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Philosophy of Mathematics
- LAST REVIEWED: 08 October 2015
- LAST MODIFIED: 10 May 2010
- DOI: 10.1093/obo/9780195396577-0069

- LAST REVIEWED: 08 October 2015
- 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: 10.1057/9780230245198Save Citation »Export Citation »E-mail Citation »

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.

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Burgess, John P., and Gideon A. Rosen.

*A Subject with No Objects: Strategies for Nominalistic Interpretation of Mathematics*. Oxford: Clarendon, 1997.Save Citation »Export Citation »E-mail Citation »

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

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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.Save Citation »Export Citation »E-mail Citation »

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

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Horsten, Leon. “Philosophy of Mathematics.” In

*The Stanford Encyclopedia of Philosophy*. Edited by Edward N. Zalta. 2008.Save Citation »Export Citation »E-mail Citation »

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).

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Irvine, Andrew D., ed.

*Philosophy of Mathematics*. Handbook of the Philosophy of Science series. Amsterdam: North Holland, 2009.Save Citation »Export Citation »E-mail Citation »

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.

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Linnebo, Øystein. “Platonism in the Philosophy of Mathematics.” In

*The Stanford Encyclopedia of Philosophy*. Edited by Edward N. Zalta. 2009.Save Citation »Export Citation »E-mail Citation »

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.

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Schirn, Matthias, ed.

*The Philosophy of Mathematics Today*. Oxford: Clarendon, 1998.Save Citation »Export Citation »E-mail Citation »

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.

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Shapiro, Stewart, ed.

*The Oxford Handbook of Philosophy of Mathematics and Logic*. New York: Oxford University Press, 2005.Save Citation »Export Citation »E-mail Citation »

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.

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## 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.Save Citation »Export Citation »E-mail Citation »

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.

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Brown, James Robert.

*Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures*. 2d ed. New York: Routledge, 2008.Save Citation »Export Citation »E-mail Citation »

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.

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Friend, Michèle.

*Introducing Philosophy of Mathematics*. Montreal: McGill-Queen’s University Press, 2007.Save Citation »Export Citation »E-mail Citation »

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.

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George, Alexander, and David J. Velleman.

*Philosophies of Mathematics*. Malden, MA: Blackwell, 2002.Save Citation »Export Citation »E-mail Citation »

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.

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Shapiro, Stewart

*Thinking about Mathematics: The Philosophy of Mathematics*. Oxford: Oxford University Press, 2000.Save Citation »Export Citation »E-mail Citation »

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.

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## 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 not usually found in other anthologies. The relation between proof and knowledge in mathematics is the topic of the papers in Detlefsen 1992a, whereas Detlefsen 1992b brings together articles focusing on proof, logic, and formalization.

Asprey, William, and Philip Kitcher, eds.

*History and Philosophy of Modern Mathematics*. Minnesota Studies in the Philosophy of Science 11. Minneapolis: University of Minnesota Press, 1988.Save Citation »Export Citation »E-mail Citation »

A collection of essays by prominent philosophers and historians of mathematics, exploring the complex connections between philosophy and the history of modern mathematics, with a focus on logic and the foundations of mathematics, reinterpretations in the history of mathematics, and several case studies in the history and philosophy of mathematics.

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Benacerraf, Paul, and Hilary Putnam, eds.

*Philosophy of Mathematics: Selected Readings*. 2d ed. Cambridge, UK: Cambridge University Press, 1983.Save Citation »Export Citation »E-mail Citation »

First published in 1964, this anthology has been a classic. The second edition incorporated work done throughout the 1970s but is now no longer up-to-date. Nonetheless, it preserved the original focus of the anthology, and reprinted primary works by leading philosophers and mathematicians reflecting on the foundations of mathematics, existence of mathematical objects, nature of mathematical truth, and concept of set.

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Detlefsen, Michael, ed.

*Proof and Knowledge in Mathematics*. London: Routledge, 1992a.Save Citation »Export Citation »E-mail Citation »

A collection of papers by prominent philosophers of mathematics focusing on the connections between proof and knowledge in mathematics, including discussions of proof as a source of truth, logicism, the concept of proof in elementary geometry, and Brouwerian intuitionism.

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Detlefsen, Michael, ed.

*Proof, Logic, and Formalization*. London: Routledge, 1992b.Save Citation »Export Citation »E-mail Citation »

A collection of papers by leading philosophers of mathematics and of logic on the relations between proof, justification, and formalization, including discussions of what is a proof, proof and epistemic structure, arithmetical truth, the impredicativity of induction, and the alleged refutation of Hilbert’s program using Gödel’s first incompleteness theorem.

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Hart, W. D., ed.

*The Philosophy of Mathematics*. New York: Oxford University Press, 1996.Save Citation »Export Citation »E-mail Citation »

A splendid collection of some of the most significant work in the philosophy of mathematics in the second half of the 20th century, featuring papers by Quine, Putnam, Dummett, Parsons, Benacerraf, Boolos, Field, Maddy, Shapiro, Tait, Isaacson, and Hart.

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Jacquette, Dale, ed.

*Philosophy of Mathematics: An Anthology*. Oxford: Blackwell, 2002.Save Citation »Export Citation »E-mail Citation »

A collection of papers in the philosophy of mathematics, focusing on what mathematics is about, the ontology of mathematics and mathematical truth, models and methods of mathematical proof, intuitionism, and philosophical foundations of set theory. Some of the papers collected in this volume are not often found in other anthologies.

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## Platonism

Platonism is among the most influential views in the philosophy of mathematics. There are distinct versions of this view, but typically it tends to be committed to (1) the existence of mathematical objects, (2) the independence of these objects from our mental processes and linguistic practices, and (3) the claim that mathematical objects are abstract (that is, these objects are causally inert and not located in space-time). Platonism raises significant epistemological worries, which Platonists have tried to address. According to Gödel 1983, we have something akin to a perception of the objects of set theory, and this perception (or intuition) plays a role in the production of mathematical knowledge. Broadly Gödelian views have been articulated further, in original and different ways, in Maddy 1992 and Parsons 2008. Quine 1960 offers a distinctive form of Platonism, emphasizing that, with regard to abstract objects, only the commitment of classes is ultimately needed. Some aspects of Quine’s proposal are developed further in Colyvan 2001 and also, to some extent, in Maddy 1992. A neo-Fregean form of Platonism, according to which mathematical posits are, in a certain sense, extremely lightweight, is defended in Hale and Wright 2001, whereas Linsky and Zalta 1995 defends a Platonized version of naturalism as a way of making sense of knowledge of mathematical truths. Finally, in the first part of Balaguer 1998, a robust, full-blooded form of Platonism is defended on the grounds that it offers the best account of the epistemology of mathematics. According to this view, all mathematical objects that logically could exist actually do exist.

Balaguer, Mark.

*Platonism and Anti-Platonism in Mathematics*. New York: Oxford University Press, 1998.Save Citation »Export Citation »E-mail Citation »

The first part of this book offers a critical examination of standard forms of Platonism and presents a sustained defense of a novel form of Platonism—full-blooded Platonism—according to which all mathematical objects that logically could exist actually do exist. It is argued that this form of Platonism provides solutions to epistemological and metaphysical problems that traditional versions of Platonism are unable to address successfully.

