This paper portrays the later Wittgenstein’s conception of contradictions and his therapeutic approach to them. I will focus on and give relevance to the Lectures on the Foundations of Mathematics, plus the Remarks on the Foundations of Mathematics. First, I will explain why Wittgenstein’s attitude towards contradictions is rooted in: a rejection of the debate about realism and anti-realism in mathematics; and Wittgenstein’s endorsement of logical pluralism. Then, I will explain Wittgenstein’s therapeutic approach towards contradictions, and why it means that (...) a contradiction is not a problem for logic and mathematics. Rather, contradictions are problematic when we do not know what to infer from them. Once a meaning is established through a new rule of inference, the contradiction becomes a usable expression like many others in our inferential apparatus. Thus, the apparent problem is dissolved. Finally, I will take three examples of dissolved contradictions from Wittgenstein to clarify further his notion. I will conclude considering why his position on contradictions led him to clash with Alan Turing, and whether the latter was convinced by the Wittgensteinian proposal. (shrink)

In the scholarship on Wittgenstein’s later philosophy of mathematics, the dominant interpretation is a theoretical one that ascribes to Wittgenstein some type of ‘ism’ such as radical verificationism or anti-realism. Essentially, he is supposed to provide a positive account of our mathematical practice based on some basic assertions. However, I claim that he should not be read in terms of any ‘ism’ but instead should be read as examining philosophical pictures in the sense of unclear conceptions. The contrast here is (...) that basic assertions that frame philosophical ‘isms’ are propositional such that they are subject to normal argumentative evaluation, while pictures in Wittgenstein’s sense are non-propositional—they lack a clear truth condition. They, therefore, need clarification rather than argumentation. In this paper, I provide a detailed analysis of Wittgenstein’s treatment of philosophical pictures with special focus on his argument on contradiction. I begin by explaining the problem with this trend of theoretical interpretation, taking Steve Gerrard’s otherwise excellent interpretation as a representative example and pointing out why it is problematic. Next, I will argue that those problems do not arise if we take Wittgenstein’s task as the clarification of philosophical pictures. I do this, first, by explaining Wittgenstein’s method using his argument concerning the Augustinian Picture in Philosophical Investigations and then pointing out that the same method can be identified in the crucial arguments in his philosophy of mathematics. Finally, in order to connect my interpretation with the current scholarship, I will explain the relation of my interpretation with those of New Wittgensteinian scholars. (shrink)

Proposition 5.122 of Wittgenstein’s Tractatus has been the source of much puzzlement among interpreters, so much so that no fully satisfactory account is yet available. This is unfortunate, if only because the containment account of logical consequence has a venerable tradition behind it. Pasquale Frascolla’s interpretation of proposition 5.122 is based on a valid argument and one true premise. However, the argument explains sense containment only in an indirect way, leaving some crucial questions unanswered. Besides, Frascolla does not address the (...) issue of how to make sense containment notationally perspicuous, an essential theme in Wittgenstein’s reflections. In this paper, we elaborate on Frascolla’s account by looking at the issue through the Tractarian notion of logical space. Our analysis shows that, for containment to be fully appreciated, one should adopt a negative perspective on the notion of sense, in line with the exclusionary theory of conceptual content, as labelled by Ian Rumfitt. Besides this, we introduce and discuss two methods—one envisaged by Wittgenstein himself—for making sense containment notationally perspicuous. (shrink)

Mi objetivo es discutir las principales dificultades que Frank P. Ramsey encontró en Principia Mathematica y la solución que, vía el Tractatus Logico-Philosophicus, propuso al respecto. Sostengo que las principales dificultades que Ramsey encontró en Principia Mathematica están, todas, relacionadas con que Russell y Whitehead desatendieron la forma lógica de las proposiciones matemáticas, las cuales, según Ramsey, deben ser tautológicas.

