Element contents
Mathematical Pluralism
Published online by Cambridge University Press: 29 March 2024
Summary
Mathematical pluralism is the view that there is an irreducible plurality of pure mathematical structures, each with their own internal logics; and that qua pure mathematical structures they are all equally legitimate. Mathematical pluralism is a relatively new position on the philosophical landscape. This Element provides an introduction to the position.
Keywords
- Type
- Element
- Information
- Online ISBN: 9781009091640Publisher: Cambridge University PressPrint publication: 18 April 2024
References
Avron, A. (2010), ‘A New Approach to Predicative Set Theory’, pp. 31–64 of Schindler, R. (ed.), Ways of Proof Theory, Heisenstamm: Ontos Verlag.CrossRefGoogle Scholar
Balaguer, M. (2018), ‘Fictionalism in the Philosophy of Mathematics’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/fictionalism-mathematics/.Google Scholar
Beall, J. (1999), ‘From Full Blooded Platonism to Really Full Blooded Platonism’, Philosophia Mathematica 7: 322–5.CrossRefGoogle Scholar
Bell, J. (2008), A Primer of Infinitesimal Analysis, 2nd ed., Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Bell, J. (2022), ‘Continuity and Infinitesimals’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/continuity/.Google Scholar
Benacerraf, P. (1965), ‘What Numbers Could not Be’, Philosophical Review 74: 47–73.CrossRefGoogle Scholar
Berto, F., French, R., Priest, G., and Ripley, D. (2018), ‘Williamson on Counterpossibles’, Journal of Philosophical Logic 47: 693–713.CrossRefGoogle ScholarPubMed
Brady, R. (1989), ‘The Non-Triviality of Dialectical Set Theory’, pp. 437–71 of Priest, G., Routley, R., and Norman, J. (eds.), Paraconsistent Logic: Essays on the Inconsistent, Munich: Philosophia Verlag.Google Scholar
Bridges, D. (2013), ‘Constructive Mathematics’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/mathematics-constructive/.Google Scholar
Bridges, D., and Richman, F. (1987), Varieties of Constructive Mathematics, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Brown, B., and Priest, G. (2004), ‘Chunk and Permeate, a Paraconsistent Inference Strategy; Part I, the Infinitesimal Calculus’, Journal of Philosophical Logic 22: 379–88.Google Scholar
Bueno, O. (2011), ‘Relativism in Set Theory and Mathematics’, pp. 553–68 of Hales, S. (ed.), A Companion to Relativism, Oxford: Wiley-Blackwell.Google Scholar
Bueno, O., and Colyvan, M. (2011), ‘An Inferential Conception of the Application of Mathematics’, Noûs 45: 345–74.CrossRefGoogle Scholar
Carnap, R. (1950), ‘Empiricism, Semantics and Ontology’, Revue Internationale de Philosophie 4: 20–40; reprinted as pp. 205–21 of Meaning and Necessity: a Study in Semantics and Modal Logic, 2nd ed., Chicago: University of Chicago Press, 1956.Google Scholar
Chakravartty, A. (2017), ‘Scientific Realism’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/scientific-realism/.Google Scholar
Clarke-Doane, J. (2022), Mathematics and Metaphilosophy, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Colyvan, M. (2001), The Indispensability of Mathematics, New York: Oxford University Press.CrossRefGoogle Scholar
Colyvan, M. (2019), ‘Indispensability Arguments in the Philosophy of Mathematics’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/mathphil-indis/.Google Scholar
Crosilla, L. (2019), ‘Set Theory: Constructive and Intuitionist ZF’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/set-theory-constructive/.Google Scholar
Curry, H. (1951), Outlines of a Formalist Philosophy of Mathematics, Amsterdam: North-Holland.Google Scholar
Da Costa, N. (1974), ‘On the Theory of Inconsistent Formal Systems’, Notre Dame Journal of Formal Logic 15: 497–509.CrossRefGoogle Scholar
Davies, E. B. (2005), ‘A Defence of Mathematical Pluralism’ , Philosophia Mathematica 13: 252–76.CrossRefGoogle Scholar
