•  223
    A Role for Mathematics in the Physical Sciences
    Noûs 41 (2): 253-275. 2007.
    Conflicting accounts of the role of mathematics in our physical theories can be traced to two principles. Mathematics appears to be both (1) theoretically indispensable, as we have no acceptable non-mathematical versions of our theories, and (2) metaphysically dispensable, as mathematical entities, if they existed, would lack a relevant causal role in the physical world. I offer a new account of a role for mathematics in the physical sciences that emphasizes the epistemic benefits of having math…Read more
  •  181
    Abstract Explanations in Science
    British Journal for the Philosophy of Science 66 (4): 857-882. 2014.
    This article focuses on a case that expert practitioners count as an explanation: a mathematical account of Plateau’s laws for soap films. I argue that this example falls into a class of explanations that I call abstract explanations.explanations involve an appeal to a more abstract entity than the state of affairs being explained. I show that the abstract entity need not be causally relevant to the explanandum for its features to be explanatorily relevant. However, it remains unclear how to uni…Read more
  •  180
    Carnap's logical structure of the world
    Philosophy Compass 4 (6): 951-961. 2009.
    This article aims to give an overview of Carnap's 1928 book Logical Structure of the World or Aufbau and the most influential interpretations of its significance. After giving an outline of the book in Section 2 , I turn to the first sustained interpretations of the book offered by Goodman and Quine in Section 3 . Section 4 explains how this empirical reductionist interpretation was largely displaced by its main competitor. This is the line of interpretation offered by Friedman and Richardson wh…Read more
  •  159
    Mathematical Structural Realism
    In Alisa Bokulich & Peter Bokulich (eds.), Scientific Structuralism, Springer Science+business Media. pp. 67--79. 2011.
    Epistemic structural realists have argued that we are in a better epistemic position with respect to the structural claims made by our theories than the non-structural claims. Critics have objected that we cannot make the structure/non-structure distinction precise. I respond that a focus on mathematical structure leads to a clearer understanding of this debate. Unfortunately for the structural realist, however, the contribution that mathematics makes to scientific representation undermines any …Read more
  •  115
    Mathematics and Scientific Representation
    Oxford University Press USA. 2012.
    Mathematics plays a central role in much of contemporary science, but philosophers have struggled to understand what this role is or how significant it might be for mathematics and science. In this book Christopher Pincock tackles this perennial question in a new way by asking how mathematics contributes to the success of our best scientific representations. In the first part of the book this question is posed and sharpened using a proposal for how we can determine the content of a scientific re…Read more
  •  114
    Russell's version of the multiple-relation theory from the "Theory of Knowledge" manuscript is presented and defended against some objections. A new problem, related to defining truth via correspondence, is reconstructed from Russell's remarks and what we know of Wittgenstein's objection to Russell's theory. In the end, understanding this objection in terms of correspondence helps to link Russell's multiple-relation theory to his later views on propositions.
  •  100
    The Applicability of Mathematics
    Internet Encyclopedia of Philosophy. 2010.
    Depending on how it is clarified, the applicability of mathematics can lie anywhere on a spectrum from the completely trivial to the utterly mysterious. At the one extreme, it is obvious that mathematics is used outside of mathematics in cases which range from everyday calculations like the attempt to balance one s checkbook through the most demanding abstract modeling of subatomic particles. The techniques underlying these applications are perfectly clear to those who have mastered them and the…Read more
  •  97
    This paper identifies one way that a mathematical proof can be more explanatory than another proof. This is by invoking a more abstract kind of entity than the topic of the theorem. These abstract mathematical explanations are identified via an investigation of a canonical instance of modern mathematics: the Galois theory proof that there is no general solution in radicals for fifth-degree polynomial equations. I claim that abstract explanations are best seen as describing a special sort of depe…Read more
  •  92
    A new perspective on the problem of applying mathematics
    Philosophia Mathematica 12 (2): 135-161. 2004.
