Mathematical Structuralism

by

Elaine Landry

• LAST REVIEWED: 26 May 2016
• LAST MODIFIED: 26 May 2016
• DOI: 10.1093/obo/9780195396577-0305

Introduction

Mathematical structuralism is the view that, in some sense, mathematics is about structures and their relations, rather than about objects and their properties. It finds philosophical expression historically, in the works of Richard Dedekind and David Hilbert (see Erich Reck’s “Dedekind’s Structuralism: An Interpretation and Partial Defense” cited under General Overviews and Elaine Landry’s “The Genetic versus the Axiomatic Method” cited under History), and weaves itself through the mathematical development of modern algebra (see Leo Corry’s Modern Algebra and the Rise of Mathematical Structures cited under General Overviews). It has, more recently, been recovered as a contemporary philosophical position in connection with Paul Benacerraf’s “What Numbers Could Not Be” argument against taking numbers as Fregean objects (cited under History). There are three standard interpretations of mathematical structuralism and several varieties of these. The first is set-theoretic, or model-theoretic, mathematical structuralism, which holds that mathematical objects are positions in a structure qua a set-structure system or model (for a critical overview of this interpretation, see Stewart Shapiro’s Philosophy of Mathematics: Structure and Ontology cited under General Overviews). The interpretation preferred by Shapiro, however, is ante rem structuralism, which holds that mathematical objects are places in a sui generis abstract structure. In contrast, the eliminativist in re structuralist (see Geoffrey Hellman’s Mathematics without Numbers: Towards a Modal-Structural Interpretation cited under Varieties and Interpretations) eliminates talk of both structures and objects and holds that mathematical objects are positions in any or all systems that possibly exist that have the same structure. What all of these interpretations and varieties have in common is the aim of “fixing the domain” for a discussion about structures or systems. The set-theoretic structuralist typically appeals to ZFC (Zermelo Fraenkel set theory plus Choice) to do this. Shapiro uses his realist structure theory for this purpose, whereas Hellman uses his modal nominalist account of possible systems. Against these accounts, Reck's “Dedekind’s Structuralism: An Interpretation and Partial Defense”(cited under General Overviews) argues that, as a logical mathematical structuralist, Dedekind did not have such metaphysical or semantic concerns. Steve Awodey's “Structure in Mathematics and Logic: A Categorical Perspective” (cited under Varieties and Interpretations), Colin McLarty's “Exploring Categorical Structuralism” (cited under Varieties and Interpretations), Elaine Landry's “How to Be a Structuralist All the Way Down” (cited under Varieties and Interpretations) is aimed at developing a fourth, category-theoretic, interpretation of mathematical structuralism. Finally, John Burgess’s Rigor and Structure (cited under General Overviews) provides an overarching account of the interplay between the search for mathematical rigor and the position of mathematical structuralism, arguing that structuralism is a “trivial truism” of the developments of 20th-century mathematics.

General Overviews

Several general overviews of mathematical structuralism are available, each varying in both detail and updates. Reck and Price 2000 and Shapiro 1997 offer very accessible and thorough surveys of the various philosophical interpretations. For a more historical perspective of the philosophical and mathematical developments, see Reck 2003 and Corry 1996. A useful and updated (as of 2012) online account of mathematical structuralism and its relation to nominalism is given in Horsten’s Structuralism and Nominalism. For a more recent comprehensive consideration of the history and philosophy of mathematical structuralism, see Burgess 2015.

• Burgess, John. Rigor and Structure. Oxford: Oxford University Press, 2015.

  DOI: 10.1093/acprof:oso/9780198722229.001.0001

  The most up-to-date resource (as of 2015). Argues that the 20th-century search for mathematical rigor leads to mathematical structuralism as a “trivial truism.” Separates out the structuralism of philosophers (ante rem, in re, set-theoretic, category-theoretic, etc.) and argues that the mathematician’s structuralism is indifferent to these varieties and interpretations. This indifferentist position is close to Reck’s account (Reck 2003) of methodological structuralism.

• Corry, Leo. Modern Algebra and the Rise of Mathematical Structures. Basel, Switzerland: Birkhäuser, 1996.

  This book traces the history of mathematical structuralism from a mathematical, not philosophical, perspective by considering the development of the structures of modern algebra, e.g., groups, rings, fields, lattices, and categories.

• Horsten, Leon. Structuralism and Nominalism. In Stanford Encyclopedia of Philosophy. Stanford, CA: Stanford University.

  Section 4 gives a useful and updated (2012) account of the philosophical position of mathematical structuralism and its relation to mathematical nominalism.

• Reck, Erich. “Dedekind’s Structuralism: An Interpretation and Partial Defense.” Synthese 137 (2003): 369–419.

  DOI: 10.1023/B:SYNT.0000004903.11236.91

  Argues that although Dedekind did have logical concerns that arose from the desire to fix conceptually a domain of interpretation, these concerns were distinct from the semantic and metaphysical concerns that underpin current varieties of mathematical structuralism. Further argues that such a minimalist position is worth philosophical reconsideration as an alternative to the current set-theoretic, ante rem, and in re varieties of mathematical structuralism.

• Reck, Erich, and Michael Price. “Structures and Structuralism in Contemporary Philosophy of Mathematics.” Synthese 125 (2000): 341–383.

  DOI: 10.1023/A:1005203923553

  Provides the starting point for the investigation of contemporary philosophical positions on mathematical structuralism. Aims to make explicit the differences and similarities between the various varieties and interpretations of mathematical structuralism by a consideration of both the semantic and metaphysical issues that underlie these positions.

• Shapiro, Stewart. Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press, 1997.

  Reviews the various philosophical accounts of mathematical structuralism and provides an axiomatic “structure-theory” to underwrite both a semantic and metaphysical account of structures as ante rem universals, in which objects are taken as “places” in an abstract structure.