Element contents
Wittgenstein's Philosophy of Mathematics
Published online by Cambridge University Press: 23 July 2021
Summary
For Wittgenstein mathematics is a human activity characterizing ways of seeing conceptual possibilities and empirical situations, proof and logical methods central to its progress. Sentences exhibit differing 'aspects', or dimensions of meaning, projecting mathematical 'realities'. Mathematics is an activity of constructing standpoints on equalities and differences of these. Wittgenstein's Later Philosophy of Mathematics (1934–1951) grew from his Early (1912–1921) and Middle (1929–33) philosophies, a dialectical path reconstructed here partly as a response to the limitative results of Gödel and Turing.
Keywords
Wittgensteinrule-followingphilosophy of mathematicshistory and philosophy of mathematicslogicismmeaning in mathematicssurveyabilityrealism in mathematicsmathematical knowledgelogical necessitymathematical necessitylanguage and mathematicsfoundations of mathematicsphilosophy of logiclogic and communicationmathematics and languagenaturalism in philosophy of mathematicsphilosophy of mathematical practiceA12B34C56D78E90
- Type
- Element
- Information
- Online ISBN: 9781108687126Publisher: Cambridge University PressPrint publication: 12 August 2021
References
Auxier, R. E., Anderson, D. R., & Hahn, L. E. 2015. The Philosophy of Hilary Putnam. Chicago, IL: Open Court.Google Scholar
Bangu, S. 2020. “Changing the Style of Thinking”: Wittgenstein on Superlatives, Revisionism, and Cantorian Set Theory. Iyyun, the Jerusalem Philosophical Quarterly, 68, 339–359.Google Scholar
Barwise, J. (ed.). 1977. Handbook of Mathematical Logic. Amsterdam; New York: North-Holland Pub. Co.Google Scholar
Baz, A. 2020. The Significance of Aspect Perception: Bringing the Phenomenal World into View (vol. 5). Switzerland: Springer International Publishing AG, Springer.Google Scholar
Benacerraf, P. 1965. What Numbers Could Not Be. The Philosophical Review, 74(1), 47–73.CrossRefGoogle Scholar
Bernays, P. 1957. Bemerkungen zum Paradoxon von Thoralf Skolem. Avhandlinger utgitt av Det Norske Videnskaps-Akademi i Oslo, I. Mat.-Naturv. Klasse, 2nd series, 3–9. (English translation by Dirk Schllimm and Steve Awodey, “Considerations regarding the paradox of Thoralf Skolem,” www.phil.cmu.edu/projects/bernays/.)Google Scholar
Bernays, P. 1959. Betrachtungen zu Ludwig Wittgensteins Bemerkungen über die Grundlagen der Mathematik/Comments on Ludwig Wittgenstein’s Remarks on the Foundations of Mathematics. Ratio, 2(1), 1–22.Google Scholar
Brouwer, L. E. J. 1929. Mathematik, Wissenschaft und Sprache. Monatshefte für Mathematik und Physik, 36(1), 153–164.CrossRefGoogle Scholar
Cantor, G. 1874. Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Journal für die reine und angewandte Mathematik, 77, 258–262.Google Scholar
Cantor, G. 1891. Über eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker-Vereinigung, I, 75–78. (English translation in Ewald, 1996, vol. 2, pp. 920–922.)Google Scholar
Cavell, S. 1988. Declining Decline: Wittgenstein as a Philosopher of Culture. Inquiry, 31(3), 253–264.CrossRefGoogle Scholar
Church, A. 1932. A Set of Postulates for the Foundation of Logic. Annals of Mathematics, 33(2), 346–366.CrossRefGoogle Scholar
