Naturalism in the Philosophy of Mathematics

by

Mary Leng

• LAST REVIEWED: 31 August 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.

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

• 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: 10.1093/0195148770.001.0001

  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).

• Papineau, David. “Naturalism.” The Stanford Encyclopedia of Philosophy (Spring 2009).

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

• Paseau, Alexander. “Naturalism in the Philosophy of Mathematics.” The Stanford Encyclopedia of Philosophy (Fall 2010).

  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.

• 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: 10.1093/0195148770.001.0001

  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.