Element contents
Definitions and Mathematical Knowledge
Published online by Cambridge University Press: 05 December 2024
Summary
This Element discusses the philosophical roles of definitions in the attainment of mathematical knowledge. It first focuses on the role of definitions in foundational programs, and then examines their major varieties, both as regards their origins, their potential epistemic roles, and their formal constraints. It examines explicit definitions, implicit definitions, and implicit definitions of primitive terms, these latter being further divided into axiomatic and abstractive. After discussing elucidations and explications, various ways in which definitions can yield mathematical knowledge are surveyed.
Keywords
- Type
- Element
- Information
- Online ISBN: 9781009091084Publisher: Cambridge University PressPrint publication: 16 January 2025
References
Antonelli, A. (1998). Definition. In Routledge Encyclopedia of Philosophy (pp. 150–154). London: Routledge. DOI: https://doi.org/10.4324/9780415249126-Y057-1.Google Scholar
Antonelli, A., & May, R. C. (2000). Frege’s New Science. Notre Dame Journal of Formal Logic, 41(3), 242–270.CrossRefGoogle Scholar
Antos, C., Friedman, S., Honzik, R., & Ternullo, C. (Eds.). (2018). The Hyperuniverse Project and Maximality. Basel: Birkhäuser.CrossRefGoogle Scholar
, Aristotle. (1984). The Complete Works of Aristotle (Vol. 1; Barnes), J.. Princeton, NJ: Princeton University Press.Google Scholar
Assadian, B. (2023). Abstraction and Semantic Presuppositions. Analysis, 15(3), 419–428.CrossRefGoogle Scholar
Balaguer, M. (1998). Platonism and Anti-Platonism in Mathematics. York, NY: Oxford University Press.CrossRefGoogle Scholar
Barton, N. (2024). Iterative Conceptions of Set. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Batitsky, V. (2002). Some Measurement-Theoretic Concerns about Hale’s “Reals by Abstraction.” Philosophia Mathematica, 10(3), 286–303.CrossRefGoogle Scholar
Beall, J., & Restall, G. (2005). Logical Pluralism. Oxford: Oxford University Press.CrossRefGoogle Scholar
Beaney, M. (2021). Analysis. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2021 ed.). https://plato.stanford.edu/archIves/spr2010/entries/frege-hilbert/index.html.Google Scholar
Belnap, N. (1993). On Rigorous Definitions. Philosophical Studies, 72(2–3), 115–146.CrossRefGoogle Scholar
Benacerraf, P. (1965). What Numbers Could Not Be. Philosophical Review, 74(1), 47–73.CrossRefGoogle Scholar
Benacerraf, P. (1973). Mathematical Truth. Journal of Philosophy, 70(19), 661–679.CrossRefGoogle Scholar
Benacerraf, P. (1981). Frege: The Last Logicist. Midwest Studies in Philosophy, 6(1), 17–36.CrossRefGoogle Scholar
Benacerraf, P. (1996). What Mathematical Truth Could Not Be – 1. In Morton, A. & Stich, S. P. (Eds.), Benacerraf and His Critics (pp. 9–59). Oxford: Clarendon Press.Google Scholar
Bentham, J. (1962). A Fragment on Ontology. In Bowring, J. (Ed.), The Works of Jeremy Bentham (Vol. VIII, pp. 193–211). New York, NY: Russell & Russell Inc.Google Scholar
Bentzen, B. (2019). Frege on Referentiality and Julius Caesar in Grundgesetze Section 10. Notre Dame Journal of Formal Logic, 60(4), 617–637.CrossRefGoogle Scholar
Beth, E. (1953). On Padoa’s Method in the Theory of Definition. Indagationes Mathematicae, 15, 330–339.CrossRefGoogle Scholar
Biagioli, F. (2023). Federigo Enriques and the Philosophical Background to the Discussion of Implicit Definitions. In Cantù, P. & Schiemer, G. (Eds.), Logic, Epistemology, and Scientific Theories? From Peano to the Vienna Circle (pp. 153–174). Cham: Springer Nature.CrossRefGoogle Scholar
Blanchette, P. A. (1994). Frege’s Reduction. History and Philosophy of Logic. 15(1), 85–103.CrossRefGoogle Scholar
Blanchette, P. A. (2007). Frege on Consistency and Conceptual Analysis. Philosophia Mathematica, 15(3), 321–346.CrossRefGoogle Scholar
Blanchette, P. A. (2018). The Frege–Hilbert Controversy. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Fall 2018 ed.). https://plato.stanford.edu/archIves/spr2010/entries/frege-hilbert/index.html.Google Scholar
Boccuni, F., & Panza, M. (2022). Frege’s Theory of Real Numbers: A Consistent Rendering. The Review of Symbolic Logic, 15(3), 624–667.CrossRefGoogle Scholar
Boccuni, F., & Sereni, A. (Eds.). (2021). Origins and Varieties of Logicism: On the Logico-Philosophical Foundations of Logicism. Abingdon: Routledge.CrossRefGoogle Scholar
Boccuni, F., & Sereni, A. (in press). The Logics of Abstraction. In Ferrari, F., Carrara, M., Hjortland, O., Sher, G., Sagi, G., and Brendel, E. (Eds.), Oxford Handbook of Philosophy of Mathematics and Logic. Oxford: Oxford University Press.Google Scholar
Boccuni, F., & Woods, J. (2020). Structuralist Neologicism. Philosophia Mathematica, 28(3), 296–316.CrossRefGoogle Scholar
Boccuni, F., & Zanetti, L. (in press). Abstractionism. Cambridge: Cambridge University Press.Google Scholar
Boddy, R. (2021). Frege on the Fruitfulness of Definitions. Journal for the History of Analytical Philosophy, 9(11), 99–114.CrossRefGoogle Scholar
Boghossian, P. A. (1996a). Analyticity. In Hale, B. & Wright, C. (Eds.), A Companion to the Philosophy of Language (pp. 331–368). Hoboken, NJ: Wiley-Blackwell.Google Scholar
Boolos, G. (1986). Saving Frege from Contradiction. Proceedings of theAristotelian Society, 87 (1986/87): 137–151.CrossRefGoogle Scholar
Boolos, G. (1987). The Consistency of Frege’s Foundations of Arithmetic. In Thomson, J. (Ed.), On Being and Saying: Essays in Honor of Richard Cartwright (pp. 3–20). Cambridge, MA: Massachusetts Institute of Technology Press.Google Scholar
Boolos, G. (1997). Is Hume’s Principle Analytic? In Heck, R. K. (Ed.), Language, Thought and Logic (pp. 245–261). Oxford: Clarendon Press (this and subsequent work originally published under the name “Richard G. Heck, Jr”).CrossRefGoogle Scholar
Boolos, G. S., Burgess, J. P., & Jeffrey, R. C. (2007). Computability and Logic (5 ed.). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Bourbaki, N. (1950). The Architecture of Mathematics. The American Mathematical Monthly, 57(4), 221–232.CrossRefGoogle Scholar
Breckenridge, W., & Magidor, O. (2012). Arbitrary Reference. Philosophical Studies, 158(3), 377–400.CrossRefGoogle Scholar
Brown, J. R. (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. New York, NY: Routledge.Google Scholar
Brun, G. (2016). Explication as a Method of Conceptual Re-Engineering. Erkenntnis, 81(6), 1211–1241.CrossRefGoogle Scholar
Brun, G. (2020). Conceptual Re-Engineering: From Explication to Reflective Equilibrium. Synthese, 197(3), 925–954.CrossRefGoogle Scholar
Burge, T. (1984). Frege on Extensions of Concepts, from 1884 to 1903. Philosophical Review, 93(1), 3–34.CrossRefGoogle Scholar
Burgess, J. P. (2005). Fixing Frege. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Burgess, J. P., & Rosen, G. (1997). A Subject with No Object. New York, NY: Oxford University Press.Google Scholar
Cantor, G. (1872). Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reichen. Mathematische Annalen, 5, 123–132.CrossRefGoogle Scholar
Cantù, P., & Luciano, E. (2021). Giuseppe Peano and His School: Axiomatics, Symbolism and Rigor. Philosophia Scientiae, 25, 3–14. https://link.springer.com/collections/difhbeaaedCrossRefGoogle Scholar
Cantù, P., & Testa, I. (Eds.). (2023). Mathematical Practice and Social Ontology (Vol. 42). Topoi, special issue. https://link.springer.com/collections/difhbeaaed.Google Scholar
Carnap, R. (1928). Der Logische Aufbau der Welt. Hamburg: Meiner Verlag. (, Eng. trans. The Logical Structure of the World, by R. A. George, , Berkeley, CA: University of California Press, 1967 ed.).Google Scholar
Carnap, R. (1947). Meaning and Necessity: A Study in Semantics and Modal Logic. Chicago, IL: University of Chicago Press.Google Scholar
Carnap, R. (1950a). Empiricism, Semantics and Ontology. Revue Internationale de Philosophie, 4(11), 20–40.Google Scholar
Carnap, R. (1950b). Logical Foundations of Probability. Chicago, IL: University of Chicago Press.Google Scholar
Carrara, M., & Zanetti, L. (2023). Thin Objects (Vol. 89) (No. 3). Theoria, special issue. https://onlinelibrary.wiley.com/doi/10.1111/theo.12463.CrossRefGoogle Scholar
Coffa, A. (1991). The Semantic Tradition From Kant to Carnap: To the Vienna Station. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
Cole, J. C. (2015). Social Construction, Mathematics, and the Collective Imposition of Function onto Reality. Erkenntnis, 80(6), 1101–1124.CrossRefGoogle Scholar
Cook, R. T. (2007). The Arché Papers on the Mathematics of Abstraction. Dordrecht: Springer.CrossRefGoogle Scholar
Cook, Roy T. (2012). Conservativeness, Stability, and Abstraction. British Journal for the Philosophy of Science 63(3), 673–696.CrossRefGoogle Scholar
Cook, R. T. (2021). Logicism, Separation and Complement. In Boccuni, F. & Sereni, A. (Eds.), Origins and Varieties of Logicism (pp. 289–308). Abingdon: Routledge.CrossRefGoogle Scholar
Cook, R. T. (2023). Frege’s Logic. In Zalta, E. N. & Nodelman, U. (Eds.), The Stanford Encyclopedia of Philosophy (Spring 2023 ed.). https://plato.stanford.edu/entries/frege-logic/.Google Scholar
Cook, R. T., & Linnebo, O. (2018). Cardinality and acceptable abstraction. Notre Dame Journal of Formal Logic, 59(1), 61–74.CrossRefGoogle Scholar
