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Naturalism in the Philosophy of Mathematics
- LAST REVIEWED: 08 October 2015
- LAST MODIFIED: 31 August 2015
- DOI: 10.1093/obo/9780195396577-0276

- LAST REVIEWED: 08 October 2015
- LAST MODIFIED: 31 August 2015
- DOI: 10.1093/obo/9780195396577-0276

## Introduction

In the context of the philosophy of mathematics, the term “naturalism” has a number of uses, covering approaches that look to be fundamentally at odds with one another. In one use, the “natural” in naturalism is contrasted with non-natural, in the sense of supernatural; in this sense, naturalism in the philosophy of mathematics appears in opposition to Platonism (the view that mathematical truths are truths about a body of abstract mathematical objects). Naturalism thus construed takes seriously the epistemological challenge to Platonism presented by Paul Benacerraf in his paper “Mathematical Truth” (cited under Ontological Naturalism). Benacerraf points out that a view of mathematics as a body of truths about a realm of abstract objects appears to rule out any (non-mystical) account of how we, as physically located embodied beings, could come to know such truths. The naturalism that falls out of acceptance of Benacerraf’s challenge as presenting a genuine problem for our claims to be able to know truths about abstract mathematical objects is sometimes referred to as “ontological naturalism,” and suggests a physicalist ontology. In a second use, the “natural” in naturalism is a reference specifically to natural science and its methods. Naturalism here, sometimes called methodological naturalism, is the Quinean doctrine that philosophy is continuous with natural science. Quine and Putnam’s indispensability argument for the existence of mathematical objects places methodological naturalism in conflict with ontological naturalism, since it is argued that the success of our scientific theories confirms the existence of the abstract mathematical objects apparently referred to in formulating those theories, suggesting that methodological naturalism requires Platonism. A final use of “naturalism” in the philosophy of mathematics is distinctive to mathematics, and arises out of consideration of the proper extent of methodological naturalism. According to Quine’s naturalism, the natural sciences provide us with the proper methods of inquiry. But, as Penelope Maddy has noted, mathematics has its own internal methods and standards, which differ from the methods of the empirical sciences, and naturalistic respect for the methodologies of successful fields requires that we should accept those methods and standards. This places Maddy’s methodological naturalism in tension with the original Quinean version of the doctrine, because, Maddy argues, letting natural science be the sole source of confirmation for mathematical theories fails to respect the autonomy of mathematics.

## General Overviews

Two articles, on naturalism in general and on naturalism in the philosophy of mathematics, provide a good place to start. The first of these, Papineau 2009, provides an excellent overview of philosophical naturalisms organized around the distinction between ontological and methodological naturalism. Paseau 2010 covers naturalism in the philosophy of mathematics, discussing a number of issues in methodological naturalism. Other helpful overviews include Maddy 2005, covering Quinean and post-Quinean naturalism, and Weir 2005, which discusses the tension between ontological and methodological naturalism and how this might be resolved. For a book length discussion and critique of naturalism in the philosophy of mathematics, see Brown 2012, which covers a number of specific naturalistic proposals, subjecting them to a non-naturalist Platonist critique.

Brown, James Robert.

*Platonism, Naturalism, and Mathematical Knowledge*. London: Routledge, 2012.Save Citation »Export Citation »E-mail Citation »

An outsider (Brown is a non-naturalist Platonist) takes a skeptical look at mathematical naturalism in all its main forms.

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Maddy, Penelope. “Three Forms of Naturalism.” In

*The Oxford Handbook of Philosophy of Mathematics and Logic*. Edited by Stewart Shapiro, 437–459. Oxford: Oxford University Press, 2005.DOI: 10.1093/0195148770.001.0001Save Citation »Export Citation »E-mail Citation »

Maddy sketches Quine’s naturalism, then her own and John P. Burgess’s developments of Quine’s views as two other forms. Maddy’s “moral of the story” is “that naturalism, even restricted to its Quinean and post‐Quinean incarnations, is a more complex position, with more subtle variants, than is sometimes supposed” (p. 438).

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Papineau, David. “Naturalism.”

*The Stanford Encyclopedia of Philosophy*(Spring 2009).Save Citation »Export Citation »E-mail Citation »

An excellent overview of philosophical naturalism, organized around the distinction between ontological and methodological naturalism, and including sections on mathematics under both headings.

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Paseau, Alexander. “Naturalism in the Philosophy of Mathematics.”

*The Stanford Encyclopedia of Philosophy*(Fall 2010).Save Citation »Export Citation »E-mail Citation »

A helpful discussion of naturalism in the philosophy of mathematics that includes a detailed discussion of a number of issues in methodological naturalism, as well as shorter discussions of naturalism as a guide to ontology and epistemology.

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Weir, Alan. “Naturalism Reconsidered.” In

*The Oxford Handbook of Philosophy of Mathematics and Logic*. Edited by Stewart Shapiro, 460–482. Oxford: Oxford University Press, 2005.DOI: 10.1093/0195148770.001.0001Save Citation »Export Citation »E-mail Citation »

Organized around the ontological/methodological distinction. Aside from some of the usual suspects (Quine and Maddy on the methodological side, Field, Hellman, Chihara, Kitcher, and Bigelow on the ontological side), Weir considers how neo-logicism could be brought to the defense of methodological naturalism, making mathematical knowledge unproblematic as a branch of logical knowledge.

