A vast amount of ink has been spilled in both the physics and the philosophy literature on the measurement problem in quantum mechanics. Important as it is, this problem is but one aspect of the more general issue of how, if at all, classical properties can emerge from the quantum descriptions of physical systems. In this paper we will study another aspect of the more general issue-the emergence of classical chaos-which has been receiving increasing attention from physicists but which has (...) largely been neglected by philosophers of science. (shrink)

This paper addresses a relatively common scientific (as opposed to philosophical) conception of intertheoretic reduction between physical theories. This is the sense of reduction in which one (typically newer and more refined) theory is said to reduce to another (typically older and coarser) theory in the limit as some small parameter tends to zero. Three examples of such reductions are discussed: First, the reduction of Special Relativity (SR) to Newtonian Mechanics (NM) as (v/c)20; second, the reduction of wave optics to (...) geometrical optics as 0; and third, the reduction of Quantum Mechanics (QM) to Classical Mechanics (CM) as0. I argue for the following two claims. First, the case of SR reducing to NM is an instance of a genuine reductive relationship while the latter two cases are not. The reason for this concerns the nature of the limiting relationships between the theory pairs. In the SR/NM case, it is possible to consider SR as a regular perturbation of NM; whereas in the cases of wave and geometrical optics and QM/CM, the perturbation problem is singular. The second claim I wish to support is that as a result of the singular nature of the limits between these theory pairs, it is reasonable to maintain that third theories exist describing the asymptotic limiting domains. In the optics case, such a theory has been called catastrophe optics. In the QM/CM case, it is semiclassical mechanics. Aspects of both theories are discussed in some detail. (shrink)

Our aim is to discover whether the notion of algorithmic orbit-complexity can serve to define “chaos” in a dynamical system. We begin with a mostly expository discussion of algorithmic complexity and certain results of Brudno, Pesin, and Ruelle (BRP theorems) which relate the degree of exponential instability of a dynamical system to the average algorithmic complexity of its orbits. When one speaks of predicting the behavior of a dynamical system, one usually has in mind one or more variables in the (...) phase space that are of particular interest. To say that the system is unpredictable is, roughly, to say that one cannot feasibly determine future values of these variables from an approximation of the initial conditions of the system. We introduce the notions of restrictedexponential instability and conditionalorbit-complexity, and announce a new and rather general result, similar in spirit to the BRP theorems, establishing average conditional orbit-complexity as a lower bound for the degree of restricted exponential instability in a dynamical system. The BRP theorems require the phase space to be compact and metrizable. We construct a noncompact kicked rotor dynamical system of physical interest, and show that the relationship between orbit-complexity and exponential instability fails to hold for this system. We conclude that orbit-complexity cannot serve as a general definition of “chaos.”. (shrink)

The notion of (deterministic) chaos is frequently used in an increasing number of scientific (as well as non-scientific) contexts, ranging from mathematics and the physics of dynamical systems to all sorts of complicated time evolutions, e.g., in chemistry, biology, physiology, economy, sociology, and even psychology. Despite (or just because of) these widespread applications, however, there seem to fluctuate around several misunderstandings about the actual impact of deterministic chaos on several problems of philosophical interest, e.g., on matters of prediction and computability, (...) and determinism and the free will. In order to clarify these points a survey of the meaning variance of the concept(s) of deterministic chaos, or the various contexts in which it is applied, is given, and its actual epistemological implications are extracted. In summary, it turns out that the various concepts of deterministic chaos do not constitute a “new science”, or a “revolutionary” change of the “scientific world picture”. Instead, chaos research provides a sort of toolbox of methods which are certainly useful for a more detailed analysis and understanding of such dynamical systems which are, roughly speaking, endowed with the property of exponential sensitivity on initial conditions. Such a property, then, implies merely one, but quantitatively strong type of limitation of long-time computability and predictability, respectively. (shrink)

Conventional wisdom has it that chaotic behavior is either strongly suppressed or absent in quantum models. Indeed, some researchers have concluded that these considerations serve to undermine the correspondence principle, thereby raising serious doubts about the adequacy of quantum mechanics. Thus, the quantum chaos question is a prime subject for philosophical analysis. The most significant reasons given for the absence or suppression of chaotic behavior in quantum models are the linearity of Schrödinger’s equation and the unitarity of the time-evolution described (...) by that equation. Both are shown in this essay to be irrelevant by demonstrating that the crucial feature for chaos is the nonseparability of the Hamiltonian. That demonstration indicates that quantum chaos is likely to be exhibited in models of open quantum systems. A measure for probing such models for chaotic behavior is developed, and then used to show that quantum mechanics has chaotic models for systems having a continuous energy spectrum. The prospects of this result for vindicating the correspondence principle (or the motivation behind it, at least) are then briefly examined. (shrink)

The Correspondence Principle of old quantum theory is commonly considered to be the requirement that quantum and classical theories converge in their empirical predictions in the appropriate asymptotic limit. That perception has persisted despite the fact that Bohr and other early proponents of CP clearly did not intend it as a mere requirement, and despite much recent historical work. In this paper, I build on this work by first giving an explicit formulation to the mentioned asymptotic requirement ) and then (...) discussing various possible formulations of CP for emission on the basis of the primary literature as well as general physical and metaphysical considerations. I shall then show that, in all of the most probable interpretations of CP that consider quantum theory as a universal theory, any system incorporating both CR and CP for emission would in fact be inconsistent. Old quantum theory measurably contradicts classical physics in the classical regime. (shrink)

Although Bohr’s Correspondence Principle (CP) played a central role in the first days of quantum mechanics, its original version seems to have no present-day relevance. The purpose of this article is to show that the CP, with no need of being interpreted in terms of the quantum-to-classical limit, still plays a relevant role in the understanding of the relationships between the classical and the quantum domains. In particular, it will be argued that a generic version of the CP is very (...) helpful in elucidating the physical meaning of the phenomenon of quantum decoherence. (shrink)

I argue that it is possible to give an interpretation of the classical ℏ→0 limit of quantum mechanics that results in a partial explanation of the success of classical mechanics. The interpretation...

