- Publisher:
- Cambridge University Press
- Online publication date:
- February 2025
- Print publication year:
- 2025
- Online ISBN:
- 9781009375139
Abstractionism
Book description
The aim of this Element is to provide an overview of abstractionism in the philosophy of mathematics. The authors distinguish between mathematical abstractionism, which interprets mathematical theories on the basis of abstraction principles, and philosophical abstractionism, which attributes a philosophical significance to mathematical abstractionism. They then survey the main semantic, ontological, and epistemological theses that are associated with philosophical abstractionism. Finally, the authors suggest that the most recent developments in the debate pull abstractionism in different directions.
References
References
Antonelli, A. (2010a). The Nature and Purpose of Numbers. Journal of Philosophy, 107(4), 191–212.
Antonelli, A. (2010b). Notions of Invariance for Abstraction Principles. Philosophia Mathematica, 18(3), 276–292.
Antonelli, A., & May, R. (2005). Frege’s Other Program. Notre Dame Journal of Formal Logic, 46(1), 1–17.
Assadian, B. (2019). Abstractionism and Mathematical Singular Reference. Philosophia Mathematica, 27(2), 177–198.
Assadian, B. (2023). Abstraction and Semantic Presuppositions. Analysis, 15(3), 419–428.
Assadian, B., & Buijsman, S. (2018). Are the Natural Numbers Fundamentally Ordinals? Philosophy and Phenomenological Research, 99(3), 564–580.
Bagaria, J. (2023). Set Theory. In Zalta, E. N., & Nodelman, U. (Eds.), The Stanford Encyclopedia of Philosophy (Spring 2023 ed.). https://plato.stanford.edu/archives/spr2023/entries/set-theory/.
Batitsky, V. (2002). Some Measurement-Theoretic Concerns about Hale’s “Reals by Abstraction.” Philosophia Mathematica, 10(3), 286–303.
Benacerraf, P. (1973). Mathematical Truth. Journal of Philosophy, 70(19), 661–679.
Blanchette, P. (2012). Frege’s Conception of Logic. Oxford University Press.
Blanchette, P. (2021). Frege on Caesar and Hume’s Principle. In Boccuni, F., & Sereni, A. (Eds.), Origins and Varieties of Logicism: On the Logico-Philosophical Foundations of Logicism (pp. 27–54). Routledge.
Boccuni, F. (2010). Plural Grundgesetze. Studia Logica, 2(96), 315–330.
Boccuni, F. (2013). Plural Logicism. Erkenntnis, 78(5), 1051–1067.
Boccuni, F., & Panza, M. (2022). Frege’s Theory of Real Numbers: A Consistent Rendering. Review of Symbolic Logic, 15(3), 624–667.
Boccuni, F., & Woods, J. (2020). Structuralist Neologicism. Philosophia Mathematica, 28(3), 296–316.
Boghossian, P. A. (1996). Analyticity Reconsidered. Noûs, 30(3), 360–391.
Boolos, G. (1984). To Be Is to Be a Value of a Variable (or to Be Some Values of Some Variables). Journal of Philosophy, 81(8), 430–449. (Reprinted in Boolos 1998b, pp. 54–72)
Boolos, G. (1985). Nominalist Platonism. Philosophical Review, 94(3), 327–344. (Reprinted in Boolos 1998b, pp. 73–87)
Boolos, G. (1987a). 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). MIT Press. (Reprinted in Boolos 1998b, pp. 183–201)
Boolos, G. (1987b). Saving Frege from Contradiction. Proceedings of the Aristotelian Society, 87, 137–151. (Reprinted in Boolos 1998b, pp. 171–182)
Boolos, G. (1989). Iteration Again. Philosophical Topics, 17(2), 5–21. (Reprinted in Boolos 1998b, pp. 88–104)
Boolos, G. (1993). Whence the Contradiction? Aristotelian Society Supplementary Volume, 67, 211–233. (Reprinted in Boolos 1998b, pp. 220–236)
Boolos, G. (1998a). Is Hume’s Principle Analytic? In Heck, R. (Ed.), Logic, Language, and Thought (pp. 245–262). Oxford University Press. (Originally published under the name “Richard G. Heck, Jr.” Reprinted in Boolos 1998b, pp. 301–314)
Boolos, G. (1998b). Logic, Logic, and Logic (Jeffrey, Richard C., Ed.). Harvard University Press.
Boolos, G., & Heck, R. K. (1998). Die Grundlagen der Arithmetik, §§82-3. In Schirn, M. (Ed.), The Philosophy of Mathematics Today (pp. 407–428). Clarendon Press. (Originally published under the name “Richard G. Heck, Jr.” Reprinted in Boolos 1998b, pp. 315–338)
Brandom, R. (1996). The Significance of Complex Numbers for Frege’s Philosophy of Mathematics. Proceedings of the Aristotelian Society, 96(1), 293–315.
