explanation why calculating microcanonical phase averages is a successful algorithm for predicting the values of thermodynamic observables. It is also well-known that this account is problematic.

This survey intends to show that ergodic theory nevertheless may have important roles to play, and it explores three other uses of ergodic theory. Particular attention is paid, firstly, to the relevance of specific interpretations of probability, and secondly, to the way in which the concern with systems in thermal equilibrium is translated into probabilistic language. With respect to the latter point, it is argued that equilibrium should not be represented as a stationary probability distribution as is standardly done; instead, a weaker definition is presented. (shrink)

I prove a theorem on the precise connection of the time and phase-space average of the Boltzmann equilibrium showing that the behaviour of a dynamical system with a stationary measure and a dominant equilibrium state is qualitatively ergodic. Explicitly, I show that given a dynamical system with a stationary measure and a region of overwhelming phase-space measure, almost all trajectories spend almost all of their time in that region. Conversely, given that almost all trajectories spend almost all of their (...) time in a certain region, that region is of overwhelming phase-space measure. In total, the time and phase-space average of the equilibrium state approximately coincide. Consequently, equilibrium can be defined equivalently in terms of the time or the phase-space average. Moreover, since the two averages are almost equal, the behaviour of the system is essentially ergodic. While this does not explain the approach to equilibrium, it provides a means to estimate the fluctuation rates. (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.

Some philosphers of science of an empiricist and pragmatist bent have proposed models of statistical explanation, but have then become sceptical of the adequacy of these models. It is argued that general considerations concerning the purpose of function of explanation in science which are usually appealed to by such philosophers show that their scepticism is not well taken; for such considerations provide much the same rationale for the search for statistical explanations, as these philosophers have characterized them, as they do (...) for lawlike explanations. But, it is further argued, a significant piece of what is frequently offered as an explanation of well known phenomena in statistical mechanics, fails to meet this general "pragmatic rationale" for statistical, or indeed any kind of, explanation. The question then arises whether the physicists have misconstrued the value of this piece of physical theorizing, ergodic theory, taking it to be explanatory when it is actually not; or whether, instead, the philosopher's account of just what is genuinely explanatory is too narrow. (shrink)

New results in ergodic theory show that averages of repeated measurements will typically diverge with probability one if there are random errors in the measurement of time. Since mean-square convergence of the averages is not so susceptible to these anomalies, we are led again to compare the mean and pointwise ergodic theorems and to reconsider efforts to determine properties of a stochastic process from the study of a generic sample path. There are also implications for models of time (...) and the interaction between observer and observable. (shrink)

I prove a theorem on the precise connection of the time and phase-space average of the Boltzmann equilibrium showing that the behaviour of a dynamical system with a stationary measure and a dominant equilibrium state is qualitatively ergodic. Explicitly, I show that given a dynamical system with a stationary measure and a region of overwhelming phase-space measure, almost all trajectories spend almost all of their time in that region. Conversely, given that almost all trajectories spend almost all of their (...) time in a certain region, that region is of overwhelming phase-space measure. In total, the time and phase-space average of the equilibrium state approximately coincide. Consequently, equilibrium can be defined equivalently in terms of the time or the phase-space average. Moreover, since the two averages are almost equal, the behaviour of the system is essentially ergodic. While this does not explain the approach to equilibrium, it provides a means to estimate the fluctuation rates. (shrink)

I begin by reviewing some recent work on the status of the geodesic principle in general relativity and the geometrized formulation of Newtonian gravitation. I then turn to the question of whether either of these theories might be said to ``explain'' inertial motion. I argue that there is a sense in which both theories may be understood to explain inertial motion, but that the sense of ``explain'' is rather different from what one might have expected. This sense of (...) explanation is connected with a view of theories---I call it the ``puzzleball view''---on which the foundations of a physical theory are best understood as a network of mutually interdependent principles and assumptions. (shrink)

