Anderson Nakano

In the Remarks on the Foundations of Mathematics, Wittgenstein discusses in detail the significance of the Gödelian “true but unprovable” proposition. Despite the negative reviews of early commentators, there have been recent attempts to provide a more positive assessment of these remarks. However, no consensus has emerged. Based on an analysis of Wittgenstein’s distinction between proof and prose and its connections with his views on indirect and inductive proofs, we approach this issue afresh, and provide a reading which not only brings to light these pivotal aspects, but shows how they are necessary for an accurate exegesis of the text and clarifies the relationship between Wittgenstein’s commentaries and Gödel’s incompleteness proof.

In the Remarks on the Foundations of Mathematics, Wittgenstein discusses in detail the significance of the Gödelian “true but unprovable” proposition. Despite the negative reviews of early commentators, there have been recent attempts to provide a more positive assessment of these remarks. However, no consensus has emerged. Based on an analysis of Wittgenstein’s distinction between proof and prose and its connections with his views on indirect and inductive proofs, we approach this issue afresh, and provide a reading which not only brings to light these pivotal aspects, but shows how they are necessary for an accurate exegesis of the text and clarifies the relationship between Wittgenstein’s commentaries and Gödel’s incompleteness proof.

An essential feature of Wittgenstein's early philosophy of logic is the conjecture of a positive solution to the question of whether logical truth in first-order logic is decidable (Entscheidungsproblem). It is often argued that the proofs of a negative solution to the Entscheidungsproblem presented in the 1930s refuted Wittgenstein's early conception of logic. However, we argue Wittgenstein's philosophy of logic does not share the principles assumed by the proof strategies of those undecidability proofs. Thus, the undecidability proofs do not refute Wittgenstein's philosophy of logic based on its own convictions. Therefore, we provide a purely logical refutation of Wittgenstein's early philosophy of logic that does not depend on assumptions that he does not accept. Wittgenstein claimed that the logical properties of first-order logic are decidable based on patterns in a suitable notation for first-order logic. We explain why this conviction is mistaken.

Turing proved the unsolvability of the decision problem for first-order logic (Entscheidungsproblem) in his famous paper On Computable Numbers, with an Application to the Entscheidungsproblem. From this proof it follows that attempts to specify a solution for the Entscheidungsproblem through pattern detection in automated theorem proving (ATP) must fail. Turing’s proof, however, merely predicts the non-existence of such solutions; it does not construct concrete examples that explain why specific attempts to solve the decision problem by pattern detection fail. ATP-search often runs in infinite loops in the case of unprovable, i.e. not refutable, formulas and one can ask why finite patterns of repeated inference steps cannot serve as criteria for unprovability. We answer this question by constructing pairs of formulas (ϕ,ϕ′) such that ϕ is provable (refutable) and ϕ′ is unprovable (satisfiable in an infinite domain) but all but the last proof step of the ATP-search for ϕ is a proper part of the endless ATP-search for ϕ′. We generate such pairs of formulas by mimicking computable sequences for a certain kind of universal Turing machine, namely, splitting Turing machines (STMs), via sequences of inference steps in ATP. In contrast to Turing’s and the textbooks’ method to formalize Turing machines, our method does not rely on further axioms and allows us to transfer the straightforward insight that the halting problem cannot be solved through pattern detection to the case of the Entscheidungsproblem. Our method is a constructive alternative to general undecidability proofs that explains why a scientific problem, namely the Entscheidungsproblem, is unsolvable in a specific way. This explanation provides a better understanding of the failure of pattern detection, which is of interest to: (i) the programmer, who is concerned with prospects and limits of pattern detection, (ii) the logician, who is interested in identifying logical properties by properties of an ideal notation, and (iii) the philosopher, who is interested in proof methods.

O objetivo deste trabalho é traçar alguns paralelos históricos que existem entre a oposição Darstellung/Beschreibung tal como ela é usada por Wittgenstein no período intermediário de seu pensamento e a mesma oposição em alguns matemáticos do século XIX, pertencentes a duas escolas distintas que divergiam quanto ao método de produção matemática. Em seguida, retornaremos aos escritos de Wittgenstein, buscando mostrar em que medida esta investigação histórica pode ser útil para iluminar alguns pontos de sua filosofia e levantar alguns problemas que surgiram neste contexto.