    This paper sets out a new framework for discussing a long-standing problem in the philosophy of mathematics, namely the connection between the physical world and a mathematical domain when the mathematics is applied in science. I argue that considering counterfactual situations raises some interesting challenges for some approaches to applications, and consider an approach that avoids these challenges.
  •  90
    Mathematical Idealization
    Philosophy of Science 74 (5): 957-967. 2007.
    Mathematical idealizations are scientific representations that result from assumptions that are believed to be false, and where mathematics plays a crucial role. I propose a two stage account of how to rank mathematical idealizations that is largely inspired by the semantic view of scientific theories. The paper concludes by considering how this approach to idealization allows for a limited form of scientific realism. ‡I would like to thank Robert Batterman, Gabriele Contessa, Eric Hiddleston, N…Read more
  •  88
    Concrete Scale Models, Essential Idealization, and Causal Explanation
    British Journal for the Philosophy of Science 73 (2): 299-323. 2022.
    This paper defends three claims about concrete or physical models: these models remain important in science and engineering, they are often essentially idealized, in a sense to be made precise, and despite these essential idealizations, some of these models may be reliably used for the purpose of causal explanation. This discussion of concrete models is pursued using a detailed case study of some recent models of landslide generated impulse waves. Practitioners show a clear awareness of the idea…Read more
  •  87
    Russell’s Influence On Carnap’s Aufbau
    Synthese 131 (1): 1-37. 2002.
    This paper concerns the debate on the nature of Rudolf Carnap'sproject in his 1928 book The Logical Structure of the Worldor Aufbau. Michael Friedman and Alan Richardson haveinitiated much of this debate. They claim that the Aufbauis best understood as a work that is firmly grounded inneo-Kantian philosophy. They have made these claims in oppositionto Quine and Goodman's ``received view'' of the Aufbau. Thereceived view sees the Aufbau as an attempt to carry out indetail Russell's external world…Read more
  •  86
    Mathematical explanations of the rainbow
    Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 42 (1): 13-22. 2011.
    Explanations of three different aspects of the rainbow are considered. The highly mathematical character of these explanations poses some interpretative questions concerning what the success of these explanations tells us about rainbows. I develop a proposal according to which mathematical explanations can highlight what is relevant about a given phenomenon while also indicating what is irrelevant to that phenomenon. This proposal is related to the extensive work by Batterman on asymptotic expla…Read more
  •  85
    On Batterman's 'On the Explanatory Role of Mathematics in Empirical Science'
    British Journal for the Philosophy of Science 62 (1). 2011.
    This discussion note of (Batterman [2010]) clarifies the modest aims of my 'mapping account' of applications of mathematics in science. Once these aims are clarified it becomes clear that Batterman's 'completely new approach' (Batterman [2010], p. 24) is not needed to make sense of his cases of idealized mathematical explanations. Instead, a positive proposal for the explanatory power of such cases can be reconciled with the mapping account.
  •  82
    Exploring the boundaries of conceptual evaluation
    Philosophia Mathematica 18 (1): 106-121. 2010.
    This is a critical notice of Mark Wilson's Wandering Significance.
  •  81
    A reserved reading of Carnap's aufbau
    Pacific Philosophical Quarterly 86 (4). 2005.
    The two most popular approaches to Carnap's 1928 Aufbau are the empiricist reading of Quine and the neo-Kantian readings of Michael Friedman and Alan Richardson. This paper presents a third "reserved" interpretation that emphasizes Carnap's opposition to traditional philosophy and consequent naturalism. The main consideration presented in favor of the reserved reading is Carnap's work on a physical construction system. I argue that Carnap's construction theory was an empirical scientific discipl…Read more
  •  78
    How to avoid inconsistent idealizations
    Synthese 191 (13): 2957-2972. 2014.