There are several ontological and consequently also methodological mistakes in contemporary mainstream economics. Among them, the so-called ergodic axiom is play significant role. It is understandable that the real economy elaborated as formalized mental model looks like dynamic system on first sight. However, that is right only of dynamical systems in mathematical formalism. Economy that is in our understanding societal and/or collective economy is complex evolving organism. If we imagine such organism in the form of dynamical system that is (...) as clear mathematical formalism, we are losing their crucial authentic character. The significant irredeemable attribute of societal economy is lying in his complex evolving network process character created by large population of people with different decision-making and complex realizing among them. Going from these imaginations the two entities in a question that is dynamical system with their ergodicity and societal economic organism as complex evolving network are qualitative very different ones. That is the reason why we cannot accede with endeavours to draw on living economy straitjacket of ergodic axiom. To articulate that cause by other words ergodic dynamical systems are applicable for physical and partly for chemical entities and only scarcely are fit for living organisms. On the other hand however, as clear method the ergodic dynamical system have good applying for didactical approaches in economics where helping in better understanding some types of complexities in dynamics. The purpose of that essay is to discuss problems around usability of ergodic dynamical system theory and methods in economics in the age of advanced ICT knowledge based society. (shrink)

This book examines the birth of the scientific understanding of motion. It investigates which logical tools and methodological principles had to be in place to give a consistent account of motion, and which mathematical notions were introduced to gain control over conceptual problems of motion. It shows how the idea of motion raised two fundamental problems in the 5th and 4th century BCE: bringing together being and non-being, and bringing together time and space. The first problem (...) leads to the exclusion of motion from the realm of rational investigation in Parmenides, the second to Zeno's paradoxes of motion. Methodological and logical developments reacting to these puzzles are shown to be present implicitly in the atomists, and explicitly in Plato who also employs mathematical structures to make motion intelligible. With Aristotle we finally see the first outline of the fundamental framework with which we conceptualise motion today. (shrink)

The foundation of statistical mechanics and the explanation of the success of its methods rest on the fact that the theoretical values of physical quantities (phase averages) may be compared with the results of experimental measurements (infinite time averages). In the 1930s, this problem, called the ergodic problem, was dealt with by ergodic theory that tried to resolve the problem by making reference above all to considerations of a dynamic nature. In the present paper, this solution will be (...) analyzed first, highlighting the fact that its very general nature does not duly consider the specificities of the systems of statistical mechanics. Second, Khinchin’s approach will be presented, that starting with more specific assumptions about the nature of systems, achieves an asymptotic version of the result obtained with ergodic theory. Third, the statistical meaning of Khinchin’s approach will be analyzed and a comparison between this and the point of view of ergodic theory is proposed. It will be demonstrated that the difference consists principally of two different perspectives on the ergodic problem: that of ergodic theory puts the state of equilibrium at the center, while Khinchin’s attempts to generalize the result to non-equilibrium states. (shrink)

I argue that Plato believes that the soul must be both the principle of motion and the subject of cognition because it moves things specifically by means of its thoughts. I begin by arguing that the soul moves things by means of such acts as examination and deliberation, and that this view is developed in response to Anaxagoras. I then argue that every kind of soul enjoys a kind of cognition, with even plant souls having a form of Aristotelian (...) discrimination (krisis), and that there is therefore no completely unintelligent, evil soul in the cosmos that can explain disorderly motions; as a result, the soul is not the principle of all motion but only motion in the cosmos after it has been ordered by the Demiurge. (shrink)

In apparent motion experiments, participants are presented with what is in fact a succession of two brief stationary stimuli at two different locations, but they report an impression of movement. Philosophers have recently debated whether apparent motion provides evidence in favour of a particular account of the nature of temporal experience. I argue that the existing discussion in this area is premised on a mistaken view of the phenomenology of apparent motion and, as a result, the space (...) of possible philosophical positions has not yet been fully explored. In particular, I argue that the existence of apparent motion is compatible with an account of the nature of temporal experience that involves a version of direct realism. In doing so, I also argue against two other claims often made about apparent motion, viz. that apparent motion is the psychological phenomenon that underlies motion experience in the cinema, and that apparent motion is subjectively indistinguishable from real motion. (shrink)

De Finetti's representation theorem is a special case of the ergodic decomposition of stationary probability measures. The problems of the interpretation of probabilities centred around de Finetti's theorem are extended to this more general situation. The ergodic decomposition theorem has a physical background in the ergodic theory of dynamical systems. Thereby the interpretations of probabilities in the cases of de Finetti's theorem and its generalization and in ergodic theory are systematically connected to each other.

