Oxford Bibliographies Online - Mathematics, Philosophy of 11/4/10 10:42 PM Oxford Bibliographies Online The World’s Largest University Press: Authority and Innovation for Research Philosophy of Mathematics by Otávio Bueno 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. Find this resource: 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. Find this resource: 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. Find this resource: A careful survey of the philosophy of mathematics focusing on some of the central proposals in the 20th century. http://www.oxfordbibliographiesonline.com/display/id/obo-9780195396577-0069 Page 1 of 16 Oxford Bibliographies Online - Mathematics, Philosophy of 11/4/10 10:42 PM 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. Find this resource: 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. Find this resource: 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. Find this resource: 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. Textbooks There are excellent textbooks in the philosophy of mathematics. The most accessible for undergraduate students with a limited background in mathematics is Friend 2007. Brown 2008 is also very accessible, and it covers topics that usually are not addressed in elementary introductions to the philosophy of mathematics (such as the epistemological significance of pictures in mathematical proofs and the importance of mathematical notation). George and Velleman 2002 offers an excellent examination of how philosophical programs in the philosophy of mathematics (such as logicism, intuitionism, and finitism) combine a philosophical vision about mathematics with particular mathematical developments. A primarily historical approach to the philosophy of mathematics is carefully presented in Bostock 2009, whereas Shapiro 2000 combines a historical and conceptual discussion of major philosophical views about mathematics (both classical and contemporary). Bostock, David. Philosophy of Mathematics: An Introduction. Oxford: Wiley-Blackwell, 2009. Find this resource: A historical introduction to the philosophy of mathematics, starting with a discussion of the works of Plato and Aristotle, going through Kant, Frege, Russell, Hilbert, and Brouwer, all the way to Gödel, Quine, Putnam, and the contemporary scene. Brown, James Robert. Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and http://www.oxfordbibliographiesonline.com/display/id/obo-9780195396577-0069 Page 2 of 16 Oxford Bibliographies Online - Mathematics, Philosophy of 11/4/10 10:42 PM Pictures. 2d ed. New York: Routledge, 2008. Find this resource: A very readable and accessible introduction to the philosophy of mathematics, covering some topics not typically discussed in elementary introductions to the field, such as the role of pictures in proofs, the significance of mathematical notation, and a refutation of the continuum hypothesis. Friend, Michèle. Introducing Philosophy of Mathematics. Montreal: McGill-Queen’s University Press, 2007. Find this resource: An extremely accessible introduction to the philosophy of mathematics for undergraduate students with a limited background in mathematics. It assesses the advantages and limitations of Platonism, logicism, formalism, constructivism, and structuralism as well as psychologism, fictionalism, and Meinongianism. George, Alexander, and David J. Velleman. Philosophies of Mathematics. Malden, MA: Blackwell, 2002. Find this resource: A careful and very readable discussion of three classical programs in the philosophy of mathematics—logicism, intuitionism, and finitism—examining how they combine a certain philosophical vision of mathematics with particular mathematical developments. Shapiro, Stewart Thinking about Mathematics: The Philosophy of Mathematics. Oxford: Oxford University Press, 2000. Find this resource: A balanced combination of historical and conceptual discussion of central proposals in the philosophy of mathematics, including an examination of the three main classical conceptions—logicism, formalism, and intuitionism—as well as central proposals in the contemporary scene—Platonism, nominalism, and structuralism. Anthologies Different anthologies in the philosophy of mathematics focus on different aspects of the field. Benacerraf and Putnam 1983 is a classic; it brings together seminal papers by preeminent philosophers and mathematicians working on the foundations of mathematics. An excellent anthology on the significant connections between history and philosophy of mathematics is Asprey and Kitcher 1988. Hart 1996 collects some of the most influential papers in the philosophy of mathematics in the second half of the 20th century, whereas Jacquette 2002 brings together essays