The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.
In his carefully argued and extensively researched article “The Implications of Recent Work in the History of Analytic Philosophy” (Preston 2005a) Aaron Preston has raised what should surely be the central methodological issue for Russell studies and the history of analytic philosophy more generally.[1] That is, what are the goals of the history of analytic philosophy and by what means can we best try to meet these goals? Preston’s main conclusion is that historical investigation into the origins of analytic philosophy (...) has made the most common answers to these questions untenable. In particular, we are encouraged to conclude that analytic philosophy is not even a genuine philosophical movement, and is in this sense “illusory”. For Preston, then, the history of analytic philosophy should reconcile itself to this fact and adjust its methods dramatically. Once we see that analytic philosophy, as traditionally conceived, never existed, then we are free to apply tools not usually deployed in the history of philosophy, e.g. memetics (Preston 2005b). (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.
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: pincock@purdue.edu. (shrink)
Christopher Pincock April 24, 2006 My goal in reviewing Soames’ book was to help readers of this journal evaluate his contribution to the history of analytic philosophy, with a special focus on his discussion of Russell.[1] Soames charges both that I misrepresent the contents of his book and that I make mistakes in the interpretation of various aspects of Russell’s philosophy. If I had committed any errors of the former sort, I would certainly apologize and thank Soames for bringing (...) such a mistake to my attention. After explaining why I do not believe I have misrepresented the contents of his book, I will turn to the one substantive issue that he raises in his reply, namely the need for unperceived sense-data in Russell’s external world program. While disagreement here is more understandable, nothing Soames says in his book or in his reply has led me to revise my original remarks. (shrink)
Christopher Pincock, Department of Philosophy, Purdue University, West Lafayette, IN 47907, USA This volume presents seventeen essays (not eleven, as the publisher inexplicably claims) by a diverse group of philosophers that arose out of a conference in Florence in 1999. As its title indicates, the focus of the conference was the contemporary significance of the topics, methods and innovations of the logical empiricists. This has led to a nicely balanced collection that combines careful historical study with an eye on (...) current debates in the philosophy of science and mind. (shrink)
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)
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)
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 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)
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.
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.
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)
Abstract: 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 in the absence of a fundamental theory. This framework (...) is illustrated using models from contemporary meteorology. (shrink)
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.
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)
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.
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)
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)
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)
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. (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.
Idealized scientific representations result from employing assumptions 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, its components are decoupled from their original physical interpretation. (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)
In her preface to this collection of 11 new essays on Ramsey, Frápolli clarifies the nonhistorical orientation of the volume: ‘Our way of honoring Ramsey has been to think with him and, wherever possible, to go beyond that, putting his ideas to work and seeing how far they can reach’ (ix). This certainly makes sense for the topics of many of these essays, building as they do on Ramsey’s rich contributions to economics and reliabilist epistemology as well as on his (...) suggestive proposals about truth, pragmatism and the content of scientific theories. Unfortunately, such an orientation leads to mixed results for the four essays that squarely focus on Ramsey’s philosophy of logic and mathematics. Here Ramsey’s contemporary significance is more debatable and the most fertile mathematical innovation that Ramsey offered, namely ‘Ramsey theory’, is not noted by the contributors to this volume. (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. 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)
This paper presents two different kinds a priori entitlements and argues that both are necessary to account for scientific knowledge. On the one hand, there are formal a priori entitlements whose existence is grounded in conditions on concept possession. On the other hand, there are material a priori entitlements that an agent accrues in virtue of practical reasoning. The discussion aims to reconcile the strengths of Christopher Peacocke’s and Michael Friedman’s recent work on the a priori, while overcoming the weaknesses (...) of their respective proposals. (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).
