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Item Da interpretação Bhk à teoria intuicionista dos tipos: a construção mental como conceito primitivo fundamental(Universidade Federal de Goiás, 2022-03-31) Albernaz, Filipe Borges; Porto, André da Silva; http://lattes.cnpq.br/3598537464598916; Porto, André da Silva; Filho, Abilio Azambuja Rodrigues; Legris, Javier; Pereira, Luiz Carlos Pinheiro Dias; Queiroz, Ruy José Guerra Barretto deIn the midst of a dispute over philosophical foundations that has lasted for over a hundred years, the intuitionistic foundations of mathematics seems ever closer to being an alternative to classical foundation. Interpretation of fundamental and primitive notions and consequences for the interpretation of logical connectives are some of the issues to be addressed in this text, in a framework that intends to show the fundamental and primitive role of the notion of mental construction in Intuitionism, from Brouwer's proposal to Martin-Löf's Intuitionistic Type Theory. The discussion of particular aspects of the Martin-Löf’s proposal does not allow us to lose sight of the fact that it is essentially a formal system, universal, however, open, but also a language for the practice of intuitionist mathematics. These and other characteristics of Martin-Löf’s formal intuitionism needed to go beyond the definitions and concepts of Brouwer's original intuitionism, until then, considered as more speculative and impractical from a practical point of view. Precisely, the conceptual deepening of the Martin-Löf’s system brought light to intuitionism and make it unique and so important, not only for mathematics, but also for logic, philosophy and even for computation. With an adequate understanding of the Intuitionistic Type Theory, especially from the fundamental intuitionist interpretation of proofs as mental constructions, we have a more accurate measure of what intuitionism is about and its main consequences. Some of them dealt with in this work are the refusal of the law of excluded middle, the interpretation of notions such as “existence”, “construction”, “proposition” and “assertion”, in addition to the compulsory constructive character for formal mathematical proofs. In the specific case of the Martin-Löf’s system, we also discuss the ideas of truth and bivalence of propositions, primitive domains and propositional domains, essential for the system and distinct from classical conceptions, despite the terminological coincidence.Item Interlocução e analogias enganadoras no The big typescript e no Livro azul de Wittgenstein(Universidade Federal de Goiás, 2020-08-13) Costa , Paulo Henrique Silva; Porto, André da Silva; http://lattes.cnpq.br/3598537464598916; Porto, André da Silva; Engelmann, Mauro Luiz; Carvalho, Marcelo Silva de; Silva, Guilherme Ghisoni da; Velloso, Araceli Rosich SoaresThe present thesis deals with the crucial role that “misleading analogies” play in relation to the formulation and dissolution of philosophical problems in Wittgenstein. For this, we will restrict our discussion to the end of the middle period and the beginning of the final period, respectively, to The Big Typescript (1932-1933) and the Blue Book (1933-1934). The reason why we will restrict the discussion to this period of transition concerns the fact that there is a growing concern of Wittgenstein about the pragmatic character of language there, which we will call in the text “interlocutory conception of language”. Therefore, we will argue that, during the transition period between The Big Typescript and the Blue Book, there is a change in respect to the conception of language in question, supported, firstly, by the idea of “grammar as a system of rules”, which we will refer to as conception of language that “operates in a vacuum” and, secondly, in the pragmatic character of language, which we will refer to as a language that describes “interlocutory situations”. In this way, we will show that the notion of “rule” behind each conception of language is distinct, namely, in The Big Typescript there is, on the one hand, an operational conception of rule, and in the Blue Book, on the other, there is a relational conception. Given that, we will argue that misleading analogies play a crucial role in the formulation and dissolution of philosophical problems due to the improper import of grammars involved. This occurs in The Big Typescript because of improper importation of different grammars and their respective systems of rule, and in the Blue Book because of the use established within an interlocutory situation.Item As bases do intuicionismo matemático de Brouwer a natureza do continuum intuicionista(Universidade Federal de Goiás, 2021-04-15) Oliveira, Paulo Júnio de; Porto, André da Silva; http://lattes.cnpq.br/3598537464598916; Pereira, Luiz Carlos Pinheiro Dias; Rodrigues Filho, Abílio Azambuja; Rezende , Cristiano Novaes de; Klotz, Hans Christian; Porto, André da SilvaThis dissertation has as its aim the philosophical presentation and discussion of the nature of the intuitionist continuum of Luitzen Egbertus Jan Brouwer (1881-1966) and of its philosophical bases. This conception of the nature of the intuitionist continuum led to the development of the notion of “real numbers” distinct