Linked bibliography for the SEP article "Philosophy of Mathematics" by Leon Horsten

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Antonutti Marfori, M., 2010. ‘Informal Rigour and Mathematical Practice’, Studia Logica, 102: 567–576. (Scholar)
Appel, K., Haken, W. & Koch, J., 1977. ‘Every Planar Map is Four Colorable’, Illinois Journal of Mathematics, 21: 429–567. (Scholar)
Avigad, J., 2021. ‘Reliability of Mathematical Inference’, Synthese, 198: 7377–7399. (Scholar)
Azzouni, J., 2004. ‘The Derivation-Indicator View of Mathematical Practice’, Philosophia Mathematica, 12: 81-105. (Scholar)
Balaguer, M., 1998. Platonism and Anti-Platonism in Mathematics, Oxford: Oxford University Press. (Scholar)
Benacerraf, P., 1965. ‘What Numbers Could Not Be’, in Benacerraf & Putnam 1983, pp. 272–294. (Scholar)
–––, 1967. ‘God, the Devil, and Gödel’, The Monist, 51: 9–32. (Scholar)
–––, 1973. ‘Mathematical Truth’, in Benacerraf & Putnam 1983, 403–420. (Scholar)
Benacerraf, P. & Putnam, H. (eds.), 1983. Philosophy of Mathematics: Selected Readings, Cambridge: Cambridge University Press, 2nd edition. (Scholar)
Bernays, P., 1935. ‘On Platonism in Mathematics’, in Benacerraf & Putnam 1983, pp. 258–271. (Scholar)
Boolos, G., 1971. ‘The Iterative Conception of Set’, in Boolos 1998, pp. 13–29. (Scholar)
–––, 1975. ‘On Second-Order Logic’, in Boolos 1998, pp. 37–53. (Scholar)
–––, 1985. ‘Nominalist Platonism’, in Boolos 1998, pp. 73–87. (Scholar)
–––, 1987. ‘The Consistency of Frege’s Foundations of Arithmetic’, in Boolos 1998, pp. 183–201. (Scholar)
–––, 1998. Logic, Logic, and Logic, Cambridge: Harvard University Press. (Scholar)
Bueno, O. , 2011. ‘Relativism in Set Theory and Mathematics’, in S. Hales (ed.), A Companion to Relativism, Oxford: Wiley-Blackwell, pp. 553–568. (Scholar)
Burge, T., 1998. ‘Computer Proofs, A Priori Knowledge, and Other Minds’, Noûs, 32: 1–37. (Scholar)
–––, 2013. Cognition Through Understanding. Self-Knowledge, Interlocution, Reasoning, Reflection. Philosophical Essays, Volume 3, Oxford: Oxford University Press. (Scholar)
Burgess, J., 2004. ‘Mathematics and Bleak House’, Philosophia Mathematica, 12: 37–53. (Scholar)
–––, 2004b, ‘Quine, Analyticity and Philosophy of Mathematics’, Philosophical Quarterly, 54: 38–55. (Scholar)
–––, 2005, Fixing Frege, Princeton: Princeton University Press. (Scholar)
–––, 2015. Rigor and Structure, Oxford: Oxford University Press. (Scholar)
Burgess, J. & Rosen, G., 1997. A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics, Oxford: Clarendon Press. (Scholar)
Button, T. & Walsh, S., 2018. Philosophy and Model Theory, Oxford: Oxford University Press. (Scholar)
Cantor, G., 1932. Abhandlungen mathematischen und philosophischen Inhalts, E. Zermelo (ed.), Berlin: Julius Springer. (Scholar)
Carnap, R., 1950. ‘Empiricism, Semantics and Ontology’, in Benacerraf & Putnam 1983, pp. 241–257. (Scholar)
Chihara, C., 1973. Ontology and the Vicious Circle Principle, Ithaca: Cornell University Press. (Scholar)
Cohen, P., 1971. ‘Comments on the Foundations of Set Theory’, in D. Scott (ed.) Axiomatic Set Theory (Proceedings of Symposia in Pure Mathematics, Volume XIII, Part 1), American Mathematical Society, pp. 9–15. (Scholar)
Colyvan, M., 2001. The Indispensability of Mathematics, Oxford: Oxford University Press. (Scholar)
Curry, H., 1958. Outlines of a Formalist Philosophy of Mathematics, Amsterdam: North-Holland. (Scholar)
Detlefsen, M., 1986. Hilbert’s Program, Dordrecht: Reidel. (Scholar)
–– (ed), 1992. Proof, logic and formalisation, London: Routledge.
