This paper examines contemporary attempts to explicate the explanatory role of mathematics in the physical sciences. Most such approaches involve developing so-called mapping accounts of the relationships between the physical world and mathematical structures. The paper argues that the use of idealizations in physical theorizing poses serious difficulties for such mapping accounts. A new approach to the applicability of mathematics is proposed.

This paper looks at emergence in physical theories and argues that an appropriate way to understand socalled “emergent protectorates” is via the explanatory apparatus of the renormalization group. It is argued that mathematical singularities play a crucial role in our understanding of at least some well-defined emergent features of the world.

I argue that a common philosophical approach to the interpretation of physical theories—particularly quantum field theories—has led philosophers astray. It has driven many to declare the quantum field theories employed by practicing physicists, so-called ‘effective field theories’, to be unfit for philosophical interpretation. In particular, such theories have been deemed unable to support a realist interpretation. I argue that these claims are mistaken: attending to the manner in which these theories are employed in physical practice, I show that interpreting effective (...) field theories yields a robust foundation for a more refined approach to scientific realism in the context of quantum field theory. The paper concludes by briefly sketching some general morals for interpretive practice in the philosophy of physics. (shrink)

This paper discusses the alleged reduction of Thermodynamics to Statistical Mechanics. It includes an historical discussion of J. Willard Gibbs' famous caution concerning the connections between thermodynamic properties and statistical mechanical properties---his so-called ``Thermodynamic Analogies.'' The reasons for Gibbs' caution are reconsidered in light of relatively recent work in statistical physics on the existence of the thermodynamic limit and the explanation of critical behavior using the renormalization group apparatus. A probabilistic understanding of the renormalization group arguments allows for a kind (...) of unification of Gibbs' approach with contemporary understanding of the reduction problem. (shrink)

Renormalization group (RG) methods are an established strategy to explain how it is possible that microscopically different systems exhibit virtually the same macro behavior when undergoing phase-transitions. I argue – in agreement with Robert Batterman – that RG explanations are non-causal explanations. However, Batterman misidentifies the reason why RG explanations are non-causal: it is not the case that an explanation is non- causal if it ignores causal details. I propose an alternative argument, according to which RG explanations are non-causal explanations (...) because their explanatory power is due to the application mathematical operations, which do not serve the purpose of representing causal relations. (shrink)

The renormalization method is specifically aimed at connecting theories describing physical processes at different length scales and thereby connecting different theories in the physical sciences.The renormalization method used today is the outgrowth of 150 years of scientific study of thermal physics and phase transitions. Different phases of matter show qualitatively different behaviors separated by abrupt phase transitions. These qualitative differences seem to be present in experimentally observed condensed-matter systems. However, the “extended singularity theorem” in statistical mechanics shows that sharp changes (...) can only occur in infinitely large systems. Abrupt changes from one phase to another are signaled by fluctuations that show correlation over infinitely long distances, and are measured by correlation functions that show algebraic decay as well as various kinds of singularities and infinities in thermodynamic derivatives and in measured system parameters.Renormalization methods were first developed in field theory to get around difficulties caused by apparent divergences at both small and large scales. However, no renormalization gives a fully satisfactory formulation of field theory.The renormalization group theory of phase transitions was put together by Kenneth G. Wilson in 1971 based upon ideas of scaling and universality developed earlier in the context of phase transitions and of couplings dependent upon spatial scale coming from field theory. Correlations among regions with fluctuations in their order underlie renormalization ideas. Wilson's theory is the first approach to phase transitions to agree with the extended singularity theorem.Some of the history of the study of these correlations and singularities is recounted, along with the history of renormalization and related concepts of scaling and universality. Applications, particularly to condensed-matter physics and particle physics, are summarized.This note is partially a summary of a talk given at the workshop “Part and Whole” in Leiden during the period March 22–26, 2010. (shrink)

Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effective field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will (...) all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispensable. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. (shrink)

Providing a precise statement of their position has long been a central challenge facing the scientific realist. This paper draws some morals about how realism ought to be formulated from the renormalization group framework in high energy physics.

