Local reduction in physics

https://doi.org/10.1016/j.shpsb.2015.02.004Get rights and content

Highlights

  • A “local” account of inter-theoretic reduction in physics is given.

  • Local reduction between theories relies on multiple context-specific derivations.

  • These derivations concern relations between models of individual systems.

  • Where the models are dynamical systems, a broadly Nagelian analysis can be given.

Abstract

A conventional wisdom about the progress of physics holds that successive theories wholly encompass the domains of their predecessors through a process that is often called “reduction.” While certain influential accounts of inter-theory reduction in physics take reduction to require a single “global” derivation of one theory׳s laws from those of another, I show that global reductions are not available in all cases where the conventional wisdom requires reduction to hold. However, I argue that a weaker “local” form of reduction, which defines reduction between theories in terms of a more fundamental notion of reduction between models of a single fixed system, is available in such cases and moreover suffices to uphold the conventional wisdom. To illustrate the sort of fixed-system, inter-model reduction that grounds inter-theoretic reduction on this picture, I specialize to a particular class of cases in which both models are dynamical systems. I show that reduction in these cases is underwritten by a mathematical relationship that follows a certain liberalized construal of Nagel/Schaffner reduction, and support this claim with several examples. Moreover, I show that this broadly Nagelian analysis of inter-model reduction encompasses several cases that are sometimes cited as instances of the “physicist׳s” limit-based notion of reduction.

Introduction

According to the most commonly told story about the progress of physics, successive theories in physics come ever closer to revealing the true, fundamental nature of reality. This convergence rests on the supposition that later theories bear a special relationship to their predecessors often called “reduction,” which minimally requires one theory to encompass the domain of application of another. More specifically, the conventional wisdom tells us that Newtonian mechanics “reduces to” special relativity,1 special relativity to general relativity, classical mechanics to quantum mechanics, quantum mechanics to relativistic quantum mechanics, relativistic quantum mechanics to quantum field theory, thermodynamics to statistical mechanics, and more. In order to assess the truth of the conventional wisdom, however, it is necessary to gain a more precise sense of what is needed in a given case to show that one theory reduces to another.

In his widely cited 1973 paper, Nickles distinguished two types of approach to reduction in physics: first, the approach commonly employed by philosophers, which originates in Ernest Nagel׳s well-known account of reduction, and second, the approach commonly employed by physicists that requires one theory to be a “limit” or “limiting case” of another (Nickles, 1973). Since Nickles׳ paper, these two accounts have tended to dominate philosophical discussion concerning issues of the general methodology of reduction in physics. As commonly presented, both strongly suggest—and in some cases, state explicitly—that reduction between theories in physics should rest on a single “global” derivation of a high-level theory׳s laws from those of a low-level theory. Here, I argue by means of a particular example that global reduction is not always available in cases where the conventional wisdom requires reduction to hold. However, I argue that it is possible to a define a weaker “local” notion of reduction in physics that suffices to uphold the conventional wisdom in that it suffices to ensure the subsumption of one theory׳s domain by another. This notion of reduction is “local” in the sense that it permits the reducing theory to account for the reduced theory׳s success through numerous context-specific derivations that are relativized to different systems in the high-level theory׳s domain. These derivations concern the specific models that the theories use to describe a single fixed system, rather than the theories as a whole.

This paper has two main goals, which are mutually supporting. The first is to motivate and develop a local account of inter-theoretic reduction in physics. Inter-theoretic reduction in physics, understood minimally as the requirement that one theory subsume the domain of another, does not require anything as strong as global reduction directly between theories; local reduction suffices, and moreover avoids difficulties that afflict global approaches. I further argue that local reduction between theories should be understood in terms of the more basic notion of reduction between models of a single fixed system. The second goal of the paper is to give an account of fixed-system, inter-model reduction—on which this local account of inter-theoretic reduction rests—in a specialized class of cases where both models of the system in question are dynamical systems, and in particular to show that such cases can be analyzed in terms of a certain local, model-based adaptation of the Nagel/Schaffner approach to reduction. I further show that this Nagelian analysis of inter-model reduction encompasses many cases that have been cited as instances of physicists׳ limit-based notion of reduction, as well as providing a more precise characterization of these cases than do existing formulations of the limit-based approach.

The present analysis of reduction is given in two parts, corresponding respectively to the two goals just described. Part I, which consists of 2 Nickles׳ two senses of reduction, 3 Local reduction in physics, is largely non-technical and concerns issues of general methodology. As suggested, its purpose is to motivate and present a certain local, model-based approach to inter-theoretic reduction and to explain how this strategy avoids certain difficulties that afflict more global approaches. In Section 2, I briefly review two approaches to reduction—global Nagelian and global limit-based—that are often taken as the focus of philosophical discussions on this topic, and highlight some of their limitations. In Section 3, I sketch a local approach to inter-theoretic reduction in physics that relies on the more basic notion of fixed-system reduction between models and respond to one major objection that such an approach is likely to elicit.