Church, A. 1936. A Note on the Entscheidungsproblem. Journal of Symbolic Logic, 1(1), 40–41.Google Scholar
Copeland, B. J. 2002. Accelerating Turing Machines. Minds and Machines, 12(2), 281–300.CrossRefGoogle Scholar
Copeland, B. J. & Proudfoot, D. 2010. Deviant Encodings and Turing’s Analysis of Computability. Studies in History and Philosophy of Science, 41(3), 247–252.Google Scholar
Costreie, S. (ed.) 2016. Early Analytic Philosophy: New Perspectives on the Tradition. New York: Springer Publishing Switzerland.CrossRefGoogle Scholar
Crocco, G., & Engelen, E.-M. (eds.) 2016. Kurt Gödel: Philosopher-Scientist (vol. 1). Aix-en-Provence: Presses Universitaires de Provence.CrossRefGoogle Scholar
Davis, M. 2017. Universality Is Ubiquitous. (In Floyd and Bokulich, 2017, pp. 153–158.)Google Scholar
Dedekind, R. 1872. Stetigkeit und irrationale Zahlen. Wiesbaden: Friedrich Vieweg und Sohn. (English translation Continuity and Irrational Numbers in Ewald, 1996, vol. 2, pp. 766–779.)Google Scholar
Diamond, C. 1991. The Realistic Spirit: Wittgenstein, Philosophy, and the Mind. Cambridge, MA: Massachusetts Institute of Technology Press.CrossRefGoogle Scholar
Dreben, B., & Floyd, J. 1991. Tautology: How Not to Use a Word. Synthese, 87(1), 23–50.Google Scholar
Dummett, M. A. E. 1959. Wittgenstein’s Philosophy of Mathematics. The Philosophical Review, LXVII, 324–348.Google Scholar
Dybjer, P., Lindström, S., Palmgren, E., & Sundholm, G. (eds.) 2012. Epistemology versus Ontology, Logic, Epistemology: Essays in Honor of Per Martin-Löf. Dordrecht: Springer Science+Business Media.Google Scholar
Ebert, P. A., & Rossberg, M. (eds.) 2019. Essays on Frege’s Basic Laws of Arithmetic. Oxford: Oxford University Press.Google Scholar
Ellis, J., & Guevara, D. (eds.) 2012. Wittgenstein and the Philosophy of Mind. New York/ Oxford,UK: Oxford University Press.CrossRefGoogle Scholar
Engelmann, M. L. 2013. Wittgenstein’s Philosophical Development: Phenomenology, Grammar, Method, and the Anthropological View. Basingstoke, UK: Palgrave Macmillan.Google Scholar
Ewald, W. (ed.). 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press (2 vols.).Google Scholar
Fisher, D., & McCarty, C. 2016. Reconstructing a Logic from Tractatus: Wittgenstein’s Variables and Formulae. (In Costreie, 2016, pp. 301–324).CrossRefGoogle Scholar
Floyd, J. 1995. Wittgenstein, Gödel and the Trisection of the Angle. (In Hintikka, 1995, pp. 373–426.)CrossRefGoogle Scholar
Floyd, J. 2001. Prose versus Proof: Wittgenstein on Gödel, Tarski and Truth. Philosophia Mathematica, 3(9), 280–307.Google Scholar
Floyd, J. 2005. Wittgenstein on Philosophy of Logic and Mathematics. (In Shapiro, 2005, pp. 75–128.)CrossRefGoogle Scholar
Floyd, J. 2010. On Being Surprised: Wittgenstein on Aspect Perception, Logic and Mathematics. (In Krebs and Day, 2010, pp. 314–337.)CrossRefGoogle Scholar
Floyd, J. 2011. Prefatory Note to the Frege–Wittgenstein Correspondence. In De Pelligrin, E. (ed.), Interactive Wittgenstein (pp. 1–14). Springer Science + Business Media B.V.Google Scholar
Floyd, J. 2012a. Das Überraschende: Wittgenstein on the Surprising in Mathematics. (In Ellis and Guevara, 2012, pp. 224–258.)Google Scholar
