Juliet Floyd

Tautology: How not to Use a Word

In 1923 C. I. Lewis wrote to F. J. Woodbridge, editor of The Journal of Philosophy: Have you looked at Wittgenstein’s new book yet? I am discouraged by Russell’s foolishness in writing the introduction to such nonsense. I fear it will be looked upon as what symbolic logic leads to. If so, it will be the death of the subject. 1

On Saying What You Really Want to Say: Wittgenstein, Gödel and the Trisection of the Angle

Wittgenstein’s remarks on the first incompleteness theorem1 have often been denounced, and mostly dismissed. Despite indirect historical evidence to the contrary,2 it is a commonplace that Wittgenstein rejected Godel’s proof because he did not, or even could not, understand it.3 Kreisel twice used the word “wild” when he reviewed Wittgenstein on Gode1.4 Dummett, in many respects an admirer of Wittgenstein’s philosophy, wrote that the remarks on Godel and on the notion of consistency are “of poor quality or contain definite errors”.5 Godel’s own comments were damning (see Section III below).

Wittgenstein’s Diagonal Argument: A Variation on Cantor and Turing

On 30 July 1947 Wittgenstein penned a series of remarks that have become well-known to those interested in his writings on mathematics. It begins with the remark “Turings ‘machines’: these machines are humans who calculate. And one might express what he says also in the form of games”. Though most of the extant literature interprets the remark as a criticism of Turing’s philosophy of mind (that is, a criticism of forms of computationalist or functionalist behaviorism, reductionism and/or mechanism often associated with Turing), its content applies directly to the foundations of mathematics. For immediately after mentioning Turing, Wittgenstein frames what he calls a “variant” of Cantor’s diagonal proof. We present and assess Wittgenstein’s variant, contending that it forms a distinctive form of proof, and an elaboration rather than a rejection of Turing or Cantor.

Recent Themes in the History of Early Analytic Philosophy

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).

Turing on “Common Sense”: Cambridge Resonances

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, Carnap, and Turing: Contrasting Notions of Analysis

Carnap never once mentions Turing in his published writings. From an historical point of view this is unsurprising. For the implications of Turing’s analysis of computability for the foundations of mathematics and physics, artificial intelligence, and the very architecture of science were to be developed and widely appreciated only after Carnap’s death. Perhaps more significant, once Carnap had developed the position articulated in The Logical Syntax of Language his principle of tolerance licensed a form of conciliatory pluralism about positions in the foundations of mathematics. Carnap’s pluralism construed debates over infinitary reasoning, impredicavity, logicism, and intuitionism as rationally tractable, but not through direct reasoning on behalf of truth claims. Rather, he proposed the development of formal axiomatizations of languages and pragmatic assessments of these. In subsequent work Carnap was not inclined to view these particular foundational debates as the primary arena for the articulation of his philosophy1. Instead, he broadened his conception of explication to account for the distinction between analytic and synthetic truth in all areas of science (Carus 2007a).

The Frege-Wittgenstein Correspondence: Interpretive Themes

Twenty-one cards and letters from Frege to Wittgenstein—the totality of the correspondence between them presently known to exist—were discovered in 1988, long after elaborate and far-reaching interpretive traditions had grown up around each philosopher.1 It is unlikely that these missives will of themselves radically reshape our understanding of either. But for historians of logic and analytic philosophy, as well as for anyone interested in German and Austrian intellectual history at the time of the First World War—and especially Wittgenstein’s and Frege’s places within it—these are significant and interesting documents.

Frege-Wittgenstein Correspondence

Lieber Herr Wittgenstein! Ich danke Ihnen bestens fur Ihren Kartengruss. Dass Sie als Kriegsfreiwilliger eingetreten sind, habe ich mit besonderer Befriedigung gelesen und bewundere es, dass Sie sich noch dabei der Wissenschaft widmen konnen. Moge es mir vergonnt sein, Sie nach dem Kriege gesund wiederzusehen, und die Unterredungen mit Ihnen weiterzufuhren. Gewiss werden wir uns dadurch zuletzt naher kommen und uns immer besser verstehen. Wir hatten hier 3 Leichtverwundete im Hause; Alfred musste dazu seine Spielstube hergeben. Sie erzahlten viel von ihren Kampfen in den Vogesen, ohne Hochachtung vor den Franzosen, denen sie sich bei gleicher Anzahl uberlegen fuhlten; aber sie hatten den Eindruck, meist gegen eine grosse Uberzahl gekampft zu haben. Ihnen alles Gute wunschend mit herzlichem Grusse

The Fluidity of Simplicity: Philosophy, Mathematics, Art

Simplicity is not simple. It wears many faces, and stands for a host of factors we use in argumentation. In the following narrative, the notion will not be analyzed. Instead, a parade of different observations about simplicity will be run through, drawing on philosophy, mathematics, and art. This run-through is designed to allow a certain ideal of simplicity to emerge. This ideal, as it happens, has been explicitly adhered to by a wide variety of modern thinkers: philosophers, mathematicians, and artists. Our aim is to characterize it.

Parikh and Wittgenstein

A survey of Parikh’s philosophical appropriations of Wittgensteinian themes, placed into historical context against the backdrop of Turing’s famous paper, “On computable numbers, with an application to the Entscheidungsproblem” (Turing in Proc Lond Math Soc 2(42): 230–265, 1936/1937) and its connections with Wittgenstein and the foundations of mathematics. Characterizing Parikh’s contributions to the interaction between logic and philosophy at its foundations, we argue that his work gives the lie to recent presentations of Wittgenstein’s so-called metaphilosophy (e.g., Horwich in Wittgenstein’s metaphilosophy. Oxford University Press, Oxford, 2012) as a kind of “dead end” quietism. From early work on the idea of a feasibility in arithmetic (Parikh in J Symb Log 36(3):494–508, 1971) and vagueness (Parikh in Logic, language and method. Reidel, Boston, pp 241–261, 1983) to his more recent program in social software (Parikh in Advances in modal logic, vol 2. CSLI Publications, Stanford, pp 381–400, 2001a), Parikh’s work encompasses and touches upon many foundational issues in epistemology, philosophy of logic, philosophy of language, and value theory. But it expresses a unified philosophical point of view. In his most recent work, questions about public and private languages, opportunity spaces, strategic voting, non-monotonic inference and knowledge in literature provide a remarkable series of suggestions about how to present issues of fundamental importance in theoretical computer science as serious philosophical issues.