Research: Christopher Pincock
Last update: October 13, 2015
My primary research project considers the role of mathematics in science. Mathematics is prominent throughout our best scientific practices, and yet the point of this central place is poorly understood and often generates conceptual confusion. I approach this daunting issue in three complementary ways.
Scientific representation: Mathematical concepts provide some of the best tools for effectively representing phenomena of scientific interest. I argue that the most important contribution that mathematics makes to representation is in terms of facilitating scientific knowledge. That is, in several different ways, using mathematics to represent target systems allows a scientist to know more about those target systems. One way that this can happen is tied to the precise way that mathematics allows one to limit and calibrate the claims being made about some phenomenon. This epistemic approach to mathematized science raises several questions. Can we develop a general account of how models represent that includes mathematical models and other kinds of models like concrete scale models? Also, what sort of scientific knowledge is possible using mathematical models, given that they are highly idealized and of limited scope? My current research on representation is focused on these issues.
Scientific explanation: One of the aims of science is to understand why some phenomenon occurs, and not just to accurately describe how it occurs. Mathematics is often central to a scientific explanation, and yet this apparently indispensable role for mathematics in explanation continues to generate philosophical controversy. I claim that there are some distinctively "abstract" scientific explanations where the mathematics is not representing causes, but instead generating an informative classification of physical systems. At the same time, I reject so-called indispensability arguments for mathematical platonism based on an analysis of the necessary conditions for inference to the best explanation (IBE). My current work on explanation builds on these commitments. I aim to develop an account of scientific explanation that permits several kinds of explanations. A second project here is to refine and improve on Lipton's model of IBE. I argue that IBE can generate knowledge only if it operates under stringent constraints. This limited form of IBE can license belief in the existence of unobservables like atoms, and yet is unlikely to generate support for abstract objects.
Scientific change: The history of science includes many non-cumulative, revolutionary changes in theories, methods and values. In some cases, mathematical concepts survive these changes, and this has suggested some special place for mathematics in understanding scientific change. I argue that scientists have improved their understanding of the costs and benefits of deploying mathematics. This means that one must consider how a mathematical tool is being used in some context in order to predict how that tool will fare in future science. This research project is informed by a series of historical case studies that illustrate the past failures and improved awareness of how mathematics is best used in science. The first case that I consider draws on the mathematical investigations of the tides after Newton.
Other interests: My two other research specializations are the philosophy of mathematics and the history of analytic philosophy. In the philosophy of mathematics I am most interested in tracing how philosophical issues arise out of mathematical practice and in response to pressing mathematical needs. In the history of analytic philosophy I focus on the philosophical development of Russell as well as the logical empiricists.
- Mathematics and Scientific Representation, Oxford University Press, 2012. (Paperback, Dec. 2014.)
Chapter 1 (with table of contents).
Some reviews, discussions, & critical notices: J. Saatsi, NDPR 2012.10.06, M. Balaguer, E. Landry & S. Bangu, Metascience 22 (2013): 247-273, S. Baron, Mind 122 (2013): 1167-1171, A. Kennedy, Int. Stud. Phil. Sci. 27 (2013): 95-98, M. Liston, Phil. Math. 21 (2013): 371-385, S. Walsh, E. Knox & A. Caulton, Phil. Sci. 81 (2014): 460-469, A. Baker, Brit. J. Phil. Sci. 66 (2015): 695-699.
- Edited Volumes
- The Unsolvability of the Quintic: A Case Study in Abstract Mathematical Explanation
Philosophers' Imprint 15 (2015): 1-19.
Abstract: 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.
- How to Avoid Inconsistent Idealizations, Synthese 191 (2014): 2957-2972.
Abstract: 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.
- Mathematical Models of Biological Patterns: Lessons from Hamilton's Selfish Herd, Biology and Philosophy 27 (2012): 481-496.
An earlier version of this article is online here.
Abstract: 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.
- Philosophy of Mathematics, in J. Saatsi & S. French (eds.), Companion to the Philosophy of Science, Continuum, 2011, 314-333.
