Christopher Pincock

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.

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

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.

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

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, …

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.

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.

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 protected]

In lieu of an abstract, here is a brief excerpt of the content:_Russell_ journal (home office): E:CPBRRUSSJOURTYPE2502\REVIEWS.252 : 2006-02-27 11:52 Reviews  HISTORY OF PHILOSOPHICAL ANALYSIS C P Philosophy / Purdue U. West Lafayette,  ,  @. Scott Soames. Philosophical Analysis in the Twentieth Century, Vol. : The Dawn of Analysis; Vol. : The Age of Meaning. Princeton: Princeton U. P., . Pp. xix, ; xxii, . . (hb), . (pb) for each volume. he last twenty years have seen an explosion in books and papers on RusTsell ’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 -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. Soames’ focus is on what he takes to be the most important and influential work of analytic philosophers, beginning with Moore’s Principia Ethica and ending in  with Kripke’s Naming and Necessity lectures. Kripke, in fact, marks the culmination of one of the two great achievements of analytic philos- _Russell_ journal (home office): E:CPBRRUSSJOURTYPE2502\REVIEWS.252 : 2006-02-27 11:52  Reviews ophy that Soames sees in this period: … the two most important achievements that have emerged from the analytic tradition in this period are (i) the recognition that philosophical speculation must be grounded in pre-philosophical thought, and (ii) the success achieved in understanding, and separating one from another, the fundamental methodological notions of logical consequence, logical truth, necessary truth, and apriori truth. (: xi) Moore is credited with the methodological innovation required by (i), as once we accept that what we think we know prior to philosophical reflection is a constraint on our epistemology, Moore’s response to scepticism, which Soames endorses, inevitably follows. But even Moore, and nearly every other figure that Soames discusses, is guilty of confusing necessity, analyticity and apriority. Soames endorses Kripke’s basic point that necessity is a metaphysical concept that can come apart from the epistemic notion of apriority and the semantic category of analytic propositions. Volume  repeats this charge several times, using it to undermine Moore’s views on ethics in Part One, Russell’s conception of analysis in Part Two, and logical positivism in Part Three. It is noteworthy that Soames takes Ayer’s Language, Truth and Logic as representative of logical positivism, ignoring contemporary scholarship on the Vienna Circle just as much as he ignores Russell scholarship. Part Four reconstructs Wittgenstein’s views in the Tractatus and Volume  ends with a discussion of Quine’s “Two Dogmas of Empiricism”. The second volume begins with a part on Wittgenstein’s Philosophical Investigations. This paves the way for Soames’ discussion of ordinary language philosophy in Parts Two and Three, which Soames sees as closely tied to the later Wittgenstein. Ryle’s Concept of Mind, Strawson’s early views on truth, Hare’s theory of goodness as well as Malcolm’s paradigm-case argument and Austin’s Sense and Sensibilia are investigated in these chapters. Part Four presents Grice’s theory of conversational implicature as the final nail in the coffin of ordinary language philosophy. In Part Five, Soames returns to Quine, this time discussing the ambitious arguments of Word and Object and the more general project of naturalized epistemology. Part Six articulates Davidson’s program for constructing a theory of meaning for natural languages along the lines of a Tarskistyle theory of truth, and Volume  ends with an extended discussion of the promise and limitations of Kripke’s Naming and Necessity. The material on Russell is entirely confined to four chapters in Volume : Chapter : “Logical Form, Grammatical Form, and the Theory of Descriptions ”, Chapter : “Logic and Mathematics: The Logicist Reduction”, Chapter : “Logical Constructions and...

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.

In lieu of an abstract, here is a brief excerpt of the content:82 Reviews c:\users\arlene\documents\rj issues\type3501\rj 3501 061 red.docx 2015-07-10 4:07 PM LOGICISM BEYOND PRINCIPIA MATHEMATICA Chris Pincock Philosophy / Ohio State U. Columbus, oh 43210–1365, usa [email protected] William Demopoulos. Logicism and Its Philosophical Legacy. Cambridge: Cambridge U. P., 2013. Pp. xii, 272. isbn: 9781107029804.£60.00; us$104.99 (hb). his book brings together eight previously published essays along with three new essays and a brief introduction. In one way or another, each essay pursues either logicism or some broader implication of logicism that a major figure like Russell or Carnap explored. In a narrow sense, logicism is just the position that grounds the concepts and claims of arithmetic (or all of mathematics) in logic. However, in Demopoulos’ hands, logicism becomes a project of much greater significance. A note added to Chapter 5, the classic 1985 article “Bertrand Russell’s The Analysis of Matter” (co-authored with Michael Friedman), indicates this broader scope and Demopoulos’ own take on logicism. Originally Demopoulos and Friedman had concluded their article with both the observation that there are serious “intellectual tensions produced by logicism’s attempt to account for both pure mathematics and applied mathematics (mathematical physics)” and the pessimistic conclusion that “it appears that we can account for the distinctive character of the one only at the expense of the other” (p. 107). Now Demopoulos indicates that even though “something close to [this] is true of Carnap’s Ramsey-sentence q= Reviews 83 c:\users\arlene\documents\rj issues\type3501\rj 3501 061 red.docx 2015-07-10 4:07 PM reconstruction of the language of science”, “the four previous chapters are an extended argument against” the earlier conjecture. For Demopoulos, the main problem is to make sense of both pure and applied mathematics in something like the original logicist form found in Frege, Russell and Carnap.The solution to this problem arises only when we have pinpointed exactly where these logicist projects failed and how their failures can be overcome. Chapter 1 focuses on Frege and his logicism for arithmetic. Demopoulos’ main contention is that Frege aims to reduce arithmetic to logic in order to demonstrate the autonomy of arithmetic from geometric intuition and empirical experience. Frege’s worry about intuition is not motivated by scepticism or by a concern about the cogency of our knowledge of arithmetic (p. 11).The point of logicism is to free arithmetic from the obscurity associated with intuition and to get a clearer grasp of the generality that is characteristic of arithmetic. (Chapters 8 and 9 develop this point in more detail and use it to motivate a brand of logicism for arithmetic that is taken to improve on current neo-Fregean approaches.) For this reason Hume’s Principle takes a central role in Frege’s reconstruction of arithmetic for it not only establishes the autonomy of arithmetic, but also secures the core use of arithmetic in application, i.e. in counting (p. 19). Unlike Dummett and other commentators Demopoulos does not think that this aspect of Frege’s logicism is flawed. Instead, flaws arise only with the accounts of applied mathematics offered by later figures like Russell and Carnap.The root of these problems is that they extend a particular logicist strategy to other areas of mathematics, such as geometry, and to our scientific knowledge more generally. Chapter 2 considers “Carnap’sThesis”, which Demopoulos summarizes as “the assertion that certain applied mathematical theories are not factual” (p. 28). Although the thesis is clearly central to Carnap throughout his career, Demopoulos ultimately argues for a non-Carnapian basis for the factual/nonfactual distinction.The root of the difference is the kind of criteria of identity that are appropriate for the objects at the heart of the theory: … the factuality of an applied mathematical theory will be shown to turn on the recognition that the criteria of identity appropriate to its objects are empirically constrained in a way that the criteria of identity appropriate to the objects of a nonfactual theory are not. (P. 28) For arithmetic, the natural numbers associated with two sortal concepts are identical just in case the objects that...

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.

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 what these models are able to achieve and how they can support a successful scientific research program. I argue that the way these models are interpreted is central to the success of such programs.

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

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.