Floyd, J. 2012b. Wittgenstein’s Diagonal Argument: A Variation on Cantor and Turing. (In Dybjer, Lindström, Palmgren, and Sundholm, 2012, pp. 25–44.)Google Scholar
Floyd, J. 2013. Turing, Wittgenstein and Types: Philosophical Aspects of Turing’s ‘The Reform of Mathematical Notation’ (1944–5). (In Turing, 2013, pp. 250–253.)Google Scholar
Floyd, J. 2016. Chains of Life: Turing, Lebensform, and the Emergence of Wittgenstein’s Later Style. Nordic Wittgenstein Review, 5(2), 7–89.CrossRefGoogle Scholar
Floyd, J. 2017. Turing on “Common Sense”: Cambridge Resonances. (In Floyd and Bokulich, 2017, pp. 103–152.)Google Scholar
Floyd, J. 2020. Aspects of the Real Numbers: Putnam, Wittgenstein, and Nonextensionalism. The Monist, 103(4), 427–441.Google Scholar
Floyd, J. 2021. Sheffer, Lewis and the “Logocentric Predicament.” (To appear in Narboux, J. P. and Wagner, H., eds., The Legacy of C. I. Lewis, Routledge Series in American Philosophy.)Google Scholar
Floyd, J., & Bokulich, A. (eds.) 2017. Philosophical Explorations of the Legacy of Alan Turing – Turing 100. Dordrecht: Springer.CrossRefGoogle Scholar
Floyd, J., & Kanamori, A. 2016. Gödel vis-à-vis Russell: Logic and Set Theory to Philosophy. (In Crocco and Engelen, 2016, pp. 243–326.)Google Scholar
Floyd, J., & Mühlhölzer, F. 2020. Wittgenstein’s Annotations to Hardy’s Course of Pure Mathematics, An Investigation of Wittgenstein’s Non-Extensionalist Understanding of the Real Numbers. Switzerland: Springer.Google Scholar
Floyd, J., & Putnam, H. 2000. A Note on Wittgenstein’s ‘Notorious Paragraph’ about the Gödel Theorem. Journal of Philosophy, 45(11), 624–632.Google Scholar
Floyd, J., & Putnam, H. 2012. Wittgenstein’s “Notorious” Paragraph about the Gödel Theorem: Recent Discussions. (In Putnam, 2012, pp. 458–481.)CrossRefGoogle Scholar
Fogelin, R. J. 1987. Wittgenstein (2nd ed.). London; New York: Routledge K. Paul. (1st ed., 1976.)Google Scholar
Fogelin, R. J. 2009. Taking Wittgenstein at His Word: A Textual Study (vol. 29). Princeton, NJ: Princeton University Press.Google Scholar
Frege, G. 1879. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Louis Nebert. (Reprinted 1964 by G. Olms, Hildesheim. English translation by Stefan Bauer-Mengelberg in van Heijenoort, 1967, pp. 5–82 and Preface and Part I by Michael Beaney in Frege, 1997, pp. 47–78.)Google Scholar
Frege, G. 1880/1881. Booles rechnende Logik und die Begriffsschrift (1880/1881. (In Frege [1969], pp. 9–52. English translation “Boole’s Logical Calculus and the Concept-script” in Frege, 1979, pp. 47–52.)Google Scholar
Frege, G. 1884. Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über der Begriff der Zahl. Breslau: W. Koebner. (English translation by J. L. Austin, The Foundations of Arithmetic: a Logico-Mathematical Enquiry into the Concept of Number, New York: Torchbooks, Harper, 2nd rev. ed., 1953.)Google Scholar
Frege, G. 1891. Funktion und Begriff. Jena: Hermann Pohle. (In Frege, 1969, pp. 125–143. English translation by Peter Geach in Frege, 1997, pp. 130–148.)Google Scholar
Frege, G. 1892. Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik, 100, 25–50. (English translation by Max Black, “On Sinn and Bedeutung,” in Frege, 1997, pp. 151–171.)Google Scholar