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)
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)
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.
write to correct errors in Christopher Pincock’s review of my discussion of IRussell. First, according to Pincock, I attempt to “undermine Moore’s views on ethics in Part One, [and] Russell’s conception of analysis in Part Two” by charging them with a pre-Kripkean conflation of necessity with apriority and analyticity. Not so. Although I do show that such conflation had negative consequences for the views of several philosophers, Moore and Russell are not among them. Moore’s error—which marred the defence (...) of his thesis that conclusions about goodness are never consequences of purely descriptive premisses—was in tacitly assuming that all necessary/a priori relations among concepts arise from definitions (see my : –). A similar problem occurs in Russell, but only tangentially in connection with one possible route to his problematic principle () in Our Knowledge of the External World, the critique of which was not as part of any attack on his general conception of analysis (. (shrink)
Chris Pincock is offended that I presumed to write a historical overview of analytic philosophy without filling it with scholarly detail provided by specialists. Instead of relying on them, I simply read the works of leading philosophers and tried to figure out for myself what they were up to. Didn’t I know that this is impossible? I myself point out in the Epilogue that the history of philosophy is now a specialized discipline. How, Pincock wonders, could I have (...) failed to recognize the implications of this lesson for my own project? Don’t try this at home! Read the original works, if you must, but don’t dare say anything about the views you find – let alone evaluate them by contemporary standards -- unless you first vet your remarks with those in the archives. History isn’t easy, you know! On the contrary, Pincock tells us, “the overriding lesson of work in the history of analytic philosophy is that history is hard.” Conveying that lesson should, he tells us, be the main goal of any historical introduction to the subject. “Above all,” he says, “I would hope that the reader would finish reading such a book with an appreciation of the difficulties inherent in the study of the history of philosophy.” This, I submit, is self-serving nonsense. Conveying its own difficulty is not an overriding goal of any worthwhile intellectual enterprise. The chief difficulty that daunts Pincock is, of course, the secondary literature produced by those like himself. According to him, any proper historical introduction “would have to build on the mountain of books and papers” – by which he means the mountain of secondary literature – and, “judiciously choose from all the proposed interpretations of those details,” carefully referencing alternative interpretations. I disagree. There are different kinds of historical work, with different goals, which make different contributions. My goal was to present analytic philosophy by identifying both its most important 2 achievements and those of its failures from which we have the most to learn.. (shrink)
CHRISTOPHER PINCOCK, Department of Philosophy, Purdue University, West Lafayette, IN 47907, USA The volume under review contains fifteen new essays by some of the most influential scholars of the history of early analytic philosophy. The focus of the essays is, as the editor says in the preface, ‘the work of Gottlob Frege and of Ludwig Wittgenstein (mostly the early Wittgenstein), as well as various ties between them’ (p. x). The essays are divided into four parts. The first part, ‘Background (...) and General Themes’, contains essays by E. Reck, G. Gabriel and S. Gerrard. The second part on Frege has contributions by H. Sluga, S. Shieh, M. Ruffino and J. Weiner. Essays on the relation between Frege and the early Wittgenstein by W. Goldfarb, D. Macbeth, T. Ricketts and C. Diamond comprise the third part. The volume concludes with essays by I. Proops, J. Floyd, M. Ostrow and J. Conant on the early Wittgenstein. This volume is an important contribution to our understanding of Frege and the early Wittgenstein and should prove a help to specialists in the history of analytic philosophy. I have chosen to briefly discuss seven of these essays with an emphasis on topics in the history and philosophy of logic. Reck’s opening essay, ‘Wittgenstein’s “Great Debt” to Frege: Biographical Traces and Philosophical Themes’, gives a helpful overview of our current knowledge of the contacts between Frege and Wittgenstein. Reck argues quite persuasively for the conclusion that Wittgenstein engaged with Frege’s work throughout his philosophical career. The depth of this engagement is in-. (shrink)
No categories
My bibliography
Export citation
Off-campus access
Using PhilPapers from home?
Click here to configure this browser for off-campus access.
Monitor this page
Be alerted of all new items appearing on this page. Choose how you want to monitor it:
