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# Philosophy of Mathematics

Edited by Øystein Linnebo (University of Oslo) Assistant editor: Sam Roberts (University of Oslo) About this topic*Summary*

The philosophy of mathematics studies the nature of mathematical truth, mathematical proof, mathematical evidence, mathematical practice, and mathematical explanation.

Three philosophical views of mathematics are widely regarded as the ‘classic’ ones. *Logicism* holds that mathematics is reducible to principles of pure logic. *Intuitionism* holds that mathematics is concerned with mental constructions and defends a revision of classical mathematics and logic. Finally, *formalism* is the view that much or all of mathematics is devoid of content and a purely formal study of strings of mathematical language.

In recent decades, some new views have entered the fray. An important newer arrival is *structuralism*, which holds that mathematics is the study of abstract structures. A *non-eliminative* version of structuralism holds that there exist such things as abstract structures, whereas an *eliminative* version tries to make do with concrete objects variously structured. *Nominalism* denies that there are any abstract mathematical objects and tries to reconstruct classical mathematics accordingly. *Fictionalism* is based on the idea that, although most mathematical theorems are literally false, there is a non-literal (or fictional) sense in which assertions of them nevertheless count as correct. *Mathematical naturalism *urges that mathematics be taken as a *sui generis* discipline in good scientific standing.

*Key works*On the more traditional views, it is hard to beat the selection of readings in Benacerraf & Putnam 1964. Non-eliminative structuralism is defended in Resnik 1997, Shapiro 1997, and Parsons 2007. A modal version of eliminative structuralism derives from Putnam 1967 and is developed in Hellman 1989. Two classic defenses of logicism are Frege 1953 and Russell 1919. A neo-Fregean programme was initiated in Wright 1983; see the essays collected in Hale & Wright 2001 and, for critical discussion, Dummett 1991. Nominalism is often driven by the epistemic challenge due to Benacerraf 1973 and Field 1989, ch. 1 and 7. Field’s classic attempt to vindicate nominalism is Field 1980. For a comprehensive overview of the subject, see Burgess & Rosen 1997. On fictionalism, see Yablo 2010. The indispensability argument derives from Quine but crystallized in Putnam 1971; for a recent defense, see Colyvan 2001. On mathematical naturalism, see Maddy 1997 and Maddy 2007.

*Introductions*

Introductory book: Shapiro 2000. Anthologies: Benacerraf & Putnam 1964, Hart 1996, Bueno & Linnebo 2009, and Marcus & McEvoy 2016 (with lots of historical material). Handbook: Shapiro 2005.

Show all references In this area Subcategories Epistemology of Mathematics (**1,107**| 198)Alan Baker Apriority in Mathematics (

