This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

Readers beware! This book is other than it first seems. Ludwig Wittgenstein’s latest philosophical offering is unlike anything that we have had from him to date. Its preface warns that the Tractatus is no textbook. This is an extreme understatement; really it is a deep puzzle—one that must be handled with great care. As the first lines signal there has been a radical change in the author’s characteristic style. Gone are the ingenious, probing explorations of topics undertaken in his highly (...) fragmentary writings that have been Wittgenstein’s trademark to date. In his new book Wittgenstein abandons the interrogative that his signature vision of and approach to philosophy found in his many masterpieces—The Blue and Brown Books, The Philosophical Investigations, and On Certainty. His previous masterworks have prosecuted a continuous philosophical enterprise designed to expose, resist and reject certain deeply compelling pictures that attract, mesmerize and blind us when it comes to thinking .. (shrink)

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)

The idea that the function of language is its use is commonly ascribed to the Later Wittgenstein. In this paper, I argue that there is textual evidence already coming from the early manuscripts proving that Wittgenstein's philosophical development is culminating in the idea of function as use around 1929–30. I interpret a passage from Ms‐107 in order to show that Wittgenstein's practical turn has sources in different stages of his philosophical development, each of which is dominated by different ideas: the (...) idea of a picture theory; the idea of a phenomenological language and the idea of function as use. (shrink)

In the preface of the Philosophical Investigations, Wittgenstein says that his “most fruitful ideas” are due to the stimulus of Sraffa's criticism, but Sraffa is not mentioned anywhere else in the book. It remains a puzzle in the literature how and why Sraffa influenced Wittgenstein. This paper presents a solution to this puzzle. Sraffa's criticism led Wittgenstein away from the calculus conception of language of the Big Typescript (arguably, an adaptation of the calculus of the Tractatus), and towards the “anthropological (...) view,” which structures both the opening sections of the Philosophical Investigations and Wittgenstein's later philosophy of mathematics. (shrink)

The philosopher Ludwig Wittgenstein cites the Gestalt psychologist Wolfgang Koehler almost as often as he cites William James in his posthumously published writings on the philosophy of psychology. Yet, few treatments of the Wittgenstein–Koehler relation in the philosophical literature could be called sustained discussions. Moreover, most of them treat Koehler as a mere whipping boy for Wittgenstein, one more opportunity to criticize the practice of psychologists. This article emphasizes how much the two thinkers agreed, and the extent to which some (...) of Wittgenstein’s work not only agreed with but also has a logical structure parallel to some of Koehler’s text. Both thinkers hold that the theoretician should strive to recognize and resist the impulse to step in and purify, distill, streamline, or exclude phenomena: common, everyday experience for Koehler and common, everyday uses of words for Wittgenstein. They both aim to counteract the tendency to discount and disparage what is ordinary and common. (shrink)

Wittgenstein’s comparison of philosophical methods to therapies has been interpreted in highly different ways. I identify the illness, the patient, the therapist and the ideal of health in Wittgenstein’s philosophical methods and answer four closely related questions concerning them that have often been wrongly answered by commentators. The results of this paper are, first, some answers to crucial questions: philosophers are not literally ill, patients of philosophical therapies are not always philosophers, not all philosophers qualify as therapists, the therapies are (...) not necessarily to be thought of as psychological therapies and the ideal of health does not consist in the end of philosophy. Second, the paper shows that the comparison has had a misleading effect, because properties of therapies have been illegitimately projected onto the philosophical methods advanced by Wittgenstein. (shrink)

The article contains a critical analysis of Wittgenstein’s theory of logical symbolism. According to an influential interpretation, Wittgenstein presented in the Tractatus a new method of solving paradoxes. This method seems a simple and effective alternative to Russell’s type theory. Wittgenstein’s theory of logical symbolism is based on the requirement of clear notation and the context principle: the type of a symbol only “shows” itself in the way we use the signs of our language. The function sign φ(φx) does not (...) express any paradox, because the syntactic rules for its use, written in clear notation, should “show” us that φ(φx) = ψ(φx). Many researchers (Davant, Ishiguro, Mounce, Ruffino, Friedlander, Jolley, Livingston, Ladov, et al.) follow this interpretation. However, the difficulty of such a view on Wittgenstein’s theory of logical symbolism is that there hides the fallacy of petitio principii. Indeed, in examples of a functional sign of the form φ(φx), we are interested not only in the question of whether the functions φ are different symbols, but also in how this functional sign φ(φx) itself excludes the symbolization of the same object by different ways. This interpretation is contrasted with the idea that Wittgenstein’s theory of logical symbolism is in fact a modified analogue of Russell’s simple theory of types. The reciprocality principle becomes the core of Wittgenstein’s theory: the combinatorial potential of the “prototype” of a functional sign is identical to the combinatorial potential of the “prototype” of an argument. According to Wittgenstein, only describing the combinatorial potential of linguistic expressions (symbols) can vanish the illusion of paradoxes. The function cannot be its argument, because the function sign φ(φx) already contains the “prototype” of its argument, “showing” us that φ(φx) = φx. The correctness of this interpretation does not exclude the possibility that the differences between Russell and Wittgenstein are in fact nothing more than façon de parler. (shrink)