⦿ In my dissertation I introduce, motivate and take the first steps in the implementation of, the project of naturalising modal metaphysics: the transformation of the field into a chapter of the philosophy of science rather than speculative, autonomous metaphysics. -/- ⦿ In the introduction, I explain the concept of naturalisation that I apply throughout the dissertation, which I argue to be an improvement on Ladyman and Ross' proposal for naturalised metaphysics. I also object to Williamson's proposal that modal metaphysics (...) --- or some view in the area --- is already a quasi-scientific discipline. -/- ⦿ Recently, some philosophers have argued that the notion of metaphysical modality is as ill defined as to be of little theoretical utility. In the second chapter I intend to contribute to such skepticism. First, I observe that each of the proposed marks of the concept, except for factivity, is highly controversial; thus, its logical structure is deeply obscure. With the failure of the "first principles" approach, I examine the paradigmatic intended applications of the concept, and argue that each makes it a device for a very specific and controversial project: a device, therefore, for which a naturalist will find no use for. I conclude that there is no well-defined or theoretically useful notion of objective necessity other than logical or physical necessity, and I suggest that naturalising modal metaphysics can provide more stable methodological foundations. -/- ⦿ In the third chapter I answer a possible objection against the in-principle viability of the project: that the concept of metaphysical modality cannot be understood through the philosophical analysis of any scientific theory, since metaphysical necessity "transcends'' natural necessity, and science only deals with the latter. I argue that the most important arguments for this transcendence thesis fail or face problems that, as of today, remain unsolved. -/- ⦿ Call the idea that science doesn't need modality, "demodalism''. Demodalism is a first step in a naturalistic argument for modal antirealism. In the fourth chapter I examine six versions of demodalism to explain why a family of formalisms, that I call "spaces of possibility'', are (i) used in a quasi-ubiquitous way in mathematised sciences (I provide examples from theoretical computer science to microeconomics), (ii) scientifically interpreted in modal terms, and (iii) used for at least six important tasks: (1) defining laws and theories; (2) defining important concepts from different sciences (I give several examples); (3) making essential classifications; (4) providing different types of explanations; (5) providing the connection between theory and statistics, and (6) understanding the transition between a theory and its successor (as is the case with quantisation). -/- ⦿ In fifth chapter I propose and defend a naturalised modal ontology. This is a realism about modal structure: my realism about constraints. The modal structure of a system are the relationships between its possible states and between its possible states and those of other systems. It is given by the plurality of restrictions to which said system is subject. A constraint is a factor that explains the impossibility of a class of states; I explain this concept further. First, I defend my point of view by rejecting some of its main rivals: constructive empiricism, Humean conventionalism, and wave function realism, as they fail to make sense of quantum chaos. This is because the field requires the notion of objective modal structure, and the mentioned views have trouble explaining the modal facts of quantum dynamics. Then, I argue that constraint realism supersedes these views in the context of Bohm's standard theory and mechanics, and underpins the study of quantum chaos. Finally, I consider and reject two possible problems for my point of view. -/- ⦿ A central concern of modal metaphysicians has been to understand the logical system that best characterises necessity. In the sixth chapter I intend to recover the logical project applied to my naturalistic modal metaphysics. Scientists and philosophers of science accept different degrees of physical necessity, ranging from purely mathematically necessary facts that restrict physical behaviour, to kinetic principles, to particular dynamical constraints. I argue that this motivates a multimodal approach to modal logic, and that the time dependence of dynamics motivates a logic of historical necessity. I propose multimodal propositional (classical) logics for Bohmian mechanics and the Everettian theory of many divergent worlds, and I close with a criticism of Williamson's approach to the logic of state spaces of dynamic systems. (shrink)

Niels Bohr’s “correspondence principle” is typically believed to be the requirement that in the limit of large quantum numbers (n→∞) there is a statistical agreement between the quantum and classical frequencies. A closer reading of Bohr’s writings on the correspondence principle, however, reveals that this interpretation is mistaken. Specifically, Bohr makes the following three puzzling claims: First, he claims that the correspondence principle applies to small quantum numbers as well as large (while the statistical agreement of frequencies is only for (...) large n); second, he claims that the correspondence principle is a law of quantum theory; and third, Bohr argues that formal apparatus of matrix mechanics (the new quantum theory) can be thought of as a precise formulation of the correspondence principle. With further textual evidence, I offer an alternative interpretation of the correspondence principle in terms of what I call Bohr’s selection rule. I conclude by showing how this new interpretation of the correspondence principle readily makes sense of Bohr’s three puzzling claims. (shrink)