An earlier version of is available here.
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 Explanations of the Rainbow, Studies in the History and Philosophy of Modern Physics 42 (2011): 13-22.
An earlier version of this article is online here.
Abstract: 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.
- Discussion Note: Batterman's "On the Explanatory Role of Mathematics in Empirical Science", British Journal for the Philosophy of Science 62 (2011): 211-217.
An earlier version of this article is online here.
Abstract: 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.
- Mathematics, Science and Confirmation Theory, Philosophy of Science (Symposium Proceedings) 77 (2010): 959-970.
An earlier version is available here.
Abstract: 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.
- Modeling Reality, Synthese 180 (2011): 19-32.
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.
- Mathematical Structural Realism
In A. Bokulich & P. Bokulich (eds.), Scientific Structuralism, Boston Studies in the Philosophy of Science, volume 281, Springer, 2011, 67-79.
An earlier version of this article is online here.
Abstract: 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.
- Applicability of Mathematics, Internet Encyclopedia of Philosophy.
In section 1 I consider one version of the problem of applicability tied to what is often called "Frege's Constraint". This is the view that an adequate account of a mathematical domain must explain the applicability of this domain outside of mathematics. Then, in section 2, I turn to the role of mathematics in the formulation and discovery of new theories. This leaves out several different potential contributions that mathematics might make to science such as unification, explanation and confirmation. These are discussed in section 3 where I suggest that a piecemeal approach to understanding the applicability of mathematics is the most promising strategy for philosophers to pursue.
- Towards a Philosophy of Applied Mathematics
In O. Bueno & O. Linnebo (eds.), New Waves in Philosophy of Mathematics, Palgrave Macmillan, 2009.
Abstract: 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.
- Carnap's Logical Structure of the World
Philosophy Compass 4/6 (2009): 951-961.
- From Sunspots to the Southern Oscillation: Confirming Models
of Large-Scale Phenomena in Meteorology
Studies in the History and Philosophy of Science 40 (2009): 45-56.
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.
- Mathematical Idealization
Philosophy of Science (Proceedings) 74 (2007): 957-967.
Abstract: 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.
- Russell's Last (and Best) Multiple-relation Theory of Judgment
Mind 117 (2008): 107-140.
Abstract: 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
- Carnap, Russell and the External World
In M. Friedman & R. Creath (eds.), Cambridge Companion to Carnap, 2008, 106-128.
Abstract: After summarizing and criticizing Quine’s own somewhat self-serving
reconstruction of the Russell-Carnap relationship, I turn to the more subtle proposal first
offered by Demopoulos and Friedman in their paper "Bertrand Russell’s
The Analysis of Matter: Its Historical Context and Contemporary Interest."
The link noted by Demopoulos and Friedman then frames my survey of the later developments
of Carnap and Russell. Here we find Russell filling out his proposals from the 1920s in a
fairly straightforward manner, while Carnap adopts a series of increasingly radical
proposals to overcome the problems of traditional philosophy. From where Carnap ends up
in the 1950 paper "Empiricism, Semantics and Ontology," Russell’s views on experience,
language and metaphysics look outdated and confused.
- A Role for Mathematics in the Physical Sciences
Nous 41 (2007): 253-275.
Abstract: 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 Limits of the Relative A Priori
Soochow Journal of Philosophical Studies (Taiwan) 16 (2007): 51-68.
Abstract: I consider Friedman's notion of relative a priori justification and conclude that even though Friedman's specific proposal does not work, a closely related notion has some chance of success.
- Accounting for the Unity of Experience in Dilthey, Rickert, Bradley and Ward
In U. Feest (ed.), Historical Perspectives on Erklären and Verstehen, Max Planck Institute for the History of Science, Preprint 324, 2007, 175-192.
- Richard Semon and Russell's Analysis of Mind
Russell 26 (2006): 101-125.
Abstract: 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.
- Overextending Partial Structures: Idealization and Abstraction
Philosophy of Science (Proceedings) 72 (2005): 1248-1259.
Abstract: 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.