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Colyvan, Mark.

*The Indispensability of Mathematics*. New York: Oxford University Press, 2001.Save Citation »Export Citation »E-mail Citation »

A careful and systematic defense of the indispensability argument, according to which we ought to be ontologically committed to all (and only) those mathematical objects that are indispensable to our best theories of the world. A broadly Quinean and naturalist view in the philosophy of mathematics is thoroughly articulated.

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Gödel, Kurt “What Is Cantor’s Continuum Problem?” In

*Philosophy of Mathematics: Selected Readings*. 2d ed. Edited by Paul Benacerraf and Hilary Putnam, 470–485. Cambridge, UK: Cambridge University Press, 1983.Save Citation »Export Citation »E-mail Citation »

This is a revised and expanded version of a paper originally published in 1947 in the

*American Mathematical Monthly*54: 515–525. Gödel argues that we have “something like a perception of the objects of set theory,” given that “the axioms force themselves upon us as being true.” Gödel’s views have been extremely influential; see, in particular, Maddy 1992 and Parsons 2008, and some reservations in Balaguer 1998.Find this resource:

Hale, Bob, and Crispin Wright.

*The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics*. Oxford: Clarendon, 2001.Save Citation »Export Citation »E-mail Citation »

A thorough defense of a neo-Fregean approach to the philosophy of mathematics, including an examination of issues related to ontology and abstraction principles, Hume’s principle, the differentiation of abstracts, and the introduction of real numbers by abstraction.

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Linsky, Bernard, and Edward N. Zalta. “Naturalized Platonism vs. Platonized Naturalism.”

*Journal of Philosophy*92 (1995): 525–555.DOI: 10.2307/2940786Save Citation »Export Citation »E-mail Citation »

A defense of Platonized naturalism, the view according to which a certain kind of Platonism is consistent with naturalism. The Platonism in question is characterized by the introduction of general comprehension principles that assert the existence of abstract objects. Knowledge of mathematical truths is linked to knowledge of such comprehension principles. Available online.

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Maddy, Penelope.

*Realism in Mathematics*. Oxford: Clarendon, 1992.Save Citation »Export Citation »E-mail Citation »

A defense of set-theoretic realism as a response to two major challenges posed by Benacerraf: How is it possible to have knowledge of mathematical objects, given that they are abstract? And how can mathematics be the study of particular objects, given that all that seems to matter from a mathematical point of view are structural relations? Set-theoretic realism builds critically on the Platonist proposals by Quine 1960 and Gödel 1983.

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Parsons, Charles.

*Mathematical Thought and Its Objects*. Cambridge, UK: Cambridge University Press, 2008.Save Citation »Export Citation »E-mail Citation »

A thorough defense of a broadly Gödelian view in the philosophy of mathematics (see Gödel 1983), with special emphasis on set theory and arithmetic, and a careful examination of the role of intuition in mathematical knowledge.

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Quine, W. V. “Ontic Decision.” In

*Word and Object*. Edited by W. V. Quine, 233–276. Cambridge, MA: MIT Press, 1960.Save Citation »Export Citation »E-mail Citation »

A clear statement of Quine’s distinctive form of Platonism. According to Quine, ontological commitment should be restricted to those entities that are indispensable to our best theories of the world. In particular, Quine argues that, with regard to abstract objects, only classes are ultimately needed. For further developments, see Colyvan 2001.

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## Structuralism

Mathematical structuralism emerged in the contemporary scene as a response to some perceived difficulties of traditional, object-oriented forms of Platonism. Benacerraf 1965 raised one of these difficulties, arguing that what matters to natural numbers are the abstract structures they determine rather than the objects that number terms may be taken to refer to. Motivated in part by considerations of this sort, distinct versions of structuralism in mathematics have been articulated in Resnik 1997, Shapiro 1997, and Parsons 2008. Despite their differences, these versions of mathematical structuralism have in common a commitment to some form of Platonism. A version of structuralism that lacks this commitment, and thus favors nominalism, has been developed in Hellman 1989 and Hellman 1996. Finally, Linnebo 2008 offers a recent critical assessment of mathematical structuralism, with particular emphasis on the notion of dependence.

Benacerraf, Paul. “What Numbers Could Not Be.”

*Philosophical Review*74 (1965): 47–73.DOI: 10.2307/2183530Save Citation »Export Citation »E-mail Citation »

Criticizes standard, object-oriented versions of Platonism, and favors a form of mathematical structuralism. It insists that what matters to natural numbers are the abstract structures they determine rather than the objects that number terms may be taken to refer to. The point is made with reference to the distinct, but equally adequate, identifications of natural numbers with sets, advanced by Zermelo and von Neumann.

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Hellman, Geoffrey.

*Mathematics without Numbers: Towards a Modal-Structural Interpretation*. Oxford: Clarendon, 1989.Save Citation »Export Citation »E-mail Citation »

A defense of a modal-structural interpretation, which combines mathematical structuralism with a version of nominalism via a modal second-order language. Each mathematical statement

*S*is translated into two components: (1) a hypothetical component, according to which if there were structures of a suitable kind,*S*would be true in such structures, and (2) a categorical component, according to which structures of that kind are possible. In this way, the commitment to mathematical objects can be avoided.Find this resource:

Hellman, Geoffrey “Structuralism without Structures.”

*Philosophia Mathematica*4 (1996): 100–123.Save Citation »Export Citation »E-mail Citation »

An extension of the modal-structural framework presented in Hellman 1989 to fourth-order number theory in order to incorporate mathematical structures of crucial importance to functional analysis, measure theory, and topology.

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Linnebo, Øystein. “Structuralism and the Notion of Dependence.”

*Philosophical Quarterly*58 (2008): 59–79.Save Citation »Export Citation »E-mail Citation »

It is argued that the notion of dependence among mathematical objects is more significant to structuralist views than generally acknowledged. A compromise view about dependence relations is then developed according to which structuralists are right about some mathematical objects (roughly, the algebraic ones), but wrong about others (particularly, sets).

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Parsons, Charles

*Mathematical Thought and Its Objects*. Cambridge, UK: Cambridge University Press, 2008.Save Citation »Export Citation »E-mail Citation »

An insightful examination of different forms of mathematical structuralism, including the use of modality in structuralist approaches. But the book contains much more than that; in particular, it provides significant discussion of intuition in mathematics, and careful philosophical reflections about set theory, arithmetic, mathematical induction, and much more.

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Resnik, Michael D.

*Mathematics as a Science of Patterns*. Oxford: Clarendon, 1997.Save Citation »Export Citation »E-mail Citation »

A defense of mathematical structuralism according to which mathematical objects are abstract positions in structures (or patterns), and whose identities are fixed only via the relations they bear to each other. The view is realist throughout. A careful explanation of how mathematical knowledge emerges from consideration of templates and patterns is provided together with a detailed account of the status and nature of mathematical structuralism.

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Shapiro, Stewart

*Philosophy of Mathematics: Structure and Ontology*. New York: Oxford University Press, 1997.Save Citation »Export Citation »E-mail Citation »

A defense of a Platonist form of mathematical structuralism, including an account of the epistemology of mathematics and reference to mathematical objects within a structuralist perspective. Challenges to rival views are raised, and an account of the application of mathematics is also sketched.