Em seu livro recente, Engelmann (2021) avançou uma leitura do Tractatus que, a seu ver, removeria as dificuldades presentes em outras leituras (em particular, nas leituras “metafísica” e “resoluta”). Neste artigo, ocupar-me-ei da crítica de Engelmann às leituras metafísicas do Tractatus. Tal crítica baseia-se na ideia de que essas leituras estão comprometidas com algumas necessidades de re, o que violaria a autonomia da lógica e a ideia Tractariana segundo a qual só há necessidade lógica. Analisarei o caso particular de uma suposta necessidade de re, a saber, a do número de objetos no mundo. Depois de descrever alguns aspectos positivos e alguns obstáculos à interpretação de Engelmann desse caso, sugerirei que Engelmann está inconscientemente atribuindo ao Tractatus uma concepção de lógica que Wittgenstein, por sua vez, criticamente atribuiu a Russell e a Ramsey e que, portanto, não pode ser a sua. Por fim, retornarei à motivação original para a interpretação de Engelmann, que é a de salvar a autonomia da lógica, mostrando de maneira breve como conciliar a autonomia da lógica com um número fixo de objetos.AbstractIn a recent book, Engelmann (2021) has put forward a reading of the Tractatus which he maintains would remove the difficulties presented in other readings, particularly the “metaphysical” and the “resolute” readings. In this paper, my concern will be with Engelmann’s criticism of the metaphysical readings of the Tractatus. This criticism is centered on the idea that such readings are committed to some de re necessities,which violates the autonomy of logic and the Tractarian idea that there is only logical necessity. I analyze the particular case of a supposed de re necessity, namely, the number of objects in the world. After describing some positive aspects and drawbacks of Engelmann’s interpretation of this case, I suggest that Engelmann is unconsciously attributing to the Tractatus a conception of logic that Wittgenstein in turncritically attributed to Russell and Ramsey; thus, this conception cannot be his own. Finally, I return to the original motivation for Engelmann’s interpretation, which is to save the autonomy of logic, and briefly show how to conciliate the autonomy of logic with a fixed number of objects.

RESUMO: O objetivo deste artigo é estruturar, em torno de uma única ideia fundamental, a saber, a de que não há atalhos pela lógica, os diversos comentários de Wittgenstein à época das conversas com Waismann e o círculo de Viena sobre o problema da consistência da aritmética. Observações notórias sobre as noções de consistência, trivialidade e negação na matemática são consideradas do ponto de vista dessa sistematização. Analisa-se também em que medida esses comentários correspondem a um desenvolvimento crítico do pensamento do autor.ABSTRACT: This work aims at structuring, around a single fundamental idea, namely, that there are no shortcuts through logic, the various comments by Wittgenstein at the time of the conversations with Waismann and the Vienna circle on the problem of the consistency of arithmetic. Notable remarks on the notions of consistency, triviality and negation in mathematics are considered from the point of view of this systematization. It is also analyzed to what extent these comments correspond to a critical development of the author's thinking.

In her book Deviant Logic, Susan Haack argued for a “pragmatist” conception of logic. This conception holds that, logic is a theory on a par with other scientific theories, differing only from such theories by its degree of generality and the choice of a particular logic is to be made based on pragmatist principles, namely, economy, coherence, and simplicity. This view was contrasted, in this book, with an “absolutist” view of logic, according to which logical laws are necessary and immune to revision. Two decades later, however, Haack acknowledged, in the Introduction to an enlarged version of the same book, that she would not approach the question of the revisability of logic in the same way she did earlier. What was missing in her first book was a distinction between the question of the necessity of the laws of logic and the question of our fallibility in recognizing which are the true laws of logic and what structures are essential to representation. She also acknowledged that this change was mainly influenced by Peirce, with whose work she had “only the most superficial acquaintance twenty years ago”. In this context, this paper has two aims: to show that, in Philosophy of Logics, we can find elements that reveal a tension between her early “pragmatic” views and her changing views on the nature of logic; to present some hypotheses about the role Peirce may have had in this change.

