Christopher Pincock

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.

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.

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.

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 program. I argue that both sidesof this debate have made errors in their interpretation ofRussell. These errors have led these interpreters to misunderstandthe connection between Russell's project and Carnap's project.Russell in fact exerted a crucial influence on Carnap in the1920s. This influence is complicated, however, due to the factthat Russell and Carnap disagreed on many philosophical issues. Iconclude that interpretations of the Aufbau that ignoreRussell's influence are incomplete.

This book offers new perspectives on the history of analytical philosophy, surveying recent scholarship on the philosophical study of mind, language, logic and reality over the course of the last 200 years. Each chapter contributes to a broader engagement with a wider range of figures, topics and disciplines outside of philosophy than has been traditionally associated with the history of analytical philosophy. The book acquaints readers with new aspects of analytical philosophy’s revolutionary past while engaging in a much needed methodological reflection. It questions the meaning associated with talk of 'analytic' philosophy and offers new perspective on its development. It offers original studies on a range of topics – including in the philosophy of language and mind, logic, metaphysics and the philosophy of mathematics – and figures whose relevance, when they is not already established as in the case of Russell, Moore and Wittgenstein, are just now beginning to become the topic of mainstream literature: Franz Brentano, William James, Susan Langer as well as the German and British logicians of the nineteenth century.

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.