Coumans, V. J. W. (2024). Definitions (and Concepts) in Mathematical Practice. In Sriraman, B. (Ed.), Handbook of the History and Philosophy of Mathematical Practice (pp. 135–157). Cham: Springer.CrossRefGoogle Scholar
De Toffoli, S. (2021). Groundwork for a Fallibilist Account of Mathematics. Philosophical Quarterly, 7(4), 823–844.Google Scholar
Decock, L. (2022). Frege’s Theorem and Mathematical Cognition. In Boccuni, F. & Sereni, A. (Eds.), Origins and Varieties of Logicism: On the Logicophilosophical Foundations of Logicism (pp. 372–394). New York, NY: Routledge.Google Scholar
Dedekind, R. (1888). Was sind und was sollen die Zahlen? Braunschweig: Vieweg (Trans. in Essays on the Theory of Numbers, The Open Court Publishing Company, Chicago, IL, 1901, pp. 31–115).Google Scholar
Dehaene, S. (2011). The Number Sense: How the Mind Creates Mathematics. New York, NY: Oxford University Press.Google Scholar
de Jong, W. R., & Betti, A. (2010). The Classical Model of Science: A Millennia-Old Model of Scientific Rationality. Synthese, 174(2), 185–203.CrossRefGoogle Scholar
deRosset, L., & Linnebo, O. (2023). Abstraction and grounding. Philosophy and Phenomenological Research, 109(1), 357–390.CrossRefGoogle Scholar
Descartes, R. (1637). Discourse and Essays. In Cottingham, J., Stoothoff, R., & Murdoch, D. (Eds.), The Philosophical Writings of Descartes, 1985 (Vol. 1, pp. 109–176). Cambridge: Cambridge University Press.Google Scholar
Donaldson, T. (2017). The (Metaphysical) Foundations of Arithmetic? Noûs, 51(2), 775–801.CrossRefGoogle Scholar
Dummett, M. (1991). Frege: Philosophy of Mathematics. Cambridge, MA: Harvard University Press.Google Scholar
Dummett, M. (1993a). Discussions: Chairman’s Address: Basic Law V. Proceedings of the Aristotelian Society, 94, 243–252.CrossRefGoogle Scholar
Dummett, M. (1993b). Origins of Analytical Philosophy. Cambridge, MA: Harvard University Press.Google Scholar
Dummett, M. (1998). Neo-Fregeans: In Bad Company? In Schirn, M. (Ed.), The Philosophy of Mathematics Today. Oxford: Clarendon Press.Google Scholar
Ebert, P. A. (2016). A Framework for Implicit Definitions and the A Priori. In Ebert, P. A. & Rossberg, M. (Eds.), Abstractionism (pp. 133–160). Oxford: Oxford University Press.CrossRefGoogle Scholar
Ebert, P. A., & Rossberg, M. (2016). Abstractionism. Oxford: Oxford University Press.CrossRefGoogle Scholar
Ebert, P. A., & Rossberg, M. (2019). Mathematical Creation in Frege’s Grundgesetze. In Ebert, P. A. & Rossberg, M. (Eds.), Essays on Frege’s Basic Laws of Arithmetic (pp. 325–342). Oxford: Oxford University Press.CrossRefGoogle Scholar
, Euclid. (1926). The Thirteen Books of the Elements. Cambridge: Cambridge University Press. (Translated with introduction and commentary by Sir Thomas L. Heath; 3 vols.)Google Scholar
Falguera, J. L., Martínez-Vidal, C., & Rosen, G. (2022). Abstract Objects. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2022 ed.). https://plato.stanford.edu/entries/abstract-objects/.Google Scholar
Ferreirós, J. (2007). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. (2nd ed.). Basel: Birkäuser.Google Scholar
Ferreirós, J. (2023). Conceptual Structuralism. Journal for General Philosophy of Science, 54(1), 125–148.CrossRefGoogle Scholar
Field, H. (1974). Quine and the Correspondence Theory. Philosophical Review, 83(2), 200–228.CrossRefGoogle Scholar
Fine, K. (2012). Guide to Ground. In Correia, F. & Schnieder, B. (Eds.), Metaphysical Grounding (pp. 37–80). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Frans, J., Coumans, V., & de Regt, H. (2022). Explanation, Understanding, and Definitions in Mathematics. Logique et Analyse, 257, 79–99.Google Scholar
Frege, G. (1879). Begriffsschrift, eine der Arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Nebert. English translation in van Heijenoort (1967).Google Scholar
Frege, G. (1884). Die Grundlagen der Arithmetik: eine logische mathematische Untersuchung über den Begriff der Zahl. Breslau: Koebner. English translation by J. Austin, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, Blackwell, Oxford, 1950/1953.Google Scholar
Frege, G. (1892). Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik, Vol. 100, 25–50.Google Scholar
Frege, G. (1893–1903). Die Grundgesetze der Arithmetick (Vol. I–II). Jena: H. Phole. English translation by P. Ebert, M Rossberg, ed. Wright, C., Basic Laws of Arithmetic, Oxford University Press, Oxford, 2016.Google Scholar
Frege, G. (1971). On the Foundations of Geometry and Formal Theories of Arithmetic (Kluge, E.-H. W., Ed.). New Haven, CT: Yale University Press.Google Scholar
Frege, G. (1980). Philosophical and Mathematical Correspondence. Oxford: Basil Blackwell.Google Scholar