    Idealized scientific representations result from employing claims that we take to be false. It is not surprising, then, that idealizations are a prime example of allegedly inconsistent scientific representations. I argue that the claim that an idealization requires inconsistent beliefs is often incorrect and that it turns out that a more mathematical perspective allows us to understand how the idealization can be interpreted consistently. The main example discussed is the claim that models of oc…Read more
  •  77
    How can a reflective scientist put forward an explanation using a model when they are aware that many of the assumptions used to specify that model are false? This paper addresses this challenge by making two substantial assumptions about explanatory practice. First, many of the propositions deployed in the course of explaining have a non-representational function. In particular, a proposition that a scientist uses and also believes to be false, i.e. an “idealization”, typically has some non-rep…Read more
  •  76
    Philosophy of Mathematics
    In J. Saatsi & S. French (eds.), Companion to the Philosophy of Science, Continuum. pp. 314-333. 2011.
    For many philosophers of science, mathematics lies closer to logic than it does to the ordinary sciences like physics, biology and economics. While this view may account for the relative neglect of the philosophy of mathematics by philosophers of science, it ignores at least two pressing questions about mathematics that philosophers of science need to be able to answer. First, do the similarities between mathematics and science support the view that mathematics is, after all, another science? Se…Read more
  •  70
    Fictions in Science: Philosophical Essays on Modeling and Idealization (review)
    International Studies in the Philosophy of Science 25 (2). 2011.
    International Studies in the Philosophy of Science, Volume 25, Issue 2, Page 196-199, June 2011
  •  69
    Mathematical models of biological patterns are central to contemporary biology. This paper aims to consider what these models contribute to biology through the detailed consideration of an important case: Hamilton’s selfish herd. While highly abstract and idealized, Hamilton’s models have generated an extensive amount of research and have arguably led to an accurate understanding of an important factor in the evolution of gregarious behaviors like herding and flocking. I propose an account of wh…Read more
  •  69
    Bas C. Van Fraassen * Scientific Representation: Paradoxes of Perspective (review)
    British Journal for the Philosophy of Science 62 (3): 677-682. 2011.
  •  64
    A pluralist about explanation posits many explanatory relevance relations, while an invariantist denies any substantial role for context in fixing genuine explanation. This article summarizes one approach to combining pluralism and invariantism that emphasizes the contrastive nature of explanation. If explanations always take contrasts as their objects and contrasts come in types, then the role for the context in which an explanation is given can be minimized. This approach is illustrated using …Read more
  •  61
    Mathematics, Science, and Confirmation Theory
    Philosophy of Science 77 (5): 959-970. 2010.
    This paper begins by distinguishing intrinsic and extrinsic contributions of mathematics to scientific representation. This leads to two investigations into how these different sorts of contributions relate to confirmation. I present a way of accommodating both contributions that complicates the traditional assumptions of confirmation theory. In particular, I argue that subjective Bayesianism does best accounting for extrinsic contributions, while objective Bayesianism is more promising for intr…Read more
  •  55
    Philosophers unacquainted with the workings of actual scientific practice are prone to imagine that our best scientific theories deliver univocal representations of the physical world that we can use to calibrate our metaphysics and epistemology. Those few philosophers who are also scientists, like Heinrich Hertz, tend to contest this assumption. As Jesper Lützen relates in his scholarly and engaging book, Hertz's Principles of Mechanics contributed to a lively debate about the content of classi…Read more
  •  45
    Ian Hacking why is there philosophy of mathematics at all? (review)
    British Journal for the Philosophy of Science 67 (3): 907-912. 2016.
  •  41
    Overextending Partial Structures: Idealization and Abstraction
    Philosophy of Science 72 (5): 1248-1259. 2005.
    The partial structures program of da Costa, French and others offers a unified framework within which to handle a wide range of issues central to contemporary philosophy of science. I argue that the program is inadequately equipped to account for simple cases where idealizations are used to construct abstract, mathematical models of physical systems. These problems show that da Costa and French have not overcome the objections raised by Cartwright and Suárez to using model-theoretic techniques i…Read more
  •  40
    Preface
    Synthese 190 (2): 187-188. 2013.