Starting from the standard quantum formalism for a single spin 1/2 system (e.g., an electron), this essay develops a model rich enough not only to afford an explication of symmetry breaking but also to frame questions about how to circumscribe physical possibility on behalf of theories that countenance symmetry breaking.

While the fundamental laws of physics are time-reversal invariant, most macroscopic processes are irreversible. Given that the fundamental laws are taken to underpin all other processes, how can the fundamental time-symmetry be reconciled with the asymmetry manifest elsewhere? In statistical mechanics, progress can be made with this question. What I dub the ‘Zwanzig–Zeh–Wallace framework’ can be used to construct the irreversible equations of SM from the underlying microdynamics. Yet this framework uses coarse-graining, a procedure that has faced much criticism. I (...) focus on two objections in the literature: claims that coarse-graining makes time-asymmetry ‘illusory’ and ‘anthropocentric’. I argue that these objections arise from an unsatisfactory justification of coarse-graining prevalent in the literature, rather than from coarse-graining itself. This justification relies on the idea of measurement imprecision. By considering the role that abstraction and autonomy play, I provide an alternative justification and offer replies to the illusory and anthropocentric objections. Finally, I consider the broader consequences of this alternative justification: the connection to debates about inter-theoretic reduction and the implication that the time-asymmetry in SM is weakly emergent. 1Introduction 1.1Prospectus2The Zwanzig–Zeh–Wallace Framework3Why Does This Method Work? 3.1The special conditions account3.2When is a density forwards-compatible?4Anthropocentrism and Illusion: Two Objections 4.1The two objections in more detail4.2Against the justification by measurement imprecision5An Alternative Justification 5.1Abstraction and autonomy5.2An illustration: the Game of Life6Reply to Illusory7Reply to Anthropocentric8The Wider Landscape: Concluding Remarks 8.1Inter-theoretic relations8.2The nature of irreversibility. (shrink)

My goal here is to explore the relationship between pure and applied mathematics and then, eventually, to draw a few morals for both. In particular, I hope to show that this relationship has not been static, that the historical rise of pure mathematics has coincided with a gradual shift in our understanding of how mathematics works in application to the world. In some circles today, it is held that historical developments of this sort simply represent changes in fashion, or in (...) social arrangements, governments, power structures, or some such thing, but I resist the full force of this way of thinking, clinging to the old school notion that we have gradually learned more about the world over time, that our opinions on these matters have improved, and that seeing how we reached the point we now occupy may help us avoid falling back into old philosophies that are now no longer viable. In that spirit, it seems to me that once we focus on the general question of how mathematics relates to science, one observation is immediate: the march of the centuries has produced an amusing reversal of philosophical fortunes. Let me begin there. In the beginning, that is, in Plato, mathematical knowledge was sharply distinguished from ordinary perceptual belief about the world. According to Plato’s metaphysics, mathematics is the study of eternal and unchanging abstract Forms,1 while science is uncertain and changeable opinion about the world of mere becoming. Indeed, in Plato’s lights, of the two, only mathematics deserves to be called ‘knowledge’ at all! Of course if sense perception cannot give us knowledge, if mathematics is not about perceivable things, then Plato owes us an account of how we ordinary humans achieve this wonderful insight into the properties of the abstract world of Forms. Plato’s answer is that we do not actually acquire mathematical knowledge at all; rather, we recollect it from a time before birth, when our souls, unencumbered by physical bodies, were free to commune with the Forms, and not just the mathematical ones, either – also Truth, Beauty, Justice, the Good, and so on.2 Whatever appeal this position may have held for the ancient Greeks, it will not begin to satisfy a contemporary, scientifically minded philosopher.. (shrink)

Two alternative accounts of quantum spontaneous symmetry breaking (SSB) are compared and one of them, the decompositional account in the algebraic approach, is argued to be superior for understanding quantum SSB. Two exactly solvable models are given as applications of our account -- the Weiss-Heisenberg model for ferromagnetism and the BCS model for superconductivity. Finally, the decompositional account is shown to be more conducive to the causal explanation of quantum SSB.