Part II, which consists of 4 Fixed-system reduction between dynamical systems models, 5 Examples of DS reduction, 6 Extending DS reduction, provides a detailed technical analysis of fixed-system, inter-model reduction in a particular set of cases where both models of the system in question are dynamical systems, as well as briefly discussing various possible expansions of this analysis to a more comprehensive account of fixed-system, inter-model reduction in physics. Section 4 describes a general mathematical relationship between dynamical systems models that serves to underwrite many real instances of fixed-system, inter-model reduction in physics. In a certain strong sense, this mathematical relationship constitutes an application of the criteria for Nagel/Schaffner reduction to the context of fixed-system, inter-model reduction between dynamical systems models. Section 5 shows how this general relationship serves to characterize reduction across a wide range of particular cases, and to subsume a number of cases that are commonly cited as examples of the “physicist׳s” limit-based notion of reduction. Section 6 briefly discusses possibilities for extending and generalizing this strategy for inter-model reduction beyond the set of cases discussed here: first, to an analysis of the relationship between symmetries of the two models involved in a reduction, and second, to an analysis of cases where one or both of the models involved in the reduction is not a dynamical system but some other kind of model (e.g., stochastic, non-dynamical, etc.).

The distinct portions of the analysis given in Parts I and II complement each other in a number of important ways. Part I serves to frame the analysis of reduction between dynamical systems given in Part II within a more general picture of inter-theoretic reduction and in particular to situate this analysis relative to the two accounts of inter-theoretic reduction in physics first distinguished by Nickles. By the same token, Part II provides a concrete illustration of the sort of fixed-system, inter-model reduction that is taken as the basis for the local approach to inter-theoretic reduction described in Part I.

Before proceeding, it is worth taking a moment to clarify several points of terminology.

Because debates about reduction are often frought with ambiguity as to what, precisely, is meant by reduction, I should clarify my use of the term here. I do not attach my usage to any specific account of reduction—for example, Nagelian, limit-based, New Wave and functionalist approaches. Rather, I use it to designate a certain general concept that, I take it, all, or most, of the many specific accounts aim to make more precise. “Reduction,” then, is taken to designate the general requirement that two descriptions of the world “dovetail” in such a manner that one description entirely encompasses the range of successful applications of the other. That is, reduction on this usage requires subsumption of one description׳s domain of applicability by the other, while the specific sense in which the two descriptions “dovetail” in order to achieve this is deliberately left vague, so as not to bias its usage toward any particular account.

As Nickles has noted, the usage of the term “reduction” most common among philosophers takes the less accurate and encompassing description in a reduction to “reduce to” the more accurate and encompassing description, whereas the usage most common among physicists takes the more accurate, encompassing description to “reduce to” the less accurate and encompassing description. In what follows, I will always adopt the philosopher׳s convention, even when discussing the physicist׳s limit-based notion of reduction, so that if theory T2 is a “limiting case” of T1, we will say that T2 “reduces to” T1.

I will also reserve the term “high-level” to refer to the description that is purportedly reduced and “low-level” to refer to the description that purportedly does the reducing. This usage generalizes another use of the “high-level/low-level” distinction, which presupposes that the high-level description is in some sense a coarse-graining of the low-level description, or that the high-level description is in some sense “macro” and the low-level description in some sense “micro.” Here, no such assumption is made. For example, where the relation between Kepler׳s and Newton׳s theories of planetary motion is concerned, Kepler׳s theory would count on our usage as the “high-level” and Newton׳s as the “low-level” theory even though Kepler׳s theory is not in any normal sense a coarse-graining of Newton׳s. While some authors have emphasized the distinction between “inter-level” reductions (e.g., thermodynamics to statistical mechanics) and “intra-level” reduction (e.g., Newtonian mechanics to special relativity, or Kepler׳s to Newton׳s theory of planetary motion), the picture of reduction presented here does not rely on this distinction and treats both kinds of reduction on a par.2

Henceforth, when I speak of “Nagelian” reduction, the reader should take this to refer specifically to the Nagel/Schaffner account of reduction, which allows for approximative derivations rather than requiring exact derivations. While Nagel/Schaffner reduction is widely framed within a syntactic view of theories—as opposed to the semantic, model-theoretic view adopted here—and is often taken to require global rather than local derivations, my use of the label “Nagelian” here does not presuppose these characteristics. Rather, what is taken to be constitutive of “Nagelian” reduction on my usage is the general requirement that it be possible to derive, on the basis of the low-level description and through the use of bridge principles, approximate versions of the laws or constraints or equations that serve to characterize the high-level description. My usage does not presuppose any view as to whether theories are understood syntactically or semantically—although the specific local approach to reduction advocated here fits much more naturally with a semantic view. Moreover, my usage does not assume any commitments as to the particular nature of these bridge principles—such as whether they are to be understood as empirically established laws or as definitions—apart from their role in enabling a translation or comparison between the frameworks of the two descriptions in question. Generally speaking, rather than taking the term “Nagelian” to designate a specific set of precisely defined formal requirements for reduction, I use it here to designate a certain broad strategy for reduction.

