Mathematical Pluralism

by

Justin Clarke-Doane, Joel David Hamkins

• LAST REVIEWED: 28 February 2017
• LAST MODIFIED: 28 February 2017
• DOI: 10.1093/obo/9780195396577-0337

Introduction

Consider the Parallel Postulate. This is true of Euclidean space, and false of, e.g., hyperbolic space. Moreover, there is an obvious sense in which these geometries are “on a metaphysical par” as pure mathematical theories. In this sense, the Parallel Postulate lacks a unique truth-value. To what extent are set-theoretic, and number-theoretic, claims similar? Does the Continuum Hypothesis have a unique truth-value in this sense? Does the Powerset axiom? Does the claim that set theory is consistent? Does the claim that T implies P? The literature on mathematical pluralism seeks to answer such questions.

General Overviews

The literature on mathematical pluralism covers a considerable range of problems, both mathematical and philosophical. Gaifman 2012 and Shapiro 2007 may help readers get their bearings.

• Gaifman, Haim. “On Ontology and Realism in Mathematics.” Review of Symbolic Logic 5.3 (2012): 480–512.

  DOI: 10.1017/S1755020311000372

  Gaifman discusses a range of positions on the “determinacy” of mathematics, beginning with the view that even certain arithmetic statements lack determinate truth-values, and ascending to the view that every statement in the language of set theory is determinate.

• Shapiro, Stewart. “The Objectivity of Mathematics.” Synthese 156.2 (2007): 337–381.

  DOI: 10.1007/s11229-005-5298-y

  Shapiro applies Crispin Wright’s criteria of Truth and Objectivity to mathematics. In this book, Wright argues that there are various “realist credentials” that an area of inquiry may enjoy. For instance, it might obey “cognitive command,” according to which all differences of opinion are attributable to some kind of cognitive shortcoming. Shapiro usefully examines the extent to which mathematics (and logic) obey such constraints.