In this paper, a criticism of the traditional theories of approximation and idealization is given as a summary of previous works. After identifying the real purpose and measure of idealization in the practice of science, it is argued that the best way to characterize idealization is not to formulate a logical model – something analogous to Hempel's D-N model for explanation – but to study its different guises in the praxis of science. A case study of it is then made (...) in thermostatistical physics. After a brief sketch of the theories for phase transitions and critical phenomena, I examine the various idealizations that go into the making of models at three difference levels. The intended result is to induce a deeper appreciation of the complexity and fruitfulness of idealization in the praxis of model-building, not to give an abstract theory of it. (shrink)

Huw Price (1996, 2002, 2003) argues that causal-dynamical theories that aim to explain thermodynamic asymmetry in time are misguided. He points out that in seeking a dynamical factor responsible for the general tendency of entropy to increase, these approaches fail to appreciate the true nature of the problem in the foundations of statistical mechanics (SM). I argue that it is Price who is guilty of misapprehension of the issue at stake. When properly understood, causal-dynamical approaches in the foundations of SM (...) offer a solution for a different problem; a problem that unfortunately receives no attention in Price’s celebrated work. (shrink)

Gases reach equilibrium when left to themselves. Why do they behave in this way? The canonical answer to this question, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer has been criticised on different grounds and is now widely regarded as flawed. In this paper we argue that some of the main arguments against Boltzmann's answer, in particular, arguments based on the KAM-theorem and the Markus-Meyer theorem, are beside the point. We then argue that something (...) close to Boltzmann's original proposal is true for gases: gases behave thermodynamic-like if they are epsilon-ergodic, i.e., ergodic on the entire accessible phase space except for a small region of measure epsilon. This answer is promising because there are good reasons to believe that relevant systems in statistical mechanics are epsilon-ergodic. (shrink)

This is a companion to another paper. Together they rebut two widespread philosophical doctrines about emergence. The first, and main, doctrine is that emergence is incompatible with reduction. The second is that emergence is supervenience; or more exactly, supervenience without reduction.In the other paper, I develop these rebuttals in general terms, emphasising the second rebuttal. Here I discuss the situation in physics, emphasising the first rebuttal. I focus on limiting relations between theories and illustrate my claims with four examples, each (...) of them a model or a framework for modelling, from well-established mathematics or physics.I take emergence as behaviour that is novel and robust relative to some comparison class. I take reduction as, essentially, deduction. The main idea of my first rebuttal will be to perform the deduction after taking a limit of some parameter. Thus my first main claim will be that in my four examples (and many others), we can deduce a novel and robust behaviour, by taking the limit N→∞ of a parameter N.But on the other hand, this does not show that the N=∞ limit is “physically real”, as some authors have alleged. For my second main claim is that in these same examples, there is a weaker, yet still vivid, novel and robust behaviour that occurs before we get to the limit, i.e. for finite N. And it is this weaker behaviour which is physically real.My examples are: the method of arbitrary functions (in probability theory); fractals (in geometry); superselection for infinite systems (in quantum theory); and phase transitions for infinite systems (in statistical mechanics). (shrink)

We introduce ‘model migration’ as a species of cross-disciplinary knowledge transfer whereby the representational function of a model is radically changed to allow application to a new disciplinary context. Controversies and confusions that often derive from this phenomenon will be illustrated in the context of econophysics and phylogeographic linguistics. Migration can be usefully contrasted with concept of ‘imperialism’, that has been influentially discussed in the context of geographical economics. In particular, imperialism, unlike migration, relies upon extension of the original model (...) via an expansion of the domain of phenomena it is taken to adequately described. The success of imperialism thus requires expansion of the justificatory sanctioning of the original idealising assumptions to a new disciplinary context. Contrastingly, successful migration involves the radical representational re-interpretation of the original model, rather than its extension. Migration thus requires ‘re-sanctioning’ of new ‘counterpart idealisations’ to allow application to an entirely different class of phenomena. Whereas legitimate scientific imperialism should be based on the pursuit of some form of ontological unification, no such requirement is need to legitimate the practice of model migration. The distinction between migration and imperialism will thus be shown to have significant normative as well as descriptive value. (shrink)

