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 representation. Several different sorts of contributions from mathematics are then articulated. Pincock argues that each contribution can be understood as broadly epistemic, so that what mathematics ultimately contributes to science is best connected with our scientific knowledge. In the second part of the book, Pincock critically evaluates alternative approaches to the role of mathematics in science. These include the potential benefits for scientific discovery and scientific explanation. A major focus of this part of the book is the indispensability argument for mathematical platonism. Using the results of part one, Pincock argues that this argument can at best support a weak form of realism about the truth-value of the statements of mathematics. The book concludes with a chapter on pure mathematics and the remaining options for making sense of its interpretation and epistemology. Thoroughly grounded in case studies drawn from scientific practice, this book aims to bring together current debates in both the philosophy of mathematics and the philosophy of science and to demonstrate the philosophical importance of applications of mathematics. (shrink)
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 unify (...) abstract and causal explanations as instances of a single sort of thing. I conclude by examining the implications of the claim that explanations require objective dependence relations. If this claim is accepted, then there are several kinds of objective dependence relations. 1 Introduction2 A Case3 Abstract and Causal Explanations4 Recent Work on Mathematical Explanation5 Explanation and Dependence6 Conclusion. (shrink)
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 mathematics around when we do (...) science. This account successfully reconciles theoretical indispensability and metaphysical dispensability and has important consequences for both advocates and critics of indispensability arguments for platonism about mathematics. (shrink)
Mark Colyvan uses applications of mathematics to argue that mathematical entities exist. I claim that his argument is invalid based on the assumption that a certain way of thinking about applications, called `the mapping account,' is correct. My main contention is that successful applications depend only on there being appropriate structural relations between physical situations and the mathematical domain. As a variety of non-realist interpretations of mathematics deliver these structural relations, indispensability arguments are invalid.
My aim in this paper is to articulate an account of scientific modeling that reconciles pluralism about modeling with a modest form of scientific realism. The central claim of this approach is that the models of a given physical phenomenon can present different aspects of the phenomenon. This allows us, in certain special circumstances, to be confident that we are capturing genuine features of the world, even when our modeling occurs independently of a wholly theoretical motivation. This framework is illustrated (...) using a recent debate from meteorology. (shrink)
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, Nicholaos Jones, and Susan Vineberg for helpful (...) discussions and encouragement. †To contact the author, please write to: Department of Philosophy, Beering Hall, Purdue University, 100 N. University Street, West Lafayette, IN 47907-2098; e-mail: email@example.com. (shrink)
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 ocean waves typically involve the false assumption (...) that the ocean is infinitely deep. While it is true that the variable associated with depth is often taken to infinity in the representation of ocean waves, I explain how this mathematical transformation of the original equations does not require the belief that the ocean being modeled is infinitely deep. More generally, as a mathematical representation is manipulated, some of its components are decoupled from their original physical interpretation. (shrink)
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 dependence relation between (...) distinct mathematical domains. This case study highlights the importance of the conceptual, as opposed to computational, turn of much of modern mathematics, as recently emphasized by Tappenden and Avigad. The approach adopted here is contrasted with alternative proposals by Steiner and Kitcher. (shrink)
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 in the philosophy of science. (...) However, my concerns arise independently of the more controversial assumptions that Cartwright and Suárez have employed. (shrink)
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.
Most contemporary philosophy of mathematics focuses on a small segment of mathematics, mainly the natural numbers and foundational disciplines like set theory. While there are good reasons for this approach, in this paper I will examine the philosophical problems associated with the area of mathematics known as applied mathematics. Here mathematicians pursue mathematical theories that are closely connected to the use of mathematics in the sciences and engineering. This area of mathematics seems to proceed using different methods and standards when (...) compared to much of mathematics. I argue that applied mathematics can contribute to the philosophy of mathematics and our understanding of mathematics as a whole. (shrink)
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 explanation with special reference to Batterman’s own discussion of the (...) rainbow. (shrink)
Science and mathematics: the scope and limits of mathematical fictionalism Content Type Journal Article Category Book Symposium Pages 1-26 DOI 10.1007/s11016-011-9640-3 Authors Christopher Pincock, University of Missouri, 438 Strickland Hall, Columbia, MO 65211-4160, USA Alan Baker, Department of Philosophy, Swarthmore College, Swarthmore, PA 19081, USA Alexander Paseau, Wadham College, Oxford, OX1 3PN UK Mary Leng, Department of Philosophy, University of York, Heslington, York, YO10 5DD UK Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
This discussion note of (Batterman ) 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 , 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.