Frege, G. 1893/1903. Grundgesetze der Arithmetik; Begriffsschriftlich Abgeleitet, Volumes I and II. Jena: Verlag von Hermann Pohle. (Edited and translated into English as Basic Laws of Arithmetic by Philip A. Ebert, Marcus Rossberg, and Crispin Wright, Oxford University Press, Oxford, 2013.)Google Scholar
Frege, G. 1897. Logik. (In Frege, 1969, pp. 137–163. English translation by Peter Long and Roger White, “Logic (1897),” in Frege, 1997, pp. 227–250.)Google Scholar
Frege, G. 1918/19. Der Gedanke. Beiträge zur Philosophie des deutschen Idealismus, I, 58–77. (English translation by Peter Geach and R. H. Stoothoff in Frege, 1984, pp. 342–362 and in Frege, 1997, pp. 325–346.)Google Scholar
Frege, G. 1969. Frege: Nachgelassene Schriften, Band I. Hamburg: Felix Meiner Verlag. (Edited by Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach.)Google Scholar
Frege, G. 1980. Philosophical and Mathematical Correspondence. Oxford: B. Blackwell. (Edited by Gottfried Gabriel, Hans Hermes, Kambartel, Christian Thiel, Albert Veraart ; abridged from the German edition by Brian McGuinness, translated by Hans Kaal.)Google Scholar
Frege, G. 1983. Wissenschaftlicher Briefwechsel (Edited by Gabriel, Gottfried, Hermes, Hans, Kambartel, Friedrich, and Christian Thiel). Hamburg, F. Meiner. (English translation in Frege, 1980.)Google Scholar
Frege, G. 1984. Collected Papers on Mathematics, Logic, and Philosophy. New York: Oxford University Press.Google Scholar
Frege, G. 1997. The Frege Reader. Malden, MA: Blackwell Publishers. (Edited by Michael Beaney.)Google Scholar
Gödel, K. 1930. Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik, 37, 349–360. (Reprinted with English translation in Gödel, 1986, 103–123.)CrossRefGoogle Scholar
Gödel, K. 1931. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter System I. Monatshefte für Mathematik und Physik, 38, 173–198. (Reprinted with English translation in Gödel, 1986, 144–195.)CrossRefGoogle Scholar
Gödel, K. 1946. Remarks Before the Princeton Bicentennial Conference on Problems in Mathematics. (Reprinted with English translation in Gödel, 1990, 150–153.)Google Scholar
Gödel, K. 1964a. On Undecideable Propositions of Formal Mathematical Systems. (Lectures given at the Institute for Advanced Study, February–May 1934. Transcribed by S. C. Kleene and J. B. Rosser. In Gödel, 1986, pp. 346–369, with a Postscriptum, namely Gödel, 1964b.)Google Scholar
Gödel, K. 1964b. Postscriptum to the 1934 Princeton Lectures, 1964. (Postscriptum added June 3, 1964 to the reprinting of Gödel, 1964a, in Gödel, 1986: 369–371.)Google Scholar
Gödel, K. 1986. Collected Works, Volume I: Publications 1929–1936. Oxford: Clarendon Press. (Edited by Solomon Feferman et al.)Google Scholar
Gödel, K. 1990. Collected Works, Volume II: Publications 1938–1974. Oxford: Clarendon Press. (Edited by Solomon Feferman et al.)Google Scholar
Goldfarb, W. 2018a. Moore’s Notes and Wittgenstein’s Philosophy of Mathematics: The Case of Mathematical Induction. (In Stern, 2018, pp. 241–252.)Google Scholar
Goodstein, R. 1957a. Critical Notice of Remarks on the Foundations of Mathematics. Mind, LXVI, 271–286.Google Scholar
Gowers, T. 2002. Mathematics: A Very Short Introduction. Oxford: Oxford University Press.CrossRefGoogle Scholar
Gowers, T. 2017a. A Dialogue Concerning the Need for the Real Number System. (At www.dpmms.cam.ac.uk/wtg10/reals.html, accessed January 30, 2017.)Google Scholar