Entropy is ubiquitous in physics, and it plays important roles in numerous other disciplines ranging from logic and statistics to biology and economics. However, a closer look reveals a complicated picture: entropy is defined differently in different contexts, and even within the same domain different notions of entropy are at work. Some of these are defined in terms of probabilities, others are not. The aim of this chapter is to arrive at an understanding of some of the most important notions (...) of entropy and to clarify the relations between them, After setting the stage by introducing the thermodynamic entropy, we discuss notions of entropy in information theory, statistical mechanics, dynamical systems theory and fractal geometry. (shrink)

One recent debate in philosophy of physics has centered whether quantum particles are individuals or not. The received view is that particles are not individuals and the standard methodology is to approach the question via the structure of quantum theory. I challenge both the received view and the standard methodology. I contend not only that the structure of quantum theory is not the right place to look for conditions of individuality that quantum particles may or may not satisfy, but also (...) that there is an important role for traditional metaphysics to play. Consequently, my work brings together the philosophy of physics and traditional metaphysics literatures to shed new light on the debate over the individuality of quantum particles. I defend a set of conditions of individuality and argue that quantum particles satisfy these conditions thereby defending the view that particles are individuals in opposition to the received view. I also challenge a second feature of the standard methodology insofar as I challenge the significance of the Principle of the Identity of Indiscernibles in terms of which much discussion in the philosophy of physics literature is framed. My work is significant in a number of additional ways as well. My work implies that the dominant explanation for quantum statistics in terms of non-individuality is incorrect and it also undermines the ontic-structural realists metaphysical underdetermination challenge to the scientific realist. (shrink)

This thesis makes the issue of reconciling the existence of thermodynamically irreversible processes with underlying reversible dynamics clear, so as to help explain what philosophers mean when they say that an aim of nonequilibrium statistical mechanics is to underpin aspects of thermodynamics. Many of the leading attempts to reconcile the existence of thermodynamically irreversible processes with underlying reversible dynamics proceed by way of discussions that attempt to underpin the following qualitative facts: (i) that isolated macroscopic systems that begin away from (...) equilibrium spontaneously approach equilibrium, and (ii) that they remain in equilibrium for incredibly long periods of time. These attempts standardly appeal to phase space considerations and notions of typicality. This thesis considers and evaluates leading typicality accounts, and, in particular, highlights their limitations. Importantly, these accounts do not underpin a large and important set of facts. They do not, for example, underpin facts about the rates in which systems approach equilibrium, or facts about the kinds of states they pass through on their way to equilibrium, or facts about fluctuation phenomena. To remedy these and other shortfalls, this thesis promotes an alternative, and arguably more important, line of research: understanding and accounting for the success of the techniques and equations physicists use to model the behaviour of systems that begin away from equilibrium. Accounting for their success would help underpin not just the qualitative facts the literature has focused on, but also many of the important quantitative facts that typicality accounts cannot. This thesis also takes steps in this promising direction. It outlines and examines a technique commonly used to model the behaviour of an interesting and important kind of system: a Brownian particle that's been introduced to an isolated homogeneous fluid at equilibrium. As this thesis highlights, the technique returns a wealth of quantitative and qualitative information. This thesis also attempts to account for the success of the model and technique, by identifying and grounding the technique's key assumptions. (shrink)

In (Weaver 2021), I showed that Boltzmann’s H-theorem does not face a significant threat from the reversibility paradox. I argue that my defense of the H-theorem against that paradox can be used yet again for the purposes of resolving the recurrence paradox without having to endorse heavy-duty statistical assumptions outside of the hypothesis of molecular chaos. As in (Weaver 2021), lessons from the history and foundations of physics reveal precisely how such resolution is achieved.