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## Ontological Naturalism

The starting point for ontological naturalism in this section is the physicalist/materialist doctrine that “material objects in space and time are the only things that exist” (Brown 2012, p. 3, cited under General Overviews), or at the very least, the slightly weaker doctrine that material objects in space in time are the only things that we have reason to believe exist. Such a doctrine may be motivated, for example, by Benacerrafian considerations about how we can come to know about objects, as presented in Benacerraf 1973. This version of naturalism obviously has difficulties with the standard view of mathematical truths as truths about abstract mathematical objects. Physicalist alternatives to this standard view can involve (a) accepting mathematical truths as truths about objects, but naturalizing those objects by bringing some of them into the material/spatiotemporal/causal realm; (b) reinterpreting mathematical truths so that they no longer appear to involve mathematical objects, but instead are explicable as knowable through natural processes; or (c) accepting the standard view of mathematical truth and concluding from this that we do not have any knowledge of mathematical truths so construed. Irvine 1990 brings together some early discussions of physicalist views as motivated by the considerations of Benacerraf 1973. Burgess and Rosen 1997 is an extended discussion of nominalist strategies of avoiding commitment to the abstract.

Benacerraf, Paul. “Mathematical Truth.”

*The Journal of Philosophy*70 (1973): 661–679. .DOI: 10.2307/2025075Save Citation »Export Citation »E-mail Citation »

Paul Benacerraf’s famous articulation of the knowledge problem for Platonism. This is the key motivation for many ontological (physicalist) naturalists, who take Benacerraf’s challenge as fatal for accounts of mathematical knowledge as knowledge of a realm of abstract objects.

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Burgess, John P., and Gideon Rosen.

*A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics*. Oxford: Clarendon, 1997.Save Citation »Export Citation »E-mail Citation »

Outlines a number of nominalist strategies for interpreting mathematics, first in the abstract, before tying them to concrete proposals (specifically, Field’s geometric strategy and Chihara’s and Hellman’s two modal strategies).

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Irvine, A. D., ed.

*Physicalism in Mathematics*. Dordrecht, The Netherlands: Kluwer, 1990.DOI: 10.1007/978-94-009-1902-0Save Citation »Export Citation »E-mail Citation »

Edited collection bringing together a number of physicalist responses to Benacerraf 1973, as well as objections to physicalist views.

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### Naturalizing Mathematical Objects

Maddy 1980 presents the view, developed further in Maddy 1990a and Maddy 1990b, that at least some mathematical objects (in particular, sets of physical objects) are spatiotemporally located (with sets being located with their members). Bringing these mathematical objects into the spatiotemporal realm renders them accessible to perception and thus explains how we can have at least some mathematical knowledge (knowledge of perceptually-accessible sets). Mathematical knowledge of more complex objects, those not accessible to perception, is theoretical and relates to our basic set theoretical knowledge, much like our knowledge of electrons relates to our knowledge of medium sized physical objects: it is confirmed by its usefulness in organizing the basic knowledge we gain from perception. Bigelow 1988 and Bigelow 1990 see mathematical objects as physical, but whereas Maddy accounts for (some) mathematical objects as physically located particulars, Bigelow views numbers and sets as universals, physically located with their instances.

Bigelow, John.

*The Reality of Numbers: A Physicalist’s Philosophy of Mathematics*. Oxford: Clarendon, 1988.Save Citation »Export Citation »E-mail Citation »

Argues that mathematical objects are universals, physically located in particulars (so that the universal “2” is located in all instances of pairs).

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Bigelow, John. “Sets Are Universals.” In

*Physicalism in Mathematics*. Edited by A. D. Irvine, 291–306. Dordrecht, The Netherlands: Kluwer, 1990.DOI: 10.1007/978-94-009-1902-0Save Citation »Export Citation »E-mail Citation »

Article-length sketch of the position of Bigelow 1988.

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Maddy, Penelope. “Perception and Mathematical Intuition.”

*The Philosophical Review*89 (1980): 163–196.DOI: 10.2307/2184647Save Citation »Export Citation »E-mail Citation »

An early presentation of Maddy’s naturalizing of Gödelian mathematical intuition as involving perception of mathematical objects in the form of physically located impure sets.

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Maddy, Penelope.

*Realism in Mathematics*. Oxford: Clarendon, 1990a.Save Citation »Export Citation »E-mail Citation »

Book-length presentation of the view initiated in Maddy 1980.

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Maddy, Penelope. “Physicalistic Platonism.” In

*Physicalism in Mathematics*. Edited by A. D. Irvine, 259–290. Dordrecht, The Netherlands: Kluwer, 1990b.DOI: 10.1007/978-94-009-1902-0Save Citation »Export Citation »E-mail Citation »

Develops and defends the apparent contradiction in terms of physicalist Platonism and compares it to Field’s nominalism as a physicalist philosophy of mathematics.