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## Nominalism

Nominalist views are typically characterized by the denial of the existence of mathematical objects, or, at least, by the insistence that no commitment to these entities is needed to make sense of mathematics. The main motivations for nominalism emerged from the perceived difficulties faced by Platonism, particularly on the epistemological front, and by attempts to block the Quine–Putnam indispensability argument. Field 1980 was responsible for revamping nominalism in the contemporary scene, by arguing that the indispensability argument could be resisted, at least in one significant case: Newtonian gravitational theory. It was argued, particularly by Malament, that Field’s nominalization strategy could not be extended to other significant physical theories, such as quantum mechanics. In response to this challenge, Balaguer 1998 attempted to extend Field’s program to accommodate quantum theory. In response, Bueno 2003 contended that Balaguer’s strategy does not rely on a nominalistically acceptable ontology. Meanwhile, other nominalist strategies provided reformulations of mathematical theories in order to avoid commitment to mathematical objects, either by invoking a modal second-order language (see discussions under Structuralism), or by employing constructability quantifiers, as articulated in Chihara 1990. More recently, some nominalist strategies have been developed that grant the indispensability of mathematics, but insist that it is perfectly coherent to quantify over mathematical objects while rejecting that they exist, as Azzouni 2004 and Melia 2000 argue for. Finally, Yablo 2005 discusses the use of pretense theory in the context of nominalist views.

Azzouni, Jody.

*Deflating Existential Consequence: A Case for Nominalism*. New York: Oxford University Press, 2004.Save Citation »Export Citation »E-mail Citation »

A thorough defense of nominalism, particularly in mathematics, by rejecting the Quinean identification of ontological commitment with those items that are indispensable to our best theories of the world. Rather, Azzouni insists, it is perfectly intelligible to quantify over objects that do not exist (such as fictional detectives). A detailed nominalist account of the application of mathematics is then offered.

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Balaguer, Mark.

*Platonism and Anti-Platonism in Mathematics*. New York: Oxford University Press, 1998.Save Citation »Export Citation »E-mail Citation »

In the second part of this book, the development of a version of nominalism, based on Field 1980, is articulated, and in particular, an attempt is made to extend Field’s program to quantum mechanics and to offer a fictionalist account of the applicability of mathematics. In Bueno 2003, some reservations are stated.

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Bueno, Otávio. “Is It Possible to Nominalize Quantum Mechanics?”

*Philosophy of Science*70 (2003): 1424–1436.DOI: 10.1086/377419Save Citation »Export Citation »E-mail Citation »

A critical assessment of Balaguer’s attempt to extend Field’s program to the nominalization of quantum mechanics. It is argued that Balaguer failed to provide a nominalistically acceptable ontology to carry out properly the proposal, and that it is unclear that such ontology can be ultimately offered.

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Chihara, Charles S.

*Constructibility and Mathematical Existence*. Oxford: Clarendon, 1990.Save Citation »Export Citation »E-mail Citation »

A reformulation of mathematical theories by invoking constructibility quantifiers is offered, and the corresponding commitment to mathematical objects is tentatively avoided, with particular emphasis on number theory and analysis. A critical assessment of structuralism, mathematical fictionalism, Platonism, and empiricism in mathematics is also presented.

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Field, Hartry H.

*Science without Numbers: A Defence of Nominalism*. Princeton, NJ: Princeton University Press, 1980.Save Citation »Export Citation »E-mail Citation »

One of the most influential works in recent philosophy of mathematics. In many ways, this book led to a revival of the philosophy of mathematics, by challenging the Quine–Putnam indispensability argument in the case of a particular, but significant, physical theory: Newtonian gravitational theory. Field argues that it is possible to reformulate the latter theory without quantification over mathematical objects (in particular, real numbers), and concludes that reference to mathematical objects is not indispensable to (at least some of) our best theories of the world.

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Melia, Joseph. “Weaseling Away the Indispensability Argument.”

*Mind*109 (2000): 455–480.DOI: 10.1093/mind/109.435.455Save Citation »Export Citation »E-mail Citation »

A strategy is introduced that allows the nominalist to quantify over abstract objects while denying their existence—a practice that is called “weaseling.” It is then argued that such a practice is coherent, unproblematic, and rational.

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Yablo, Stephen. “The Myth of the Seven.” In

*Fictionalism in Metaphysics*. Edited by Mark Eli Kalderon, 88–115. Oxford: Clarendon, 2005.Save Citation »Export Citation »E-mail Citation »

A provocative discussion of nominalism in the philosophy of mathematics in the context of pretense theory.

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## Neo-Fregeanism and Neo-Logicism

A distinctive form of Platonism is advanced by the neo-Fregean approach of Hale and Wright 2001, which adapts, revises, and extends some of the central moves developed by Frege in the philosophy of mathematics. Cook 2007 is an excellent and up-to-date collection of papers focusing on the mathematics of abstraction, including both defenses and criticism of neo-Fregeanism as well as developments of broadly neo-logicist views (which need not be so tied to specific Fregean doctrines). Further critical assessment of the neo-Fregean approach is offered in Boolos 1998, Rayo 2003, and Linnebo and Uzquiano 2009. Zalta 2000 and Linsky and Zalta 2006 develop a neo-logicist approach in object theory, whereas Heck 2005 presents a detailed discussion of the notorious Julius Caesar problem.

Boolos, George.

*Logic, Logic, and Logic*. Cambridge, MA: Harvard University Press, 1998.Save Citation »Export Citation »E-mail Citation »

Several papers reprinted here deal explicitly with Frege and neo-Fregeanism, and they offer some thoughtful challenges to the latter. Some responses can be found in Hale and Wright 2001.

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Cook, Roy T., ed.

*The Arché Papers on the Mathematics of Abstraction*. Dordrecht, The Netherlands: Springer, 2007.Save Citation »Export Citation »E-mail Citation »

An up-to-date and thorough collection of papers on the mathematics of abstraction by the main players in the debate, including discussions of the philosophy and mathematics of Hume’s principle, the logic of abstraction, the use of abstraction in real analysis, and neo-Fregean and neo-logicist approaches to set theory.

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Hale, Bob, and Crispin Wright.

*The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics*. Oxford: Clarendon, 2001.Save Citation »Export Citation »E-mail Citation »

A thorough defense of a neo-Fregean approach in the philosophy of mathematics, including examination of issues related to ontology and abstraction principles, Hume’s principle, the differentiation of abstracts, and the introduction of real numbers by abstraction.

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Heck, Richard G., Jr. “Julius Caesar and Basic Law V.”

*Dialectica*59 (2005): 161–178.Save Citation »Export Citation »E-mail Citation »

A detailed discussion of the Julius Caesar problem: roughly speaking, the problem of providing, based on Hume’s principle, a suitable conception of number according to which it can be known, for example, that Julius Caesar is not the number zero. Available online.

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Linnebo, Øystein., and Gabriel Uzquiano. “Which Abstraction Principles Are Acceptable? Some Limitative Results.”

