"Wittgenstein's Remarks on Gödel?"Abstract:"1) A survey of historical evidence currently available concerning Wittgenstein's attitude toward, and knowledge of, Gödel's first incompleteness theorem, including his discussions with Turing, Watson and others in 1937-1939, and later testimony of Goodstein and Kreisel, as well as some recently discovered 1941 remarks of Wittgenstein's on w-inconsistency and "self-reference"; 2) Discussion of the philosophical and historical importance of Wittgenstein's attitude toward Gödel's and other theorems in mathematical logic; 3) Replies to an instructive criticism of my early 1995 treatment of topics 1) and 2) by Mark Steiner, focussing on the importance of Tarski's semantical work, both for our understanding of Wittgenstein¹s remarks on Gödel, and our understanding of Gödel's theorem itself. 4) Comments on recent work with Putnam on the philosophical significance of Wittgenstein's remarks on Gödel."
"The Rule of the Mathematical: Wittgenstein's Later Discussions."
PhD Dissertation, Harvard University, 1990.
Abstract in Dissertation Abstracts International (June 1991), 51(12A):4146-A.
Abstract:"If we consider Wittgenstein's career as a whole, it appears that he wrote more on the philosophy of logic and mathematics than any other subject. Yet his writings on these subjects have exerted little influence. Indeed, the tide of response to Remarks on the Foundations of Mathematics, which contains the bulk of his latest views of mathematics, has been for the most part overwhelmingly negative. Given his later emphasis on the context-bound character of language, mathematics and logic--where language apparently operates in an maximally precise, clear and general way--represent two of the most difficult cases for Wittgenstein to confront. My thesis aims to defend Wittgenstein from the charges of benighted arrogance traditionally levelled against him. I argue that Wittgenstein's later discussions of mathematics form a central part of a larger philosophical project, internally related to Philosophical Investigations and shaping in specific ways Wittgenstein's reaction to both scepticism and accounts of the nature of logical and mathematical truth. I see Wittgenstein criticizing the unreflective use of mathematical tools in philosophy, not offering a competing philosophy of mathematics. Too few of Wittgenstein's readers (whether interpreters or critics) have been willing to offer detailed exegesis of his writing (especially in the case of Remarks on the Foundations of Mathematics). This has made it difficult to understand how the aphoristic (and, to many, troubling) style of his late writing bears on the philosophical problems being discussed. On my view, the quality of Wittgenstein's writing is intrinsic to his later conception of the nature of logic, mathematics and philosophy. Line by line engagement with his texts is thus imperative in order to achieve an understanding of his philosophical objectives and criticisms. My dissertation offers as a paradigm of such reading a detailed exegesis and criticism of the opening five sections of Remarks on the Foundations of Mathematics. I briefly explore in some detail the origin and genesis of Wittgenstein's interest in rule-following, from its roots in the Tractatus's conception of logical syntax. This background illuminates Wittgenstein's later conception of logical truth and his criticisms of Frege's and Russell's arguments for logicism. To explore these criticisms, I focus on Wittgenstein' s discussions of the Frege-Russell definition of 'Number' in 'logical' terms. Placed in their appropriate philosophical and historical context, Wittgenstein's seemingly outrageous remarks about the nature of proof, mathematical logic and the foundations of mathematics do not simply betray his ignorance. Nor do they commit him to a revisionist attitude toward mathematical practice. Rather, they raise fundamental questions about the philosophical presuppositions lying behind attempts to bring particular mathematical results to bear in philosophy."
(with Burton Dreben.) "Tautology: How Not to Use a Word." Synthese (April 1991), 87(1):23-50.
"Wittgenstein on 2, 2, 2...: The Opening of Remarks on the Foundations of Mathematics."
Synthese (1991), 87(1):143-180.
Abstract:"I use a detailed, line by line reading of the first three sections of Wittgenstein's IRemarks on the Foundation of MathematicsD to illustrate the point and importance of his discussions of mathematics and logic. The manuscript basis of IRemarks on the Foundation of MathematicsD is set into historical context, and I emphasize the relation of Wittgenstein's dialectical style of writing to his aims and achievements in criticizing the traditional categories of a priori and necessary truth. I maintain that his remarks on the following of a rule for an elementary arithmetical series place our thinking about logical inference in a new and original light."
"Wittgenstein on 2,2,2....: On the Opening Remarks on the Foundations of Mathematics." In Jaakko Hintikka, ed., Wittgenstein in Florida. Dordrecht: Kluwer, 1991.
Review of Ted Cohen, Paul Guyer, and Hilary Putnam, eds. Pursuits of Reason: Essays in Honor of Stanley Cavell. Ethics (October 1994), 105(1):236-237.
"On Saying What You Really Want to Say: Wittgenstein, Gödel and the Trisection of the Angle." In Jaakko Hintikka, ed., From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, pp. 373-426. Synthese Library, 251. Dordrecht & Norwell, MA: Kluwer, 1995.
Review of Michael Nedo, ed., Ludwig Wittengenstein's Wiener Ausgabe. Band 1. Philosophische Bemerkungen. Journal of the History of Philosophy (July 1996), 34(3):475-477.
"Frege, Semantics and the Double-Definition Stroke." In Anat Biletzki and Anat Matar, eds., The Story of Analytic Philosophy: Plot and Heroes, pp. 141-166. Routledge Studies in Twentieth-Century Philosophy. London & New York: Routledge 1998.
"Frege, Semantics and the Double-Definition Stroke." (In French) In Mathieu Marion and Alain Voizard, eds., Frege, Logique et Philosophie. Collection Tradition semantique. Montreal: L'Harmattan, 1998.
"Heautonomy and the Critique of Sound Judgment: Kant on Reflective Judgment and Systematicity." In Herman Parret, ed., Kants Ästhetik/Kant's Aesthetics/L'Esthétique de Kant. Berlin & New York: de Gruyter, 1998.
"The Uncaptive Eye: Solipsism in Wittgenstein's Tractatus." In Leroy S. Rouner, ed., Loneliness, pp. 79-108. Boston University Studies in Philosophy and Religion, v. 19. Notre Dame, Ind.: University of Notre Dame Press, 1998.
Review of Avrum Stroll's Moore and Wittgenstein on Certainty. Philosophical Quarterly (July 1999), 49(196):412.
(with Hilary Putnam.) "A Note on Wittgenstein's 'Notorious Paragraph' about the Gödel Theorem." Journal of Philosophy (November 2000), 45(11):624-632.
"Wittgenstein, Mathematics and Philosophy." In Alice Crary and Ruper Read, eds., The New Wittgenstein, pp. 232-261. London & New York: Routledge, 2000.
Wittgenstein sur Gödel et les Mathématiques." In Élisabeth Rigal, ed., Mathématiques chez Wittgenstein. Trans. Élise Domenach. Paris: T.E.R. [Trans-Europ-repress], 2000.
Edited, with an Introduction. (with Sanford Shieh.) Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy. Oxford & New York: Oxford University Press, 2001.
"Number and Ascriptions of Number in the Tractatus." In Juliet Floyd and Sanford Shieh, eds., Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy, pp. 145-191. Oxford & New York: Oxford University Press, 2001.
"Prose versus Proof: Wittgenstein on Gödel, Tarski, and Truth" Philosophia Mathematica (2001), 9:901-928.
"Number and Ascriptions of Number in the Tractatus." In Erich H. Reck, ed., From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy, pp. 308-352. New York: Oxford University Press, 2002.
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