Gowers, T. 2017b. What Is So Wrong with Thinking of Real Numbers as Infinite Decimals? (At www.dpmms.cam.ac.uk/wtg10/decimals.html, accessed January 30, 2017.)Google Scholar
Haller, R., & Puhl, K. (eds.). 2002. Wittgenstein and the Future of Philosophy. A Reassessment after 50 Years. Vienna: öbv&hpt.Google Scholar
Hardy, G. H. 1941. A Course of Pure Mathematics (8th ed.). Cambridge: Cambridge University Press. (10th ed. [1952, with index] republished as Hardy [2008].)Google Scholar
Hardy, G. H. 2008. A Course of Pure Mathematics (10th [1952, with index] ed.). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Harrington, B., Shaw, D., & Beaney, M. (eds.). 2018. Aspects after Wittgenstein: Seeing-As and Novelty. New York: Routledge.CrossRefGoogle Scholar
Hellman, G., & Cook, R. T. 2018. Hilary Putnam on Logic and Mathematics (vol. 9). Cham: Springer International Publishing AG Springer.CrossRefGoogle Scholar
Herken, R. (ed.). 1988. The Universal Turing Machine: A Half-Century Survey. New York: Oxford University Press.Google Scholar
Hintikka, J. (ed.). 1995. From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics. Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
Hobson, E. W. 1927. The Theory of Functions of a Real Variable and the Theory of Fourier’s Series (vol. I). Cambridge, UK: Cambridge University Press. (Revised and enlarged 3rd ed.; 1st ed., 1907, 2nd ed., 1921.)Google Scholar
Hrbacek, K., & Jech, T. J. 1999. Introduction to Set Theory (3rd revised and expanded ed.). Boca Raton: CRC / Taylor and Francis.Google Scholar
Kennedy, J. 2017. Turing, Gödel and the “Bright Abyss.” (In Floyd and Bokulich, 2017, pp. 63–92.)Google Scholar
Kennedy, J. 2020. Gödel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Krebs, V., & Day, W. (eds.). 2010. Seeing Wittgenstein Anew: New Essays on Aspect Seeing. New York: Cambridge University Press.Google Scholar
Kripke, S. 1982. Wittgenstein on Rules and Private Language: An Elementary Exposition. Cambridge, MA: Harvard University Press.Google Scholar
Kuusela, O., & McGinn, M. (eds.). 2012. The Oxford Handbook of Wittgenstein. Oxford University Press.Google Scholar
Lillehammer, H., Mellor, D. H., & Mellor, D. H. 2005. Ramsey’s Legacy. Oxford: Oxford University Press.CrossRefGoogle Scholar
Maddy, P. 1997. Naturalism in Mathematics. Oxford, New York: Clarendon Press; Oxford University Press.Google Scholar
Maddy, P. 2007. Second Philosophy: A Naturalistic Method. New York: Oxford University Press.CrossRefGoogle Scholar
Maddy, P. 2011. Defending the Axioms: On the Philosophical Foundations of Set Theory. New York: Oxford University Press.CrossRefGoogle Scholar
Maddy, P. 2014. The Logical Must: Wittgenstein on Logic. New York: Oxford University Press.CrossRefGoogle Scholar
Makovec, D., & Shapiro, S. 2019. Friedrich Waismann: The Open Texture of Analytic Philosophy. Cham: Springer International Publishing AG Palgrave Macmillan.Google Scholar
Mancosu, P. 2008. The Philosophy of Mathematical Practice. New York: Oxford University Press.CrossRefGoogle Scholar
Marion, M. 1998. Wittgenstein, Finitism and the Foundations of Mathematics. New York: Oxford, Clarendon Press.Google Scholar