*British Journal for the Philosophy of Science*60 (2009): 239–252.DOI: 10.1093/bjps/axp001Save Citation »Export Citation »E-mail Citation »

Abstraction principles are crucial for the neo-Fregean. But which principles are acceptable? The notion of stability offers a promising response; roughly speaking, an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. Two counterexamples are then offered to stability as a sufficient condition for acceptability, and it is then argued that only with major changes in the neo-Fregean approach can these counterexamples be avoided.

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Linsky, Bernard, and Edward N. Zalta. “What Is Neologicism?”

*Bulletin of Symbolic Logic*12 (2006): 60–99.DOI: 10.2178/bsl/1140640944Save Citation »Export Citation »E-mail Citation »

A new version of neo-logicism is developed, within third-order nonmodal object theory, and it is argued that this theory provides a version of neo-logicism that most closely satisfies the central goals of the original logicist program.

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Rayo, Agustín. “Success by Default?”

*Philosophia Mathematica*11 (2003): 305–322.Save Citation »Export Citation »E-mail Citation »

It is argued that neo-Fregean accounts of arithmetical knowledge rely on a thesis to the effect that prima facie we are justified in thinking that some stipulations are successful. Given that neo-Fregeans have not yet offered a defense of this thesis, their account has a significant gap. A naturalistic remedy is then provided. Available online.

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Zalta, Edward N. “Neo-Logicism? An Ontological Reduction of Mathematics to Metaphysics.”

*Erkenntnis*53 (2000): 219–265.DOI: 10.1023/A:1005614102033Save Citation »Export Citation »E-mail Citation »

Argues that mathematical objects can be reduced to abstract objects, which in turn can be systematically formulated in a particular axiomatic metaphysical theory that does not presuppose any mathematics. These abstract objects are, in a certain sense, logical objects, introduced by a comprehension principle that seems to be an analytic truth. A new kind of logicism then emerges.

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## Empiricism

There are different forms of empiricism in the philosophy of mathematics. Some emphasize the methodological similarities between mathematics and the natural sciences; others stress the non–a priori status of mathematics. Some do both. This is the case in Kitcher 1983, which has advanced the most developed empiricist view in the contemporary scene, focusing on the nature of mathematical knowledge and the analogies between mathematics and science. The methodological similarities are also emphasized in Lakatos 1978, where a broadly Popperian framework is adopted. Bueno 2000 identifies certain patterns of theory change in mathematics and in science, and indicates the relevant similarities between them. A critical assessment of experimental mathematics, and an important challenge for such views, is presented in Baker 2008. Baker attempts to refute the idea that experimental mathematics makes essential use of electronic computers; another insists that it uses inductive support for mathematical hypotheses.

Baker, Alan. “Experimental Mathematics.”

*Erkenntnis*68 (2008): 331–344.DOI: 10.1007/s10670-008-9109-ySave Citation »Export Citation »E-mail Citation »

A critical examination of experimental mathematics, and a discussion of whether experimental mathematics really challenges the traditional understanding of mathematics as an a priori, nonempirical enterprise. An alternative formulates experimental mathematics in terms of the calculation of instances of some general hypothesis. However, Baker notes, this characterization is compatible with the traditional understanding of mathematics as an a priori discipline.

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Bueno, Otávio. “Empiricism, Mathematical Change, and Scientific Change.”

*Studies in History and Philosophy of Science*31 (2000): 269–296.DOI: 10.1016/S0039-3681(99)00037-0Save Citation »Export Citation »E-mail Citation »

Offers an empiricist and historically informed model of theory change in which the similarities between theory change in mathematics and in science are articulated and developed.

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Kitcher, Philip.

*The Nature of Mathematical Knowledge*. New York: Oxford University Press, 1983.Save Citation »Export Citation »E-mail Citation »

Defends the view that mathematical knowledge is not a priori, and that mathematics is ultimately an empirical science that evolves in time bearing close similarities to the development of the natural sciences. A detailed and historically informed account of how mathematical languages are modified, why certain mathematical questions become prominent, and how standards of proof change is also offered.

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Lakatos, Imre.

*Philosophical Papers*. Vol. 2,*Mathematics, Science, and Epistemology*. Cambridge, UK: Cambridge University Press, 1978.Save Citation »Export Citation »E-mail Citation »

Particularly in the first part of this collection of papers, careful and thoughtful similarities between the methodology of mathematics and of the natural sciences are explored, within a broadly Popperian framework. In particular, it is argued that mathematics is quasi-empirical and that there are “potential falsifiers” in mathematics.

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## Naturalism

There are many naturalist views in the philosophy of mathematics. Some emphasize the continuity between mathematics and science, whereas others stress the importance of being sensitive to mathematical practice (to mention just two possibilities). The most thoroughly developed form of naturalism in the philosophy of mathematics has been articulated in Maddy 1997, with particular attention to set theory, and Maddy 2007, which also includes a naturalist treatment of logic. Several naturalist views combine the defense of naturalism with Platonism. This is the case in Colyvan 2001, as part of the discussion of the indispensability argument, and Linsky and Zalta 1995, which defends Platonized naturalism. Baker 2001 develops a provocative naturalist defense of the indispensability argument (see The Indispensability Argument), arguing for the significance of mathematics in the development of new theories and in the discovery of new results.

Baker, Alan. “Mathematics, Indispensability, and Scientific Progress.”

*Erkenntnis*55 (2001): 85–116.DOI: 10.1023/A:1011977604848Save Citation »Export Citation »E-mail Citation »

An indirect defense of naturalism, offered on the grounds that even if there were nominalist views that manage to capture the physical consequences of our best scientific theories, this would not be enough to undermine the indispensability argument. After all, in order to develop new theories and discover new results, more mathematical resources are typically required than to derive known theorems. More work is needed to reject the indispensability argument.

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Colyvan, Mark.

*The Indispensability of Mathematics*. New York: Oxford University Press, 2001.Save Citation »Export Citation »E-mail Citation »

A broadly Quinean and naturalist view in the philosophy of mathematics is articulated, with particular emphasis on the indispensability argument. According to the latter, we ought to be ontologically committed to mathematical objects, given that they are indispensable to our best theories of the world.

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Linsky, Bernard, and Edward N. Zalta. “Naturalized Platonism vs. Platonized Naturalism.”

*Journal of Philosophy*92 (1995): 525–555.DOI: 10.2307/2940786Save Citation »Export Citation »E-mail Citation »

A defense of Platonized naturalism, the view according to which a certain kind of Platonism is consistent with naturalism. The Platonism in question is characterized by the introduction of general comprehension principles that assert the existence of abstract objects. Knowledge of mathematical truths is linked to knowledge of such comprehension principles.

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Maddy, Penelope.

*Naturalism in Mathematics*. Oxford: Clarendon, 1997.Save Citation »Export Citation »E-mail Citation »

A searching critique of realism in mathematics, with particular emphasis on set theory, and the development of a naturalist alternative, which is then applied to the selection and justification of set-theoretic axioms. Maddy has an impressive command of the relevant literature in both set theory and in the philosophy and history of mathematics. A pleasure to read.

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Maddy, Penelope.

*Second Philosophy: A Naturalistic Method*. Oxford: Clarendon, 2007.Save Citation »Export Citation »E-mail Citation »

The most thoroughly developed naturalist view (or “second philosophy,” as Maddy prefers to call it) in the philosophy of mathematics and logic, including careful accounts of the role of mathematics in application, as well as the methodology and the epistemology of mathematics.