- A Reserved Reading of Carnap's Aufbau
Pacific Philosophical Quarterly 86 (2005): 518-543.
Abstract: 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. The
main consideration presented in favor of the reserved reading is
Carnap's work on a physical construction system. Once we detach
Carnap's project from the more usual philosophical movements, his
aims and failures come into sharper focus.
- Conditions on the Use of the One-dimensional Heat Equation
In G. Sica (ed.), Essays on the Foundations of Mathematics and Logic, Vol. 2, Polimetrica, 2005, 67-79.
Abstract: 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.
- A New Perspective on the Problem of Applying Mathematics
Philosophia Mathematica 12 (2004): 135-161.
Abstract: This paper sets out a new framework
for discussing a long-standing problem in the philosophy of
mathematics, namely the connection between the physical world and
a mathematical domain when that mathematics is applied in science.
I argue that considering counterfactual situations raises some
interesting challenges for some approaches to applications, and
consider an approach that avoids these challenges.
- A Revealing Flaw in Colyvan's Indispensability Argument
Philosophy of Science 71 (2004), 61-79.
Abstract: 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.
- Carnap and the Unity of Science: 1921-1928
In T. Bonk (ed.), Language, Truth and Knowledge: Contributions to the Philosophy of Rudolf Carnap, Vienna Circle Institute Library, Volume 2, Kluwer, 2003, pp. 87-96.
- Russell's Influence on Carnap's Aufbau
Synthese 131 (2002), 1-37.
Abstract: 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.
- Reviews and Critical Notices
- Book Symposium: A. Casullo, Essays on A Priori Knowledge and Justification
Philosophical Studies, forthcoming.
- Review of I. Hacking, Why is There Philosophy of Mathematics At All?
British Journal for the Philosophy of Science, forthcoming. (doi: 10.1093/bjps/axv044)
- Review of W. Demopoulos, Logicism and its Philosophical Legacy
Russell 35 (2015): 82-87.
- Review of J. R. Brown, Platonism, Naturalism and Mathematical Knowledge
Mind 123 (2014): 1174-1177.
- Critical Notice: S. Bangu, The Applicability of Mathematics in Science: Indispensability and Ontology
Philosophia Mathematica 22 (2014): 401-412.
- Book Symposium: H. J. Glock, What is Analytic Philosophy?
Journal for the History of Analytical Philosophy, 2/2 (2013): 6-10.
- With P. Mancosu, Mathematical Explanation
Oxford Bibliographies in Philosophy, D. Pritchard (ed.), 2012.
- Book Symposium: M. Leng, Mathematics and Reality
Metascience 21 (2012): 269-275.
- Review of B. van Fraassen, Scientific Representation: Paradoxes of Perspective
British Journal for the Philosophy of Science, 62 (2011): 677-682.
- Review of M. Suárez (ed.), Fictions in Science: Philosophical Essays on Modeling and Idealization
International Studies in the Philosophy of Science 25 (2011): 196-199.
- Critical Notice of Mark Wilson's Wandering Significance: An Essay on Conceptual Behavior
Philosophia Mathematica 18 (2010): 106-121.
- Review of B. Gold & R. A. Simons (ed.), Proof and Other Dilemmas: Mathematics and Philosophy
Association for Computing Machinery SIGACT News, 41 (2010): 28-33.
- Review of D. Reed, The Origins of Analytic Philosophy: Kant and Frege
History and Philosophy of Logic 30 (2009): 308-309.
- Review of Luetzen, Mechanistic Images in Geometric Form: Heinrich Hertz's Principles of Mechanics
Philosophia Mathematica 16 (2008): 140-144.
- Preston on the Illusory Character of Analytic Philosophy
Bertrand Russell Society Quarterly 136 (2007): 40-47.
- Review of Miah, Russell's Theory of Perception (1905-1919), Continuum, 2006.
Notre Dame Philosophical Reviews, 2007.03.08.
- Reply to Soames
Russell 26 (summer 2006): 77-86. (Here is what I am replying to. See below for the related exchange at the 2006 Pacific APA.)