Burgess, J. P. (1998). On a Consistent Subsystem of Frege’s Grundgesetze. Notre Dame Journal of Formal Logic, 39(2), 274–278.
Burgess, J. P. (2005). Fixing Frege. Princeton University Press.
Clark, M. J., & Liggins, D. (2012). Recent Work on Grounding. Analysis Reviews, 72(4), 812–823.
Cocchiarella, N. (1985). Frege’s Double Correlation Thesis and Quine’s Set Theories NF and ML. Journal of Philosophical Logic, 14(1), 1–39.
Cocchiarella, N. (1992). Cantor’s Power-Set Theorem versus Frege’s Double-Correlation Thesis. History and Philosophy of Logic, 13(2), 179–201.
Conti, L. (2020). Russell’s Paradox and Free Zig Zag Solutions. Foundations of Science, 28(1), 1–19.
Cook, R. (2003). Iteration One More Time. Notre Dame Journal of Formal Logic, 44(2), 63–92.
Cook, R. (2017). Abstraction and Four Kinds of Invariance. Philosophia Mathematica, 25(1), 3–25.
Cook, R. (2019). Frege’s Little Theorem and Frege’s Way Out. In Ebert, P., & Rossberg, M. (Eds.), Essays on Frege’s Basic Laws of Arithmetic (pp. 384–410). Oxford University Press.
Cook, R. (2021a). Abstractionism in Mathematics. In Internet Encyclopedia of Philosophy. https://iep.utm.edu/abstractionism/.
Cook, R. (2021b). Logicism, Separation, and Complement. In Boccuni, F., & Sereni, A. (Eds.), Origins and Varieties of Logicism: On the Logico-Philosophical Foundations of Logicism (pp. 289–308). Routledge.
Cook, R. (2023). Frege’s Logic. In Zalta, E. N., & Nodelman, U. (Eds.), The Stanford Encyclopedia of Philosophy (Spring 2023 ed.). https://plato.stanford.edu/archives/spr2023/entries/frege-logic/.
Cook, R., & Ebert, P. (2005). Abstraction and Identity. Dialectica, 59(2), 121–139.
Cook, R., & Linnebo, Ø. (2018). Cardinality and Acceptable Abstraction. Notre Dame Journal of Formal Logic, 59(1), 61–74.
Correia, F., & Schnieder, B. (2012). Grounding: An Opinionated Introduction. In Correia, F., & Schnieder, B. (Eds.), Metaphysical Grounding: Understanding the Structure of Reality (pp. 1–30). Oxford University Press.
Demopoulos, W., & Bell, W. (1993). Frege’s Theory of Concepts and Objects and the Interpretation of Second-Order Logic. Philosophia Mathematica, 1(2), 139–156.
deRosset, L., & Linnebo, Ø. (2023). Abstraction and Grounding. Philosophy and Phenomenological Research, 109(1), 357–390.
Doherty, F. (2021). The Ontology of Abstraction, from Neo-Fregean to Neo-Dedekindian Logicism. In Boccuni, F., & Sereni, A. (Eds.), Origins and Varieties of Logicism: On the Logico-Philosophical Foundations of Logicism (pp. 349–371). Routledge.
Donaldson, T. (2017). The (Metaphysical) Foundations of Arithmetic? Noûs, 51(4), 775–801.
Dummett, M. (1973). Frege: Philosophy of Language. Duckworth.
Dummett, M. (1991). Frege: Philosophy of Mathematics. Harvard University Press.
Dummett, M. (1993). What Is Mathematics about? In George, A. (Ed.), The Seas of Language (pp. 429–445). Oxford University Press.
Dummett, M. (1994). Chairman’s Address: Basic Law V. Proceedings of the Aristotelian Society, 94, 243–251.
Dummett, M. (1998). Neo-Fregeans: In Bad Company? In Schirn, M. (Ed.), The Philosophy of Mathematics Today (pp. 369–388). Clarendon Press.
Ebels-Duggan, S. (2015). The Nuisance Principle in Infinite Settings. Thought: A Journal of Philosophy, 4(4), 263–268.
Ebert, P., & Rossberg, M. (2016). Introduction to Abstractionism. In Ebert, P., & Rossberg, M. (Eds.), Abstractionism: Essays in Philosophy of Mathematics (pp. 3–33). Oxford University Press.
Ebert, P., & Shapiro, S. (2009). The Good, the Bad and the Ugly. Synthese, 170(3), 415–441.