Doherty, F. (2017), ‘Hilbert on Consistency as a Guide to Mathematical Reality’, Logique et Analyse 237: 107–28.Google Scholar
Dummett, M. (2000), Elements of Intuitionism, 2nd ed., Oxford: Oxford University Press.CrossRefGoogle Scholar
Eklund, M. (2019), ‘Fictionalism’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/fictionalism/.Google Scholar
Engeler, E., and Röhrl, H. (1969), ‘On the Problems of Foundations of Category Theory’, Dialectica 23: 58–66.CrossRefGoogle Scholar
Field, H. (1980), Science without Numbers, Oxford: Oxford University Press; 2nd ed., 2016.Google Scholar
Field, H. (1989), ‘Introduction: Fictionalism, Epistemology, and Modality’, ch. 1 of Realism, Mathematics, and Modality, Oxford: Basil Blackwell.Google Scholar
French, S. (2019), ‘Identity and Individuality in Quantum Theory’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/qt-idind/.Google Scholar
Friend, M. (2014), Pluralism in Mathematics: A New Position in Philosophy of Mathematics, Dordrecht: Springer.Google Scholar
Gregory, H. (2015), Language and Logics: An Introduction to the Logical Foundations of Language, Edinburgh: Edinburgh University Press.CrossRefGoogle Scholar
Hájek, P., and Haniková, Z. (2003), ‘A Development of Set Theory in Fuzzy Logic’, pp. 273–85 of Fitting, M. and Orłowska, E. (eds.), Beyond Two: Theory and Applications of Multiple-Valued Logic, Heidelberg: Springer.Google Scholar
Hallett, M. (2013), ‘Zermelo’s Axiomatization of Set Theory’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/zermelo-set-theory/.Google Scholar
Hamkins, J. (2012), ‘The Set-Theoretic Multiverse’, Review of Symbolic Logic 5: 416–49.CrossRefGoogle Scholar
Hamkins, J. (2020), Lectures on the Philosophy of Mathematics, Cambridge, MA: Massachusetts Institute of Technology Press.Google Scholar
Hamkins, J. (2023), ‘Skolem’s Paradox’, Infinitely More, https://www.infinitelymore.xyz/p/skolems-paradox.Google Scholar
Hamkins, J., and Yang, R. (2013), ‘Satisfaction is Not Absolute’, arXiv: 1312.0670, https://arxiv.org/abs/1312.0670.Google Scholar
Hellman, G., and Bell, J. (2006), ‘Pluralism and the Foundations of Mathematics’ , pp. 64–79 of Waters, C., Longino, H., and Kellert, S. (eds.), Scientific Pluralism, Minneapolis: University of Minnesota Press.Google Scholar
Holmes, M. (2017), ‘Alternative Axiomatic Set Theories’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/settheory-alternative/#AckeSetTheo.Google Scholar
Horsten, L. (2017), ‘Philosophy of Mathematics’, Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/philosophy-mathematics.Google Scholar
Hylton, P., and Kemp, G. (2019), ‘Willard Van Orman Quine’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/quine/.Google Scholar
Iemhoff, R. (2013), ‘Intuitionism in the Philosophy of Mathematics’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/intuitionism/.Google Scholar
Irvine, A. (2015), ‘Principia Mathematica’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/principia-mathematica/.Google Scholar
Jacob, P. (2023), ‘Intentionality’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/intentionality/.Google Scholar
Koellner, P. (2009), ‘Truth in Mathematics: The Question of Pluralism’, pp. 80–116 of Bueno, O. and Linnebo, Ø (eds.), New Waves in Philosophy of Mathematics: New Waves in Philosophy, London: Palgrave Macmillan.CrossRefGoogle Scholar
Koellner, P. (2010), ‘Independence and Large Cardinals’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/independence-large-cardinals/.Google Scholar
Koellner, P. (2013), ‘Large Cardinals and Determinacy’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/large-cardinals-determinacy/.Google Scholar
Kroon, F., and Voltolini, A. (2023), ‘Fictional Entities’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/fictional-entities/.Google Scholar
Kunen, K. (1980), Set Theory: An Introduction to Independence Proofs, Amsterdam: North Holland.Google Scholar
Maddy, P. (2007), Second Philosophy: A Naturalistic Method, Oxford: Oxford University Press.CrossRefGoogle Scholar