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## The Indispensability Argument

Significant work in the philosophy of mathematics has been motivated by the indispensability argument. Quine originally formulated the argument, which was then more fully developed in Putnam 1979. Colyvan 2001 offered the most thorough and the first book-length treatment of the indispensability argument, giving the latter an explicit formulation: we ought to be ontologically committed to all (and only) those entities that are indispensable to our best theories of the world; mathematical entities are indispensable to our best theories; therefore, we ought to be ontologically committed to these entities. In an attempt to resist the argument, several responses were advanced. Maddy 1992 argued that the indispensability argument turned out to be incompatible with scientific and mathematical practice. Sober 1993 insisted that, on an account of confirmation based on the likelihood principle, mathematical statements are not confirmed by the observations that support the scientific theories in which such statements occur. To resist these objections, Resnik 1995 offered a pragmatic version of the indispensability argument that does not rely on scientific realism. Azzouni 1997 raised a different challenge to the indispensability argument on the grounds that existential quantification need not require ontological commitment. Melia 2000 identified a practice (that he calls “weaseling”) according to which it is perfectly coherent to quantify over mathematical objects while denying their existence. Baker 2001, in turn, offered a defense of the indispensability argument by noting that in order to develop new theories and discover new results, mathematical objects turn out to be indispensable.

Azzouni, Jody “Applied Mathematics, Existential Commitment, and the Quine-Putnam Indispensability Thesis.”

*Philosophia Mathematica*5 (1997): 193–209.Save Citation »Export Citation »E-mail Citation »

A critique of the indispensability argument is advanced on the grounds that when scientific theories are regimented, existential quantification—even if the latter is interpreted objectually rather than substitutionally—does not entail ontological commitment. A thorough development of this critique is presented in Azzouni 2004 (cited under Nominalism).

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Baker, Alan. “Mathematics, Indispensability, and Scientific Progress.”

*Erkenntnis*55 (2001): 85–116.DOI: 10.1023/A:1011977604848Save Citation »Export Citation »E-mail Citation »

Defends the indispensability argument by arguing that even if there were nominalist views that manage to capture the physical consequences of our best scientific theories, this would not be enough to undermine the indispensability argument. After all, according to Baker, in order to develop new theories and discover new results, we need more mathematical resources than to derive known theorems. More work is then required to reject the indispensability argument.

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Colyvan, Mark.

*The Indispensability of Mathematics*. New York: Oxford University Press, 2001.Save Citation »Export Citation »E-mail Citation »

A careful and systematic defense of the indispensability argument, according to which we ought to be ontologically committed to only those objects that are indispensable to our best theories of the world; given that mathematical objects are indispensable, we ought to be ontologically committed to them. A broadly Quinean and naturalist view in the philosophy of mathematics is then thoroughly articulated.

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Maddy, Penelope. “Indispensability and Practice.”

*Journal of Philosophy*89 (1992): 275–289.DOI: 10.2307/2026712Save Citation »Export Citation »E-mail Citation »

A provocative critique of the indispensability argument on the grounds that it is ultimately inconsistent with scientific and mathematical practice. For further developments, see Maddy 1997 (cited under Naturalism).

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Melia, Joseph. “Weaseling Away the Indispensability Argument.”

*Mind*109 (2000): 455–480.DOI: 10.1093/mind/109.435.455Save Citation »Export Citation »E-mail Citation »

The indispensability argument is resisted via a strategy that allows the nominalist to quantify over abstract objects while denying their existence—a practice that is called “weaseling.” It is then argued that such a practice is coherent, unproblematic, and rational.

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Putnam, Hilary. “Philosophy of Logic.” In

*Philosophical Papers*. Vol. 2,*Mathematics, Matter, and Method*. 2d ed. Edited by Hilary Putnam, 323–357.Cambridge, UK: Cambridge University Press, 1979.Save Citation »Export Citation »E-mail Citation »

A clear statement of the indispensability argument, insisting that given that quantification over mathematical objects is indispensable to science, the commitment to the existence of these objects follows. As Putnam notes, Quine originally formulated the argument.

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Resnik, Michael D. “Scientific vs. Mathematical Realism: The Indispensability Argument.”

*Philosophia Mathematica*3 (1995): 166–174.DOI: 10.1093/philmat/3.2.166Save Citation »Export Citation »E-mail Citation »

A pragmatic indispensability argument is formulated in order to overcome the objections that were raised by Sober 1993 and Maddy 1992 to the confirmational indispensability argument. The pragmatic argument favors mathematical realism independently of scientific realism. For further developments, see Resnik 1997 (cited under Structuralism).

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Sober, Elliott “Mathematics and Indispensability.”

*Philosophical Review*102 (1993): 35–57.DOI: 10.2307/2185652Save Citation »Export Citation »E-mail Citation »

A searching critique of the indispensability argument by invoking contrastive empiricism and the likelihood principle, according to which an observation

*O*favors a hypothesis_{H}_{1}over_{H}_{2}if the conditional probability of*O*given_{H}_{1}is higher than the conditional probability of*O*given_{H}_{2}. It is then argued that mathematical statements are not confirmed by the observations that support the scientific theories in which such statements occur.Find this resource:

## The Application of Mathematics

The application of mathematics has generated significant philosophical work. Azzouni 2000 gave consideration to the effect that there is no genuine philosophical problem in the success of applied mathematics. Colyvan 2001 defended the opposing view, insisting that the application of mathematics does yield a genuine problem, which neither fictionalists (such as Field) nor platonists (such as Quine) are able to solve. This disagreement notwithstanding, nominalist accounts of the application of mathematics have been offered in Azzouni 2004 and Bueno 2005. Pincock 2004 examines the strengths and weaknesses of so-called mapping accounts of the application of mathematics, and Batterman 2009 argues that mapping accounts are ultimately unable to accommodate idealizations in physical theorizing. Finally, Pincock 2007 advances a new account of a role for mathematics in the physical sciences, emphasizing the epistemic benefits of mathematics in scientific theorizing.

Azzouni, Jody. “Applying Mathematics: An Attempt to Design a Philosophical Problem.”

*Monist*83 (2000): 209–227.Save Citation »Export Citation »E-mail Citation »

Consideration is given to the effect that there is no genuine philosophical problem in the success of applied mathematics (for an opposing view, see Colyvan 2001). Once particular attention is given to implicational opacity—our inability to see, before a proof is offered, the consequences of various mathematical statements—much of the alleged surprise in the successful application of mathematics should vanish.

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Azzouni, Jody.

*Deflating Existential Consequence: A Case for Nominalism*. New York: Oxford University Press, 2004.Save Citation »Export Citation »E-mail Citation »

The second part of this book offers an extremely original account of applied mathematics, including an examination of the epistemic burdens that posits bear, the connections between posits and existence, as well as a careful discussion of two models of applying mathematics, and the relations between applied mathematics and ontology.

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Batterman, Robert W. “On the Explanatory Role of Mathematics in Empirical Science.”

