Juliet Floyd

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’.

An overview of essays in this volume, with an emphasis on the philosophical legacy of Turing’s work, specifically the ways in which it bridges not only the gap between the sciences and the humanities, but also foundational and practical aspects of science and everyday life. Three sections of the volume are outlined, framing the overarching structure of Turing’s intellectual development: (i) Turing on the foundations of mathematics, incompleteness, the limits of analysis; (ii) Turing’s Universal Machine, implying the ubiquity of computational processes in our world, exemplified by applications in the early history of voice encryption, the history of computer music, the frontiers of computation, and the topic of emergence; (iii) Turing’s work on machines and mind, including his famed “Turing test” as a societal mechanism, the nature of perception as cognition, his views on freedom of the will and the integration of human and machine intelligence, and the developing idea of social algorithms.

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 ” (1936/1937) and Turing’s dissertation (1939), but in his exchanges with Wittgenstein in 1939 and in two later papers, “The Reform of Mathematical Phraseology and Notation” (1944/1945) and “Solvable and Unsolvable Problems” (1954).

Wittgenstein’s explicitly interlocutory style of philosophizing and writing, along with his use of the notion of Lebensform, first appeared in 1936-1937. Here we give an account of why. Lebensform plays an important elucidatory role in Wittgenstein’s later conception of philosophy, and is distinguished from the notion of Lebenswelt familiar in phenomenology. In utilizing the notion of “form”, rather than “world”, Wittgenstein indicates his preoccupation with the question, “What is the nature of the logical?” It is argued that the analysis of logic contained in Turing’s “On computable numbers, with an application to the Entscheidungsproblem” (1936/7) stimulated Wittgenstein’s writing of Philosophical Investigations, leading him to deepen the notion of a “languagegame”, to eliminate the idea of Kultur as foundational, and to focus on the general idea of rule-following. Turing’s paper, in turn, was indebted to Wittgenstein’s conception of philosophical method, especially the idea of comparing ordinary human calculative behavior with words to the mechanical workings of a calculus. Wittgenstein’s mature philosophical method, expressed in the Investigation’s multilogue shifts in voice, is, we argue, both logically and philosophically necessitated. And his conception of Lebensformen makes an important and novel intervention in philosophy that is relevant for our times.

This article is a survey of Wittgenstein’s writings on logic and mathematics; an analytical bibliography of contemporary articles on rule-following, social constructivism, Wittgenstein, Gödel, and constructivism is appended. Various historical accounts of the nature of mathematical knowledge have 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 Ramsey took extensionalism to result. Wittgenstein’s interest in rule-following emerged through his reactions to these attempted appropriations.

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

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.

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.

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.

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

This book presents a wide range of thinking about how the discipline of philosophy has engaged and might in the future engage with the profound questions raised by rapidly shifting methods of communication. Although social media and telecommunications have dramatically altered the daily lives of people, and no technology has enjoyed the same rapid success as that of the mobile phone, the philosophical underpinnings and consequences of these changes have been insufficiently explored. Clustered in six sections, this book addresses major questions of interest to the thoughtful professional. Section I considers ontology: How should we consider emerging media and what are the categories through which we can organize and understand this new realm of social reality. Section II investigates human nature and its relationship to the social transformations that are occurring through new media. It also considers and compares historical methods of communication such as paper. In Section III, questions of time, narrative voice, and emotions are assessed in relation to emerging media. General issues concerning emergence of meaning, philosophy of mind, artificial intelligence, and agency are the focus of Section IV. Symbols and speech acts—including the philosophical ramifications of the communication process itself as viewed through emerging technologies—represent the key concerns of Section V. Section VI probes various interpersonal, institutional, and political/power dimensions stemming from new systems of communication. A coda points forward to outstanding open questions, calling for further research to be done. The book represents a seminal contribution assembling original thoughts from both leading and rising scholars of the field.

BOOK REVIEWS 475 mean the traditional notion of arche, i.e., a present homogeneous origin/kingdom" . The significance of this Ursprung is that of the event of world pretheoretically yielding itself itself and things . Van Buren's basic contention is that Heidegger's demythologized question about being in the early Freiburg period is overtaken by remythologizing tendencies, from the existential-transcendental configuration of the question in Being and Time to various mythopoetic reconfigurations. The latter developments are at odds with Heidegger's earlier Freiburg elaboration of the concrete historicity and the personal, lived, anarchic character of the question of being. One intriguing and far-reaching implication of van Buren's interpretation is that Being and Time amounts to an Abweg, a mistaken by-way, from which Heidegger, despite the Kehre, never fully recovers. The choice of the meta- phor of an Abwegis revealing and, in this connection, perhaps not entirely felicitous. For, if Being and Time is truly "off track," then there is presumably a right way and it must be possible to demonstrate both that right way and the nature of Being and Time's deviation from it. The terms of this crucial argument have been expertly set by van Buren in his monumental..