Eklund, M. (2006). Neo-Fregean Ontology. Philosophical Perspectives, 20(1), 95–121.
Felka, K. (2014). Number Words and Reference to Numbers. Philosophical Studies, 168(1), 261–282.
Ferreira, F. (2018). Zigzag and Fregean Arithmetic. In Tahiri, H. (Ed.), The Philosophers and Mathematics: Festschrift for Roshdi Rashed (pp. 81–100). Springer.
Ferreira, F., & Wehmeier, K. F. (2002). On the Consistency of the -CA Fragment of Frege’s Grundgesetze. Journal of Philosophical Logic, 31(4), 301–311.
Field, H. (1980). Science without Numbers: A Defence of Nominalism. Princeton University Press.
Field, H. (1989). Realism, Mathematics, and Modality. Blackwell.
Field, H. (2005). Recent Debates about the A Priori. In Gendler, T. S., & Hawthorne, J. (Eds.), Oxford Studies in Epistemology Volume 1 (pp. 69–88). Oxford University Press.
Fine, K. (1995a). The Logic of Essence. Journal of Philosophical Logic, 24(3), 241–273.
Fine, K. (1995b). Senses of Essence. In Sinnott-Armstrong, W., Raffman, D., & Asher, N. (Eds.), Modality, Morality, and Belief: Essays in Honor of Ruth Barcan Marcus (pp. 53–73). Cambridge University Press.
Fine, K. (2002). The Limits of Abstraction. Oxford University Press.
Fine, K. (2005). Our Knowledge of Mathematical Objects. In Gendler, T. Z., & Hawthorne, J. (Eds.), Oxford Studies in Epistemology (pp. 89–109). Clarendon Press.
Fine, K. (2012). Guide to Ground. In Correia, F., & Schnieder, B. (Eds.), Metaphysical Grounding (pp. 37–80). Cambridge University Press.
Florio, S., & Leach-Krouse, G. (2017). What Russell Should Have Said to Burali-Forti. Review of Symbolic Logic, 10(4), 682–718.
Florio, S., & Linnebo, Ø. (2021). The Many and the One: A Philosophical Study of Plural Logic. Oxford University Press.
Frege, G. (1879). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Louis Nebert. (Translated as Concept Script, a formal language of pure thought modelled upon that of arithmetic, by Bauer-Mengelberg, S. in van Heijenoort, J. (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, 1967, pp. 1–82)
Frege, G. (1884). Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. (Translated as The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, by J. L. Austin, Oxford: Blackwell, 1980)
Frege, G. (1891). Funktion und Begriff. Jena: Hermann Pohle. (Translated as Function and Concept by P. Geach in Geach, and , Black (eds. and trans.) Translations from the Philosophical Writings of Gottlob Frege, Oxford: Blackwell, 1980, pp. 21–41)
Frege, G. (1893/1903). Grundgesetze der Arithmetik: Band I/II. Jena: Verlag Hermann Pohle. (Complete translation by P. Ebert and M. Rossberg (with C. Wright) as Basic Laws of Arithmetic: Derived Using Concept-Script, Oxford University Press, 2013)
Ganea, M. (2007). Burgess’ PV Is Robinson’s Q. Journal of Symbolic Logic, 72(2), 619–624.
Hale, B. (2000). Reals by Abstraction. Philosophia Mathematica, 8(2), 100–123. (Reprinted in Hale and Wright 2001a, pp. 399–420)
Hale, B. (2001a). A Response to Potter and Smiley: Abstraction by Recarving. Proceedings of the Aristotelian Society, 101(3), 339–358.
Hale, B. (2001b). Singular Terms (1). In The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics (pp. 31–47). Clarendon Press.
Hale, B. (2001c). Singular Terms (2). In The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics (pp. 48–71). Clarendon Press.
Hale, B. (2002). Real Numbers, Quantities, and Measurement. Philosophia Mathematica, 10(3), 304–323.
Hale, B. (2005). Real Numbers and Set Theory. Extending the Neo-Fregean Programme beyond Arithmetic. Synthese, 147(1), 21–41.
Hale, B. (2013). Necessary Beings: An Essay on Ontology, Modality, and the Relations between Them. Oxford University Press.
Hale, B. (2018). Essence and Definition by Abstraction. Synthese, 198(8), 2001–2017.
Hale, B. (2019). Second-Order Logic: Properties, Semantics, and Existential Commitments. Synthese, 196(7), 2643–2669.
Hale, B. (2020). Ordinals by Abstraction. In Leech, J. (Ed.), Essence and Existence: Selected Essays by Bob Hale (pp. 240–255). Oxford University Press. (Reprinted in Boccuni, F. and Sereni, A. (Eds.), Origins and Varieties of Logicism: On the Logico-Philosophical Foundations of Logicism. Ch. 11. Routledge. 2021.)