Following the experimental discovery of the Higgs boson, physicists explained the discovery to the public by appealing to analogies with condensed matter physics. The historical root of these analogies is the analogies to models of superconductivity that inspired the introduction of spontaneous symmetry breaking into particle physics in the early 1960s. We offer a historical and philosophical analysis of the analogies between the Higgs model of the electroweak interaction and the Ginsburg-Landau and Bardeen-Cooper-Schrieffer models of superconductivity, respectively. The conclusion of (...) our analysis is that both sets of analogies are purely formal in virtue of the fact that they are accompanied by substantial physical disanalogies. In particular, the formal analogies do not map the temporal, causal, or modal structures of SSB in superconductivity to temporal, causal, or modal structures in the Higgs model. These substantial physical disanalogies mean that analogies to models of superconductivity cannot supply the basis for the physical interpretation of EW SSB; however, an appreciation of the contrast between the physical interpretations of SSB in superconductivity and the Higgs model does help to clarify some foundational issues. Unlike SSB in superconductivity, SSB in the Higgs sector of the Standard Model is neither a temporal nor a causal process. We discuss the implications for the `eating' metaphor for mass gain in the Higgs model. Furthermore, the distinction between the phenomenological GL model and the dynamical BCS model does not carry over to EW models, which clarifies the desiderata for so-called `dynamical' models of EW SSB. Finally, the development of the Higgs model is an illuminating case study for philosophers of science because it illustrates how purely formal analogies can play a fruitful heuristic role in physics. (shrink)

Most philosophical discussion of the particle concept that is afforded by quantum field theory has focused on free systems. This paper is devoted to a systematic investigation of whether the particle concept for free systems can be extended to interacting systems. The possible methods of accomplishing this are considered and all are found unsatisfactory. Therefore, an interacting system cannot be interpreted in terms of particles. As a consequence, quantum field theory does not support the inclusion of particles in our ontology. (...) In contrast to much of the recent discussion on the particle concept derived from quantum field theory, this argument does not rely on the assumption that a particulate entity be localizable. (shrink)

By Parallel Reasoning is the first comprehensive philosophical examination of analogical reasoning in more than forty years designed to formulate and justify standards for the critical evaluation of analogical arguments. It proposes a normative theory with special focus on the use of analogies in mathematics and science. In recent decades, research on analogy has been dominated by computational theories whose objective has been to model analogical reasoning as a psychological process. These theories have devoted little attention to normative questions. In (...) this book Bartha proposes that a good analogical argument must articulate a clear relationship that is capable of generalization. This idea leads to a set of distinct models for the critical analysis of prominent forms of analogical argument. The same core principle makes it possible to relate analogical reasoning to norms and values of scientific practice. Reasoning by analogy is justified because it strikes an optimal balance between conservative values, such as simplicity and coherence, and progressive values, such as fruitfulness and theoretical unification. Analogical arguments are also justified by appeal to symmetry--like cases are to be treated alike. In elaborating the connection between analogy and these broad epistemic principles, By Parallel Reasoning offers a novel contribution to explaining how analogies can play an important role in the confirmation of scientific hypotheses. (shrink)

The book examines issues related to the way modeling and simulation enable us to reconstruct aspects of the world we are investigating. It also investigates the processes by which we extract concrete knowledge from those reconstructions and how that knowledge is legitimated.

An Introduction to Quantum Field Theory is a textbook intended for the graduate physics course covering relativistic quantum mechanics, quantum electrodynamics, and Feynman diagrams. The authors make these subjects accessible through carefully worked examples illustrating the technical aspects of the subject, and intuitive explanations of what is going on behind the mathematics. After presenting the basics of quantum electrodynamics, the authors discuss the theory of renormalization and its relation to statistical mechanics, and introduce the renormalization group. This discussion sets the (...) stage for a discussion of the physical principles that underlie the fundamental interactions of elementary particle physics and their description by gauge field theories. (shrink)

In this work, Paul Bartha proposes a normative theory of analogical arguments and raises questions and proposes answers regarding the criteria for evaluating analogical arguments, the philosophical justification for analogical reasoning, and the place of scientific analogies in the context of theoretical confirmation.

This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis--the success of mathematical physics ...

Robert Batterman examines a form of scientific reasoning called asymptotic reasoning, arguing that it has important consequences for our understanding of the scientific process as a whole. He maintains that asymptotic reasoning is essential for explaining what physicists call universal behavior. With clarity and rigor, he simplifies complex questions about universal behavior, demonstrating a profound understanding of the underlying structures that ground them. This book introduces a valuable new method that is certain to fill explanatory gaps across disciplines.