*British Journal for the Philosophy of Science*(2009): 1–29.Save Citation »Export Citation »E-mail Citation »

A critical assessment of contemporary attempts to make sense of the explanatory role of mathematics in empirical science, and a critique of the so-called mapping accounts of the relations between mathematical structures and the physical world. It is argued that these accounts are unable to accommodate the use of idealizations in physical theorizing. A new approach to the application of mathematics is then advanced. Available online.

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Bueno, Otávio. “Dirac and the Dispensability of Mathematics.”

*Studies in History and Philosophy of Modern Physics*36 (2005): 465–490.DOI: 10.1016/j.shpsb.2005.03.002Save Citation »Export Citation »E-mail Citation »

A case is made for the dispensability of mathematics in the context of applied mathematics, by adapting some of the resources presented in Azzouni 2004, with particular emphasis on the various uses of mathematics that Dirac articulated in his work in quantum mechanics.

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Colyvan, Mark. “The Miracle of Applied Mathematics.”

*Synthese*127 (2001): 265–277.DOI: 10.1023/A:1010309227321Save Citation »Export Citation »E-mail Citation »

It is argued that the application of mathematics to science presents a genuine problem (for an opposing view, see Azzouni 2000). In particular, it is argued that two major philosophical accounts of mathematics—Field’s mathematical fictionalism and Quine’s Platonist realism—are unable to explain the problem. It is then suggested that the problem cuts across the realism/antirealism debate in the philosophy of mathematics.

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Pincock, Christopher “A New Perspective on the Problem of Applying Mathematics.”

*Philosophia Mathematica*12 (2004): 135–161.Save Citation »Export Citation »E-mail Citation »

A framework for discussing the problem of the application of mathematics is presented, and an account in terms of this framework advanced. In particular, the framework offers resources to assess the strengths and weaknesses of an approach to the application of mathematics in terms of mappings between the physical world and a mathematical domain. Available online.

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Pincock, Christopher “A Role for Mathematics in the Physical Sciences.”

*Noûs*41 (2007): 253–275.Save Citation »Export Citation »E-mail Citation »

A new account of a role for mathematics in the physical sciences is offered, and the epistemic benefits of mathematics in scientific theorizing are emphasized. In particular, the account brings together the theoretical indispensability of mathematics (the latter’s significance in scientific theorizing) and the metaphysical dispensability of mathematical objects (given that the latter play no causal role in the physical world).

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## Pictures and Proofs in Mathematics

Proofs play a crucial role in mathematics. Detlefsen 1992a offers a collection of papers that explore the connections between proofs and mathematical knowledge, whereas Detlefsen 1992b brings together essays on the relations between proofs, justification, and formalization. A highly original account of mathematical proof (the derivation-indicator view) is developed in Azzouni 2006. A significant related issue is the role of pictures in proofs. Brown 1997 argues that pictures have a positive evidential role, and supports the point with historical considerations and striking examples. The most thorough philosophical study of visual thinking in mathematics is presented in Giaquinto 2007, whereas Mancosu, et al. 2005 collects some papers on visualization and mathematical reasoning (including some work by Brown and Giaquinto).

Azzouni, Jody.

*Tracking Reason: Proof, Consequence, and Truth*. New York: Oxford University Press, 2006.Save Citation »Export Citation »E-mail Citation »

The second part of this book offers a highly original account of mathematical proof, including an examination of what makes mathematics unique as a social practice, the development of the derivation-indicator view of mathematical practice, and an account of how to nominalize mathematical formalism.

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Brown, James Robert. “Proofs and Pictures.”

*British Journal for the Philosophy of Science*48 (1997): 161–180.DOI: 10.1093/bjps/48.2.161Save Citation »Export Citation »E-mail Citation »

It is argued that pictures have a positive evidential role in mathematics, and that they are more than just psychologically or heuristically useful. Historical considerations and striking examples are offered in support of this view.

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Detlefsen, Michael, ed.

*Proof and Knowledge in Mathematics*. London: Routledge, 1992a.Save Citation »Export Citation »E-mail Citation »

A collection of papers by prominent philosophers of mathematics focusing on the connections between proof and knowledge in mathematics, including discussions of proof as a source of truth, logicism, the concept of proof in elementary geometry, and Brouwerian intuitionism.

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Detlefsen, Michael, ed.

*Proof, Logic, and Formalization*. London: Routledge, 1992b.Save Citation »Export Citation »E-mail Citation »

A collection of papers by leading philosophers of mathematics and of logic on the relations between proof, justification, and formalization, including discussions of what is a proof, proof and epistemic structure, arithmetical truth, the impredicativity of induction, and the alleged refutation of Hilbert’s program using Gödel’s first incompleteness theorem.

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Giaquinto, Marcus.

*Visual Thinking in Mathematics: An Epistemological Study*. Oxford: Clarendon, 2007.Save Citation »Export Citation »E-mail Citation »

A thorough and insightful account of visual thinking in mathematics. Giaquinto argues that visual thinking often has an epistemological role in mathematics, and in some cases it offers a means of discovery. The proposal is supported by case studies from geometry, algebra, arithmetic, and real analysis, and it draws on philosophical work on the nature of concepts, as well as empirical studies of visual perception, numerical cognition, and mental imagery.

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Mancosu, Paolo, Klaus Frovin Jørgensen, and Stig Andur Pedersen, eds.

*Visualization, Explanation, and Reasoning Styles in Mathematics*. Dordrecht, The Netherlands: Springer, 2005.DOI: 10.1007/1-4020-3335-4Save Citation »Export Citation »E-mail Citation »

The first part of this excellent collection has some papers on visualization and mathematical reasoning, which examine visualization in logic and mathematics (in particular, geometry), as well as the significance of pictures and proofs to platonic intuitions, and a thoughtful account of mathematical activity.

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## Mathematical Practice

An important trend in the philosophy of mathematics emerged from the examination of philosophical issues that arise from mathematical practice. Lakatos 1976 offers a particularly insightful treatment of mathematical practice in the context of the dynamics of proofs and refutations. An extremely original approach to various puzzles that emerge from mathematical practice is developed in Azzouni 1994. Mancosu 1996 offers a careful discussion of mathematical practice in the 17th century, whereas the papers collected in Mancosu 2008 and Mancosu, et al. 2005 clearly illustrate the rich source of insight that mathematical practice offers when close attention is paid to it.

Azzouni, Jody.

*Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences*. Cambridge, UK: Cambridge University Press, 1994.Save Citation »Export Citation »E-mail Citation »

A highly original and insightful treatment of some puzzles raised by mathematical practice, and a careful examination of the role played by mathematical terms and empirical terms in actual scientific and mathematical practice.

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Lakatos, Imre.

*Proofs and Refutations: The Logic of Mathematical Discovery*. Cambridge, UK: Cambridge University Press, 1976.Save Citation »Export Citation »E-mail Citation »

A careful study of the vagaries of a mathematical conjecture regarding the classification of polyhedra (first formulated by Euler), and the series of proofs and refutations that it engendered. The main text is written as a dialogue in an imaginary classroom, while the footnotes reconstruct some aspects of the actual history. Along the way, the struggles of actual mathematical practice are insightfully rendered and analyzed.

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Mancosu, Paolo.

*Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century*. New York: Oxford University Press, 1996.Save Citation »Export Citation »E-mail Citation »

A careful and detailed examination of philosophy of mathematics and mathematical practice in the 17th century.