Marion, M. 2007. Interpreting Arithmetic: Russell on Applicability and Wittgenstein on Surveyability. Travaux de logique, 18, 167–184.Google Scholar
Marion, M. 2011. Wittgenstein on the Surveyability of Proofs. (In Kuusela and McGinn, 2012, pp. 138–161.)Google Scholar
Marion, M., & Okada, M. 2018. Wittgenstein, Goodstein and the Origin of the Uniqueness Rule. (In Stern, 2018, pp. 253–271.)Google Scholar
Martin, C. (ed.). 2018. Language, Form(s) of Life, and Logic: Investigations after Wittgenstein. Berlin: de Gruyter.Google Scholar
Menger, K. 1994. Reminiscences of the Vienna Circle and the Mathematical Colloquium (Vol. 20). Dordrecht: Kluwer Academic/Springer Science+Business Media.Google Scholar
Monk, R. 1990. Ludwig Wittgenstein: The Duty of Genius. New York/London: Free Press/Jonathan Cape.Google Scholar
Moyal-Sharrock, D. 2015. Wittgenstein on Forms of Life, Patterns of Life, and Ways of Living. Nordic Wittgenstein Review, Special Issue on Forms of Life, 21–42.Google Scholar
Mühlhölzer, F. 2002. Wittgenstein and Surprises in Mathematics. (In Haller and Puhl, 2002, pp. 306–315.)Google Scholar
Mühlhölzer, F. 2006. “A Mathematical Proof Must Be Surveyable”: What Wittgenstein Meant by This and What It Implies. Grazer Philosophische Studien, 71, 57–86.Google Scholar
Mühlhölzer, F. 2010. Braucht die Mathematik eine Grundlegung? Ein Kommentar des Teils III von Wittgensteins BEMERKUNGEN ÜBER DIE GRUNDLAGEN DER MATHEMATIK. Frankfurt am Main: Vittorio Klostermann. (English translation to appear in the Anthem Studies in Wittgenstein Series.)Google Scholar
Mühlhölzer, F. 2012. Wittgenstein and Metamathematics. (In Stekeler-Weithofer, 2012, pp. 103–128.)CrossRefGoogle Scholar
Mühlhölzer, F. 2014. How Arithmetic Is about Numbers: A Wittgensteinian Perspective. Grazer Philosophische Studien, 89, 39–59.Google Scholar
Narboux, J.-P. 2014. Showing, the Medium Voice, and the Unity of the Tractatus. Philosophical Topics, 42(2).Google Scholar
Olszewski, A., Wolenski, J., & Janusz, R., (eds.). 2006. Church’s Thesis After 70 Years. Frankfurt/Paris/Ebikon/Lancaster/New Brunswick: Ontos Verlag.CrossRefGoogle Scholar
Paris, J., & Harrington, L. 1977. A Mathematical Incompleteness in Peano Arithmetic. (In Barwise, 1977, pp. 1133–1142.)Google Scholar
Parsons, C. 1983. Frege’s Theory of Number. In Mathematics in philosophy (p. 150–175). Ithaca, NY: Cornell University Press.Google Scholar
Potter, M. 2000. Reason’s Nearest Kin: Philosophies of Arithmetic from Kant to Carnap. New York: Oxford University Press.Google Scholar
Potter, M. 2005. Ramsey’s Transcendental Argument. (In Lillehammer, Mellor, and Mellor, 2005, pp. 71–82.)Google Scholar
Putnam, H. 1962. It Ain’t Necessarily So. The Journal of Philosophy, LIX, 658–671.CrossRefGoogle Scholar
Putnam, H. 1999. The Threefold Cord: Mind, Body and World (vol. 112). Columbia University Press.Google Scholar
Putnam, H. 2012. Philosophy in an Age of Science: Physics, Mathematics, and Skepticism. Cambridge, MA: Harvard University Press. (Edited by Mario De Caro and David Macarthur.)Google Scholar
Putnam, H. 2016. Naturalism, Realism, and Normativity. Cambridge, MA: Harvard University Press. (Edited by Mario De Caro. )CrossRefGoogle Scholar
Quinon, P. 2020. Implicit and explicit examples of the phenomenon of deviant encodings. Studies in Logic, Grammar and Rhetoric, 63(76), 53–67.CrossRefGoogle Scholar