Marquis, J.-P. (2014), ‘Category Theory’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/category-theory/.Google Scholar
McCarty, D. (1991), ‘Incompleteness in Intuitionist Mathematics’ , Notre Dame Journal of Formal Logic 32: 323–58.CrossRefGoogle Scholar
Miller, A., and Sultanescu, O. (2022), ‘Rule Following and Intentionality’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/rule-following/.Google Scholar
Mortensen, C. (2017), ‘Inconsistent Mathematics’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/mathematics-inconsistent/.Google Scholar
Moss, L. (2018), ‘Non-wellfounded Set Theory’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/nonwellfounded-set-theory/.Google Scholar
Petersen, U. (2000), ‘Logic without Contraction and Based on Unrestricted Abstraction’, Studia Logica 64: 365–403.CrossRefGoogle Scholar
Pincock, C. (2004), ‘A New Perspective on the Problem of Applying Mathematics’, Philosophia Mathematica 12: 135–61.CrossRefGoogle Scholar
Pincock, C. (2014), ‘How to Avoid Inconsistent Idealizations’, Synthese 191: 2957–72.CrossRefGoogle Scholar
Posy, C. (2020), Mathematical Intuitionism, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Priest, G. (1973), ‘A Bedside Reader’s Guide to the Conventionalist Philosophy of Mathematics’, pp. 115–32 of Bell, J., Cole, J., Priest, G., and Slomson, A. (eds.), Proceedings of the Bertrand Russell Memorial Logic Conference, Denmark 1971, Leeds: University of Leeds.Google Scholar
Priest, G. (1997a), ‘Sylvan’s Box’, Notre Dame Journal of Formal Logic 38: 573–82; reprinted as §6.6 of Priest (2005).Google Scholar
Priest, G. (1997b), ‘Inconsistent Models of Arithmetic, I: Finite Models’, Journal of Philosophical Logic 26: 1519–25.CrossRefGoogle Scholar
Priest, G. (1998), ‘Number’, pp. 47–54, Vol. 7, of Craig, E. (ed.), Encyclopedia of Philosophy, London: Routledge.Google Scholar
Priest, G. (2000), ‘Inconsistent Models of Arithmetic, II: The General Case’, Journal of Symbolic Logic 65: 223–35.CrossRefGoogle Scholar
Priest, G. (2003), ‘On Alternative Geometries, Arithmetics, and Logics: A Tribute to Łukasiewicz’, Studia Logica 74: 441–68.CrossRefGoogle Scholar
Priest, G. (2005), Towards Non-Being, Oxford: Oxford University Press; 2nd ed., 2016.CrossRefGoogle Scholar
Priest, G. (2006a), In Contradiction, 2nd ed., Oxford: Oxford University Press.CrossRefGoogle Scholar
Priest, G. (2006b), ‘ Logical Pluralism’, ch. 12 of Doubt Truth to be a Liar, Oxford: Oxford University Press.Google Scholar
Priest, G. (2010), ‘Quine: Naturalism Unravelled’, pp. 19–30 of Dumitru, M. and Stoenescu, C. (eds.), Cuvinte, Teorii si Lucruri: Quine in Perspectiva, Bucharest: Editura Pelican.Google Scholar
Priest, G. (2013b), ‘Indefinite Extensibility: Dialetheic Style’ , Studia Logica 101: 1263–75.CrossRefGoogle Scholar
Priest, G. (2014), ‘Revising Logic’, ch. 12 of Rush, P. (ed.), The Metaphysics of Logic, Cambridge: Cambridge University Press.Google Scholar
Priest, G. (2016), ‘Logical Disputes and the a Priori’, Logique et Analyse 236: 347–66.Google Scholar
Priest, G. (2017), ‘What If? The Exploration of an Idea’, Australasian Journal of Logic 14(1): Article 4, https://ojs.victoria.ac.nz/ajl/article/view/4028/3574.CrossRefGoogle Scholar
Priest, G. (2019a), ‘From the Foundations of Mathematics to Mathematical Pluralism’ , pp. 363–80 of Centrone, S., Kant, D., and Sarikaya, D. (eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts, New York: Springer.Google Scholar
Priest, G. (2019b), ‘Gödel’s Theorem and Paraconsistency’, in Almeida, E., Costa-Leite, A., and Freire, R. (eds.), Lógica no Avião, http://lna.unb.br/lna_n01_01_gpriest.pdf.Google Scholar
Priest, G. (2021a), ‘Logical Abductivism and Non-Deductive Inference’ , Synthese 199: 3207–17.CrossRefGoogle Scholar
Priest, G. (2021b), ‘A Note on Mathematical Pluralism and Logical Pluralism’, Synthese 198: 4937–46.CrossRefGoogle Scholar