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). Clarendon Press.
Hale, B., & Wright, C. (2001a). The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics. Clarendon Press.
Hale, B., & Wright, C. (2001b). To Bury Caesar…. In The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics (pp. 335–396). Clarendon Press.
Hale, B., & Wright, C. (2002). Benacerraf’s Dilemma Revisited. European Journal of Philosophy, 10(1), 101–129.
Hale, B., & Wright, C. (2008). Abstraction and Additional Nature. Philosophia Mathematica, 16(2), 182–208.
Hale, B., & Wright, C. (2009a). Focus Restored: Comments on John MacFarlane. Synthese, 170(3), 457–482.
Hale, B., & Wright, C. (2009b). The Metaontology of Abstraction. In Manley, R. W. David Chalmers David (Ed.), Metametaphysics: New Essays on the Foundations of Ontology (pp. 178–212). Oxford University Press.
Hallett, M. (1984). Cantorian Set Theory and Limitation of Size. Clarendon Press.
Hawley, K. (2007). Neo-Fregeanism and Quantifier Variance. Aristotelian Society Supplementary Volume, 81(1), 233–249.
Heck, R. K. (1996). The Consistency of Predicative Fragments of Frege’s Grundgesetze der Arithmetik. History and Philosophy of Logic, 17(1–2), 209–220. (Originally published under the name “Richard G. Heck, Jr.”)
Heck, R. K. (1997a). Finitude and Hume’s Principle. Journal of Philosophical Logic, 26(6), 589–617. (Originally published under the name “Richard G. Heck, Jr.”)
Heck, R. K. (1997b). The Julius Caesar Objection. In Heck, R. K. (Ed.), Language, Thought, and Logic: Essays in Honour of Michael Dummett (pp. 273–308). Oxford University Press. (Originally published under the name “Richard G. Heck, Jr.”)
Heck, R. K. (2000). Cardinality, Counting, and Equinumerosity. Notre Dame Journal of Formal Logic, 41(3), 187–209. (Originally published under the name “Richard G. Heck, Jr.”)
Heck, R. K. (2011a). Frege’s Theorem. Oxford University Press.
Heck, R. K. (2011b). A Logic for Frege’s Theorem. In Frege’s Theorem: An Introduction (pp. 267–296). Oxford University Press. (Originally published under the name “Richard G. Heck, Jr.”)
Hellman, G., & Shapiro, S. (2018). Mathematical Structuralism. Cambridge University Press.
Hodes, H. T. (1984). Logicism and the Ontological Commitments of Arithmetic. Journal of Philosophy, 81(3), 123–149.
Hodes, H. T. (1990). Ontological Commitments, Thick and Thin. In Boolos, G. (Ed.), Method, Reason and Language: Essays in Honor of Hilary Putnam (pp. 235–260). Cambridge University Press.
Hofweber, T. (2005). Number Determiners, Numbers, and Arithmetic. Philosophical Review, 114(2), 179–225.
Hofweber, T. (2023). Refocusing Frege’s Other Puzzle: A Response to Snyder, Samuels, and Shapiro. Philosophia Mathematica, 31(2), 216–235.
Horsten, L., & Leitgeb, H. (2009). How Abstraction Works. In Hieke, A., & Leitgeb, H. (Eds.), Reduction – Abstraction – Analysis (Vol. 11, pp. 217–226). Ontos Verlag.
Iemhoff, R. (2020). Intuitionism in the Philosophy of Mathematics. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Fall 2020 ed.). https://plato.stanford.edu/archives/fall2020/entries/intuitionism/.
Jané, I., & Uzquiano, G. (2004). Well- and Non-Well-Founded Fregean Extensions. Journal of Philosophical Logic, 33(5), 437–465.
Knowles, R. (2015). What “The Number of Planets Is Eight” Means. Philosophical Studies, 172(10), 2757–2775.
Leach-Krouse, G. (2015). Structural-Abstraction Principles. Philosophia Mathematica, 25(1), 45–72.
Linnebo, Ø. (2009a). Bad Company Tamed. Synthese, 170(3), 371–391.
Linnebo, Ø. (2009b). Introduction. Synthese, 170(3), 321–329.
Linnebo, Ø. (2010). Pluralities and Sets. Journal of Philosophy, 107(3), 144–164.
Linnebo, Ø. (2011). Some Criteria for Acceptable Abstraction. Notre Dame Journal of Formal Logic, 52(3), 331–338.
Linnebo, Ø. (2013). The Potential Hierarchy of Sets. Review of Symbolic Logic, 6(2), 205–228.