Ramsey, F. P. 1923. Review of Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Mind, 32(128), 465–478.CrossRefGoogle Scholar
Ramsey, F. P. 1926. The Foundations of Mathematics. Proceedings of the London Mathematical Society, Series 2, 25(1), 338–384.CrossRefGoogle Scholar
Ramsey, F. P. 1930. On a Problem of Formal Logic. Proceedings of the London Mathematical Society, Series 2, 2–3 (1), 264–286.Google Scholar
Reck, E. H. (ed.). 2013. Logic, Philosophy of Mathematics, and Their History: Essays in Honor of W. W. Tait (Vol. 36). Milton Keynes, UK: College Publications.Google Scholar
Ricketts, T. 2014. Analysis, Independence, Simplicity and the General Sentence-Form. Philosophical Topics, 42(2), 263–288.Google Scholar
Rodych, V. 2002. Wittgenstein on Gödel: The Newly Published Remarks. Erkenntnis, 56(3), 379–397.Google Scholar
Rogers, B., & Wehmeier, K. 2012. Tractarian First-Order Logic: Identity and the N-Operator. Review of Symbolic Logic, 5(4), 538–573.CrossRefGoogle Scholar
Russell, B. 1903. The Principles of Mathematics. Cambridge/New York: Cambridge University Press/Norton. (1st ed., 1903; 2nd ed., 1938.)Google Scholar
Russell, B. 1914. Our Knowledge of the External World as a Field for Scientific Method in Philosophy. Chicago: Open Court Publishing Co.Google Scholar
Russell, B. 1920. Introduction to Mathematical Philosophy (2nd ed.). London/New York: Allen and Unwin/Macmillan. (1st ed., 1919.)Google Scholar
Russell, B. 1936. The Limits of Empiricism. Proceedings of the Aristotelian Society, 36, 131–150.Google Scholar
Shanker, S. 1987. Wittgenstein and the Turning-Point in the Philosophy of Mathematics. Albany, NY: State University of New York Press.Google Scholar
Shapiro, S. 1982. Acceptable Notation. Notre Dame Journal of Formal Logic, 23(1), 14–20.Google Scholar
Shapiro, S. (ed.). 2005. The Oxford Handbook to the Philosophy of Logic and Mathematics. New York: Oxford University Press.CrossRefGoogle Scholar
Shapiro, S. 2008. Identity, Indiscernibiity, and ante rem Structuralism: the Tale of i and −i. Philosophia Mathematica, 16, 285–309.Google Scholar
Shapiro, S. 2012. An “i” for an “i”: Singular Terms, Uniqueness and Reference. The Review of Symbolic Logic, 5, 380–415.Google Scholar
Shapiro, S. 2018. Changing the Subject: Quine, Putnam and Waismann on Meaning-Change, Logic, and Analyticity. (In Hellman and Cook, 2018, pp. 115–126.)CrossRefGoogle Scholar
Sheffer, H. M. 1913. A set of five independent postulates for Boolean Algebras, with Application to Logical Constants. Transactions of the American Mathematical Society, 14 (4),481–488.Google Scholar
Shieh, S. 2019. Necessity Lost. Modality and Logic in Early Analytic Philosophy. Volume I (1st ed.). Oxford: Oxford University Press.CrossRefGoogle Scholar
Skolem, T. 1923. The foundations of elementary arithmetic established by means of the recursive mode of thought, without the use of apparent variables ranging over infinite domains. (In van Heijenoort, 1967, pp. 302–333)Google Scholar
Sluga, H., & Stern, D. (eds.). 2018. The Cambridge Companion to Wittgenstein (2nd rev. ed.). New York: Cambridge University Press.Google Scholar