Frege, G. (1984). Collected Papers on Mathematics, Logic and Philosophy. Oxford: Basil Blackwell.Google Scholar
Frege, G., & Carnap, R. (2003). Frege’s Lectures on Logic: Carnap’s Student Notes, 1910–1914 (Reck, E. H. and Awodey, S., Eds.). Chicago, IL: Open Court.Google Scholar
Gabriel, G. (1978). Implizite Definitionen: eine Verwechselungsgeschichte. Annals of Science, 35(4), 419–423.CrossRefGoogle Scholar
Gergonne, J. D. (1818–9). Essai sur la théorie des definitions. Annales de Mathématiques Pures et Appliquée, 9, 1–35.Google Scholar
Giaquinto, M. (2007). Visual Thinking in Mathematics. New York: Oxford University Press.CrossRefGoogle Scholar
Gideon Rosen, S. Y. (2020). Solving the Caesar Problem – with Metaphysics. In Logic, Language, and Mathematics: Themes from the Philosophy of Crispin Wright (pp. 116–132). Oxford: Oxford University Press.CrossRefGoogle Scholar
Giovannini, E. N., & Schiemer, G. (2019). What Are Implicit Definitions? Erkenntnis, 86(6), 1661–1691.CrossRefGoogle Scholar
Gödel, K. (1964 (1st ed. 1947)). What Is Cantor’s Continuum Problem (1964 Version). Journal of Symbolic Logic, (2), 116–117.Google Scholar
Goldman, A. I. (1967). A Causal Theory of Knowing. Journal of Philosophy, 64(12), 357–372.CrossRefGoogle Scholar
Goodman, N. (1954/1983). Fact, Fiction, and Forecast (4th ed.). Cambridge, MA: Harvard University Press.Google Scholar
Graham, P., & Pedersen, N. J. L. L. (Eds.). (2020). Epistemic Entitlement. Oxford: Oxford University Press.CrossRefGoogle Scholar
Gupta, A., & Mackereth, S. (2023). Definitions. In Zalta, E. N. & Nodelman, U. (Eds.), The Stanford Encyclopedia of Philosophy (Fall 2023 ed.). https://plato.stanford.edu/entries/definitions/.Google Scholar
Hale, B. (1994). Dummett’s Critique of Wright’s Attempt to Resuscitate Frege. In Hale, B. & Wright, C. (Eds.), Reason’s Proper Study (pp. 189–213). Oxford: Oxford University Press.Google Scholar
Hale, B. (1997). Grundlagen §64. Proceedings of the Aristotelian Society, 97(3), 243–261.CrossRefGoogle Scholar
Hale, B. (2000). Reals by Abstraction. Philosophia Mathematica (III), 8, 100–123.CrossRefGoogle Scholar
Hale, B., & Wright, C. (2000). Implicit Definition and the A Priori. In Boghossian, P. & Peacocke, C. (Eds.), New Essays on the A Priori (pp. 286–319). Oxford: Oxford University Press.CrossRefGoogle Scholar
Hale, B., & Wright, C. (2001a). Reason’s Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford: Oxford University Press.CrossRefGoogle Scholar
Hale, B., & Wright, C. (2001b). To Bury Caesar ... In The Reason’s Proper Study (pp. 335–396). New York, NY: Oxford University Press.CrossRefGoogle Scholar
Hale, B., & Wright, C. (2002). Benacerraf’s Dilemma Revisited. European Journal of Philosophy, 10(1), 101–8211.CrossRefGoogle Scholar
Hale, B., & Wright, C. (2008). Abstraction and Additional Nature. Philosophia Mathematica, 16(2), 182–208.CrossRefGoogle Scholar
Hale, B., & Wright, C. (2009a). Focus Restored: Comments on John MacFarlane. Synthese, 170(3), 457–482.CrossRefGoogle Scholar
Hale, B., & Wright, C. (2009b). The Metaontology of Abstraction. In Chalmers, D., Manley, D., & Wasserman, R. (Eds.), Metametaphysics (pp. 178–212). Oxford: Oxford University Press.CrossRefGoogle Scholar
Hallett, M. (2019). Frege on Creation. In Ebert, P. A. & Rossberg, M. (Eds.), Essays on Frege’s Basic Laws of Arithmetic (pp. 285–324). Oxford: Oxford University Press.CrossRefGoogle Scholar
Hallett, M. (2021). Frege and Hilbert on Conceptual Analysis and Foundations. In Boccuni, F. & Sereni, A. (Eds.), Origins and Varieties of Logicism. Abingdon: Routledge.Google Scholar
Hamkins, J. D. (2012). The Set-Theoretic Multiverse. The Review of Symbolic Logic, 5(3), 416–449.CrossRefGoogle Scholar
Heck, R. K. (1993). The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik. Journal of Symbolic Logic, 58(2), 579–601.CrossRefGoogle Scholar
Heck, R. K. (1999). Grundgesetze der Arithmetic I §10. Philosophia Mathematica, 7(3), 258–292.CrossRefGoogle Scholar
Heck, R. K. (2000). Cardinality, Counting, and Equinumerosity. Notre Dame Journal of Formal Logic, 41(3), 187–209.CrossRefGoogle Scholar
Hellman, G., & Shapiro, S. (2018). Mathematical Structuralism. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Hilbert, D. (1899). Die Grundlagen der Geometrie. Leipzig: Teubner. English translation in Hilbert, D. The Foundations of Geometry. Chicago: Open Court, 1902.Google Scholar
Hilbert, D. (1900). On the Concept of Number. In Ewald, W. B. (Ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics (1996) (pp. 1089–1095). Oxford: Oxford University Press.Google Scholar