Linnebo, Ø. (2016). Impredicativity in the Neo-Fregean Programme. In Ebert, P., & Rossberg, M. (Eds.), Abstractionism: Essays in Philosophy of Mathematics (pp. 247–268). Oxford University Press.
Linnebo, Ø. (2018). Thin Objects: An Abstractionist Account. Oxford University Press.
Linnebo, Ø. (2022). Plural Quantification. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2022 ed.). https://plato.stanford.edu/archives/spr2022/entries/plural-quant/.
Linnebo, Ø. (2023a). 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/archives/sum2023/entries/platonism-mathematics/.
Linnebo, Ø. (2023b). Replies. Theoria, 89(3), 393–406.
Linnebo, Ø. , & Pettigrew, R. (2014). Two Types of Abstraction for Structuralism. Philosophical Quarterly, 64(255), 267–283.
Linnebo, Ø. , & Shapiro, S. (2017). Actual and Potential Infinity. Noûs, 53(1), 160–191.
Linsky, B., & Zalta, E. N. (2006). What Is Neologicism? Bulletin of Symbolic Logic, 12(1), 60–99.
Litland, J. E. (2022). Collective Abstraction. Philosophical Review, 131(4), 453–497.
MacBride, F. (2000). On Finite Hume. Philosophia Mathematica, 8(2), 150–159.
MacBride, F. (2003). Speaking with Shadows: A Study of Neo-logicism. British Journal for the Philosophy of Science, 54(1), 103–163.
MacBride, F. (2006). The Julius Caesar Objection: More Problematic than Ever. In MacBride, F. (Ed.), Identity and Modality (pp. 174–202). Oxford University Press.
MacBride, F. (2016). Neofregean Metaontology. In Ebert, P., & Rossberg, M. (Eds.), Abstractionism: Essays in Philosophy of Mathematics (pp. 94–112). Oxford: Oxford University Press.
MacFarlane, J. (2017). Logical Constants. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Winter 2017 ed.). https://plato.stanford.edu/archives/win2017/entries/logical-constants/.
Mackereth, S. (in press). Neo-Logicism and Conservativeness. Journal of Philosophy.
Mackereth, S., & Avigad, J. (2022). Two-Sorted Frege Arithmetic Is Not Conservative. Review of Symbolic Logic, 16(4), 1199–1232.
Mancosu, P. (2016). Abstraction and Infinity. Oxford University Press.
Moltmann, F. (2013). Reference to Numbers in Natural Language. Philosophical Studies, 162(3), 499–536.
Moltmann, F. (2016). The Number of Planets, a Number-Referring Term? In Ebert, P., & Rossberg, M. (Eds.), Abstractionism: Essays in Philosophy of Mathematics (pp. 113–129). Oxford University Press.
Moretti, L., & Wright, C. (2023). Epistemic Entitlement, Epistemic Risk and Leaching (1st ed.). Philosophy and Phenomenological Research, 106(3), 566–580.
Nolt, J. (2021). Free Logic. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Fall 2021 ed.). https://plato.stanford.edu/archives/fall2021/entries/logic-free/.
Nutting, E. S. (2018). The Limits of Reconstructive Neologicist Epistemology. Philosophical Quarterly, 68(273), 717–738.
Panza, M., & Sereni, A. (2013). Plato’s Problem: An Introduction to Mathematical Platonism. Palgrave MacMillan.
Panza, M., & Sereni, A. (2019). Frege’s Constraint and the Nature of Frege’s Foundational Program. Review of Symbolic Logic, 12(1), 97–143.
Parsons, T. (1987). On the Consistency of the First-Order Portion of Frege’s Logical System. Notre Dame Journal of Formal Logic, 28(1), 161–168.
Paseau, A. C. (2015). Did Frege Commit a Cardinal Sin? Analysis, 75(3), 379–386.
Payne, J. (2013a). Abstraction Relations Need Not Be Reflexive. Thought: A Journal of Philosophy, 2(2), 137–147.
Payne, J. (2013b). Expansionist Abstraction (Unpublished doctoral dissertation). University of Sheffield.
Pedersen, N. J. (2016). Hume’s Principle and Entitlement: On the Epistemology of the Neo-Fregean Program. In Ebert, P., & Rossberg, M. (Eds.), Abstractionism: Essays in Philosophy of Mathematics (pp. 161–185). Oxford University Press.
Picardi, E. (2017). Michael Dummett’s Interpretation of Frege’s Context Principle: Some Reflections. In Frauchiger, M. (Ed.), Truth, Meaning, Justification, and Reality: Themes from Dummett (pp. 29–62). De Gruyter. (Reprinted in Picardi 2022, Ch. 4)
Picardi, E. (2022). Frege on Language, Logic and Psychology: Selected Essays (Coliva, A., Ed.). Oxford University Press.
