This essay accounts for the notion of _Lebensform_ by assigning it a _logical _role in Wittgenstein’s later philosophy. Wittgenstein’s additions of the notion to his manuscripts of the _PI_ occurred during the initial drafting of the book 1936-7, after he abandoned his effort to revise _The Brown Book_. It is argued that this constituted a substantive step forward in his attitude toward the notion of simplicity as it figures within the notion of logical analysis. Next, a reconstruction of his later (...) remarks on _Lebensformen_ is offered which factors in his reading of Alan Turing’s “On computable numbers, with an application to the _Entscheidungsproblem_“, as well as his discussions with Turing 1937-1939. An interpretation of the five occurrences of _Lebensform_ in the PI is then given in terms of a logical “regression” to _Lebensform_ as a fundamental notion. This regression characterizes Wittgenstein’s mature answer to the question, “What is the nature of the logical?”. (shrink)
A survey of current evidence 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; 2) Discussion of the philosophical and historical importance of Wittgenstein's attitude toward Gödel's and other theorems in mathematical logic, contrasting this attitude with that of, e.g., Penrose; 3) Replies to an instructive criticism of my 1995 paper by Mark Steiner which assesses the importance of Tarski's semantical (...) work, both for our understanding of Wittgenstein's remarks on Gödel, and our understanding of Gödel's theorem itself. (shrink)
In his Tractatus Logico-Philosophicus, Wittgenstein conveyed the idea that ethics cannot be located in an object or self-standing subject matter of propositional discourse, true or false. At the same time, he took his work to have an eminently ethical purpose, and his attitude was not that of the emotivist. The trajectory of this conception of the normativity of philosophy as it developed in his subsequent thought is traced. It is explained that and how the notion of a ‘form of life’ (...) (Lebensform) emerged only in his later thought, in 1937, earmarking a significant step forward in his philosophical method. We argue that the concept of Lebensform represents a way of domesticating logic itself, the very idea of a claim or reason, supplementing the idea of a ‘language game’, which it deepens. Lebensform is contrasted with the phenomenologists’ Lebenswelt through a reading of the notions of ‘I’, ‘world’ and ‘self’ as they were treated in the Tractatus, The Blue and Brown Books and Philosophical Investigations. Finally, the notion of Lebensform is shown to have replaced the notion of culture (Kultur) in Philosophical Investigations. Wittgenstein’s spring 1937 ‘domestication’ of the nature of logic is shown to have been fully consonant with the idea that he was influenced by his reading Alan Turing’s 1936/1937 paper, ‘On computable numbers, with an application to the Entscheidungsproblem’. (shrink)
This collection of previously unpublished essays presents a new approach to the history of analytic philosophy--one that does not assume at the outset a general characterization of the distinguishing elements of the analytic tradition. Drawing together a venerable group of contributors, including John Rawls and Hilary Putnam, this volume explores the historical contexts in which analytic philosophers have worked, revealing multiple discontinuities and misunderstandings as well as a complex interaction between science and philosophical reflection.
I defend Putnam’s modal structuralist view of mathematics but reject his claims that Wittgenstein’s remarks on Dedekind, Cantor, and set theory are verificationist. Putnam’s “realistic realism” showcases the plasticity of our “fitting” words to the world. The applications of this—in philosophy of language, mind, logic, and philosophy of computation—are robust. I defend Wittgenstein’s nonextensionalist understanding of the real numbers, showing how it fits Putnam’s view. Nonextensionalism and extensionalism about the real numbers are mathematically, philosophically, and logically robust, but the two (...) perspectives are often confused with one another. I separate them, using Turing’s work as an example. (shrink)
Turing was a philosopher of logic and mathematics, as well as a mathematician. His work throughout his life owed much to the Cambridge milieu in which he was educated and to which he returned throughout his life. A rich and distinctive tradition discussing how the notion of “common sense” relates to the foundations of logic was being developed during Turing’s undergraduate days, most intensively by Wittgenstein, whose exchanges with Russell, Ramsey, Sraffa, Hardy, Littlewood and others formed part of the backdrop (...) which shaped Turing’s work. Beginning with a Moral Sciences Club talk in 1933, Turing developed an “anthropological” approach to the foundations of logic, influenced by Wittgenstein, in which “common sense” plays a foundational role. This may be seen not only in “On Computable Numbers” and Turing’s dissertation ), but in his exchanges with Wittgenstein in 1939 and in two later papers, “The Reform of Mathematical Phraseology and Notation” and “Solvable and Unsolvable Problems”. (shrink)
A survey of Wittgenstein's writings on logic and mathematics; an analytical bibliography of contemporary articles on rule-following, social constructivism, Wittgenstein, Godel, and constructivism is appended. Various historical accounts of the nature of mathematical knowledge glossed over the effects of linguistic expression on our understanding of its status and content. Initially Wittgenstein rejected Frege's and Russell's logicism, aiming to operationalize the notions of logical consequence, necessity and sense. Vienna positivists took this to place analysis of meaning at the heart of philosophy, (...) while Ramey took extensionalism to result. Wittgenstein's interest in rule-following emerged through his reactions to these attempted appropriations. (shrink)