Hodges, W. (2023). Model Theory. In Zalta, E. N. & Nodelman, U. (Eds.), The Stanford Encyclopedia of Philosophy (Fall 2023 ed.). https://plato.stanford.edu/entries/model-theory/.Google Scholar
Hume, D. (1739–40). A Treatise of Human Nature. London: John Noon (Books I–II) and Thomas Longman (Book III). (3 vols.).Google Scholar
Jeshion, R. (2001). Frege’s Notions of Self-Evidence. Mind, 110(440), 937–976.CrossRefGoogle Scholar
Jeshion, R. (2004). Frege: Evidence for Self-Evidence. Mind, 113(449), 131–138.CrossRefGoogle Scholar
Kitcher, P. (1983). The Nature of Mathematical Knowledge. Oxford: Oxford University Press.Google Scholar
Korbmacher, J., & Schiemer, G. (2018). What Are Structural Properties? Philosophia Mathematica, 26(3), 295–323.CrossRefGoogle Scholar
Lakatos, I. (Ed.). (1976). Proofs and Refutations. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
Landry, E. (2013). The Genetic versus the Axiomatic Method: Responding to Feferman 1977. Review of Symbolic Logic, 6(1), 24–51.CrossRefGoogle Scholar
Leach-Krouse, G. (2017). Structural-Abstraction Principles. Philosophia Mathematica, 25(1), 45–72.Google Scholar
Lésniewski, S. (1931). Über Definitionen in der sogenannten Theorie der Deduktion. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie (Classe 3), 24.Google Scholar
Liggins, D. (2010). Epistemological Objections to Platonism. Philosophy Compass, 5(1), 67–77.CrossRefGoogle Scholar
Linnebo, Ø. (2003). Frege’s Conception of Logic: From Kant to Grundgesetze. Manuscrito, 26(2), 235–252.Google Scholar
Linnebo, Ø. (2004). Frege’s Proof of Referentiality. Notre Dame Journal of Formal Logic, 45(2), 73–98.CrossRefGoogle Scholar
Linnebo, Ø. (2006). Epistemological Challenges to Mathematical Platonism. Philosophical Studies, 129(3), 545–574.CrossRefGoogle Scholar
Linnebo, Ø. (2008). Structuralism and the Notion of Dependence. Philosophical Quarterly, 58(230), 59–79.Google Scholar
Linnebo, Ø. (2016). Impredicativity in the Neo-Fregean Program. In Ebert, P. A. & Rossberg, M. (Eds.), Abstractionism (pp. 247–268). Oxford: Oxford University Press.CrossRefGoogle Scholar
Linnebo, Ø. (2017). Philosophy of Mathematics. Princeton, NJ: Princeton University Press.Google Scholar
Linnebo, Ø. (2018a). Dummett on Indefinite Extensibility. Philosophical Issues, 28(1), 196–220.CrossRefGoogle Scholar
Linnebo, Ø. (2022). Plural Quantification. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2022 ed.). https://plato.stanford.edu/entries/plural-quant/.Google Scholar
Linnebo, Ø. (2023). Platonism in the Philosophy of Mathematics. In Zalta, E. N. & Nodelman, U. (Eds.), The Stanford Encyclopedia of Philosophy (Summer 2023 ed.). https://plato.stanford.edu/entries/platonism-mathematics/.Google Scholar
Linnebo, Ø. , & Pettigrew, R. (2014). Two Types of Abstraction for Structuralism. Philosophical Quarterly, 64(255), 267–283.CrossRefGoogle Scholar
Locke, J. (1690). An Essay Concerning Human Understanding (Nidditch, P. H., Ed.). Oxford: Oxford University Press.CrossRefGoogle Scholar
MacFarlane, J. (2002). Frege, Kant, and the Logic in Logicism. The Philosophical Review, 111(1), 25–65.CrossRefGoogle Scholar
MacFarlane, J. (2009). Double Vision: Two Questions about the Neo-Fregean Program. Synthese, 170(3), 443–456.CrossRefGoogle Scholar
Mackereth, S. (in press). Neologicism and Conservativeness. Journal of Philosophy. https://philpapers.org/versions/MACNAC-6Google Scholar
Maddy, P. (2011). Defending the Axioms: On the Philosophical Foundations of Set Theory. Oxford: Oxford University Press.CrossRefGoogle Scholar
Mancosu, P. (2016). Abstraction and Infinity. Oxford: Oxford University Press.CrossRefGoogle Scholar
Manders, K. (2008). The Euclidean Diagram. In Mancosu, P. (Ed.), The Philosophy of Mathematical Practice (pp. 80–133). Oxford: Oxford University Press.CrossRefGoogle Scholar
May, R. C., & Wehmeier, K. F. (2019). The Proof of Hume’s Principle. In Essays on Frege’s Basic Laws of Arithmetic. Oxford: Oxford University Press.Google Scholar
McGee, V. (1997). How We Learn Mathematical Language. Philosophical Review, 106(1), 35–68.CrossRefGoogle Scholar
McKenzie, K. (2022). Fundamentality and Grounding. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
McLarty, C. (1993). Numbers Can Be Just What They Have to. Noûs, 27(4), 487–498.CrossRefGoogle Scholar
Menger, K. (1943). What Is Dimension? The American Mathematical Monthly, 50(1), 2–7.CrossRefGoogle Scholar
Mill, J. S. (1843). A System of Logic, Ratiocinative and Inductive. Cambridge: Cambridge University Press.Google Scholar
Mueller, I. (1981). Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. Cambridge, MA: Massachusetts Institute of Technology Press.Google Scholar