Priest, G. (202+b), ‘Logic as Applied Mathematics: With Application to the Notion of Logical Form’, to appear.Google Scholar
Priest, G., Berto, F., and Weber, Z. (2022), ‘Dialetheism’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/dialetheism/.Google Scholar
Priest, G., Tanaka, K., and Weber, Z. (2022), ‘Paraconsistent Logic’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/logic-paraconsistent/.Google Scholar
Quine, W. (1951), ‘Two Dogmas of Empiricism’, Philosophical Review 60: 20–43; reprinted as ch. 2 of From a Logical Point of View, Cambridge, MA: Harvard University Press.Google Scholar
Reck, E., and Schiemer, G. (2019), ‘Structuralism in the Philosophy of Mathematics’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/structuralism-mathematics/.Google Scholar
Reicher, M. (2019a), ‘Alexius Meinong’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/meinong/.Google Scholar
Reicher, M. (2019b), ‘Non-Existent Objects’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/nonexistent-objects/.Google Scholar
Russell, G. (2019), ‘Logical Pluralism’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/logical-pluralism/.Google Scholar
Sambin, G. (2011), ‘A Minimalist Foundation at Work’, ch. 4 of DeVidi, D., Hallett, M., and Clarke, P. (eds.), Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in Honour of John L. Bell, Heidelberg: Springer.Google Scholar
Shapiro, S. (2004), ‘Foundations of Mathematics: Ontology, Epistemology, Structure’, Philosophical Quarterly 54: 16–37.CrossRefGoogle Scholar
Stei, E. (2023), Logical Pluralism and Logical Consequence, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Sweeney, D. (2014), ‘Chunk and Permeate: The Infinitesimals of Isaac Newton’, History and Philosophy of Logic 35: 1–23.CrossRefGoogle Scholar
Takeuti, G. (1981), ‘Quantum Set Theory’, pp. 303–22 of Beltrametti, E. and van Fraassen, B. (eds.), Current Issues in Quantum Logic, New York: Plenum.Google Scholar
Tal, E. (2020), ‘Measurement in Science’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/measurement-science/.Google Scholar
Tennant, N. (2017), ‘Logicism and Neologicism’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/logicism/.Google Scholar
Van Atten, M. (2017), ‘The Development of Intuitionist Logic’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/intuitionistic-logic-development/.Google Scholar
Warren, J. (2015), ‘Conventionalism, Consistency, and Consistency Sentences’, Synthese 192: 1351–71.CrossRefGoogle Scholar
Weber, Z. (2010), ‘Transfinite Numbers in Paraconsistent Set Theory’, Review of Symbolic Logic 3: 71–92.CrossRefGoogle Scholar
Weber, Z. (2012), ‘Transfinite Cardinals in Paraconsistent Set Theory’, Review of Symbolic Logic 5: 269–93.CrossRefGoogle Scholar
Weber, Z. (2021), Paradoxes and Inconsistent Mathematics, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Weber, Z. (2022), Paraconsistency in Mathematics, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Weir, A. (2019), ‘Formalism in the Philosophy of Mathematics’, Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/formalism-mathematics/.Google Scholar
Wigner, E. (1960), ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, Communications on Pure and Applied Mathematics 13: 1–14.CrossRefGoogle Scholar
Williamson, T. (2018), ‘Alternative Logics and Applied Mathematics’, Philosophical Issues 28: 399–424.CrossRefGoogle Scholar
Wittgenstein, L. (1967), Remarks on the Foundations of Mathematics, Cambridge: Massachusetts Institute of Technology Press.Google Scholar
Zach, R. (2023), ‘Hilbert’s Program’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/hilbert-program/.Google Scholar
Zalta, E. (2016), ‘Frege’, in Zalta, E. (ed.), Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/frege/.Google Scholar
Zalta, E. (2023), ‘Mathematical Pluralism’, Noûs, https://onlinelibrary.wiley.com/doi/full/10.1111/nous.12451.CrossRefGoogle Scholar
- 1
- Cited by