•  1356
    As dizimas periódicas na filosofia da matemática de Wittgenstein
    Philósophos - Revista de Filosofia 8 (2). 2003.
    O presente artigo tem como tema as extensas discussões de Wittgenstein sobre uma das formas mais simples e elementares de infinitude em matemática: as dízimas periódicas. Tentamos organizar os vários argumentos do autor em uma única exposição continuada. No final do artigo, introduzimos, ainda que de forma breve, o famoso argumento sobre “execução de regras” de Wittgenstein, bem como a idéia de interpretações nãostandard de processos infinitos
  •  664
    Semantical Mutation, Algorithms and Programs
    Dissertatio (S1): 44-76. 2015.
    This article offers an explanation of perhaps Wittgenstein’s strangest and least intuitive thesis – the semantical mutation thesis – according to which one can never answer a mathematical conjecture because the new proof alters the very meanings of the terms involved in the original question. Instead of basing our justification on the distinction between mere calculation and proofs of isolated propositions, characteristic of Wittgenstein’s intermediary period, we generalize it to include conject…Read more
  •  425
    À Maneira de Um Colar de Pérolas?
    Revista Portuguesa de Filosofia 73 (3-4): 1381-1404. 2017.
    This paper offers an overview of various alternative formulations for Analysis, the theory of Integral and Differential Calculus, and its diverging conceptions of the topological structure of the continuum. We pay particularly attention to Smooth Analysis, a proposal created by William Lawvere and Anders Kock based on Grothendieck’s work on a categorical algebraic geometry. The role of Heyting’s logic, common to all these alternatives is emphasized.
  •  424
    Wittgenstein Sobre as Provas Indutivas
    Dois Pontos 6 (2). 2009.
    This paper offers a reconstruction of Wittgenstein's discussion on inductive proofs. A "algebraic version" of these indirect proofs is offered and contrasted with the usual ones in which an infinite sequence of modus pones is projected.
  •  402
    Wittgenstein on Mathematical Identities
    Disputatio 4 (34): 755-805. 2012.
    This paper offers a new interpretation for Wittgenstein`s treatment of mathematical identities. As it is widely known, Wittgenstein`s mature philosophy of mathematics includes a general rejection of abstract objects. On the other hand, the traditional interpretation of mathematical identities involves precisely the idea of a single abstract object – usually a number –named by both sides of an equation.
  •  343
    Rule-following and Functions
    O Que Nos Faz Pensar 33 95-141. 2013.
    This paper presents a new reconstruction of Wittgenstein’s famous (and controversial) rule-following arguments. Two are the novel features offered by our reconstruction. In the first place, we propose a shift of the central focus of the discussion, from the general semantics and the philosophy of mind to the philosophy of mathematics and the rejection of the notion of a function. The second new feature is positive: we argue that Wittgenstein offers us a new alternative notion of a rule (to repla…Read more
  •  331
    Considerações sobre a Noção Construtiva de Verdade
    with Luiz Carlos Pereira
    O Que Nos Faz Pensar 17 107-123. 2003.
    This paper deals with the recent Swedish proposals of a Intuitionistic notion of Truth, by Dag Prawitz and Per Martin-Löf.
  •  320
    Wittgenstein E a medida da circunferência
    Philósophos - Revista de Filosofia 12 (2). 2007.
    Wittgenstein’s philosophy of mathematics involves two highly controversial theses: the idea that mathematical propositions are not about (abstract) objects and the idea that no mathematical conjecture is ever answered as such, because the advent of the proof always determines a semantical shift of the meanings of the terms involved in the conjecture. The present article offers a reconstruction of Wittgenstein’s arguments supporting these theses within a very restricted setting: Archimedes’ disco…Read more
  •  291
    Composicionalismo semântico, predicação E o automorfismo de Quine
    Philósophos - Revista de Filosofia 10 (2). 2005.
    Este artigo oferece uma nova reconstrução para os argumentos do famoso segundo capítulo de Word and Object de Quine e sua idéia da Tradução Radical . De acordo com essa abordagem, o maior alvo de Quine é a noção de composicionalidade como sendo o elemento fundamental para qualquer teoria do significado. Em poucas palavras, não poderia haver nenhuma “teoria do significado”, para Quine, simplesmente porque a noção de composicionalidade deveria ser rejeitada como a concepção central da semântica. A…Read more
  •  263
    Singularité et perception visuelle1
    Philosophiques 39 (1): 75-100. 2012.