Nodelman, U., & Zalta, E. N. (2024). Number Theory and Infinity Without Mathematics. Journal of Philosophical Logic, DOI: https://doi.org/10.1007/s10992-024-09762-7.CrossRefGoogle Scholar
Novaes, C. D., & Reck, E. H. (2017). Carnapian Explication, Formalisms as Cognitive Tools, and the Paradox of Adequate Formalization. Synthese, 194(1), 195–215.CrossRefGoogle Scholar
Nutting, E. S. (2018). The Limits of Reconstructive Neologicist Epistemology. Philosophical Quarterly, 68(273), 717–738.CrossRefGoogle Scholar
Nutting, E. S. (2024). The Benacerraf Problem of Mathematical Truth and Knowledge. The Internet Encyclopedia of Philosophy, https://iep.utm.edu.Google Scholar
Otero, M. H. (1970). Gergonne on Implicit Definition. Philosophy and Phenomenological Research, 30(4), 596–599.CrossRefGoogle Scholar
Otte, M., & Panza, M. (Eds.). (1997). Analysis and Synthesis in Mathematics. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
Padoa, A. (1900). Logical Introduction to Any Deductive Theory. In From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press.Google Scholar
Pantsar, M. (2024). Numerical Cognition and the Epistemology of Arithmetic. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Panza, M. (2016). Abstraction and Epistemic Economy. In Costreie, S. (ed.) Early Analytic Philosophy – New Perspectives on the Tradition. The Western Ontario Series in Philosophy of Science, vol. 80. Cham: Springer. DOI: https://doi.org/10.1007/978-3-319-24214-9_17.Google Scholar
Panza, M., & Sereni, A. (2013). Plato’s Problem. An Introduction to Mathematical Platonism. London: Palgrave Macmillan.Google Scholar
Panza, M., & Sereni, A. (2019). Frege’s Constraint and the Nature of Frege’s Logicism. Review of Symbolic Logic, 12(1), 97–143.CrossRefGoogle Scholar
Pascal, B. (1658). De l’Esprit géométrique et de l’Art de persuader. In Oeuvres complètes. Paris: Seuill.Google Scholar
Pasch, M. (1882). Vorlesungen über neuere Geometrie (2nd Edition 1926). Leipzig: Teubner.Google Scholar
Paseau, A. C., & Wrigley, W. (2024). The Euclidean Programme. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Peano, G. (1889). Arithmetices Principia, Nova Methodo Exposita (in van Heijenohoort (1967), pp. 83–97). Turin: Fratelli Bocca.Google Scholar
Pedersen, N. J. L. L. (2016). Hume’s Principle and Entitlement: On the Epistemology of the Neo-Fregean Program. In Ebert, P. & Rossberg, M. (Eds.), Abstractionism (pp. 186–202). Oxford: Oxford University Press.CrossRefGoogle Scholar
Picardi, E. (2022). Frege, Peano, and Russell on the Primitive Ideas of Logic. In Coliva, A. (Ed.), Frege on Language, Logic and Psychology (pp. 101–130). New York, NY: Oxford University Press.CrossRefGoogle Scholar
Pollard, S. (Ed.) (2010), Essays on the Foundations of Mathematics by Moritz Pasch, Dordrecht, Springer.CrossRefGoogle Scholar
Posy, C. J. (2020). Mathematical Intuitionism. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Potter, M. D. (2004). Set Theory and Its Philosophy. Oxford: Oxford University Press.CrossRefGoogle Scholar
Potter, M. D., & Sullivan, P. (2005). What Is Wrong with Abstraction? Philosophia Mathematica, 13(2), 187–193.CrossRefGoogle Scholar
Priest, G. (2024). Mathematical Pluralism. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Quine, W. V. O. (1936). Truth by Convention. In Whitehead, A. N. (Ed.), Philosophical Essays for Alfred North Whitehead. New York, NY: Longman, Green, & Company Inc.Google Scholar
Quine, W. V. O. (1951). Two Dogmas of Empiricism. Philosophical Review, 60(1), 20–43.CrossRefGoogle Scholar
Quine, W. V. O. (1960). Word and Object. Cambridge, MA: Massachusetts Institute of Technology Press.Google Scholar
Quine, W. V. O. (1969). Epistemology Naturalized. In Ontological Relativity and Other Essays. New York, NY: Columbia University Press.CrossRefGoogle Scholar
Quine, W. V. O. (1970). Philosophy of Logic (2nd Edition 1986). Cambridge, MA: Harvard University Press.Google Scholar
Raatikainen, P. (2022). Gödel’s Incompleteness Theorems. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2022 ed.). https://plato.stanford.edu/archives/spr2022/entries/goedel-incompleteness/.Google Scholar
Ramsey, F. P. (1931). Theories. In Braithwaite, R. B. (Ed.), The Foundations of Mathematics and Other Logical Essays. London: Routledge & Kegan Paul.Google Scholar
Rayo, A. (2013). The Construction of Logical Space. Oxford: Oxford University Press.CrossRefGoogle Scholar
Reck, E. H. (2007). Frege–Russell Numbers: Analysis or Explication? In Beaney, M. (Ed.), The Analytic Turn (pp. 33–50). New York, NY: Routledge.Google Scholar