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### Reinterpreting Mathematical Truths

Drawing on J. S. Mill’s empiricism, Philip Kitcher develops a view of mathematical truths as truths about (idealized) human collecting and segregating abilities. Kitcher 1983 views individual mathematical knowledge as grounded in the knowledge of a community of mathematicians that can be traced back, through a series of rational transitions, ultimately, to perceptual knowledge of our ancestors. Arithmetic knowledge, for example, is rooted, according to Kitcher, in our knowledge of some basic truths about collecting, segregating, and matching activities, once these activities are idealized to avoid obvious limitations (they involve truths about collections we could make given world enough and time). Mathematical knowledge is thus, for Kitcher, knowledge of the idealized activities of ideal agents. Kitcher 1988 discusses the naturalistic epistemology standing behind the author’s account of mathematical knowledge. A difficulty for Kitcher’s view is explaining the status of the ideal agents appealed to in his account of mathematical knowledge. This challenge is taken up in Hoffman 2004, which develops a fictionalist account of Kitcher’s ideal agents. An alternative reinterpretation of mathematical truths is provided by nominalist versions of structuralism, such as the modal versions of structuralism provided in Hellman 1989, Hellman 1990, and Chihara 2003. Chihara 2003 is itself a structuralist reworking of Chihara 1990, which aims to interpret mathematical truths as truths about constructability.

Chihara, Charles S.

*Constructibility and Mathematical Existence*. Oxford: Clarendon, 1990.Save Citation »Export Citation »E-mail Citation »

Presents a nominalistic reconstruction of a basic set theory (the simple theory of types) in terms of modal claims about the possibility of individuals satisfying open sentence tokens. The central idea is to replace claims about the existence of mathematical objects with claims about what constructions are possible.

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Chihara, Charles S.

*A Structural Account of Mathematics*. Oxford: Clarendon, 2003.Save Citation »Export Citation »E-mail Citation »

Develops Chihara’s previous constructibilist account in a structuralist vein. This leads to a nominalist version of structuralism that differs from the main nominalist structuralist alternative (Hellman 1989). Specifically, Chihara adds to his nominalist reconstruction of type theory a structuralist interpretation that sees mathematical claims within axiomatic theories as implicitly about what’s true in all set theoretic models of the axioms.

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Hellman, Geoffrey.

*Mathematics without Numbers: Towards a Modal Structural Interpretation*. Oxford: Clarendon, 1989.Save Citation »Export Citation »E-mail Citation »

Although not explicitly presented here in physicalist/naturalist terms, Hellman’s reinterpretation of mathematical theories as bodies of truths about what follows from our mathematical axioms (intuitively speaking, about what

*would*be true*were there any*objects satisfying our mathematical theories), as a form of nominalism, provides one way of avoiding concerns about mathematical knowledge as knowledge of abstracta, without denying the truth of mathematics.Find this resource:

Hellman, Geoffrey. “Modal-Structural Mathematics.” In

*Physicalism in Mathematics*. Edited by A. D. Irvine, 307–330. Dordrecht, The Netherlands: Kluwer, 1990.DOI: 10.1007/978-94-009-1902-0Save Citation »Export Citation »E-mail Citation »

Article-length sketch of the position of Hellman 1989.

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Hoffman, Sarah. “Kitcher, Ideal Agents, and Fictionalism.”

*Philosophia Mathematica*12 (2004): 3–17.DOI: 10.1093/philmat/12.1.3Save Citation »Export Citation »E-mail Citation »

Argues that Kitcher’s account of mathematical knowledge as grounded in our knowledge of the activities of idealized human agents is best developed as a kind of fictionalism, where the fictional entities are ideal agents rather than abstract mathematical objects.

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Kitcher, Philip.

*The Nature of Mathematical Knowledge*. New York: Oxford University Press, 1983.Save Citation »Export Citation »E-mail Citation »

Kitcher naturalizes mathematical knowledge by presenting a reworking of J. S. Mill’s empiricism, which grounds mathematical knowledge in knowledge of natural processes, and ultimately, in our knowledge of the idealized activities of ideal agents.

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Kitcher, Philip. “Mathematical Naturalism.” In

*History and Philosophy of Modern Mathematics*. Vol. 11. Edited by William Aspray and Philip Kitcher, 293–328. Minnesota Studies in Philosophy of Science. Minneapolis: University of Minnesota Press, 1988.Save Citation »Export Citation »E-mail Citation »

Develops and defends the naturalistic view of epistemology that stands behind Kitcher 1983, particularly as involving the rejection of the a priori, and the understanding of knowledge gathering as a communal, human enterprise with a history.

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### Rejecting Mathematical Knowledge

The radical solution to Benacerraf’s dilemma for mathematical knowledge involves accepting both the standard view of mathematical truth (as truth about abstract objects) and the claim that knowledge of such truths is humanly impossible. This approach to mathematics is taken by mathematical fictionalists, who hold that mathematics does not have to be true to be good. The main challenge to fictionalism is to explain the value of mathematics, and particularly its use in natural science, if our mathematical theories are not true. Field 1980 and Field 1989 take up this challenge, arguing that (a) mathematics is not confirmed by its role in our best scientific theories, since we can dispense with reference to mathematical objects in formulating those theories (replacing standard Platonist scientific theories with nominalist stated alternatives, which avoid reference to abstracta); and (b) the value of mathematics in empirical science is in enabling us to draw out the consequences of our nominalistically stated scientific theories. Field’s fictionalism is also endorsed in Papineau 1993. More recently, Leng 2010 suggests that we can account for the role of mathematics in empirical science without dispensing with mathematics in the formulation of our scientific theories. Balaguer 1998 also develops a defense of fictionalism along these lines, though he is not himself a fictionalist. Such attempts to reject the existence of mathematical objects while continuing to speak (in doing science) as if there such things aim to travel what Colyvan 2010 calls an “easy road to nominalism” (as compared with Field’s “hard road”). Colyvan criticizes those who wish to travel this easy road, arguing that they will inevitably find themselves back on the hard road of dispensing with mathematics. Finally, Lakoff and Núñez 2000 (discussed as a version of naturalism in Brown 2012, cited under General Overviews) presents a somewhat unclear position on mathematical truth and knowledge (perhaps unsurprisingly given that the authors are cognitive scientists rather than philosophers). However, in presenting an account of mathematics in terms of metaphor, Lakoff and Núñez’s naturalistic account of the development of mathematical thinking as sketched in this work is perhaps best interpreted as rejecting the existence of mathematical knowledge standardly construed.