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 general confidence we might have in the structural claims made by (...) our theories. Thinking about the role of mathematics in science may also complicate other versions of realism. (shrink)
This paper explores the conditions under which scientists are warranted in adding the one-dimensional heat equation to their theories and then using the equation to describe particular physical situations. Summarizing these derivation and application conditions motivates an account of idealized scientiﬁc representation that relates the use of mathematics in science to interpretative questions about scientiﬁc theories.
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 discipline and that the basic relations of its construction systems need (...) not be eliminated. (shrink)
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? Second, does the central (...) role of mathematics in science shed any light on traditional philosophical debates about science like scientific realism, the nature of explanation or reduction? When faced with these kinds of questions many philosophers of science have little to say. Unfortunately, most philosophers of mathematics also fail to engage with questions about the relationship between mathematics and science and so a peculiar isolation has emerged between philosophy of science and philosophy of mathematics. In this introductory survey I aim to equip the interested philosopher of science with a roadmap that can guide her through the often intimidating terrain of contemporary philosophy of mathematics. I hope that such a survey will make clear how fruitful a more sustained interaction between philosophy of science and philosophy of mathematics could be. (shrink)
This paper concerns the debate on the nature of Rudolf Carnap's project in his 1928 book "The Logical Structure of the World or Aufbau". Michael Friedman and Alan Richardson have initiated much of this debate. They claim that the "Aufbau" is best understood as a work that is firmly grounded in neo-Kantian philosophy. They have made these claims in opposition to Quine and Goodman's "received view" of the "Aufbau". The received view sees the "Aufbau" as an attempt to carry out (...) in detail Russell's external world program. I argue that both sides of this debate have made errors in their interpretation of Russell. These errors have led these interpreters to misunderstand the connection between Russell's project and Carnap's project. Russell in fact exerted a crucial influence on Carnap in the 1920s. This influence is complicated, however, due to the fact that Russell and Carnap disagreed on many philosophical issues. I conclude that interpretations of the "Aufbau" that ignore Russell's influence are incomplete. (shrink)
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 intrinsic contributions.
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 there seems to be (...) little for the philosopher to say about such cases. At the same time, moving to the other extreme, scientists and philosophers have often remarked on the remarkable power that mathematics provides to the scientist, especially in the formulation of new scientific theories. Most famously, Wigner claimed that The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve (Wigner 1960, p. 14). But Wigner is far from an isolated case. According to Kant, in any special doctrine of nature there can be only as much proper science as there is mathematics therein (Kant 1786, p 6), and others seem to agree that there is some significant tie between mathematics and modern science. (shrink)
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 which focuses on issues of objectivity. In Section 5 , I turn to two more recent interpretations that can be thought of as emphasizing Carnap's concern with rational reconstruction. Finally, the article concludes by noting some current work by Leitgeb that aims to develop and update some aspects of the Aufbau project for contemporary epistemology. (shrink)
Forthcoming, Studies in the History and Philosophy of Science Abstract: The epistemic problem of assessing the support that some evidence confers on a hypothesis is considered using an extended example from the history of meteorology. In this case, and presumably in others, the problem is to develop techniques of data analysis that will link the sort of evidence that can be collected to hypotheses of interest. This problem is solved by applying mathematical tools to structure the data and connect it (...) to the competing hypotheses. I conclude that mathematical innovations provide crucial epistemic links between evidence and theories precisely because the evidence and theories are mathematically described. (shrink)
Most contemporary philosophy of mathematics focuses on a small segment of mathematics, mainly the natural numbers and foundational disciplines like set theory. While there are good reasons for this approach, in this paper I will examine the philosophical problems associated with the area of mathematics known as applied mathematics.