This paper systematically outlines Wittgenstein's ab-notation. The purpose of this notation is to provide a proof procedure in which ordinary logical formulas are converted into ideal symbols that identify the logical properties of the initial formulas. The general ideas underlying this procedure are in opposition to a traditional conception of axiomatic proof and are related to Peirce's iconic logic. Based on Wittgenstein's scanty remarks concerning his ab-notation, which almost all apply to propositional logic, this paper explains how to extend his (...) method to a subset of first-order formulas, namely, formulas that do not contain dyadic sentential connectives within the scope of any quantifier. (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)

Some interpreters have ascribed to Wittgenstein the view that mathematical statements must have an application to extra-mathematical reality in order to have use and so any statements lacking extra-mathematical applicability are not meaningful (and hence not bona fide mathematical statements). Pure mathematics is then a mere signgame of questionable objectivity, undeserving of the name mathematics. These readings bring to light that, on Wittgenstein’s offered picture of mathematical statements as rules of description, it can be difficult to see the role of (...) mathematical statements which relate to concepts that are not employed in empirical propositions e.g. set-theoretic concepts. I will argue that Wittgenstein’s picture is more flexible than might at first be thought and that Wittgenstein sees such statements as serving purposes not directly related to empirical description. Whilst this might make such systems fringe cases of mathematics, it does not bring their legitimacy as mathematical systems into question. (shrink)

The analysis of logical predication has long philosophical tradition in which one of the central subjects of study is the analysis of the logical form of the proposition. We contemporaneously can say that the way more well-finished of logic predication is propositional function. Historically, the propositional function arises as a logical analysis of the proposition scheme resulting from the convergence of mathematics and logic between the XIX and XX centuries. Two of the main responsible for this convergence were Gottlob Frege (...) (1848 - 1925) and Bertrand Russell (1872-1970). Influenced by Frege's ideas and direct heir of Russell, having been a disciple of this in Cambridge, Ludwig Wittgenstein (1889 - 1951) has become, in view of the originality of his thought, one of the most discussed and commented thinkers of the twentieth century, especially with the publication of his work entitled "Tractatus Logico-Philosophicus" (1921). It's in this work that Wittgenstein more considers himself debtor from Frege and Russell, having left explicit mention to them. Then the question that guides the development of this thesis is: what is the meaning of the propositional function of Frege and Russell in the Wittgenstein's Tractatus (1921)? Our goal is, in this sense, to investigate the meaning of the concept of propositional function in the Tractatus (1921). Our starting point is to understand the meaning of the propositional function in Frege and Russell to understand its meaning in the Tractatus (1921). We will focus our analysis on what Wittgenstein calls "propositional variable" (Satzvariable), term that best approximates, in our view, of the propositional function of Frege and Russell. In this sense, our interpretative question was formulated as: what is the meaning of the concept of propositional variable in the Tractatus (1921)? We defend the thesis that the role of the propositional function in Frege and Russell corresponds to the role of the propositional variable in Wittgenstein. (shrink)

In the penultimate remark of the Tractatus Logico-Philosophicus Wittgenstein declares that anyone who understands him judges the book to be nonsense. The immediate reaction to this paradoxical statement is to reject the insights of the book that this assessment is based on; that is, to reject the book’s theories of logic and language. Commentators have tried to save the book’s fundamental philosophical ideas by blocking this immediate response. In this thesis I characterise and explore different attempts to do so. I (...) discuss attempts of Russell, Carnap, Max Black, Malcolm, Hacker’s Ineffability interpretation and Conant’s Therapeutic interpretation. I argue that the Therapeutic reading is the most promising attempt in its main ideas. Nonetheless, current versions of the Therapeutic readings do not seem successful. I borrow ideas from Grice’s pragmatic theory of conversation and Davidson’s account of metaphor to explain how the book is to be read therapeutically. I argue that the book is a long conversation between Wittgenstein and his audience which eventually turns out to be a pointless series of remarks. The book, however, works metaphorically in such a way that it affects its readers and helps them to divest themselves of the inclination to do philosophy. (shrink)

Departing from and closing with reflections on issues regarding teaching practices of philosophy of mathematics, I propose a comparison between the main features of the Leibnizian notion of symbolic knowledge and some passages from the Tractatus on arithmetic. I argue that this reading allows (i) to shed a new light on the specificities of the Tractarian definition of number, compared to those of Frege and Russell; (ii) to highlight the understanding of the nature of mathematical knowledge as symbolic or formal (...) knowledge that Wittgenstein mobilizes in his book; (iii) to offer reasons for the claim that Wittgenstein can be considered the philosopher of mathematical practice avant la lettre. The paper ends with an overview, a return to the initial reflection on the connections between research and teaching, and a defense of the reading key used here in terms of its potential for the research in philosophy of mathematics. (shrink)