Balaguer, Mark.

*Platonism and Anti-Platonism in Mathematics*. New York: Oxford University Press, 1998.Save Citation »Export Citation »E-mail Citation »

In a bold dialectic approach, Balaguer sets out to show that both Platonism (in its full-blooded form, involving a highly non-physicalist commitment to as many abstract mathematical objects as are logically possible) and anti-Platonism (in the form of fictionalism) are defensible against the standard objections to these views, before concluding that there is no way of choosing between them, and that there is therefore no fact of the matter about which is right.

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Colyvan, Mark. “There Is No Easy Road to Nominalism.”

*Mind*119 (2010): 285–306. .DOI: 10.1093/mind/fzq014Save Citation »Export Citation »E-mail Citation »

Colyvan discusses three attempts (by Jody Azzouni, Joseph Melia, and Stephen Yablo) to defend a nominalist interpretation of mathematics without following Hartry Field down the hard road of dispensing with mathematics. He argues that, in all cases, these nominalistic detours ultimately lead back to Field’s hard road of dispensing with mathematics in explanatory contexts.

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Field, Hartry.

*Science without Numbers: A Defence of Nominalism*. Princeton, NJ: Princeton University Press, 1980.Save Citation »Export Citation »E-mail Citation »

Field accepts the Quinean claim that we are committed to the existence of those objects that are indispensably posited by our best scientific theories, but he argues that mathematical posits can be dispensed with in our best theories. He argues that standard Platonist scientific theories can be accounted for as being useful, without being literally true, on the grounds that they are conservative extensions of nominalistically stated alternatives.

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Field, Hartry.

*Realism, Mathematics, and Modality*. Oxford: Blackwell, 1989.Save Citation »Export Citation »E-mail Citation »

Collects a number of Field’s essays on these topics. “Introduction: Fictionalism, Epistemology and Modality” characterizes the account of Field 1980 as a form of fictionalism. “Realism and Anti-Realism about Mathematics” gives a good overview of the approach of Field 1980. “Is Mathematical Knowledge Just Logical Knowledge?” and “On Conservativeness and Incompleteness” respond to concerns about Field’s modal assumptions.

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Lakoff, George, and Rafael Núñez.

*Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being*. New York: Basic Books, 2000.Save Citation »Export Citation »E-mail Citation »

As cogntive scientists, Lakoff and Núñez take themselves to be providing a “cognitive science of mathematics,” explaining how mathematical ideas arise out of human experience, particularly through metaphorical transitions. As naturalist anti-Platonists, they propose an “embodied realism” with respect to the “mind-based mathematics” they describe, but their realism concerns the objectivity of mathematical claims, not the existence of mathematical objects.

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Leng, Mary.

*Mathematics and Reality*. Oxford: Oxford University Press, 2010.DOI: 10.1093/acprof:oso/9780199280797.001.0001Save Citation »Export Citation »E-mail Citation »

Develops and defends the idea (see Balaguer 1998) that it is coherent in light of the successes of science to adopt a fictionalist attitude to the mathematical posits assumed in our scientific theorizing, even if those posits cannot be dispensed with.

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Papineau, David.

*Philosophical Naturalism*. Oxford: Blackwell, 1993.Save Citation »Export Citation »E-mail Citation »

An extended discussion and defense of physicalist naturalism. In his illuminating chapter on mathematics, “Mathematics and Other Non-Natural Subjects,” Papineau endorses Field-style mathematical fictionalism, having first set aside a number of initially plausible physicalist alternatives.

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## Quinean Methodological Naturalism

Methodological naturalism involves the abandonment of “first philosophy,” and in particular the abandonment of the Cartesian idea that we can set aside (as in Descartes’ Meditations on First Philosophy) the beliefs and conceptual frameworks we have inherited from the natural sciences and can build up a superior body of knowledge by applying purely philosophical principles from scratch. Quine’s naturalist retort to this Cartesian model of inquiry invokes Neurath’s metaphor of the inquirer as a sailor who must fix his boat while out at sea. “The naturalist philosopher begins his reasoning within the inherited world theory as a going concern. He tentatively believes all of it, but believes also that some unidentified portions are wrong. He tries to improve, clarify, and understand the system from within. He is the busy sailor adrift on Neurath’s boat (Quine 1981a, p. 72, cited under Naturalism in Quine’s Work).” Quine characterizes his methodological doctrine as “the recognition that it is within science itself, and not in some prior philosophy, that reality is to be identified and described” (Quine 1981b, p. 21, cited under Naturalism in Quine’s Work). Philosophy has no distinctive methods of inquiry that are superior to those of the natural sciences. So in our metaphysical inquiries into the nature of reality, we cannot stand outside of science, but instead must consider what we can learn from our best scientific theories about such questions. Although starting as a methodological doctrine, Quine’s naturalism leads to ontological conclusions, and these appear to be in tension with the physicalist understanding of ontological naturalism outlined above. In particular, in Quine’s view, the question of whether we should accept any kind of object becomes the question of whether we find it most useful to quantify over such objects in our best scientific theories (Quine 1948, Quine 1951, both cited under Naturalism in Quine’s Work). If it turns out (as seems likely) that we find ourselves quantifying over mathematical objects in our best theories, and if (as many think) our best understanding of these objects is as abstracta, then the ontological conclusion of Quine’s methodological naturalism is anti-physicalist: mathematical Platonism. This is Quine’s indispensability argument, from methodological naturalism to mathematical Platonism. A great deal of contemporary naturalist philosophy of mathematics involves defending, or responding to, this indispensability argument.