Plebani, M., San Mauro, L., & Venturi, G. (2023). Thin Objects Are Not Transparent. Theoria, 89(3), 314–325.
Potter, M., & Smiley, T. (2001). Abstraction by Recarving. Proceedings of the Aristotelian Society, 101(3), 327–338.
Potter, M., & Smiley, T. (2002). Recarving Content: Hale’s Final Proposal. Proceedings of the Aristotelian Society, 102(3), 301–304.
Pregel, F. (2023). Neo-Logicism and Gödelian Incompleteness. Mind, 131(524), 1055–1082.
Quine, W. V. O. (1955). On Frege’s Way Out. Mind, 64(254), 145–159.
Quine, W. V. O. (1970). Philosophy of Logic. Harvard University Press.
Rayo, A. (2002). Frege’s Unofficial Arithmetic. Journal of Symbolic Logic, 67(4), 1623–1638.
Rayo, A. (2007). Ontological Commitment. Philosophy Compass, 2(3), 428–444.
Rayo, A. (2013). The Construction of Logical Space. Oxford University Press.
Rayo, A. (2014). Reply to Critics. Inquiry: An Interdisciplinary Journal of Philosophy, 57(4), 498–534.
Reck, E. (2018). On Reconstructing Dedekind Abstraction Logically. In Reck, E. (Ed.), Logic, Philosophy of Mathematics, and Their History: Essays in Honor of W.W. Tait (pp. 113–138). College Publications.
Reck, E. (2020). Dedekind’s Contributions to the Foundations of Mathematics. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Winter 2020 ed.). https://plato.stanford.edu/archives/win2020/entries/dedekind-foundations/.
Reck, E. (2021). Dedekind’s Logicism: A Reconsideration and Contextualization. In Boccuni, F., & Sereni, A. (Eds.), Origins and Varieties of Logicism: On the Logico-Philosophical Foundations of Logicism (pp. 119–146). Routledge.
Reck, E., & Schiemer, G. (2023). Structuralism in the Philosophy of Mathematics. In Zalta, E. N., & Nodelman, U. (Eds.), The Stanford Encyclopedia of Philosophy (Spring 2023 ed.). https://plato.stanford.edu/archives/spr2023/entries/structuralism-mathematics/.
Roeper, P. (2016). A Vindication of Logicism. Philosophia Mathematica, 24(3), 360–378.
Roeper, P. (2020). Reflections on Frege’s Theory of Real Numbers. Philosophia Mathematica, 28(2), 236–257.
Rosen, G. (1993). The Refutation of Nominalism (?). Philosophical Topics, 21(2), 149–186.
Rosen, G. (2010). Metaphysical Dependence: Grounding and Reduction. In Hale, B., & Hoffmann, A. (Eds.), Modality: Metaphysics, Logic, and Epistemology (pp. 109–136). Oxford University Press.
Rosen, G. (2011). The Reality of Mathematical Objects. In Polkinghorne, J. (Ed.), Meaning in Mathematics (pp. 113–131). Oxford University Press.
Rosen, G. (2016). Mathematics and Metaphysical Naturalism. In Clark, K. (Ed.), The Blackwell Companion to Naturalism (pp. 277–288). Wiley Blackwell.
Rosen, G., & Yablo, S. (2020). Solving the Caesar Problem – with Metaphysics. In Miller, A. (Ed.), Logic, Language, and Mathematics: Themes from the Philosophy of Crispin Wright (pp. 116–132). Oxford University Press.
Rumfitt, I. (2003). Singular Terms and Arithmetical Logicism. Philosophical Books, 44(3), 193–219.
Rumfitt, I. (2018). Neo-Fregeanism and the Burali-Forti Paradox. In Rivera, I. F., & Leech, J. (Eds.), Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale (pp. 188–223). Oxford University Press.
Schiemer, G. (2021). Logicism in Logical Empiricism. In Boccuni, F., & Sereni, A. (Eds.), Origins and Varieties of Logicism: On the Logico-Philosophical Foundations of Logicism (pp. 243–266). Routledge.
Schiemer, G., & Wigglesworth, J. (2019). The Structuralist Thesis Reconsidered. British Journal for the Philosophy of Science, 70(4), 1201–1226.
Schindler, T. (2021). Steps towards a Minimalist Account of Numbers. Mind, 131(523), 863–891.
Schirn, M. (2002). Second-Order Abstraction, Logicism and Julius Caesar. Diálogos. Revista de Filosofía de la Universidad de Puerto Rico, 37(79), 319–372.