Reck, E. H. (2013). Frege, Dedekind, and the Origins of Logicism. History and Philosophy of Logic, 34(3), 242–265.CrossRefGoogle Scholar
Reck, E. H. (2021). Dedekind’s Logicism: A Reconsideration and Contextualization. In Boccuni, F. & Sereni, A. (Eds.), Origins and Varieties of Logicism (pp. 119–146). New York: Routledge.CrossRefGoogle Scholar
Reck, E. H. (2024). Carnapian Explication: Origins and Shifting Goals. In Richardson, A. W. & Tuboly, A. T. (Eds.), Interpreting Carnap (pp. 127–149). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Reck, E. H., & Price, M. P. (2000). Structures and Structuralism in Contemporary Philosophy of Mathematics. Synthese, 125(3), 341–383.CrossRefGoogle Scholar
Reck, E. H., & Schiemer, G. (2023). Structuralism in the Philosophy of Mathematics. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2023 ed.). https://plato.stanford.edu/entries/structuralism-mathematics/.Google Scholar
Rosen, G. (2010). Metaphysical Dependence: Grounding and Reduction. In Hale, B. & Hoffmann, A. (Eds.), Modality: Metaphysics, Logic, and Epistemology (pp. 109–136). Oxford: Oxford University Press.CrossRefGoogle Scholar
Russo, L. (1998). The Definitions of Fundamental Geometric Entities Contained in Book I of Euclid’s Elements. Archive for History of Exact Sciences, 52(3), 195–219.CrossRefGoogle Scholar
Samuels, R., & Snyder, E. (2024). Number Concepts: An Interdisciplinary Inquiry. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Schwartzkopff, R. (2011). Numbers as Ontologically Dependent Objects: Hume’s Principle Revisited. Grazer Philosophische Studien, 82, 353–373.CrossRefGoogle Scholar
Schwartzkopff, R. (2016). Singular Terms Revisited. Synthese, 193(3), 909–936. DOI: https://doi.org/10.1007/s11229-015-0777-2.CrossRefGoogle Scholar
Sereni, A. (2016). A Dilemma for Benacerraf’s Dilemma? In Pataut, F. (Ed.), Truth, Objects, Infinity: New Perspectives on the Philosophy of Paul Benacerraf (pp. 93–125). Cham: Springer.CrossRefGoogle Scholar
Sereni, A. (2019). On the Philosophical Significance of Frege’s Constraint. Philosophia Mathematica, 27(2), 244–275.CrossRefGoogle Scholar
Sereni, A., Sforza Fogliani, M. P., & Zanetti, L. (2023). For Better and for Worse: Abstractionism, Good Company, and Pluralism. Review of Symbolic Logic, 16(1), 268–297.CrossRefGoogle Scholar
Sereni, A., & Zanetti, L. (2023). Minimalism, Trivialism, Aristotelianism. Theoria, 89(3), 280–297.CrossRefGoogle Scholar
Shapiro, S. (1991). Foundations without Foundationalism: A Case for Second-Order Logic. New York, NY: Oxford University Press.Google Scholar
Shapiro, S. (1997). Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press.Google Scholar
Shapiro, S. (2000). Frege Meets Dedekind: A Neologicist Treatment of Real Analysis. Notre Dame Journal of Formal Logic, 4, 317–421.Google Scholar
Shapiro, S. (2004). Foundations of Mathematics: Metaphysics, Epistemology, Structure. The Philosophical Quarterly, 54, 16–37.CrossRefGoogle Scholar
Shapiro, S. (2009). We Hold These Truths to Be Self-Evident: But What Do We Mean by That? Review of Symbolic Logic, 2(1), 175–207.CrossRefGoogle Scholar
Shapiro, S. (2011). Epistemology of Mathematics: What Are the Questions? What Count as Answers? Philosophical Quarterly, 61(242), 130–150.CrossRefGoogle Scholar
Shapiro, S. (2012). Objectivity, Explanation, and Cognitive Shortfall. In Wright, C. & Coliva, A. (Eds.), Mind, Meaning, and Knowledge: Themes from the Philosophy of Crispin Wright. Oxford: Oxford University Press.Google Scholar
Shapiro, S., & Roberts, C. (2021). Open Texture and Mathematics. Notre Dame Journal of Formal Logic, 62(1), 173–191.CrossRefGoogle Scholar
Shapiro, S., & Wright, C. (2006). All Things Indefinitely Extensible. In Rayo, A. & Uzquiano, G. (Eds.), Absolute Generality (pp. 255–304). Oxford: Clarendon Press.CrossRefGoogle Scholar
Simons, P. M. (1987). Frege’s Theory of Real Numbers. History and Philosophy of Logic, 8(1), 25–44. DOI: https://doi.org/10.1080/01445348708837106.CrossRefGoogle Scholar
Snyder, E., Samuels, R., & Shapiro, S. (2018). Neologicism, Frege’s Constraint, and the Frege–Heck Condition. Noûs, 54(1), 54–77.CrossRefGoogle Scholar
Studd, J. P. (2016). Abstraction Reconceived. British Journal for the Philosophy of Science, 67(2), 579–615. DOI: https://doi.org/10.1093/bjps/axu035.CrossRefGoogle Scholar
Tanswell, F. S. (2018). Conceptual Engineering for Mathematical Concepts. Inquiry: An Interdisciplinary Journal of Philosophy, 61(8), 881–913.CrossRefGoogle Scholar
Tanswell, F. S. (2024). Mathematical Rigour and Informal Proof. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Tappenden, J. (2008). Mathematical Concepts: Fruitfulness and Naturalness. In Mancosu, P. (Ed.), The Philosophy of Mathematical Practice (pp. 276–301). Oxford: Oxford University Press.CrossRefGoogle Scholar