### Naturalism in Quine’s Work

The Quinean methodological doctrine that sees philosophy as continuous with natural science, and recasts ontological questions as questions about the ontological commitments of our best scientific theories, permeates most of Quine’s writings. In particular, Quine 1948 and Quine 1951, both reproduced in Quine 1961, present, in turn, Quine’s criterion for ontological commitment (in the form of the slogan “To be is to be the value of a variable” (p. 15), and Quine’s argument for taking seriously all of the ontological commitments of our best scientific theories, based on anti-reductionist/holistic considerations. Quine 1951 presents confirmational holism as the view that “our statements about the external world face the tribunal of sense experience not individually but only as a corporate body” (p. 41); and the author’s non-physicalist version of ontological naturalism is drawn out as a corollary and applied specifically to mathematics: “Ontological questions, under this view, are on a par with questions of natural science. Consider the question whether to countenance classes as entities. This, as I have argued elsewhere, is the question whether to quantify with respect to variables which take classes as values” (45). Quine 1981a and Quine 1981b, both found in Quine 1981c, provide further characteristic presentations of Quine’s naturalism: “The scientific system, ontology and all, is a conceptual bridge of our own making, linking sensory stimulation to sensory stimulation. . . . But I also expressed, at the beginning, my unswerving belief in external things—people, nerve endings, sticks, stones. This I reaffirm. I believe also, if less firmly, in atoms and electrons and in classes. Now how is all this robust realism to be reconciled with the barren scene that I have just been depicting? The answer is naturalism: the recognition that it is within science itself, and not in some prior philosophy, that reality is to be identified and described” (Quine 1981b, p. 21).

Quine, W. V. “On What There Is.”

*Review of Metaphysics*2 (1948): 21–36.Save Citation »Export Citation »E-mail Citation »

Presents Quine’s criterion of ontological commitment, noting that this criterion does not answer the question of what there is, but can only answer the question of what a given doctrine

*says*there is. Quine adds to this the naturalistic claim that the question of what there is a matter of what we say there is when doing science.Find this resource:

Quine, W. V. “Two Dogmas of Empiricism.”

*Philosophical Review*60 (1951): 21–43.DOI: 10.2307/2181906Save Citation »Export Citation »E-mail Citation »

Most famous as the article where Quine presents his rejection of the analytic/synthetic distinction. Having thus ruled out the traditional empiricist assumption that mathematics can be set aside from empirical science as analytic, “Two Dogmas” also contains Quine’s argument for the location of mathematics as a part of natural science, and for the confirmation of mathematics through its role in natural science.

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Quine, W. V.

*From a Logical Point of View*. 2d ed. Cambridge, MA: Harvard University Press, 1961.Save Citation »Export Citation »E-mail Citation »

The main themes of this collection, which includes Quine 1948 and Quine 1951, are, in Quine’s words: “the problem of meaning, particularly as involved in the notion of an analytic statement,” and “the notion of ontological commitment, particularly as involved in the problem of universals” (p. xi).

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Quine, W. V. “Five Milestones of Empiricism.” In

*Theories and Things*. By W. V. Quine, 67–72. Cambridge, MA: Harvard University Press, 1981a.Save Citation »Export Citation »E-mail Citation »

The milestones of empiricism are “points where empiricism has taken a turn for the better” (67), at least in Quine’s view. The fifth of these milestones is “naturalism: abandonment of the goal of a first philosophy. It sees natural science as an inquiry into reality, fallible and corrigible but not answerable to any supra-scientific tribunal, and not in need of any justification beyond observation and the hypothetico-deductive method” (p. 72).

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Quine, W. V. “Things and their Places in Theories.” In

*Theories and Things*. By W. V. Quine, 1–23. Cambridge, MA: Harvard University Press, 1981b.Save Citation »Export Citation »E-mail Citation »

Presents the view that, though theoretical posits are just introduced as part of a convenient conceptual scheme, we should take them—all of them, people, atoms, classes—seriously, as amongst our ontological commitments.

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Quine, W. V.

*Theories and Things*. Cambridge, MA: Harvard University Press, 1981c.Save Citation »Export Citation »E-mail Citation »

This collection reproduces the paper originally presented at a symposium in 1975 and presents Quine 1981a and Quine 1981b.

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### The Indispensability Argument and Platonism

Arising out of Quine’s naturalism and his criterion of ontological commitment, the Quine-Putnam indispensability argument for the existence of mathematical objects is stated explicitly in Putnam 1971 and Putnam 1979. Objections to Quine’s placing of mathematics as continuous with natural science (and confirmed by its role therein) include Parsons 1979–1980, Parsons 1983, and Maddy 1992 (cited under Naturalist Objections to Quine’s Confirmational Holism), leading to the development of Maddy’s mathematical naturalism (see Post-Quinean Mathematical Naturalism). Quine 1986 responds to Parsons’ objections. Colyvan 2001 is a more recent reassessment and defense of the Quine-Putnam indispensability argument and its Platonist consequences.

