Philosophy of Mathematics

by

Otávio Bueno

• LAST REVIEWED: 10 May 2010
• LAST MODIFIED: 10 May 2010
• DOI: 10.1093/obo/9780195396577-0069

Introduction

Philosophy of mathematics is arguably one of the oldest branches of philosophy, and one that bears significant connections with core philosophical areas, particularly metaphysics, epistemology, and (more recently) the philosophy of science. This entry focuses on contemporary developments, which have yielded novel approaches (such as new forms of Platonism and nominalism, structuralism, neo-Fregeanism, empiricism, and naturalism) as well as several new issues (such as the significance of the application of mathematics, the role of visualization in mathematical reasoning, particular attention to mathematical practice and to the nature of mathematical explanation). Excellent work has also been done on particular philosophical issues that arise in the context of specific branches of mathematics, such as algebra, analysis, and geometry, as well as particular mathematical theories, such as set theory and category theory. Due to limitations of space, this work goes beyond the scope of the present entry.

General Overviews

There are several general overviews of the philosophy of mathematics, varying in how detailed or up-to-date they are. Horsten 2008 and Detlefsen 1996 are very readable and thoughtful surveys of the field. The former is up-to-date and freely available online; the latter offers more detailed coverage of the issues it addresses. Longer treatments of particular topics in the philosophy of mathematics may be found in the papers collected in Shapiro 2005, Irvine 2009, Bueno and Linnebo 2009, and Schirn 1998. An excellent and up-to-date survey of Platonism in the philosophy of mathematics, which is also freely available online, is given in Linnebo 2009. Burgess and Rosen 1997 offers a critical survey of some nominalist views, but the work is no longer up-to-date.

• Bueno, Otávio, and Øystein Linnebo, eds. New Waves in Philosophy of Mathematics. Basingstoke, UK: Palgrave Macmillan, 2009.

  DOI: 10.1057/9780230245198

  A collection of thirteen essays by promising young researchers that offers an up-to-date picture of contemporary philosophy of mathematics, including a reassessment of orthodoxy in the field, the question of realism in mathematics, relations between mathematical practice and the methodology of mathematics, and connections between philosophical logic and the philosophy of mathematics.

• Burgess, John P., and Gideon A. Rosen. A Subject with No Objects: Strategies for Nominalistic Interpretation of Mathematics. Oxford: Clarendon, 1997.

  A critical examination of major nominalist interpretations of mathematics by two authors who do not defend nominalism.

• Detlefsen, Michael. “Philosophy of Mathematics in the Twentieth Century.” In Philosophy of Science, Logic and Mathematics in the Twentieth Century. Routledge History of Philosophy 9. Edited by Stuart G. Shanker, 50–123. New York: Routledge, 1996.

  A careful survey of the philosophy of mathematics focusing on some of the central proposals in the 20th century.

• Horsten, Leon. “Philosophy of Mathematics.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2008.

  A useful and up-to-date survey of the philosophy of mathematics, including a discussion of four classic approaches (logicism, intuitionism, formalism, and predicativism) as well as more recent proposals (Platonism, structuralism, and nominalism) and some special topics (philosophy of set theory, categoricity, and computation and proof).

• Irvine, Andrew D., ed. Philosophy of Mathematics. Handbook of the Philosophy of Science series. Amsterdam: North Holland, 2009.

  An up-to-date survey of the philosophy of mathematics composed by fifteen specially commissioned essays that cover central issues and conceptions in the field, with emphasis on realism and antirealism, empiricism, Kantianism, as well as logicism, formalism, and constructivism. Philosophical issues that emerge in set theory, probability theory, computability theory in addition to inconsistent and applied mathematics are also examined.

• Linnebo, Øystein. “Platonism in the Philosophy of Mathematics.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2009.

  An up-to-date survey of the main forms of mathematical Platonism as well as the central arguments for this conception and the main objections that have been raised against it.

• Schirn, Matthias, ed. The Philosophy of Mathematics Today. Oxford: Clarendon, 1998.

  A comprehensive panorama of the philosophy of mathematics given by twenty specially commissioned essays by leading philosophers of mathematics, which examine a range of issues from the nature of mathematical knowledge and the existence of mathematical objects through the characterization of the concepts of set and natural number to logical consequence and abstraction.

• Shapiro, Stewart, ed. The Oxford Handbook of Philosophy of Mathematics and Logic. New York: Oxford University Press, 2005.

  A comprehensive and up-to-date overview of the philosophy of mathematics, written by some of the main contributors to the field, and that also explores significant connections with the philosophy of logic. Balanced treatments of logicism, formalism, intuitionism, empiricism, naturalism, nominalism, structuralism, as well as the application of mathematics, concepts of logical consequence and relevance, and higher-order logic, are offered.