from classical analysis. Such a notion of “real numbers” does not accept the principle of excluded middle as universally valid and, as a result, it would not be possible to accept, for example, the law of trichotomy. The refusal of the principle of excluded middle does not arise from vacuum, and it is not the central focus of mathematical intuitionism. It is, as one would put it, a consequence of the conception of intuitionist “continuum”. The notions of “continuum” and “mathematical entity” are, as one would put it, the main focus of Brouwer’s analysis. From his point of view, the continuum is not a collection of absolutely individualized already given discrete points, which could in some way be extracted and used. The mathematical points in intuitionism are, so to speak, mentally constructed as sequences of infinitely converging intervals. Thus, no interval has an absolutely segmented, intrinsically formed, and/or absolutely individualized “limit”. For any “stage” of the interval there is always a space of infinite possibilities of other intervals. Hence there would be no point of infinitely distant cumulation of the intervals that could be found by the mathematician, as if the continuum were mappable such as a city is mappable by a geographer. From this point of view, the classical perspective would need to subscribe, even if not intentionally, to the existence of absolutely individualized discrete points in order to be able to make some kind of sense of its analysis of real numbers, of calculus, in short, of classical mathematics. This intuitionist notions of “continuum” and “real numbers” qua “infinitely converging intervals” became possible because of the type of philosophical bases that precede them. Such bases were developed from two fundamental acts that are understood in the context of an idealistic philosophy of Kantian inspiration. Thus, such acts are comprehended from a “Neo-Kantian” framework, in some sense or, more accurately, to use Brouwer’s expression, from an “up-to-date Kantism”. They are: (i) the act of mental recognition of the distinction between mathematical entities and linguistic entities and (ii) the act of mental recognition of the possibility of new mathematical entities. Both acts are interconnected and presuppose the same bases. In fact, the first act is connected with the radically intuitive aspect of the mental construction of mathematical entities, and the second act brings to light particularly important notes of the nature of the intuitionist continuum through the possibility of always emerging new mathematical entities, i.e., the continuum is a type of entity that is never determinable, it does not ever have an absolute form determined by a collection of discrete points that are absolutely individualizable. Therefore, the continuum is definitely indeterminate. In other words, the second act acknowledges that mathematical entities are intensionally “expansible” through the notion of “choice sequences”. Thus, these sequences are fundamental to the intuitionist continuum. In this dissertation, we present these philosophical bases in the first chapters and apply them to some specific mathematical contexts in the last chapters so as to try to elucidate the philosophical and mathematical nature of the intuitionist continuum and/or real numbers through the explicitness of properties of cohesion, viscosity, and infinitely converging intervals.Item “Apenas setas”: a teoria das categorias como linguagem para uma matemática estruturalista(Universidade Federal de Goiás, 2023-10-20) Saraiva, Igor Souza; Porto, André da Silva; http://lattes.cnpq.br/3598537464598916; Porto, André da Silva; Esteban Coniglio, Marcelo; Freire, Rodrigo de Alvarenga; Queiroz, Ruy José Guerra Barretto de; Santos, César Frederico dosFor some years now, there has been a philosophical debate about the relationship between two distinct intellectual movements. Within mathematical practice, the Theory of Categories emerged in the 1940s. Initially without any major foundational pretensions, little by little the theory gained in scope and came to be considered, at the very least, a very useful language for characterizing and studying abstract mathematical structure. Almost simultaneously, philosophers concerned with questions about the nature of mathematical objects proposed a structuralist program, based on the idea of shaping an understanding of the nature of mathematics that takes into account that each area of this science describes the formal factors common to various structured systems. The affinity between Category Theory and a structuralist philosophy of mathematics is almost obvious, leading naturally to the question of the possibility of the theory being employed as an autonomous conceptual framework, capable of articulating a peculiar view of mathematics, without any kind of dependence on other foundational approaches, such as Set Theory or Type Theory. This possibility has been denied by some philosophers of mathematics, giving rise to a dispute without a unanimous solution. This thesis presents a panoramic view of this whole scenario and tries to show that those who reject the autonomy of category theory have epistemological presuppositions that are not unanimously accepted.