Colyvan, Mark.

*The Indispensability of Mathematics*. Oxford University Press, 2001.DOI: 10.1093/019513754X.001.0001Save Citation »Export Citation »E-mail Citation »

Book-length elaboration and defense of the Quine-Putnam indispensability argument, including a chapter, “The Eleatic Principle,” discussing Quinean naturalism and an explicit rejection (pp. 39–65) of the equation of naturalism with physicalism.

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Parsons, Charles. “Mathematical Intuition.”

*Proceedings of the Aristotelian Society, New Series*80 (1979–1980): 145–168.Save Citation »Export Citation »E-mail Citation »

Primarily presenting a positive account of mathematical intuition, but notable in this context for Parson’s memorable complaint against Quine’s methodological naturalism, on the grounds that it fails to account for the sheer “

*obviousness*of elementary mathematics” (p. 151).Find this resource:

Parsons, Charles. “Quine on the philosophy of mathematics.” In

*Mathematics in Philosophy: Selected Essays*. By Charles Parsons, 176–205. Ithaca, NY: Cornell University Press, 1983.Save Citation »Export Citation »E-mail Citation »

This chapter was originally written for the

*Philosophy of W. V. Quine*(La Salle, IL: Open Court, 1986). Parsons objects to the position Quine’s philosophy gives to mathematics (as continuous with empirical science) on the grounds that it fails to respect the autonomy of mathematics and its apparent immunity to empirical refutation.Find this resource:

Putnam, Hilary.

*Philosophy of Logic*. New York: Harper and Row, 1971.Save Citation »Export Citation »E-mail Citation »

Putnam’s very short book on the philosophy of logic gives indispensability arguments for the existence of mathematical objects both in metalogic and empirical science, memorably attributing the form of argument to “Quine, who has for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes” (p. 347). Reprinted in Putnam 1979.

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Putnam, Hilary. “What Is Mathematical Truth?”

*Historia Mathematica*2 (1975): 529–543.DOI: 10.1016/0315-0860(75)90116-0Save Citation »Export Citation »E-mail Citation »

Reiterates the indispensability argument of Putnam 1971, and distinguishes between internal mathematical reasons to believe our mathematical theories, which give us reason to believe in their consistency, and empirical reasons for believing them to be true (due to their role in science). This paper is also the source for Putnam’s oft-quoted “no miracles” argument for scientific realism. Reprinted in Putnam 1979.

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Putnam, Hilary.

*Mathematics, Matter and Method: Philosophical Papers*. Vol. 1. 2d ed. Cambridge, UK: Cambridge University Press, 1979.DOI: 10.1017/CBO9780511625268Save Citation »Export Citation »E-mail Citation »

Collection includes Putnam 1971 and Putnam 1975.

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Quine, W. V. “Reply to Charles Parsons.” In

*The Philosophy of W. V. Quine*. Edited by L. Hahn and P. Schilpp, 396–403. La Salle, IL: Open Court, 1986.Save Citation »Export Citation »E-mail Citation »

Response to Parsons 1983. Quine allows for the independent development of mathematics without an eye to its empirical confirmation, but argues that, until that confirmation has been received, mathematicians should be viewed as being engaged in “mathematical recreation . . . without ontological rights” (p. 400).

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### Naturalist Platonists

Platonists endorsing the indispensability considerations presented in Quine’s naturalism include Resnik 1997 and Shapiro 1997, of the structuralist stripe, as well as Burgess 1990, a more traditional object-Platonist work. Burgess 1990 argues that the holistic epistemology provided by Quine’s philosophy neutralizes Benacerraf’s knowledge problem for Platonism, though other Platonist works sympathetic to naturalism, such as Linsky and Zalta 1995, take it that the challenge of explaining our knowledge of abstract mathematical objects needs an answer distinct from Quine’s holism. Linsky and Zalta 1995 presents plenitudinous (or full-blooded) Platonists (see also Balaguer 1998, cited under Rejecting Mathematical Knowledge, for a defense of full-blooded Platonism against Benacerraf’s knowledge problem), and argues, in response to Benacerraf, that we as physically located humans can know the contents of the abstract realm without having any connection to that realm via our grasp of logical possibility, so long as we assume that it contains all the mathematical objects that are logically possible.

Burgess, John P. “Epistemology and Nominalism” In

*Physicalism in Mathematics*. Edited by A. D. Irvine, 1–16. Dordrecht, The Netherlands: Kluwer, 1990.DOI: 10.1007/978-94-009-1902-0Save Citation »Export Citation »E-mail Citation »

Argues that Quine’s holistic approach to confirmation neutralises Benacerraf’s epistemological objection to Platonism.

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Linsky, Bernard, and Edward N. Zalta. “Naturalized Platonism vs. Platonized Naturalism.”

*The Journal of Philosophy*92 (1995): 525–555.DOI: 10.2307/2940786Save Citation »Export Citation »E-mail Citation »

The “Platonized Naturalism” preferred by Linsky and Zalta takes the mathematical realm to be abstract and plenitudinous. This, they argue, is consistent with the naturalistic approach to ontology, and preferable to Quine’s piecemeal Platonism in respecting mathematical methodology and avoiding Benacerrafian that seem to arise out of a model of mathematical objects as on a par with physical objects.