**67**)Alexander C. R. Oldemeier Mathematics and the Causal Theory of Knowledge (

**36**) Mathematical Intuition (

**128**)Alexander C. R. Oldemeier Mathematical Proof (

**447**| 245)Jordan Bohall Godel's Theorem (

**87**)Jordan Bohall Computer Proof (

**30**)Jordan Bohall Probabilistic Proof (

**8**)Jordan Bohall Undecidability (

**31**)Jordan Bohall Mathematical Proof, Misc (

**46**)Jordan Bohall Revisability in Mathematics (

**6**) Visualization in Mathematics* (

**69**)Silvia De Toffoli Phenomenology of Mathematics* (

**65**) Mathematical Methodology (

**35**) Nondeductive Methods in Mathematics (

**24**) Debunking Arguments about Mathematics* (

**2**) Epistemology of Mathematics, Misc (

**166**) Ontology of Mathematics (

**2,294**| 299)Rafal Urbaniak Mathematical Fictionalism (

**82**)Rafal Urbaniak Mathematical Nominalism (

**208**)Rafal Urbaniak Mathematical Platonism (

**401**)Rafal Urbaniak Mathematical Psychologism (

**22**)Rafal Urbaniak Mathematical Structuralism (

**310**)Rafal Urbaniak Mathematical Neo-Fregeanism (

**369**)Rafal Urbaniak Indeterminacy in Mathematics (

**18**)Rafal Urbaniak Debunking Arguments about Mathematics (

**2**) Indispensability Arguments in Mathematics (

**229**)Rafal Urbaniak Numbers (

**354**)Rafal Urbaniak The Nature of Sets* (

**238**| 97)Rafal Urbaniak Mathematical Cognition (

**314**| 4)Lieven Decock Mathematical Intuition* (

**128**)Alexander C. R. Oldemeier Visualization in Mathematics (

**69**)Silvia De Toffoli Mathematical Cognition, Misc (

**12**) Phenomenology of Mathematics (

**65**) Numerical Cognition (

**165**)Oliver Marshall Mathematical Truth (

**194**| 9)Mark Balaguer Analyticity in Mathematics (

**22**) Axiomatic Truth (

**47**) Objectivity Of Mathematics (

**55**)Daniel Waxman Mathematical Truth, Misc (

**61**) Set Theory (

**2,043**| 374)Toby Meadows The Nature of Sets (

**238**| 97)Rafal Urbaniak The Iterative Conception of Set (

**43**)Rafal Urbaniak Ontology of Sets (

**47**)Rafal Urbaniak The Nature of Sets, Misc (

**51**)Rafal Urbaniak Axioms of Set Theory (

**796**| 617) Axiomatic Truth* (

**47**) The Axiom of Choice (

**64**) The Axiom of Constructibility (

**5**) The Axiom of Determinacy (

**6**) The Axiom of Infinity (

**17**) New Axioms in Set Theory (

**22**) Large Cardinals* (

**56**) Nonstandard Axiomatizations (

**22**) Independence Results in Set Theory (

**27**) Axioms of Set Theory, Misc (

**16**) Cardinals and Ordinals (

**440**| 323) The Continuum Hypothesis (

**46**) Large Cardinals* (

**56**) Cardinals and Ordinals, Misc (

**15**) Set Theory as a Foundation (

**195**| 34) Russell's Paradox (

**90**)Gianluca Longa Set Theory and Logicism (

**13**)Gianluca Longa Set-Theoretic Constructions (

**11**) Set Theory as a Foundation, Misc (

**47**) Areas of Mathematics (

**8,377**| 7,375)Dirk Schlimm Algebra (

**58**) Analysis (

**43**) Category Theory (

**377**)Shay Logan Geometry (

**177**) Logic and Philosophy of Logic* (

**83,148**)Aleksandra Samonek Number Theory (

**95**) Set Theory* (

**2,043**| 374)Toby Meadows Topology (

**180**) Areas of Mathematics, Misc (

**72**) Theories of Mathematics (

**840**| 49)Roy T. Cook Logicism in Mathematics (

**166**)Gianluca Longa Formalism in Mathematics (

**70**) Intuitionism and Constructivism (

**332**) Predicativism in Mathematics (

**24**) Mathematical Naturalism (

**59**) Mathematical Finitism (

**25**) Theories of Mathematics, Misc (

**115**) History: Philosophy of Mathematics (

**388**)Erich Reck Philosophy of Mathematics, Miscellaneous (

**3,089**| 964)Sorin Bangu Explanation in Mathematics (

**73**)Gianluca Longa The Infinite (

**292**)Erich Reck The Application of Mathematics (

**228**)Nora Berenstain History of Mathematics (

**196**) Mathematical Practice (

**267**)Silvia De Toffoli Philosophy of Mathematics, General Works (

**452**) Mathematical Explanation* (

**123**)Gianluca Longa Philosophy of Mathematics, Misc (

**617**)

### History/traditions: Philosophy of Mathematics

- History: Philosophy of Mathematics (
**388**) - Ancient Greek and Roman Philosophy of Mathematics (
**7**) - Medieval Philosophy of Mathematics (
**17**) - Hobbes: Philosophy of Mathematics (
**25**) - Hume: Philosophy of Mathematics (
**57**) - Locke: Philosophy of Mathematics (
**28**) - Kant: Philosophy of Mathematics (
**386**) - 17th/18th Century Philosophy of Mathematics (
**11**| 11) - 19th Century Philosophy of Mathematics (
**15**) - Frege: Philosophy of Mathematics (
**277**) - 20th Century Philosophy of Mathematics (
**6**) - Husserl: Philosophy of Mathematics (
**108**)

- sufficient condition and necessary condition, posted 2016-10-18 by Fabrizio Ranzani
- Definition of verisimilitude I suggest, posted 2016-10-05 by Chenguang Lu
- Gödel and Turing: Al Ala Akbar (الآلة أكبر The Machine is the Greatest), posted 2016-07-07 by Hachem El Ouggouti
- Parthood vs Membership, posted 2016-05-13 by Andrea Marchesi
- Challenge to the Mathematic and Logic Community, posted 2016-03-29 by Hachem El Ouggouti
**Go to forum**

- Matemáticas y Platonismo(S).J. Ferreiros - 1999 -
*Gaceta de la Real Sociedad Matemática Española*2 (446):473.details - Mathematics and the Theory of Multiplicities: Badiou and Deleuze Revisited.Daniel W. Smith - 2003 -
*Southern Journal of Philosophy*41 (3):411-449.details - Aristotelian Finitism.Tamer Nawar - 2015 -
*Synthese*192 (8):2345-2360.details - Cónicas y Superficies Cuádricas.Jonathan Taborda & Jaime Chica - manuscriptdetails
- Intuition in Mathematics.Elijah Chudnoff - 2014 - In Barbara Held & Lisa Osbeck (eds.),
*Rational Intuition*. Cambridge University Press.details - Logic in the Tractatus.Max Weiss - 2017 -
*Review of Symbolic Logic*10 (1):1-50.details - The Case Against Infinity.Kip Sewell - manuscriptdetails
- Wittgenstein on Gödelian 'Incompleteness', Proofs and Mathematical Practice: Reading Remarks on the Foundations of Mathematics, Part I, Appendix III, Carefully.Wolfgang Kienzler & Sebastian Sunday Grève - 2016 - In Sebastian Sunday Grève & Jakub Mácha (eds.),
*Wittgenstein and the Creativity of Language*. Basingstoke, UK: Palgrave Macmillan. pp. 76-116.details - Frege on Truth, Beauty and Goodness.Simon Evnine - 2003 -
*Manuscrito*26 (2):315-330.details - Programming Planck Units From a Virtual Electron; a Simulation Hypothesis (Summary).Malcolm Macleod - 2018 -
*Eur. Phys. J. Plus*133:278.details

**New York University Abu Dhabi**Visiting Professor of Philosophy – Open Rank

**All Souls College, University of Oxford**Five-year Post-Doctoral Research Fellowship

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**Princeton University**Lecturer

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