Deutsch, D., Ekert, A. & Luppacchini, R., 2000. ‘Machines, Logic and Quantum Physics’, Bulletin of Symbolic Logic, 6: 265–283. (Scholar)
Di Toffoli, S., 2021. ‘Reconciling Rigor and Intuition’. Erkenntnis, 86: 1783–1802. (Scholar)
Essenin-Volpin, A., 1961. ‘Le Programme Ultra-intuitionniste des fondements des mathématiques’, in Infinistic Methods, New York: Pergamon Press, pp. 201-223. (Scholar)
Feferman, S., 1988. ‘Weyl Vindicated: Das Kontinuum seventy years later’, reprinted in S. Feferman, In the Light of Logic, New York: Oxford University Press, 1998, pp. 249–283. (Scholar)
–––, 2005. ‘Predicativity’, in S. Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford: Oxford University Press, pp. 590–624. (Scholar)
Field, H., 1980. Science without Numbers: a defense of nominalism, Oxford: Blackwell. (Scholar)
–––, 1989. Realism, Mathematics & Modality, Oxford: Blackwell. (Scholar)
Fine, K., 2002. The Limits of Abstraction, Oxford: Oxford University Press. (Scholar)
Frege, G., 1884. The Foundations of Arithmetic. A Logico-mathematical Enquiry into the Concept of Number, J.L. Austin (trans.), Evanston: Northwestern University Press, 1980. (Scholar)
Friend, M. 2013. ‘Pluralism and “Bad” Mathematical Theories: Challenging our Prejudices’, in K. Tanaka, et al. (eds.), Paraconsistency: Logic and Application, Dordrecht: Springer, pp. 277–307. (Scholar)
Gentzen, G., 1938. ‘Die gegenwärtige Lage in der mathematischen Grundlagenforschung. Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie’, in Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften (Neue Folge/Heft 4), Leipzig: Hirzel. (Scholar)
Gödel, K., 1931. ‘On Formally Undecidable Propositions in Principia Mathematica and Related Systems I’, in van Heijenoort 1967, pp. 596–616. (Scholar)
–––, 1944. ‘Russell’s Mathematical Logic’, in Benacerraf & Putnam 1983, pp. 447–469. (Scholar)
–––, 1947. ‘What is Cantor’s Continuum Problem?’, in Benacerraf & Putnam 1983, pp. 470–485. (Scholar)
Goodman, N. & Quine, W., 1947. ‘Steps Towards a Constructive Nominalism’, Journal of Symbolic Logic, 12: 97–122. (Scholar)
Halbach, V. & Horsten, L., 2005. ‘Computational Structuralism’, Philosophia Mathematica, 13: 174–186. (Scholar)
Hale, B. & Wright, C., 2001. The Reason’s Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics, Oxford: Oxford University Press. (Scholar)
Hallett, M., 1984. Cantorian Set Theory and Limitation of Size, Oxford: Clarendon Press. (Scholar)
Hodes, H., 1984. ‘Logicism and the Ontological Commitments of Arithmetic’, Journal of Philosophy, 3: 123–149. (Scholar) (Scholar)
Hamkins, J., 2015. ‘Is the dream solution of the continuum hypothesis attainable?’, Notre Dame Journal of Formal Logic, 56: 135–145. (Scholar)
Hellman, G., 1989. Mathematics without Numbers, Oxford: Clarendon Press. (Scholar)
Hilbert, D., 1925. ‘On the Infinite’, in Benacerraf & Putnam 1983, pp. 183–201. (Scholar)
Hodes, H., 1984. ‘Logicism and the Ontological Commitments of Arithmetic’, Journal of Philosophy, 3: 123–149. (Scholar)
Horsten, L., 2012. ‘Vom Zählen zu den Zahlen: on the relation between computation and mathematical structuralism’, Philosophia Mathematica, 20: 275–288. (Scholar)
Horsten, L. & Welch, P. (eds.), 2016. Gödel’s disjunction: the scope and limits of mathematical knowledge, Oxford: Oxford University Press.