Since the local approach to inter-theoretic reduction that I describe here is grounded in a certain account of inter-model reduction, I should say something about what I take to be the relationship between theories and models. For the purposes of this discussion, it will suffice to note that the manner in which any theory serves to successfully describe a physical system in its domain is through some particular model of that theory. Moreover, the specification of any such model entails much narrower commitments than those that serve to characterize the theory itself. For example, specification of a particular quantum or classical model of a material object׳s behavior requires commitments to a particular form for the Hamiltonian (or force law or Lagrangian), including particular values for quantities like mass and charge, while the theories of quantum and classical mechanics themselves are compatible with many functional forms of the Hamiltonian and many values for these parameters. One of the central points of my discussion here is that it is sometimes not just the more general specifications that serve to characterize the high- and low-level theories in question that are relevant to underwriting the success of the high-level theory in a given case, but also the narrower, more context-dependent specifications that characterize the particular models of the two theories.

Section snippets

Part I: local reduction in physics

In the literature on reduction across the sciences, and especially in philosophy of mind, concerns about multiple realization have led many philosophers to espouse a more “local” form of reduction that allows a low-level description to account for a high-level description׳s successes through many context-specific derivations that employ context-specific bridge principles. As a whole, the literature on reduction specifically within physics—which merits focused attention in part because of the

Nickles׳ two senses of reduction

Since Nickles׳ 1973 paper, most philosophical discussion about the general methodology of reduction in physics has revolved around the Nagelian and limit-based approaches. Conventional presentations of both approaches treat inter-theory reduction as a matter of deriving one theory from another (whether through Nagelian deduction or a limiting process). This manner of framing the issue strongly suggests that inter-theoretic reduction is being taken in these presentations to be a matter of

Local reduction in physics

It was largely in response to critiques of global Nagel/Schaffner reduction that a number of authors, mostly in the philosophy of mind literature, were prompted to advocate for a more local approach to reduction in which a lower-level description accounts for the successes of a higher-level description through many context-specific derivations employing context-specific bridge principles, rather than through a single global derivation employing the same set of bridge principles for all systems

Part II: the case of dynamical systems

My purpose in the second half of this paper is to illustrate more precisely, through a specific class of cases, what is meant by “fixed-system, inter-model reduction” (“reductionM”) in the local picture of inter-theoretic reduction (“reductionT”) spelled out in Section 3.2. In the class of cases that I consider here, both models of the system in question are dynamical systems models and, moreover, are assumed to share a common time parameter. In Section 4, I describe a general mathematical

Fixed-system reduction between dynamical systems models

Dynamical systems models occur widely throughout physics: in Hamiltonian models of non-relativistic and relativistic classical mechanics and non-relativistic and relativistic classical field theory, in Schrodinger picture models of non-relativistic quantum mechanics, relativistic quantum mechanics and quantum field theory, and in heat diffusion equations in thermodynamics, to name a few cases.

The notion of a dynamical system has been defined in a number of distinct ways that vary both in

Examples of DS reduction

In this section, I show that the conditions for DS reduction are satisfied by a range of different model pairs. I do so specifically by showing that the relation (6) holds for a certain choice of bridge map and a certain domain of states in the low-level state space, offering qualitative remarks concerning the timescale over which the relation (6) continues to hold. My presentation of each example will follow a common outline: (a) specification of the state spaces and dynamical equations of the

Extending DS reduction

In this section, I consider possible extensions of DS reduction, first to include an analysis of the relations between symmetries of different dynamical systems models and then to an analysis of the relation reductionM outside the class of cases to which reductionDS applies.

Conclusion

I have argued against a general, if not always explicit, tendency in the literature on inter-theory relations in physics to conceive of inter-theoretic reduction as an exclusively global affair—that is, as primarily a matter of deriving, in a completely general way that is insensitive to the particularities of different systems, the laws of one-theory from those of another. Taking a cue from certain authors primarily in the philosophy of mind literature, I have advocated a more local approach

Acknowledgments

I would like to thank David Wallace, Simon Saunders, Christopher Timpson, Jeremy Butterfield, John Norton and an anonymous referee for invaluable comments on earlier drafts of this work, and Jos Uffink, Cian Dorr and David Albert for helpful discussions. Thanks also to audiences in Munich, Pittsburgh, Amersfoort and Minnesota, where earlier versions of this article were presented. This work was supported by the University of Pittsburgh's Center for Philosophy of Science.

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