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Mancosu, Paolo, ed.

*The Philosophy of Mathematical Practice*. Oxford: Clarendon, 2008.Save Citation »Export Citation »E-mail Citation »

An excellent collection of papers on various philosophical aspects of mathematical practice by leading philosophers in the field. Some of the issues examined include diagrammatic reasoning and visualization in mathematics, mathematical explanation, the ideal of purity and mathematical proof, the role of computers in mathematical inquiry, and the philosophical import of recent developments in mathematical physics.

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Mancosu, Paolo, Klaus Frovin Jørgensen, and Stig Andur Pedersen, eds.

*Visualization, Explanation, and Reasoning Styles in Mathematics*. Dordrecht, The Netherlands: Springer, 2005.DOI: 10.1007/1-4020-3335-4Save Citation »Export Citation »E-mail Citation »

All the papers in this excellent collection illustrate the insights that can be gained about mathematics when close attention is given to mathematical practice, from the connections between visualization and mathematical reasoning through different proof styles to the significance that mathematical explanation and understanding have to mathematical activity.

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## Mathematical Explanation

Explanations in mathematics offer an important source of philosophical reflection. Careful and up-to-date surveys of mathematical explanation can be found in Mancosu 2008a and Mancosu 2008b; the latter is freely available online. One significant issue is whether there are genuine mathematical explanations in science. Baker 2005 defends the claim that there are, and offers a detailed example from evolutionary biology in support of his case. Colyvan 2002 also supports the existence of mathematical explanations, and argues that the use of complex numbers unifies not only the mathematical theory of differential equations but also the various physical theories that use such equations. Hafner and Mancosu 2008 offers a critique of the unification account of mathematical explanation, whereas Sandborg 1998 criticizes van Fraassen’s account of explanation by invoking examples of mathematical explanation. An excellent collection of papers that deal, in part, with mathematical explanation is Mancosu, et al. 2005.

Baker, Alan. “Are There Genuine Mathematical Explanations of Physical Phenomena?”

*Mind*114 (2005): 223–238.DOI: 10.1093/mind/fzi223Save Citation »Export Citation »E-mail Citation »

It is argued that there are genuine mathematical explanations in science. To support this point, a detailed example from evolutionary biology, involving periodical cicadas, is given. Consequences that favor the indispensability argument are then drawn.

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Colyvan, Mark. “Mathematics and Aesthetic Considerations in Science.”

*Mind*111 (2002): 69–74.DOI: 10.1093/mind/111.441.69Save Citation »Export Citation »E-mail Citation »

A case is made for the claim that mathematics unifies a great deal of scientific theorizing, and thus it has a significant explanatory role. In support of this case, the use of complex numbers to unify not only the mathematical theory of differential equations but also the various physical theories that use such equations is explored.

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Hafner, Johannes, and Paolo Mancosu. “Beyond Unification.” In

*The Philosophy of Mathematical Practice*. Edited by Paolo Mancosu, 151–178. Oxford: Clarendon, 2008.Save Citation »Export Citation »E-mail Citation »

A critique of the unification account of mathematical explanation, on the grounds that such an account makes predictions about explanatory power that conflict with certain cases from mathematical practice.

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Mancosu, Paolo “Mathematical Explanation: Why It Matters.” In

*The Philosophy of Mathematical Practice*. Edited by Paolo Mancosu, 134–150. Oxford: Clarendon, 2008a.Save Citation »Export Citation »E-mail Citation »

An insightful and up-to-date discussion of mathematical explanation and its significance.

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Mancosu, Paolo “Explanation in Mathematics.” In

*The Stanford Encyclopedia of Philosophy*. Edited by Edward N. Zalta. 2008b.Save Citation »Export Citation »E-mail Citation »

A careful and up-to-date survey of explanation in mathematics, covering two main problems: (1) whether mathematics plays an explanatory role in the empirical sciences, and (2) whether mathematical explanations occur within mathematics. The connections between the two problems and their relevance are also explored.

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Mancosu, Paolo, Klaus Frovin Jørgensen, and Stig Andur Pedersen, eds.

*Visualization, Explanation, and Reasoning Styles in Mathematics*. Dordrecht, The Netherlands: Springer, 2005.DOI: 10.1007/1-4020-3335-4Save Citation »Export Citation »E-mail Citation »

Some of the papers collected in the second part of this volume deal explicitly with mathematical explanation—in particular, the connections between explanation, proof style, and understanding in mathematics are examined, and different kinds of mathematical explanation are discussed and assessed.

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Sandborg, David. “Mathematical Explanation and the Theory of Why-Questions.”

*British Journal for the Philosophy of Science*49 (1998): 603–624.DOI: 10.1093/bjps/49.4.603Save Citation »Export Citation »E-mail Citation »

A provocative critique of van Fraassen’s account of explanation in terms of why-questions based on examples of mathematical explanation. In particular, it is argued that van Fraassen’s account cannot recognize mathematical proofs as explanatory, and an example is given of an explanation that seems to be explanatory even though it is unable to answer the why-question that motivated it.