The epistemic problem of assessing the support that some evidence confers on a hypothesis is considered using an extended example from the history of meteorology. In this case, and presumably in others, the problem is to develop techniques of data analysis that will link the sort of evidence that can be collected to hypotheses of interest. This problem is solved by applying mathematical tools to structure the data and connect them to the competing hypotheses. I conclude that mathematical innovations provide (...) crucial epistemic links between evidence and theories precisely because the evidence and theories are mathematically described.Keywords: J. N. Lockyer; Gilbert T. Walker; ENSO; Models; Bayesianism. (shrink)
Russell’s study of the biologist and psychologist Richard Semon is traced to contact with the experimental psychologist Adolf Wohlgemuth and dated to the summer of 1919. This allows a new interpretation of when Russell embraced neutral monism and presents a case-study in Russell’s use of scientific results for philosophical purposes. Semon’s distinctive notion of mnemic causation was used by Russell to clarify both how images referred to things and how the existence of images could be reconciled with a neutral monist (...) metaphysics. (shrink)
The last twenty years have seen an explosion in books and papers on Russell’s philosophy and its contemporary significance. There is good reason to think that this will continue as the contents of the Collected Papers are digested by Russell scholars and as more specialists contribute to the history of analytic philosophy more generally. Given all this good news, it is disconcerting to find a 100 page discussion of Russell, in a well-reviewed book by a first-rate philosopher, repeating many of (...) the errors and misconceptions about Russell that scholars have worked so hard against. Soames’ discussion of Russell in the volumes under review is in fact so distressing that it alone compromises the book as a suitable introduction to the history of analytic philosophy. After briefly reviewing the outline of the two volumes, I discuss the errors concerning Russell, and conclude by drawing some lessons for Russell scholarship. (shrink)
Scott Soames has given us a clear, engaging but ultimately unsatisfying introduction to the history of analytic philosophy. Based on Soames’ impressive work in the philosophy of language, when these two volumes appeared I had high hopes that he would be successful. There is certainly a need for an introductory survey of the history of analytic philosophy. Currently, there is no resource for the beginning student or the amateur historian that will summarize our current understanding of the origins and development (...) of analytic philosophy. In what respects, then, do I find Soames’ attempt to fill this gap to be unsuccessful? The fundamental problem is that he has not succeeded in presenting what we now know about analytic philosophy and its history. Instead of drawing on the work of specialists in the field, it seems that he simply read the most famous works of the most famous philosophers and tried to figure out for himself what these philosophers were up to. Readers of Soames’ papers and other books will not be surprised to hear that this always ends in a carefully presented argument for a clearly articulated conclusion. Still, at least for the major figures considered in volume one, the interpretations offered fly in face of contemporary scholarship. I will try to justify these charges shortly by considering a few specific cases, but before I get to that, it is worth emphasizing why such an approach to the history of analytic philosophy is flawed, and why it is especially inappropriate in an introductory work. (shrink)
Many philosophers would concede that mathematics contributes to the abstractness of some of our most successful scientific representations. Still, it is hard to know what this abstractness really comes to or how to make a link between abstractness and success. I start by explaining how mathematics can increase the abstractness of our representations by distinguishing two kinds of abstractness. First, there is an abstract representation that eschews causal content. Second, there are families of representations with a common mathematical core that (...) is variously interpreted. The second part of the paper makes a connection between both kinds of abstractness and success by emphasizing confirmation. That is, I will argue that the mathematics contributes to the confirmation of these abstract scientific representations. This can happen in two ways which I label "direct" and "indirect". The contribution is direct when the mathematics facilitates the confirmation of an accurate representation, while the contribution is indirect when it helps the process of disconfirming an inaccurate representation. Establishing this conclusion helps to explain why mathematics is prevalent in some of our successful scientific theories, but I should emphasize that this is just one piece of a fairly daunting puzzle. (shrink)
After reviewing some different indispensability arguments, I distinguish several different ways in which mathematics can make an important contribution to a scientific explanation. Once these contributions are highlighted it will be possible to see that indispensability arguments have little chance of convincing us of the existence of abstract objects, even though they may give us good reason to accept the truth of some mathematical claims. However, in the concluding part of this paper, I argue that even though there is a (...) valid indispensability argument for realism about some mathematical claims, this argument is problematic as it begs the question at issue. This challenge to indispensability arguments is then used to suggest that if mathematics is making these sorts of contributions to science, then it may be the case that mathematical claims receive some non-empirical support prior to their application in scientific explanation. (shrink)
Example: which mathematical truths concerning the real numbers play a role in using real numbers to represent temperature? “temperature and other scalar ﬁelds used in physics are assumed to be continuous, and this guarantees that if point x has temperature ψ(x) and point z has temperature ψ(z) and r is a real number between ψ(x) and ψ(z), then there will be a point y spatio-temporally between x and z such that ψ(y ) = r ” (Field 1980, 57).
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 classical mechanics and what, if anything, this (...) highly successful scientific theory told us about the physical world. Lützen provides an in-depth reconstruction of how Hertz reacted to the foundational problems within the physics of his day and then used these problems to motivate his influential philosophical reflections on the nature of science and scientific theorizing. While giving a thorough portrait of how Hertz brought together science and philosophy, Lützen himself offers an excellent example of the benefits of combining philosophy, the history of science, and the history of mathematics. Lützen convincingly argues that Hertz's most influential innovation was in bringing geometrical concepts to bear on mechanics in a novel and productive fashion. In his preface he motivates his book by noting that most of the work on Hertz's philosophy of science fails to engage with what Hertz does after the Introduction of his Principles. The bulk of Lützen's book, then, concerns the physical and mathematical content of Hertz's image of mechanics. This places certain demands on the reader who is otherwise unacquainted with analytic mechanics, but is sure to repay those who are willing to work carefully through the more technical details of Lützen's reconstruction.1Hertz is well known for his conception of scientific theories as images [Bilder] and for the fact that his preferred image of mechanics takes only space, …. (shrink)