Isaacson, D., 1987. ‘Arithmetical Truth and Hidden Higher-Order Concepts’, in The Paris Logic Group (eds.), Logic Colloquium ‘85, Amsterdam: North-Holland, pp. 147–169. (Scholar)
Kanamori, A., 2009. The Higher Infinite. Large cardinals from their beginnings, Berlin: Springer. (Scholar)
Koellner, P., 2009. ‘On Reflection Principles’, Annals of Pure and Applied Logic, 157: 206–219. (Scholar)
–––, 2009b. ‘Truth in Mathematics: The Question of Pluralism’, in O. Bueno & Ø. Linnebo (eds.), New Waves in Philosophy of Mathematics, Basingstoke: Palgrave Macmillan, 80–116. (Scholar)
Kreisel, G., 1967. ‘Informal Rigour and Completeness Proofs’, in I. Lakatos (ed.), Problems in the Philosophy of Mathematics, Amsterdam: North-Holland. (Scholar)
Lakatos, I., 1976. Proofs and Refutations, New York: Cambridge University Press. (Scholar)
Lavine, S., 1994. Understanding the Infinite, Cambridge, MA: Harvard University Press. (Scholar)
Leng, M., 2010. Mathematics and Reality, Oxford: Oxford University Press. (Scholar)
Linnebo, Ø., 2003. ‘Plural Quantification Exposed’, Noûs, 37: 71–92. (Scholar)
–––, 2013. ‘The Potential Hierarchy of Sets’, Review of Symbolic Logic, 6: 205–228. (Scholar)
–––, 2017. Philosophy of Mathematics, Princeton: Princeton University Press. (Scholar)
–––, 2018. Thin Objects: an abstractionist account, Oxford: Oxford University Press. (Scholar)
Linsky, B. & Zalta, E., 1995. ‘Naturalized Platonism vs. Platonized Naturalism’, Journal of Philosophy, 92: 525–555. (Scholar)
Lucas, J.R., 1961. ‘Minds, Machines, and Gödel’, Philosophy, 36: 112–127. (Scholar)
Maddy, P., 1988. ‘Believing the Axioms I, II’, Journal of Symbolic Logic, 53: 481–511, 736–764. (Scholar)
–––, 1990. Realism in Mathematics, Oxford: Clarendon Press. (Scholar)
–––, 1997. Naturalism in Mathematics, Oxford: Clarendon Press. (Scholar)
–––, 2007. Second Philosophy: a Naturalistic Method, Oxford: Oxford University Press. (Scholar)
Mancosu, P., 2008. The Philosophy of Mathematical Practice, Oxford: Oxford University Press. (Scholar)
Manders, K., 1989. ‘Domain Extensions and the Philosophy of Mathematics’, Journal of Philosophy, 86: 553–562. (Scholar)
Martin, D.A., 1998. ‘Mathematical Evidence’, in H. Dales & G. Oliveri (eds.), Truth in Mathematics, Oxford: Clarendon Press, pp. 215–231. (Scholar)
–––, 2001. ‘Multiple Universes of Sets and Indeterminate Truth Values’ Topoi, 20: 5–16. (Scholar)
McGee, V., 1997. ‘How we Learn Mathematical Language’, Philosophical Review, 106: 35–68. (Scholar)
McLarty, C., 2004. ‘Exploring Categorical Structuralism’, Philosophia Mathematica, 12: 37–53. (Scholar)
Moore, A., 2001. The Infinite, second edition, New York: Routledge. (Scholar)
Moore, G., 1982. Zermelo’s Axiom of Choice: Its Origins, Development, and Influence, New York: Springer Verlag. (Scholar)
Mormann, T., 2002. ‘Towards an evolutionary account of conceptual change in mathematics’, in G. Kampis et al (eds.), Appraising Lakatos: Mathematics, Methodology and the Man, Dordrecht: Kluwer, pp. 139–156. (Scholar)
Myhill, J., 1960. ‘Some Remarks on the Notion of Proof’, Journal of Philosophy, 57: 461–471. (Scholar)
Niebergall, K., 2000. ‘On the Logic of Reducibility: axioms and examples’, Erkenntnis, 53: 27–62. (Scholar)
Parsons, C., 1980. ‘Mathematical Intuition’, Proceedings of the Aristotelian Society, 80: 145–168. (Scholar)
–––, 1983. Mathematics in Philosophy: Selected Essays, Ithaca: Cornell University Press. (Scholar)
–––, 1990a. ‘The Structuralist View of Mathematical Objects’, Synthese, 84: 303–346. (Scholar)
–––, 1990b. ‘The Uniqueness of the Natural Numbers’, Iyyun 13: 13–44. (Scholar)
–––, 2008. Mathematical Thought and Its Objects, Cambridge: Cambridge University Press. (Scholar)
Penrose, R., 1989. The Emperor’s New Mind. Oxford: Oxford University Press. (Scholar)
–––, 1994. Shadows of the Mind. Oxford: Oxford University Press. (Scholar)
Pettigrew, R., 2008. ‘Platonism and Aristotelianism in Mathematics’, Philosophia Mathematica 16: 310–332. (Scholar)
Potter, M., 2004. Set Theory and Its Philosophy: a Critical Introduction, Oxford: Oxford University Press. (Scholar)
Pour-El, M., 1999. ‘The Structure of Computability in Analysis and Physical Theory’, in E. Griffor (ed.), Handbook of Computability Theory, Amsterdam: Elsevier, pp. 449–471. (Scholar)