- Review of S. Soames, Philosophical Analysis in the Twentieth Century, Princeton, 2003.
Russell 25 (winter 2005-2006): 167-172.
- Review of Frápolli (ed.), F. P. Ramsey: Critical Reassessments, Continuum, 2005.
History and Philosophy of Logic 27 (2006): 81-82.
- Review of Corfield, Towards a Philosophy of Real Mathematics, Cambridge University Press, 2003.
Philosophy of Science 72 (2005): 632-634.
- Critical Notice for Torsten Wilholt, Zahl und Wirklichkeit [Number and Reality], Mentis, 2004.
Philosophia Mathematica 13 (2005): 329-337.
- Review of Awodey and Klein (eds.), Carnap Brought
Home: The View from Jena, Open Court, 2004.
M. Galavotti (ed.), Cambridge and Vienna. Ramsey and the Vienna Circle. Institute Vienna Circle Yearbook 12, Kluwer, 2005, 213-218.
- Review of Milkov,
A Hundred Years of English Philosophy, Kluwer, 2003.
Notre Dame Philosophical Reviews, 2004.10.06.
- Review of Parrini, Salmon and Salmon (eds.), Logical Empiricism: Historical and Contemporary Perspectives, University of Pittsburgh Press, 2003.
F. Stadler (ed.), Induction and Deduction
in the Sciences, Institute Vienna Circle Yearbook 11, Kluwer,
- Review of Wittgenstein & Waismann, The Voices of Wittgenstein: The Vienna Circle, G. Baker (ed.), Routledge, 2003.
History and Philosophy of Logic 25 (2004): 156-157.
- Review of M. Ostrow, Wittgenstein's Tractatus: A Dialectical Interpretation, Cambridge, 2002.
Notre Dame Philosophical Reviews, 2003.01.14.
- Review of E. Reck (ed.), From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy, Oxford, 2002.
History and Philosophy of Logic 23 (2002): 297-300.
- Abstract Explanations in Science
British Journal for the Philosophy of Science, forthcoming.
A previous version of this paper is online here.
Abstract: This paper 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. Abstract 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.
- Logical Empiricism
J. Hawthorne, H. Cappelen and T. Szabo Gendler (eds.), Oxford Handbook of Philosophical Methodology, forthcoming.
Abstract: At different times logical empiricists engaged one another in debates about the proper problems and methods for philosophy or its successor discipline. The most pressing problem focused on how to coordinate the abstract statements of the sciences with what can be experienced and tested. While the new logic was the main tool for coordination for Moritz Schlick, Hans Reichenbach, and Rudolf Carnap, there was no agreement on the nature of logic or its role in coordination. Otto Neurath and Philipp Frank countered with a sophisticated alternative that emphasized the social and political context within which science is done. All told, one finds in logical empiricism a high level of methodological awareness as well as a healthy skepticism about the appropriate aims and methods of philosophy.
Work in Progress
- "Explaining with Idealized Causal Models"
- "Neutral Monism", for R. Wahl (ed.), Bloosmbury Companion to Russell, Bloomsbury Press.
- "Accommodating Explanatory Pluralism", for A. Reutlinger and J. Saatsi (eds.), Explanation Beyond Causation, Oxford University Press.
- "Mathematical Explanation Requires Mathematical Truth", for S. Dasgupta and B. Weslake (eds.), Current Controversies in Philosophy of Science, Routledge.
- "Ernest Nagel's Naturalism: A Microhistory of the American Reception of Logical Empiricism", for A. Preston (ed.), Analytic Philosophy: An Interpretive History, Routledge.
- Edited volume. Co-edited with S. Lapointe. Innovations in the History of Analytical Philosophy, Palgrave Macmillan. Under contract.
'Classics' from the Archives
- Author Meets Critics: Scott Soames, Philosophical Analysis in the Twentieth Century, Pacific APA (2006)
My comments on volume 1. Other comments were by Michael Kremer, Paul Horwich and Thomas Hurka.
Here are Soames' comments from the conference.
See above for the related exchange in Russell.