    Cet article poursuit un double but : d’une part, situer dans le parcours philosophique de Wittgenstein une partie de l’histoire du « problème du champ visuel », thème clé de sa période intermédiaire ; d’autre part, mettre en lumière sa critique de l’idée d’un champ visuel (et celle de l’idée d’un objet interne). Nous croyons que ses arguments sont nouveaux, pénétrants, et ainsi leur intérêt dépasse les limites d’un exposé purement exégétique.
  •  247
    Formalization and infinity
    Manuscrito 31 (1): 25-43. 2008.
    This article discusses some of Chateaubriand’s views on the connections between the ideas of formalization and infinity, as presented in chapters 19 and 20 of Logical Forms. We basically agree with his criticisms of the standard construal of these connections, a view we named “formal proofs as ultimate provings”, but we suggest an alternative way of picturing that connection based on some ideas of the late Wittgenstein.
  •  141
    Observações filosóficas
    Philósophos - Revista de Filosofia 14 (2): 209-217. 2009.
    Review of the translation of Ludwig Wittgenstein book:  Observações Filosóficas.  São Paulo: Edições Loyola, 2005. Tradução (do inglês) por Adail Sobral e Maria Stela Gonçalves
  •  114
    The objective of this article is to try to elucidate Wittgenstein’s ex-travagant thesis that each and every mathematical advancement involves some “semantical mutation”, i.e., some alteration of the very meanings of the terms involved. To do that we will argue in favor of the idea of a “modal incompati-bility” between the concepts involved, as they were prior to the advancement, and what they become after the new result was obtained. We will also argue that the adoption of this thesis profoundly…Read more
  •  110
    The God-given Naturals, Induction and Recursion
    with Paulo Veloso
    O Que Nos Faz Pensar 29 (49): 115-156. 2021.
    We discuss some basic issues underlying the natural numbers: induction and recursion. We examine recursive formulations and their use in establishing universal and particular properties.
  •  86
    O que é um Objeto para um Lógico?
    In Lia Levy, Carolina Araújo, Ethel Menezes Rocha, Markos Klemz Guerrero & Fábio Ferreira de Almeida (eds.), Substância Na História da Filosofia, Nepfil Online. pp. 495-512. 2023.
    Trata-se de capítulo de um livro sobre a noção de "substância" na história da filosofia e contém uma discussão sobre a noção ordinária de "corpo" do ponto de vista da filosofia analítica
  •  73
    A Concepção Estândar de Prova e o Problema de Kant
    In Gisele Secco (ed.), De Mathematicae atque Philosophiae Elegantia, College Publications. pp. 150-168. 2021.
    The objective of this paper will be to present a critical evaluation of the so-called “standard conception of a proof”. According to this conception, a text could only be called a “demonstration of a certain mathematical proposition” if we could find a completely formalized version of that demonstration, its “corresponding formal proof”. We will compare these ideas with the treatment of the same topic within two markedly different contexts, that of contemporary Swedish intuitionism and that of m…Read more
  •  32
    SINGULARITY AND VISUAL PERCEPTION
    Dissertatio 58 218-246. 2023.
    This paper deals with the mutations in Wittgenstein’s treatment of the notions of “generality” and of “singularity”, from his first philosophy, in the Tractatus Logico-Philosophicus, to his later mature philosophy represented by the Philosophical Investigations. As we shall see, Wittgenstein’s philosophical handling of the notion of “visual perception” plays a key role in those conceptual transformations.
  •  9
    Wittgenstein on Mathematical Progress
    Philósophos - Revista de Filosofia 28 (1). 2023.
    O objetivo deste artigo é tentarmos elucidar a extravagante tese de Wittgenstein de que todo e qualquer avanço matemático envolve alguma “mutação semântica”, ou seja, alguma alteração nos próprios significados dos termos envolvidos. Para isso, argumentaremos a favor da ideia de uma “incompatibilidade modal” entre os conceitos envolvidos, como eram antes do avanço, e o que se tornam após a obtenção do novo resultado. Também argumentaremos que a adoção dessa tese altera profundamente nossa maneira…Read more