In a seminal essay, Hidé Ishiguro argued that names in the Tractatus are ‘like dummy names’ and that a simple object is to be conceived of as ‘an instantiation of an irreducible predicate’. In this paper, I argue that Ishiguro’s view is incompatible with other claims made in the Tractatus and should be abandoned for this reason. To this end, I adopt a two-step strategy. First, I show that Ishiguro’s view implies the adoption of an anti-haecceitistic position. Then I show that the Tractatus allows for the possibility of there being haecceitistic differences between two complete descriptions of the world. Since anti-haecceitism will be characterized as the doctrine that haecceitistic differences are meaningless, this provides a refutation of Ishiguro’s view. In the final section, I make a few remarks on the relation between anti-haecceitism and the context principle.

In the Foundations of Mathematics, Ramsey attempted to amend Principia Mathematica’s logicism to meet serious objections raised against it. While Ramsey’s paper is well known, some questions concerning Ramsey’s motivations to write it and its reception still remain. This paper considers these questions afresh. First, an account is provided for why Ramsey decided to work on his paper instead of simply accepting Wittgenstein’s account of mathematics as presented in the Tractatus. Secondly, evidence is given supporting that Wittgenstein was not moved by Ramsey’s objection against the Tractarian account of arithmetic, and a suggestion is made to explain why Wittgenstein reconsidered Ramsey’s account in the early thirties on several occasions. Finally, a reading is formulated to understand the basis on which Wittgenstein argues against Ramsey’s definition of identity in his 1927 letter to Ramsey.

Since the publication of the Remarks on the Foundations of Mathematics, Wittgenstein’s interpreters have endeavored to reconcile his general constructivist/anti-realist attitude towards mathematics with his confessed anti-revisionary philosophy. In this article, the author revisits the issue and presents a solution. The basic idea consists in exploring the fact that the so-called “non-constructive results” could be interpreted so that they do not appear non-constructive at all. The author substantiates this solution by showing how the translation of mathematical results, given by the possibility of translation between logics, can be seen as a tool for partially implementing the solution Wittgenstein had in mind.

ABSTRACT In a recent essay, Sören Stenlund tries to align Wittgenstein’s approach to the foundations and nature of mathematics with the tradition of symbolic mathematics. The characterization of symbolic mathematics made by Stenlund, according to which mathematics is logically separated from its external applications, brings it closer to the formalist position. This raises naturally the question whether Wittgenstein holds a formalist position in philosophy of mathematics. The aim of this paper is to give a negative answer to this question, defending the view that Wittgenstein always thought that there is no logical separation between mathematics and its applications. I will focus on Wittgenstein’s remarks about arithmetic during his middle period, because it is in this period that a formalist reading of his writings is most tempting. I will show how his idea of autonomy of arithmetic is not to be compared with the formalist idea of autonomy, according to which a calculus is “cut off” from its applications. The autonomy of arithmetic, according to Wittgenstein, guarantees its own applicability, thus providing its own raison d’être. RESUMO Em um recente artigo, Sören Stenlund procura alinhar a abordagem de Wittgenstein em relação aos fundamentos e à natureza da matemática com a tradição da matemática simbólica. A caracterização da matemática simbólica feita por Stenlund, de acordo com a qual a matemática é logicamente separada de suas aplicações externas, a aproxima da posição formalista. Isto naturalmente levanta a questão de se Wittgenstein defende uma posição formalista em filosofia da matemática. O objetivo deste artigo é dar uma resposta negativa a esta questão, ao defender que Wittgenstein sempre pensou não haver separação lógica entre a matemática e suas aplicações. Atenção especial será dada às observações de Wittgenstein sobre a aritmética pertencentes ao seu período intermediário, pois é neste período que uma leitura formalista de seus escritos é mais sedutora. Mostrarei como sua ideia de autonomia da aritmética não deve ser comparada à ideia formalista de autonomia, segundo a qual um cálculo é “cortado” de suas aplicações. A autonomia da aritmética, para Wittgenstein, garante ela própria sua aplicabilidade, provendo, assim, sua própria raison d’être.

It is often held that Wittgenstein had to introduce numbers in elementary propositions due to problems related to the so-called colour-exclusion problem. I argue in this paper that he had other reasons for introducing them, reasons that arise from an investigation of the continuity of visual space and what Wittgenstein refers to as ‘intensional infinity’. In addition, I argue that the introduction of numbers by this route was prior to introducing them _via_ the colour-exclusion problem. To conclude, I discuss two problems that Wittgenstein faced in the writings before _Some Remarks on Logical Form _, problems that are independent of the colour-exclusion problem but dependent on the introduction of numbers in elementary propositions.