Schirn, M. (2013). Frege’s Approach to the Foundations of Analysis (1874–1903). History and Philosophy of Logic, 34(3), 266–292.
Schirn, M. (2023). Frege on the Introduction of Real and Complex Numbers by Abstraction and Cross-Sortal Identity Claims. Synthese, 201(6), 1–18.
Schroeder-Heister, P. (1987). A Model-Theoretic Reconstruction of Frege’s Permutation Argument. Notre Dame Journal of Formal Logic, 28(1), 69–79.
Schwartzkopff, R. (2011). Numbers as Ontologically Dependent Objects: Hume’s Principle Revisited. Grazer Philosophische Studien, 82(1), 353–373.
Schwartzkopff, R. (2015). Number Sentences and Specificational Sentences: Reply to Moltmann. Philosophical Studies, 173(8), 2173–2192.
Schwartzkopff, R. (2016). Singular Terms Revisited. Synthese, 193(3), 909–936.
Sereni, A. (2019). On the Philosophical Significance of Frege’s Constraint. Philosophia Mathematica, 27(2), 244–275.
Shapiro, S. (1991). Foundations without Foundationalism: A Case for Second-Order Logic (Vol. 17). Clarendon Press.
Shapiro, S. (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University Press.
Shapiro, S. (2000). Frege Meets Dedekind: A Neologicist Treatment of Real Analysis. Notre Dame Journal of Formal Logic, 41(4), 335–364.
Shapiro, S. (2003). Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility. British Journal for the Philosophy of Science, 54(1), 59–91.
Shapiro, S. (2009). The Measure of Scottish Neo-Logicism. In Lindström, S., Palmgren, E., Segerberg, K., & Stoltenberg-Hansen, V. (Eds.), Logicism, Intuitionism, and Formalism: What Has Become of Them? (pp. 69–90). Springer.
Shapiro, S., & Linnebo, Ø. (2015). Frege Meets Brouwer (or Heyting or Dummett). The Review of Symbolic Logic, 8(3), 540–552.
Shapiro, S., & Uzquiano, G. (2016). Ineffability within the Limits of Abstraction Alone. In Ebert, P., & Rossberg, M. (Eds.), Abstractionism: Essays in Philosophy of Mathematics (pp. 283–307). Oxford University Press.
Shapiro, S., & Weir, A. (2000). “Neo-Logicist” Logic Is Not Epistemically Innocent. Philosophia Mathematica, 8(2), 160–189.
Shapiro, S., & Wright, C. (2006). All Things Indefinitely Extensible. In Rayo, A., & Uzquiano, G. (Eds.), Absolute Generality (pp. 255–304). Oxford University Press.
Sider, T. (2007). Neo-Fregeanism and Quantifier Variance. Aristotelian Society Supplementary Volume, 81(1), 201–232.
Simons, P. M. (1987). Frege’s Theory of Real Numbers. History and Philosophy of Logic, 8(1), 25–44.
Snyder, E. (2017). Numbers and Cardinalities: What’s Really Wrong with the Easy Argument for Numbers? Linguistics and Philosophy, 40(4), 373–400.
Snyder, E., Samuels, R., & Shapiro, S. (2018). Neologicism, Frege’s Constraint, and the Frege-Heck Condition. Noûs, 54(1), 54–77.
Snyder, E., Samuels, R., & Shapiro, S. (2022a). Hofweber’s Nominalist Naturalism. In Oliveri, G., Ternullo, C., & Boscolo, S. (Eds.), Objects, Structures, and Logics (pp. 31–62). Springer.
Snyder, E., Samuels, R., & Shapiro, S. (2022b). Resolving Frege’s Other Puzzle. Philosophia Mathematica, 30(1), 59–87.
Snyder, E., & Shapiro, S. (2019). Frege on the Real Numbers. In Ebert, P., & Rossberg, M. (Eds.), Essays on Frege’s Basic Laws of Arithmetic (pp. 343–383). Oxford University Press.
Studd, J. P. (2016). Abstraction Reconceived. British Journal for the Philosophy of Science, 67(2), 579–615.
Studd, J. P. (2023). The Caesar Problem – a Piecemeal Solution. Philosophia Mathematica, 31(2), 236–267.
Tappenden, J. (1995). Geometry and Generality in Frege’s Philosophy of Arithmetic. Synthese, 102(3), 319–361.
Tappenden, J. (2019). Infinitesimals, Magnitudes, and Definition in Frege. In Ebert, P., & Rossberg, M. (Eds.), Essays on Frege’s Basic Laws of Arithmetic (pp. 235–263). Oxford University Press.
Tennant, N. (1987). Anti-Realism and Logic: Truth as Eternal. Oxford University Press.