A survey of the emergence of early analytic philosophy as a subfield of the history of philosophy. The importance of recent literature on Frege, Russell, and Wittgenstein is stressed, as is the widening interest in understanding the nineteenth-century scientific and Kantian backgrounds. In contrast to recent histories of early analytic philosophy by P.M.S. Hacker and Scott Soames, the importance of historical and philosophical work on the significance of formalization is highlighted, as are the contributions made by those focusing on systematic (...) treatments of individual philosophers, traditions, and periods in relation to contemporary issues (rule-following, neo-Fregeanism, contextualism, theory of meaning). (shrink)
1) A survey of current evidence 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; 2) Discussion of the philosophical and historical importance of Wittgenstein's attitude toward Gödel's and other theorems in mathematical logic, contrasting this attitude with that of, e.g. , Penrose; 3) Replies to an instructive criticism of my 1995 paper by Mark Steiner which assesses the importance of (...) Tarski's semantical work, both for our understanding of Wittgenstein's remarks on Gödel, and our understanding of Gödel's theorem itself. (shrink)
A German translation with 2017 postscript of Floyd, Juliet. 2012. "Wittgenstein's Diagonal Argument: A Variation on Cantor and Turing." In Epistemology versus Ontology, Logic, Epistemology: Essays in Honor of Per Martin-Löf, edited by P. Dybjer, S. Lindström, E. Palmgren and G. Sundholm, 25-44. Dordrecht: Springer Science+Business Media. An analysis of philosophical aspects of Turing's diagonal argument in his (136) "On computable numbers, with an application to the Entscheidungsproblem" in relation to Wittgenstein's writings on Turing and Cantor.
An investigation of the concept of “surveyability” as traced through the thought of Hilbert, Wittgenstein, and Turing. The communicability and reproducibility of proof, with certainty, are seen as earmarked by the “surveyability” of symbols, sequences, and structures of proof in all these thinkers. Hilbert initiated the idea within his metamathematics, Wittgenstein took up a kind of game formalism in the 1920s and early 1930s in response. Turing carried Hilbert’s conception of the “surveyability” of proof in metamathematics through into his analysis (...) of what a formal system (what a step in a computation) is in “On computable numbers, with an application to the Entscheidungsproblem” (1936). Wittgenstein’s 1939 investigations of the significance of surveyability to the concept of “proof “in Principia Mathematica were influenced, both by Turing’s remarkable everyday analysis of the Hilbertian idea, and by conversations with Turing. Although Turing does not use the word “surveyability” explicitly, it is clear that the Hilbertian idea plays a recurrent role in his work, refracted through his engagement with Wittgenstein’s idea of a “language-game”. This is evinced in some of his later writings, where the “reform” of mathematical notation for the sake of human surveyability (1944/45) may be seen to draw out the Hilbertian idea. For Turing, as for Wittgenstein, the need for “surveyability” earmarks the evolving culture of humans located in an evolving social and scientific world, just as it had for Hilbert. (shrink)
This Tractatus’s engagement with the issue of the nature of truth and falsity emerged from engagement with Russell. This engagement reverberated through the Vienna Circle and in particular affected Gödel. The Tractatus’s “elementary sentences” must be seen against the backdrop of Russell’s “multiple relation theory of judgment”, his theory of truth in Principia Mathematica, which Wittgenstein discussed at length with Russell in 1912–1913 and Gödel studied in 1929–1932. Russell’s approach was directed against both Idealism and William James’s pragmatist view of (...) truth. It aimed at a direct treatment of the distinction between truth and falsity in terms of particular, logically simple beliefs (judgments lacking in truth-functional and quantification complexity). Schlick rejected Russell’s view in favor of his more holistic correspondence theory, one which, however, tipped easily into pragmatism, conventionalism and verificationism. The Tractatus begins, rather, with Russell’s bottom-up approach truth, and then draws in two further ideas: (1) The need for a medium of representation and (2) The importance of modality (possibility and necessity) to logic. This approach was developed further in his later work, i.e., Philosophical Investigations.Aware of the Tractatus and Russell’s engagement with Wittgenstein on truth, Gödel continued to engage with Russell’s multiple relation theory of truth and Principia philosophically up through 1944. The parallel yet distinct engagements of Gödel and Wittgenstein with Russell on truth (and Vienna positivism) show that each regarded Russell’s view as requiring amendment. However, their philosophical differences with one another are not merely to be understood in terms of the dichotomy between conventionalism (the usual view of Wittgenstein) and Platonism (the usual view of Gödel). They must rather be seen to emerge from the original approach to truth we find in Russell. (shrink)