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Resnik, Michael D.

*Mathematics as a Science of Patterns*. Oxford: Oxford University Press, 1997.Save Citation »Export Citation »E-mail Citation »

Very much a Quinean naturalist (convinced of holism and the indispensability argument), Resnik defends a Platonist versions of structuralism, arguing that the uses made of the mathematical posits we structure in empirical science confirms their existence.

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Shapiro, Stewart.

*Philosophy of Mathematics: Structure and Ontology*. Oxford: Oxford University Press, 1997.Save Citation »Export Citation »E-mail Citation »

Like Resnik 1997, defends a Platonist structuralism, though Shapiro is less explicitly Quinean.

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### Naturalist Objections to Quine’s Confirmational Holism

In recent years a number of philosophers sympathetic to naturalism have attempted to resist the conclusion of the indispensability argument by challenging Quine’s confirmational holism on the grounds of the distinctive role played by mathematical posits in our scientific theories. Maddy 1992, Maddy 1997, Sober 1993, and Vineberg 1996 set the ball rolling in challenging Quine’s holism. Melia 2000 (see also Balaguer 1998 and Leng 2010, both cited under Rejecting Mathematical Knowledge) questions the Quinean assumption that mathematics is confirmed by its role in empirical science, suggesting that its role is to make more things sayable about the concrete world, and that this is a role that can be played by our uses of mathematics even if there are no mathematical objects. Colyvan 2002 (see also Colyvan 2010, cited under Rejecting Mathematical Knowledge) and Baker 2005 respond on behalf of naturalists who see mathematics as confirmed by its role in empirical science by arguing that mathematics is confirmed through its use in mathematical explanations of empirical phenomena. These responses have started a thriving debate amongst methodological naturalists on the question of whether mathematics plays a genuine explanatory role in empirical science, and, if so, whether we ought to take the mathematics involved in such explanations as confirmed by their contribution to theoretical explanations.

Baker, Alan. “Are there Genuine Mathematical Explanations of Empirical Phenomena?”

*Mind*114 (2005): 223–237.DOI: 10.1093/mind/fzi223Save Citation »Export Citation »E-mail Citation »

Much-discussed paper presenting the example of the prime life cycle length of North American periodical

*Magicicada*cicadas as a case of a mathematical explanation of an empirical phenomenon.Find this resource:

Colyvan, Mark. “Mathematics and Aesthetic Considerations in Science.”

*Mind*111 (2002): 69–74.DOI: 10.1093/mind/111.441.69Save Citation »Export Citation »E-mail Citation »

Responds to the claim, in Melia 2000, that mathematics is not confirmed by its role in our scientific theories. Colyvan argues that mathematics plays more than a merely representational role. In particular, the mathematics in our empirical theories can have unificatory, and perhaps explanatory, power.

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Maddy, Penelope. “Indispensability and Practice.”

*Journal of Philosophy*89 (1992): 275–289.DOI: 10.2307/2026712Save Citation »Export Citation »E-mail Citation »

Argues for a conflict between Quinean naturalism and mathematical and scientific practice. Despite Quine’s confirmational holism, neither mathematicians nor scientists appear to act as though they view mathematics as being confirmed by its role in empirical science.

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Maddy, Penelope.

*Naturalism in Mathematics*. Oxford: Clarendon, 1997.Save Citation »Export Citation »E-mail Citation »

Argues for a parallel to Quine’s scientific naturalism to be applied to mathematics, noting the tension in Quine’s naturalism between naturalistic respect for scientific methodologies (which should include the methodologies of mathematics) and Quine’s privileging of natural science. Maddy argues that both internal mathematical practices and scientific practices seem to speak against the Quinean assumption that the mathematics used in our scientific theories is empirically confirmed.

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Melia, Joseph. “Weaseling Away the Indispensability Argument.”

*Mind*109 (2000): 455–479.DOI: 10.1093/mind/109.435.455Save Citation »Export Citation »E-mail Citation »

Argues that we need not accept the existence of mathematical objects despite the indispensability of mathematical posits in formulating our scientific theories, since the contribution made by mathematics to our empirical theories is fundamentally different from the contribution made by other theoretical posits.

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Sober, Elliot. “Mathematics and Indispensability.”

*The Philosophical Review*102 (1993): 35–57.DOI: 10.2307/2185652Save Citation »Export Citation »E-mail Citation »

Presents his own contrastive account of confirmation as an alternative to Quine’s confirmational holism and argues that, on this account, the mathematics used in our scientific theories does not receive empirical confirmation.

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Vineberg, Susan. “Confirmation and the Indispensability of Mathematics to Science.”

*Philosophy of Science*63 (1996): S256–S263.DOI: 10.1086/289959Save Citation »Export Citation »E-mail Citation »

Argues that contemporary theories of confirmation require that, for evidence to confirm a hypothesis, there has to be the possibility of disconfirmation. Vineberg claims that this does not exist for the mathematical assumptions of our scientific theories, so mathematics does not receive confirmation.