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### Article

- A Priori Knowledge
- Abortion
- Action
- Adorno, Theodor
- Aesthetics, Analytic Approaches to
- Aesthetics, Continental
- Aesthetics, History of
- Alexander, Samuel
- Analytic/Synthetic Distinction
- Anarchism, Philosophical
- Animal Rights
- Anscombe, G. E. M.
- Anthropic Principle, The
- Applied Ethics
- Aquinas, Thomas
- Argument Mapping
- Art and Knowledge
- Art and Morality
- Artifacts
- Assertion
- Atheism
- Aurelius, Marcus
- Austin, J. L.
- Autonomy
- Bacon, Francis
- Bayesianism
- Beauty
- Belief
- Bergson, Henri
- Berkeley, George
- Bolzano, Bernard
- British Idealism
- Buber, Martin
- Buddhist Philosophy
- Camus, Albert
- Carnap, Rudolf
- Causation
- Certainty
- Chemistry, Philosophy of
- Chinese Philosophy
- Cognitive Ability
- Cognitive Phenomenology
- Cognitive Science, Philosophy of
- Coherentism
- Color
- Communitarianism
- Computational Science
- Comte, Auguste
- Concepts
- Conceptual Role Semantics
- Conditionals
- Confirmation
- Confucius
- Connectionism
- Consciousness
- Consequentialism, Ethical
- Constructive Empiricism
- Contemporary African Philosophy
- Contemporary Hylomorphism
- Contextualism
- Contrastivism
- Culture and Cognition
- Daoism and Philosophy
- Davidson, Donald
- de Beauvoir, Simone
- de Montaigne, Michel
- Death
- Decision Theory
- Deleuze, Gilles
- Democracy
- Depiction
- Derrida, Jacques
- Descartes, René
- Descriptions
- Dialetheism
- Disability
- Disagreement, Epistemology of
- Disjunctivism
- Dispositions
- Doing and Allowing
- Dummett, Michael
- Dutch Book Arguments
- Early Modern Philosophy, 1600-1750
- Education, Philosophy of
- Emotion
- Environmental Philosophy
- Epicurus
- Epistemic Basing Relation
- Epistemic Injustice
- Epistemic Justification
- Epistemic Philosophy of Logic
- Epistemology
- Epistemology and Active Externalism
- Epistemology, Feminist
- Epistemology, Moral
- Epistemology Of Education
- Epistemology, Reasons in
- Ethical Intuitionism
- Ethics of Engineering, Philosophy and
- Eugenics and Philosophy
- Events, The Philosophy of
- Evidence
- Evidence-Based Medicine, Philosophy of
- Evidential Support Relation In Epistemology, The
- Evil
- Evolutionary Debunking Arguments in Ethics
- Evolutionary Epistemology
- Experimental Philosophy
- Explanations of Religion
- Extended Mind Thesis, The
- Externalism and Internalism in the Philosophy of Mind
- Fatalism
- Feminist Metaphysics
- Feminist Philosophy
- Feyerabend, Paul
- Fiction
- Fictionalism
- Fictionalism in the Philosophy of Mathematics
- Film, Philosophy of
- Foreknowledge
- Forgiveness
- Formal Epistemology
- Foucault, Michel
- Free Will
- Frege, Gottlob
- Gadamer, Hans-Georg
- Georg Wilhelm Friedrich Hegel: Aesthetics
- Georg Wilhelm Friedrich Hegel: Metaphysics
- God and Possible Worlds
- God, Arguments for the Existence of
- God, The Existence and Attributes of
- Grice, Paul
- Heaven and Hell
- Hegel, Georg Wilhelm Friedrich: Philosophy of History
- Hegel, Georg Wilhelm Friedrich: Philosophy of Politics
- Heidegger, Martin: Early Works
- Hermeneutics
- Higher Education, Philosophy of
- History, Philosophy of
- Hobbes, Thomas
- Horkheimer, Max
- Human Rights
- Hume, David: Moral and Political Philosophy
- Husserl, Edmund
- Idealizations in Science
- Identity in Physics
- Imagination
- Imagination and Belief
- Impossible Worlds
- Incommensurability in Science
- Indian Philosophy
- Indispensability of Mathematics
- Inductive Reasoning
- Infinitism
- Instruments in Science
- Intellectual Humility
- Intuitions
- James, William
- Kant and the Laws of Nature
- Kant, Immanuel: Aesthetics and Teleology
- Kant, Immanuel: Ethics
- Kant, Immanuel: Theoretical Philosophy
- Kierkegaard, Søren
- Knowledge
- Knowledge-How
- Kuhn, Thomas S.
- Lacan, Jacques
- Lakatos, Imre
- Language of Thought
- Language, Philosophy of
- Latin American Philosophy
- Laws of Nature
- Legal Philosophy
- Legal Positivism
- Leibniz, Gottfried Wilhelm
- Levinas, Emmanuel
- Liberty
- Literature, Philosophy of
- Locke, John
- Locke, John: Identity, Persons, and Personal Identity
- Logic
- Lottery and Preface Paradoxes, The
- Lucretius
- Machiavelli, Niccolò
- Marx, Karl
- Mathematical Explanation
- Mathematical Pluralism
- Mathematical Structuralism
- Mathematics, Philosophy of
- McDowell, John
- McTaggart, John
- Meaning of Life, The
- Medically Assisted Dying
- Medicine, Contemporary Philosophy of
- Medieval Logic
- Medieval Philosophy
- Memory
- Mereology
- Meta-epistemological Skepticism
- Metaepistemology
- Metaethics
- Metametaphysics
- Metaphilosophy
- Metaphor
- Metaphysics, Contemporary
- Mind, Metaphysics of
- Modal Epistemology
- Modality
- Models and Theories in Science
- Modularity
- Montesquieu
- Moore, G. E.
- Moral Contractualism
- Moral Naturalism and Nonnaturalism
- Moral Responsibility
- Multiculturalism
- Music, Analytic Philosophy of
- Nationalism
- Natural Kinds
- Naturalism in the Philosophy of Mathematics
- Naïve Realism
- Neo-Confucianism
- Nietzsche, Friedrich
- Nonexistent Objects
- Normative Ethics
- Normative Foundations, Philosophy of Law:
- Normativity and Social Explanation
- Occasionalism
- Olfaction
- Ontological Dependence
- Ontology of Art
- Ordinary Objects
- Other Minds
- Pacifism
- Pain
- Panpsychism
- Paradoxes
- Pascal, Blaise
- Paternalism
- Peirce, Charles Sanders
- Perception, Cognition, Action
- Perception, The Problem of
- Persistence
- Personal Identity
- Phenomenal Concepts
- Phenomenal Conservatism
- Phenomenology
- Philosophy of Biology
- Philosophy of Risk
- Physicalism
- Physicalism and Metaphysical Naturalism
- Physics, Experiments in
- Plato
- Plotinus
- Political Obligation
- Political Philosophy
- Pornography and Objectification, Analytic Approaches to
- Practical Knowledge
- Practical Moral Skepticism
- Practical Reason
- Pragmatics
- Pragmatism
- Problem of Divine Hiddenness, The
- Problem of Evil, The
- Propositions
- Psychology, Philosophy of
- Punishment
- Pyrrhonism
- Qualia
- Quietism
- Quine, W.V.O.
- Race
- Rationalism
- Rationality
- Rawls: Moral and Political Philosophy, John
- Realism and Anti-Realism
- Realization
- Reductionism in Biology
- Reference, Theory of
- Reid, Thomas
- Relativism
- Reliabilism
- Religion, Philosophy of
- Religious Belief, Epistemology of
- Religious Experience
- René Descartes: Sensory Representations
- Ricoeur, Paul
- Rights
- Rorty, Richard
- Rousseau, Jean-Jacques
- Rule-Following
- Russell, Bertrand
- Sartre, Jean-Paul
- Schopenhauer, Arthur
- Science and Religion
- Scientific Explanation
- Scientific Progress
- Scientific Realism
- Scientific Representation
- Scientific Revolutions
- Scotus, Duns
- Self-Knowledge
- Sellars, Wilfrid
- Semantic Externalism
- Semantic Minimalism
- Semiotics
- Seneca
- Singular Thought
- Situationism and Virtue Theory
- Skepticism, Contemporary
- Skepticism, History of
- Social Aspects of Scientific Knowledge
- Social Epistemology
- Space and Time
- Speech Acts
- Spinoza, Baruch
- Structural Realism
- Suicide
- Supererogation
- Supervenience
- Tarski, Alfred
- Technology, Philosophy of
- Testimony, Epistemology of
- Thomas Aquinas' Philosophy of Religion
- Thought Experiments
- Time and Tense
- Time Travel
- Toleration
- Torture
- Transcendental Arguments
- Tropes
- Trust
- Truth
- Truth and the Aim of Belief
- Truthmaking
- Turing Test
- Two-Dimensional Semantics
- Understanding
- Vagueness
- Value of Knowledge
- Vienna Circle
- Virtue Epistemology
- Virtue Ethics
- War
- Weakness of Will
- Well-Being
- William of Ockham
- Williams, Bernard
- Wisdom
- Wittgenstein, Ludwig: Early Works
- Wittgenstein, Ludwig: Later Works
- Wittgenstein, Ludwig: Middle Works
- Wollstonecraft, Mary