In robustness analysis, hypotheses are supported to the extent that a result proves robust, and a result is robust to the extent that we detect it in diverse ways. But what precise sense of diversity is at work here? In this paper, I show that the formal explications of evidential diversity most often appealed to in work on robustness – which all draw in one way or another on probabilistic independence – fail to shed light on the notion of diversity (...) relevant to robustness analysis. I close by briefly outlining a promising alternative approach inspired by Horwich’s (1982) eliminative account of evidential diversity. (shrink)

The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is (...) Heisenberg's `quantum-theoretical Umdeutung (reinterpretation) of classical observables', which lies at the basis of quantization theory. Similarly, Bohr's correspondence principle (in somewhat revised form) and Schroedinger's wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely in the limit h -> 0 of small Planck's constant (in a finite system), in the limit N goes to infinity of a large system with $N$ degrees of freedom (at fixed h), and through decoherence and consistent histories. The first limit is closely related to modern quantization theory and microlocal analysis, whereas the second involves methods of C*-algebras and the concepts of superselection sectors and macroscopic observables. In these limits, the classical world does not emerge as a sharply defined objective reality, but rather as an approximate appearance relative to certain ``classical" states and observables. Decoherence subsequently clarifies the role of such states, in that they are ``einselected", i.e. robust against coupling to the environment. Furthermore, the nature of classical observables is elucidated by the fact that they typically define (approximately) consistent sets of histories. This combination of ideas and techniques does not quite resolve the measurement problem, but it does make the point that classicality results from the elimination of certain states and observables from quantum theory. Thus the classical world is not created by observation (as Heisenberg once claimed), but rather by the lack of it. (shrink)

I extract some philosophical morals from some aspects of Lagrangian mechanics. One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been largely ignored by philosophers, since it falls between their accustomed levels---``laws of nature'' and ``models''. Another main moral concerns ontology: the ontology of Lagrangian mechanics is both more subtle and more problematic than philosophers often realize. The treatment of Lagrangian mechanics provides an introduction to the subject for philosophers, and is technically elementary. (...) In particular, it is confined to systems with a finite number of degrees of freedom, and for the most part eschews modern geometry. (shrink)

Roughly speaking, classical statistical physics is the branch of theoretical physics that aims to account for the thermal behaviour of macroscopic bodies in terms of a classical mechanical model of their microscopic constituents, with the help of probabilistic assumptions. In the last century and a half, a fair number of approaches have been developed to meet this aim. This study of their foundations assesses their coherence and analyzes the motivations for their basic assumptions, and the interpretations of their central concepts. (...) The most outstanding foundational problems are the explanation of time-asymmetry in thermal behaviour, the relative autonomy of thermal phenomena from their microscopic underpinning, and the meaning of probability. A more or less historic survey is given of the work of Maxwell, Boltzmann and Gibbs in statistical physics, and the problems and objections to which their work gave rise. Next, we review some modern approaches to (i) equilibrium statistical mechanics, such as ergodic theory and the theory of the thermodynamic limit; and to (ii) non-equilibrium statistical mechanics as provided by Lanford's work on the Boltzmann equation, the so-called Bogolyubov-Born-Green-Kirkwood-Yvon approach, and stochastic approaches such as `coarse-graining' and the `open systems' approach. In all cases, we focus on the subtle interplay between probabilistic assumptions, dynamical assumptions, initial conditions and other ingredients used in these approaches. (shrink)

A comparison is made of the traditional Loschmidt and Zermelo objections to Boltzmann's H-theorem, and its simplified variant in the Ehrenfests' 1912 wind-tree model. The little-cited 1896 objection of Zermelo is also analysed. Significant differences between the objections are highlighted, and several old and modern misconceptions concerning both them and the H-theorem are clarified. We give particular emphasis to the radical nature of Poincare's and Zermelo's attack, and the importance of the shift in Boltzmann's thinking in response to the objections (...) as a whole. (shrink)

This paper, part II of a two-part project, continues to explore the meaning of spontaneous symmetry breaking (SSB) by applying and expanding the general notion we obtained in part I to some more complex and, from the physics point of view, more important models (in condensed matter physics and in quantum field theories).