This volume presents an historical and philosophical revisiting of the foundational character of Turing's conceptual contributions and assesses the impact of the work of Alan Turing on the history and philosophy of science. Written by experts from a variety of disciplines, the book draws out the continuing significance of Turing's work. The centennial of Turing's birth in 2012 led to the highly celebrated "Alan Turing Year", which stimulated a world-wide cooperative, interdisciplinary revisiting of his life and work. Turing is widely (...) regarded as one of the most important scientists of the twentieth century: He is the father of artificial intelligence, resolver of Hilbert's famous Entscheidungsproblem, and a code breaker who helped solve the Enigma code. His work revolutionized the very architecture of science by way of the results he obtained in logic, probability and recursion theory, morphogenesis, the foundations of cognitive psychology, mathematics, and cryptography. Many of Turing's breakthroughs were stimulated by his deep reflections on fundamental philosophical issues. Hence it is fitting that there be a volume dedicated to the philosophical impact of his work. One important strand of Turing's work is his analysis of the concept of computability, which has unquestionably come to play a central conceptual role in nearly every branch of knowledge and engineering. (shrink)
Wittgenstein's treatment of number words and arithmetic in the Tractatus reflects central features of his early conception of philosophy. In rejecting Frege's and Russell's analyses of number, Wittgenstein rejects their respective conceptions of function, object, logical form, generality, sentence, and thought. He, thereby, surrenders their shared ideal of the clarity a Begriffsschrift could bring to philosophy. The development of early analytic philosophy thus evinces far less continuity than some readers of Wittgenstein, from Russell and the Vienna positivists to many contemporary (...) readers of the Tractatus, have supposed. (shrink)
Throughout his philosophy of mathematics, Steiner’s views bear affinities and contrasts with those of Wittgenstein. From his early insistence on an intelligible notion of mathematical “explanation”, to his remarks on the necessity of certain extensions of mathematical concepts, to his analysis of the ideas of logicism, “surveyability” and “applicability” in mathematics and his reading of Kripke on rule-following, Steiner probed fundamental issues in contemporary philosophy of mathematics. This essay responds to Steiner’s interpretations of Wittgenstein, focusing especially on his Humean reading (...) of rules as “hardened” regularities. (shrink)
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 have been willing to offer detailed exegesis of his writing . This has made it difficult to understand how the aphoristic 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. (shrink)
In 1969 Stanley Cavell's Must We Mean What We Say? revolutionized philosophy of ordinary language, aesthetics, ethics, tragedy, literature, music, art criticism, and modernism. This volume of new essays offers a multi-faceted exploration of Cavell's first and most important book, fifty years after its publication. The key subjects which animate Cavell's book are explored in detail: ordinary language, aesthetics, modernism, skepticism, forms of life, philosophy and literature, tragedy and the self, the questions of voice and audience, jazz and sound, Wittgenstein, (...) Austin, Beckett, Kierkegaard, Shakespeare. The essays make Cavell's complex style and sometimes difficult thought accessible to a new generation of students and scholars. They offer a way into Cavell's unique philosophical voice, conveying its seminal importance as an intellectual intervention in American thought and culture, and showing how its philosophical radicality remains of lasting significance for contemporary philosophy, American philosophy, literary studies, and cultural studies. (shrink)
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 grew from his Early and Middle philosophies, a dialectical path reconstructed here partly as a response to the limitative results of Gödel and Turing.
O presente artigo procede, em primeiro lugar, a um exame das evidências disponíveis referentes à atitude de Wittgenstein em relação ao, bem como conhecimento do, primeiro teorema da incompletude de Gödel, incluindo as suas discussões com Turing, Watson e outros em 1937-1939, e o testemunho posterior de Goodstein e Kreisel Em segundo lugar, o artigo discute a importância filosófica e histórica da atitude de Wittgenstein em relação ao teorema de Gödel e outros teoremas da lógica matemática, contrastando esta atitude com (...) a de, por exemplo, Penrose. Finalmente, a autora responde também a criticas instrutivas feitas por Mark Steiner a um artigo seu publicado em 1995, as quais estabelecem a importância do trabalho semântico de Tarski, quer para o nosso entendimento das observacoes de Wittgenstein. /// This paper presents, first, a survey of current evidence 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 Secondly, the article discusses the philosophical and historical importance of Wittgenstein's attitude toward Gödal's and other theorems in mathematical logic, contrasting this attitude with that of, e. g., Penrose. Finally, the author also replies to an instructive criticism of her 1995 paper by Mark Steiner which assesses the importance ofTarskih semantical work, both for our understanding of Wittgenstein's remarks on Godel, and our understanding ofGodeVs theorem itself. (shrink)