The pointwise ergodic theorem is nonconstructive. In this paper, we examine origins of this non-constructivity, and determine the logical strength of the theorem and of the auxiliary statements used to prove it. We discuss properties of integrable functions and of measure preserving transformations and give three proofs of the theorem, though mostly focusing on the one derived from the mean ergodic theorem. All the proofs can be carried out in ACA₀; moreover, the pointwise ergodic theorem is equivalent (...) to (ACA) over the base theory RCA₀. (shrink)

We propose an "explanation scheme" for why the Gibbs phase average technique in classical equilibrium statistical mechanics works. Our account emphasizes the importance of the Khinchin-Lanford dispersion theorems. We suggest that ergodicity does play a role, but not the one usually assigned to it.

V’yugin has shown that there are a computable shift-invariant measure on $2^{\mathbb{N}}$ and a simple function $f$ such that there is no computable bound on the rate of convergence of the ergodic averages $A_{n}f$ . Here it is shown that in fact one can construct an example with the property that there is no computable bound on the complexity of the limit; that is, there is no computable bound on how complex a simple function needs to be to approximate (...) the limit to within a given $\varepsilon$. (shrink)

Margaret Cavendish is widely known as a materialist. However, since Cavendishian matter is always in motion, “matter” and “motion” are equally important foundational concepts for her natural philosophy. In Philosophical Letters (1664), she takes to task her materialist rival Thomas Hobbes by assaulting his account of accidents in general and his concept of “rest” in particular. In this article, I argue that Cavendish defends her continuous-motion view in two ways: first, she claims that her account avoids seeing (...) accidents as capable of generation and annihilation, which she argues is inconceivable; and second, she contends that according to Hobbes’s own view “rest” is an absurd conception since it cannot be drawn from experience. Beyond its function as a defense, I claim that Cavendish’s focused criticism of “rest” shows that she is a perceptive reader of Hobbes’s natural philosophy, insofar as her criticisms undercut the two a priori principles of Hobbesian physics. Finally, I show how her views developed in more detail in Philosophical and Physical Opinions (1663) and Observations upon Experimental Philosophy ([1666] 2001) avoid the worries she raises for Hobbesian materialism. (shrink)

De Finetti's representation theorem is a special case of the ergodic decomposition of stationary probability measures. The problems of the interpretation of probabilities centred around de Finetti's theorem are extended to this more general situation. The ergodic decomposition theorem has a physical background in the ergodic theory of dynamical systems. Thereby the interpretations of probabilities in the cases of de Finetti's theorem and its generalization and in ergodic theory are systematically connected to each other.

The concept of Substantial motion (حركت جوهرى) is fundamentally flawed and severely muddled. Aristotle and Mulla Sadra’s conception of motion, substance (جوهر) and substantial form صورت نوعيه)) were all based on a severe misunderstanding of nature as later was established by the scientists and philosophers that came after them. Here, by recalling the established facts of modern science, particularly the universally accepted scientific fact that, properties of objects are reducible to the motion of their electrons and there’s (...) no such thing as ‘substantial form’ much less substance, the author puts an end to a 400-year-old glorious error called ‘substantial motion’. Naturally any attempt to reconcile the paradigms of modern physics with Aristotle/Sadra paradigms of form/matter and substance/accidents, and to try to draw parallel between the two and revive their philosophy would be an exercise in futility. (shrink)

There is a longstanding definition of instantaneous velocity. It saysthat the velocity at t 0 of an object moving along a coordinate line is r if and only if the value of the first derivative of the object's position function at t 0 is r. The goal of this paper is to determine to what extent this definition successfully underpins a standard account of motion at an instant. Counterexamples proposed by Michael Tooley (1988) and also by John Bigelow and (...) Robert Pargetter (1990) are reinforced and illuminated by considering the presence or absence of changes to the object's motion. (shrink)

In the Metaphysical Foundations of Natural Science, Kant presents an argument for the centrality of <motion> to our concept <matter>. This argument has long been considered either irredeemably obscure or otherwise defective. In this paper I provide an interpretation which defends the argument’s validity and clarifies the sense in which it aims to show that <motion> is fundamental to our conception of matter.

I prove several natural preservation theorems for the countable support iteration. This solves a question of Rosłanowski regarding the preservation of localization properties and greatly simplifies the proofs in the area.

In the texts of the middle years (roughly, the 1680s and 90s), Leibniz appears to endorse two incompatible approaches to motion, one a realist approach, the other a phenomenalist approach. I argue that once we attend to certain nuances in his account we can see that in fact he has only one, coherent approach to motion during this period. I conclude by considering whether the view of motion I want to impute to Leibniz during his middle years (...) ranks as a kind of realism or rather as some kind of phenomenalism or idealism. (shrink)