Tennant, N. (2022). The Logic of Number. Oxford University Press.
Thomasson, A. (2013). Fictionalism versus Deflationism. Mind, 122(488), 1023–1051.
Thomasson, A. (2014). Ontology Made Easy. Oxford University Press.
Urbaniak, R. (2010). Neologicist Nominalism. Studia Logica, 96(2), 149–173.
Uzquiano, G. (2009). Bad Company Generalized. Synthese, 170(3), 331–347.
Uzquiano, G. (2019). Impredicativity and Paradox. Thought: A Journal of Philosophy, 8(3), 209–221.
Väänänen, J. (2021). Second-Order and Higher-Order Logic. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Fall 2021 ed.). https://plato.stanford.edu/archives/fall2021/entries/logic-higher-order/.
Visser, A. (2009). The Predicative Frege Hierarchy. Annals of Pure and Applied Logic, 160(2), 129–153.
Walsh, S. (2016). The Strength of Abstraction with Predicative Comprehension. Bulletin of Symbolic Logic, 22(1), 105–120.
Walsh, S., & Ebels-Duggan, S. (2015). Relative Categoricity and Abstraction Principles. Review of Symbolic Logic, 8(3), 572–606.
Wehmeier, K. F. (1999). Consistent Fragments of Grundgesetze and the Existence of Non-Logical Objects. Synthese, 121(3), 309–328.
Weir, A. (2003). Neo-Fregeanism: An Embarrassment of Riches? Notre Dame Journal of Formal Logic, 44(1), 13–48.
Wetzel, L. (1990). Dummett’s Criteria for Singular Terms. Mind, 99(394), 239–254.
Wigglesworth, J. (2018). Non-Eliminative Structuralism, Fregean Abstraction, and Non-Rigid Structures. Erkenntnis, 86(1), 113–127.
Woods, J. (2014). Logical Indefinites. Logique Et Analyse, 227, 277–307.
Wright, C. (1983). Frege’s Conception of Numbers as Objects. Aberdeen University Press.
Wright, C. (1990). Field & Fregean Platonism. In Irvine, A. D. (Ed.), Physicalism in Mathematics (pp. 73–93). Springer.
Wright, C. (1998). On the Harmless Impredicativity of N= (“Hume’s Principle”). In Schirn, M. (Ed.), The Philosophy of Mathematics Today (pp. 339–368). Clarendon Press. (Reprinted in Hale and Wright 2001a, §10)
Wright, C. (1999). Is Hume’s Principle Analytic? Notre Dame Journal of Formal Logic, 40(1), 6–30. (Reprinted in Hale and Wright 2001a, pp. 307–32)
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.
Wright, C. (2004a). Intuition, Entitlement and the Epistemology of Logical Laws. Dialectica, 58(1), 155–175.
Wright, C. (2004b). Warrant for Nothing (and Foundations for Free)? Aristotelian Society Supplementary Volume, 78(1), 167–212.
Wright, C. (2007). On Quantifying into Predicate Position: Steps towards a New(Tralist) Perspective. In Leng, M., Paseau, A., & Potter, M. (Eds.), Mathematical Knowledge (pp. 150–174). Oxford University Press.
Wright, C. (2014). On Epistemic Entitlement (II): Welfare State Epistemology. In Dodd, D., & Zardini, E. (Eds.), Scepticism and Perceptual Justification (pp. 213–247). Oxford University Press.
Wright, C. (2016). Abstraction and Epistemic Entitlement: On the Epistemological Status of Hume’s Principle. In Ebert, P., & Rossberg, M. (Eds.), Abstractionism: Essays in Philosophy of Mathematics (pp. 134–161). Oxford University Press.
Wright, C. (2020). Frege and Logicism: Replies to Demopoulos, Rosen and Yablo, Edwards, Boolos, and Heck. In Miller, A. (Ed.), Logic, Language, and Mathematics: Themes from the Philosophy of Crispin Wright (pp. 279–353). Oxford University Press.
Yablo, S. (2001). Go Figure: A Path through Fictionalism. Midwest Studies in Philosophy, 25(1), 72–102.
Yablo, S. (2005). The Myth of the Seven. In Kalderon, M. E. (Ed.), Fictionalism in Metaphysics (pp. 88–115). Clarendon Press.
Zalta, E. N. (1983). Abstract Objects: An Introduction to Axiomatic Metaphysics. D. Reidel.
Zalta, E. N. (2023). Frege’s Theorem and Foundations for Arithmetic. In Zalta, E. N., & Nodelman, U. (Eds.), The Stanford Encyclopedia of Philosophy (Spring 2023 ed.). https://plato.stanford.edu/archives/spr2023/entries/frege-theorem/.
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