Tappenden, J. (2012). Fruitfulness as a Theme in the Philosophy of Mathematics. The Journal of Philosophy, 109(1/2), 204–219.CrossRefGoogle Scholar
Tarski, A. (1933/1956). The Concept of Truth in Formalized Languages. In Tarski, A. (Ed.), Logic, Semantics, Metamathematics (pp. 152–278). Oxford: Clarendon Press.Google Scholar
Tennant, N. (1987). Anti-Realism and Logic: Truth as Eternal. New York, NY: Oxford University Press.Google Scholar
Thomasson, A. L. (2014). Ontology Made Easy. New York, NY: Oxford University Press.CrossRefGoogle Scholar
S̆ikić, Z. (2022). On Definitions in Mathematics. Publications De L’Institut Mathémathique, 112(125), 41–52.CrossRefGoogle Scholar
Uzquiano, G. (2015). Varieties of Indefinite Extensibility. Notre Dame Journal of Formal Logic, 56(1), 147–166.CrossRefGoogle Scholar
van Heijenoort, J. (Ed.). (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press.Google Scholar
Waismann, F. (1945). Verifiability. Proceedings of the Aristotelian Society, XIX, 101–164.Google Scholar
Waismann, F. (1951). Introduction to Mathematical Thinking. Mineola, NY: Dover Publications.Google Scholar
Weber, Z. (2022). Paraconsistency in Mathematics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Weiner, J. (1984). The Philosopher behind the Last Logicist. Philosophical Quarterly, 34(136), 242–264.CrossRefGoogle Scholar
Weyl, H. (1949). Philosophy of Mathematics and Natural Science (Helmer, O., Ed.). Princeton, NJ: Princeton University Press.Google Scholar
Whitehead, A. N., & Russell, B. (1910–13). Principia Mathematica. Cambridge: Cambridge University Press. (3 vols.) Second edition 1925–27.Google Scholar
Wittgenstein, L. (1969). On Certainty. Anscombe, G. E. M. and von Wright, G. H. (eds.), New York, NY: Harper & Row.Google Scholar
Wright, C. (1983). Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen University Press.Google Scholar
Wright, C. (1997). On the Philosophical Significance of Frege’s Theorem. In Heck, R. K. (Ed.), Language, Thought, and Logic: Essays in Honour of Michael Dummett (pp. 201–244). Oxford: Oxford University Press.CrossRefGoogle Scholar
Wright, C. (1998). On the Harmless Impredicativity of N=(‘Hume’s Principle’). In Schirn, M. (Ed.), The Philosophy of Mathematics Today (pp. 339–368). Oxford: Clarendon Press.CrossRefGoogle Scholar
Wright, C. (1999). Is Hume’s Principle Analytic? Notre Dame Journal of Formal Logic, 40(1), 6–30.CrossRefGoogle Scholar
Wright, C. (2000). Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’s Constraint. Notre Dame Journal of Formal Logic, 41(4), 317–334.CrossRefGoogle Scholar
Wright, C. (2004a). Intuition, Entitlement and the Epistemology of Logical Laws. Dialectica, 58(1), 155–175.CrossRefGoogle Scholar
Wright, C. (2004b). Warrant for Nothing (and Foundations for Free)? Aristotelian Society Supplementary Volume, 78(1), 167–212.CrossRefGoogle Scholar
Wright, C. (2007). On Quantifying into Predicate Position: Steps towards a New(Tralist) Perspective. In Leng, M., Paseau, A., & Potter, M. D. (Eds.), Mathematical Knowledge (pp. 150–174). Oxford: Oxford University Press.CrossRefGoogle Scholar
Wright, C. (2016). Abstraction and Epistemic Entitlement: On the Epistemological Status of Hume’s Principle. In Ebert, P. & Rossberg, M. (Eds.), Abstractionism (pp. 161–185). Oxford: Oxford University Press.CrossRefGoogle Scholar
Wright, C. (2020). Replies. In Miller, A. (Ed.), Logic, Language and Mathematics: Essays for Crispin Wright (pp. 277–432). Oxford: Oxford University Press.CrossRefGoogle Scholar
Yablo, S. (2005). The Myth of the Seven. In Kalderon, M. E. (Ed.), Fictionalism in Metaphysics (pp. 88–115). Oxford: Clarendon Press.CrossRefGoogle Scholar
Yap, A. (2014). Dedekind and Cassirer on Mathematical Concept Formation. Philosophia Mathematica, 25(3), 369–389.CrossRefGoogle Scholar
Zach, R. (2023). Hilbert’s Program. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Winter 2023 ed.). https://plato.stanford.edu/entries/hilbert-program/.Google Scholar
Zalta, E. N. (1999). Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege’s Grundgesetze in Object Theory. Journal of Philosophical Logic, 28(6), 619–60.CrossRefGoogle Scholar
Zalta, E. N. (2023). Frege’s Theorem and Foundations for Arithmetic. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2023 ed.). https://plato.stanford.edu/entries/frege-theorem/.Google Scholar
Zalta, E. N. (1999). Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege’s Grundgesetze in Object Theory. Journal of Philosophical Logic, 28(6), 619–660.CrossRefGoogle Scholar