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## Post-Quinean Mathematical Naturalism

Penelope Maddy (Maddy 2005, cited under General Overviews) considers herself and John P. Burgess as two post-Quinean naturalists whose views diverge somewhat from Quine and from each other. Burgess 2004 (and see Burgess 1990, cited under Naturalist Platonists) accepts Quine’s position that ontological questions are questions for the sciences to answer, but holds that Quine’s own account of science is too narrowly based on physical science. According to Burgess, mathematics is a science in its own right: if we accept as our ontology the ontological commitments of the sciences, then pure mathematics already provides us with a vast plethora of mathematical commitments. We do not, in Burgess’s view, need to restrict our commitments grudgingly to the minimal amount of mathematics required for empirical science. Maddy, on the other hand, follows Quine in reading ‘science’ as natural science, rather than viewing mathematics as a science in its own right. Mathematics is, in Maddy’s view, something that naturalists need to account for because of its role in empirical science. However, according to Maddy, this warrants a naturalistic/scientific study of mathematics and mathematical practice, and such a study reveals that there is more to mathematics than Quine’s subordinating account suggests. A naturalistic study of mathematics reveals it to have its own internal methodology independent of the methodologies of empirical science, and its own internal standards for acceptable mathematics. Unlike Burgess, Maddy thinks that the ontological status of mathematical posits should be answered by considering the role of mathematics in empirical science, but unlike Quine, she does not think that consideration of that role supports Platonism (See Maddy 1992 Maddy 1997, both cited under Naturalist Objections to Quine’s Confirmational Holism and Maddy 1995. The author’s view on this has changed since the more Quinean perspective of Maddy 1990a (cited under Naturalizing Mathematical Objects). Maddy 1997 (cited under Naturalist Objections to Quine’s Confirmational Holism), Maddy 2007, and Maddy 2011 outline and develop the author’s naturalist position. A major objection to the naturalism of Burgess and Maddy is that extending Quine’s naturalistic respect for disciplines in any way beyond natural science leads to a problem of demarcation: how to draw the line between disciplines worthy of naturalistic respect (and evaluation on their own terms) and those that can be rationally criticized. Rosen 2001 presents this as the “authority problem” for naturalism(see also Dieterle 1999). Paseau 2005 raises another problem—the place of philosophy in post-Quinean naturalism, which the author criticizes as too deferential to mathematics and natural science.

Burgess, John P. “Mathematics and

*Bleak House*.”*Philosophia Mathematica*12 (2004): 18–36.DOI: 10.1093/philmat/12.1.18Save Citation »Export Citation »E-mail Citation »

Defends naturalistic realism as “the refusal to apologize while doing philosophy for what is said while doing mathematics or science” (p. 19).

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Dieterle, J. M. “Mathematical, Astrological, and Theological Naturalism.”

*Philosophia Mathematica*7 (1999): 129–135.DOI: 10.1093/philmat/7.2.129Save Citation »Export Citation »E-mail Citation »

Puts pressure on Maddy to distinguish between those disciplines to which we ought to extend naturalistic respect and those that are open to rational criticism. Argues that Maddy’s naturalism is either too permissive, opening the way to theological and astrological naturalism alongside mathematical naturalism, or otherwise just collapses to Quinean naturalism.

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Maddy, Penelope. “Naturalism and Ontology.”

*Philosophia Mathematica*3 (1995): 248–270.DOI: 10.1093/philmat/3.3.248Save Citation »Export Citation »E-mail Citation »

Questions Quine’s indispensability argument as dependent on “a blinkered view of scientific methodology,” but argues that if we look instead to mathematical methodology to answer ontological questions we will find no guidance there. This does not mean that ontological questions about mathematics cannot be answered, just that the answer we give will be irrelevant to questions of mathematical methodology.

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Maddy, Penelope.

*Second Philosophy: A Naturalistic Method*. Oxford: Clarendon, 2007.DOI: 10.1093/acprof:oso/9780199273669.001.0001Save Citation »Export Citation »E-mail Citation »

Maddy describes her aim in

*Second Philosophy*as “to provide a philosophical backdrop for the methodological views of*Naturalism in Mathematics*” (p. 6). The second philosopher starts with common sense and sees herself as trying to understand the world in the same kinds of ways as do empirical scientists. Mathematics is included as an example of this approach in action.Find this resource:

Maddy, Penelope.

*Defending the Axioms*. Oxford: Oxford University Press, 2011.DOI: 10.1093/acprof:oso/9780199596188.001.0001Save Citation »Export Citation »E-mail Citation »

Maddy’s most recent book applies the approach of second philosophy to the question of axiom choice in set theory. She argues that a metaphysically thin version of realism and an “arealism” that sees mathematics as good without being true, are not really in conflict, but are just different ways of characterizing the objectivity of mathematics.

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Paseau, Alexander. “Naturalism in Mathematics and the Authority of Philosophy.”

*British Journal for the Philosophy of Science*56 (2005): 377–396.DOI: 10.1093/bjps/axi123Save Citation »Export Citation »E-mail Citation »

Critique of the deferential attitude to mathematics proposed by naturalists such as Maddy, holding that no good reasons have been given to adopt such an approach.

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Rosen, Gideon. “Nominalism, Naturalism, Epistemic Relativism.”

*Philosophical Perspectives 15: Metaphysics*(2001): 69–91.Save Citation »Export Citation »E-mail Citation »

Describes a community, the Bedrockers, who do science as we do, but have well thought out reservations as to the truth of their theories, believing only their nominalist content. This raises a version of what Rosen calls the “Authority Problem” for naturalism. Set alongside a community of realist scientists, the question arises for naturalists—whose norms should we take as authoritative?

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