Snyder, E., & Shapiro, S. 2019. Frege on the Real Numbers. (In Ebert and Rossberg, 2019, pp. 343–383.)CrossRefGoogle Scholar
Stekeler-Weithofer, P. (ed.). 2012. Wittgenstein: Zu Philosophie und Wissenschaft. Hamburg: Verlag Felix Meiner.Google Scholar
Stern, D. G. (ed.). 2018. Wittgenstein in the 1930s: Between the Tractatus and the Investigations. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Sullivan, P. 1995. Wittgenstein on The Foundations of Mathematics, June 1927. Theoria, 61, 105–142.Google Scholar
Sundholm, G. 1992. The General Form of the Operation in Wittgenstein’s Tractatus. Grazer Philosophische Studien, 42, 57–76.Google Scholar
Travis, C. 2006. Thought’s Footing: A Theme in Wittgenstein’s Philosophical Investigations. Oxford/New York: Oxford University Press.Google Scholar
Turing, A. 1936. On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, Series 2, 1936/7(42), 230–265. Corrections in Turing, 1937.Google Scholar
Turing, A. 1937. On computable numbers, with an application to the Entscheidungsproblem: A Correction. Proceedings of the London Mathematical Society, Series 2, 1937(43), 544–546.Google Scholar
Turing, A. 1942–44. The Reform of Mathematical Notation and Phraseology [1944-45]. (Unpublished manuscript. In Turing, 2013, pp. 245–249; online at www.turingarchive.org/search/, King’s College Online Archive.)Google Scholar
Turing, A. 1948. Intelligent Machinery. (Unpublished report for the National Physical Laboratory. In Turing, 2013, pp. 501–516.)Google Scholar
Turing, A. 2013. Alan Turing – HisWork and Impact. Amsterdam/Burlington,MA: Elsevier. (Edited by S. Barry Cooper and Jan van Leeuwen ; collected papers of Turing with commentary by experts.)Google Scholar
van Heijenoort, J. (ed.). 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press. (Reprinted 2002)Google Scholar
Van Heijenoort, J. 1967. Logic as Calculus and Logic as Language. Synthese, 17, 324–330.Google Scholar
Waismann, F. 1936a. Einführung in das mathematische Denken. Vienna: Springer. (English translation by Theodore J. Benac, Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics. New York: Harper, 1959; Dover reprint 2003, references to this edition.)Google Scholar
Waismann, F. 1976/1997. The Principles of Linguistic Philosophy (2nd ed.). London: Macmillan.Google Scholar
Weyl, H. 1918. Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis. Leipzig: Verlag von Veit und Comp. (English translation by S. Pollard and T. Bole, The Continuum, Dover Publications, New York: 1994. References are to this translation.)Google Scholar
Whitehead, A. N., & Russell, B. 1910. Principia Mathematica. Cambridge: University Press. (3 vols., 2nd ed., 1927.)Google Scholar
Wittgenstein, L. 1974. Philosophical Grammar. Oxford: Basil Blackwell. (Edited by Rush Rhees, English translation by Anthony Kenny.)Google Scholar
Wittgenstein, L., & Waismann, F. 2003. The Voices of Wittgenstein, the Vienna Circle: Original German Texts and English Translations. London: Routledge. (Edited by Gordon P. Baker.)Google Scholar
Wright, C. 1980. Wittgenstein on the Foundations of Mathematics. Harvard University Press.Google Scholar
Wright, C. 2001. Rails to Infinity: Essays on Themes from Wittgenstein’s Philosophical Investigations. Cambridge, MA: Harvard University Press.Google Scholar
- 12
- Cited by