Putnam, H., 1967. ‘Mathematics without Foundations’, in Benacerraf & Putnam 1983, 295–311. (Scholar)
–––, 1972. Philosophy of Logic, London: George Allen & Unwin. (Scholar)
Quine, W.V.O., 1969. ‘Epistemology Naturalized’, in W.V.O. Quine, Ontological Relativity and Other Essays, New York: Columbia University Press, pp. 69–90. (Scholar)
–––, 1970. Philosophy of Logic, Cambridge, MA: Harvard University Press, 2nd edition. (Scholar)
Rav, Y., 1999. ‘Why do we prove theorems?’, Philosophia Mathematica, 9: 5–41. (Scholar)
Reck, E. & Price, M., 2000. ‘Structures and Structuralism in Contemporary Philosophy of Mathematics’, Synthese, 125: 341–383. (Scholar)
Resnik, M., 1974. ‘The Frege-Hilbert Controversy’, Philosophy and Phenomenological Research, 34: 386–403. (Scholar)
–––, 1997. Mathematics as a Science of Patterns, Oxford: Clarendon Press. (Scholar)
Russell, B., 1902. ‘Letter to Frege’, in van Heijenoort 1967, 124–125. (Scholar)
Shapiro, S., 1983. ‘Conservativeness and Incompleteness’, Journal of Philosophy, 80: 521–531. (Scholar)
–––, 1991. Foundations without Foundationalism: A Case for Second-order Logic, Oxford: Clarendon Press. (Scholar)
–––, 1997. Philosophy of Mathematics: Structure and Ontology, Oxford: Oxford University Press. (Scholar)
–––, 2000. Thinking about Mathematics, Oxford: Oxford University Press. (Scholar)
Sieg, W., 1994. ‘Mechanical Procedures and Mathematical Experience’, in A. George (ed.), Mathematics and Mind, Oxford: Oxford University Press. (Scholar)
Studd, J., 2019. Everything, more or less. A defence of generality relativism. Oxford: Oxford University Press. (Scholar)
Tait, W., 1981. ‘Finitism’, reprinted in Tait 2005, pp. 21–42. (Scholar)
–––, 2005. The Provenance of Pure Reason: Essays in the Philosophy of Mathematics and its History, Oxford: Oxford University Press. (Scholar)
Tatton-Brown, O., forthcoming. ‘Rigour and Proof’, Review of Symbolic Logic, first online 21 October 2020. doi:10.1017/s1755020320000398 (Scholar)
Troelstra, A. & van Dalen, D., 1988. Constructivism in Mathematics: An Introduction (Volumes I and II), Amsterdam: North-Holland. (Scholar)
Turing, A., 1936. ‘On Computable Numbers, with an Application to the Entscheidungsproblem’, reprinted in M. Davis (ed.), The Undecidable: Basic Papers on Undecidable Propositions and Uncomputable Functions, Hewlett: Raven Press, 1965, pp. 116–151. (Scholar)
Tymoczko, T., 1979. ‘The Four-Color Problem and its Philosophical Significance’, Journal of Philosophy, 76: 57–83. (Scholar)
van Atten, M., 2004. On Brouwer, London: Wadsworth. (Scholar)
van Heijenoort, J., 1967. From Frege to Gödel: A Source Book in Mathematical Logic (1879–1931), Cambridge, MA: Harvard University Press. (Scholar)
Warren, J., 2015. ‘Conventionalism, Consistency, and Consistency Sentences’, Synthese, 192: 1351–1371. (Scholar)
Weir, A., 2003. ‘Neo-Fregeanism: An Embarrassment of Riches’, Notre Dame Journal of Formal Logic, 44: 13–48. (Scholar)
Welch, P. & Horsten, L. 2016. ‘Reflecting on Absolute Infinity.’, Journal of Philosophy, 113: 89–111. (Scholar)
Weyl, H., 1918. The Continuum: A Critical Examination of the Foundation of Analysis, S. Pollard and T. Bole (trans.), Mineola: Dover, 1994. (Scholar)
Woodin, H., 2001a. ‘The Continuum Hypothesis. Part I’, Notices of the American Mathematical Society, 48: 567–578. (Scholar)
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–––, 2011. ‘The Realm of the Infinite’, in H. Woodin & M. Heller (eds.), Infinity: New Research Frontiers, New York: Cambridge University Press, pp. 89–118. (Scholar)
Wright, C., 1983. Frege’s Conception of Numbers as Objects (Scots Philosophical Monographs, Volume 2), Aberdeen: Aberdeen University Press. (Scholar)
Yablo, S. 2014. Aboutness Princeton: Princeton University Press. (Scholar)
Zach, R., 2006. ‘Hilbert’s Program Then and Now’, in D. Jacquette (ed.), Philosophy of Logic (Handbook of the Philosophy of Science, Volume 5), Amsterdam: Elsevier, pp. 411–447. (Scholar)
Zermelo, E., 1930. ‘On Boundard Numbers and Domains of Sets’, translated by M. Hallett, in W. Ewald (ed.), From Kant to Hilbert: A Source Book in Mathematics (Volume 2), Oxford: Oxford University Press, 1996, pp. 1208–1233. (Scholar)

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