ABSTRACT And concrete diction Carl Jonas Love Almqvist’s Drottningens juvelsmycke (The Queen's Tiara; 1834) is, along with Amorina, the work primarily associated with the ”poetic fugue” – a concept the author develops in ”Om enheten av epism och dramatism; en aning om den poetiska fugan” (”On the unity of epism and dramatism; a notion of the poetic fugue”; 1821); an essay often considered vague and theoretical by researchers in the field. The meaning of the poetic fugue has been regarded unclear, but mainly considered as some kind of synthesis of epic and dramatic writing. This essay argues that that is not the case, and that this one-dimensional approach both limits the interpretations of the essay and the poetic fugue as a whole. From a multidisciplinary perspective, with myself and my own reader as a part of the fugue itself, the aim of this essay is to highlight a very important overseen aspect of the poetic fugue, and Almqvist’s writing in general – the connections to mathematics, the analogies between abstract and concrete levels, and how these are deeply intertwined. The results in this essay are derived from a close reading technique based on mathematical problem solving called the ideotic method (den ideotiska metoden), and analyzed with Douglas Hofstadter's theory of Strange loops in Gödel, Escher, Bach – an eternal golden braid (1979). This analysis shows that this analogy is not just about the composition of a poetic piece of art, a synthesis of epic and dramatic writing, or the relation between music and text. Instead the results do point to an alternative interdisciplinary interpretation, where the relations between parts and units, realities and fictions, readers and texts, make the poetic fugue more of an analogy for the universe as a whole – a living and breathing ”animal coeleste” in contrast to the Newtonian ”mechanical coeleste”. An analogy which, thanks to its mathematical construction and way of looking at time as non-linear, is connected to both Einstein’s theory of relativity and quantum theory – the science of the very big and the very small, parts and units, of everything, including ourselves.

Submitted by ROSEMEIRE DE FATIMA BATISTELA null (rosebatistela@hotmail.com) on 2017-02-11T02:22:43Z No. of bitstreams: 1 tese finalizada 10 fevereiro 2017 com a capa.pdf: 2263896 bytes, checksum: 413948c6a47fb47a21e1587275d29c03 (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2017-02-15T16:56:58Z (GMT) No. of bitstreams: 1 batistela_rf_dr_rcla.pdf: 2263896 bytes, checksum: 413948c6a47fb47a21e1587275d29c03 (MD5) / Made available in DSpace on 2017-02-15T16:56:58Z (GMT). No. of bitstreams: 1 batistela_rf_dr_rcla.pdf: 2263896 bytes, checksum: 413948c6a47fb47a21e1587275d29c03 (MD5) Previous issue date: 2017-02-02 / Apresentamos nesta tese uma proposta de inserção do tema teorema da incompletude de Gödel em cursos de Licenciatura em Matemática. A interrogação norteadora foi: como sentidos e significados do teorema da incompletude de Gödel podem ser atualizados em cursos de Licenciatura em Matemática? Na busca de elaborarmos uma resposta para essa questão, apresentamos o cenário matemático presente à época do surgimento deste teorema, expondo-o como a resposta negativa para o projeto do Formalismo que objetivava formalizar toda a Matemática a partir da aritmética de Peano. Além disso, trazemos no contexto, as outras duas correntes filosóficas, Logicismo e Intuicionismo, e os motivos que impossibilitaram o completamento de seus projetos, que semelhantemente ao Formalismo buscaram fundamentar a Matemática sob outras bases, a saber, a Lógica e os constructos finitistas, respectivamente. Assim, explicitamos que teorema da incompletude de Gödel aparece oferecendo resposta negativa à questão da consistência da aritmética, que era um problema para a Matemática na época, estabelecendo uma barreira intransponível para a demonstração dessa consistência, da qual dependia o sucesso do Formalismo e, consequentemente, a fundamentação completa da Matemática no ideal dos formalistas. Num segundo momento, focamos na demonstração deste teorema expondo-a em duas versões distintas, que para nós se nos mostraram apropriadas para serem trabalhadas em cursos de Licenciatura em Matemática. Uma, como possibilidade de conduzir o leitor pelos meandros da prova desenvolvida por Gödel em 1931, ilustrando-a, bem como, as ideias utilizadas nela, aclarando a sua compreensão. Outra, como opção que valida o teorema da incompletude apresentando-o de maneira formal, portanto, com endereçamentos e objetivos distintos, por um lado, a experiência com a numeração de Gödel e a construção da sentença indecidível, por outro, com a construção formal do conceito de método de decisão de uma teoria. Na sequência, apresentamos uma discussão focada na proposta de Bourbaki para a Matemática, por compreendermos que a atitude desse grupo revela a forma como o teorema da incompletude de Gödel foi acolhido nessa ciência e como ela continuou após este resultado. Nessa exposição aparece que o grupo Bourbaki assume que o teorema da incompletude não impossibilita que a Matemática prossiga em sua atividade, ele apenas sinaliza que o aparecimento de proposições indecidíveis, até mesmo na teoria dos números naturais, é inevitável. Finalmente, trazemos a proposta de como atualizar sentidos e significados do teorema da incompletude de Gödel em cursos de Licenciatura em Matemática, aproximando o tema de conteúdos agendados nas ementas, propondo discussão de aspectos desse teorema em diversos momentos, em disciplinas que julgamos apropriadas, culminando no trabalho com as duas demonstrações em disciplinas do último semestre do curso. A apresentação é feita tomando como exemplar um curso de Licenciatura em Matemática. Consideramos por fim, a importância do trabalho com um resultado tão significativo da Lógica Matemática que requer atenção da comunidade da Educação Matemática, dado que as consequências deste teorema se relacionam com a concepção de Matemática ensinada em todos os níveis escolares, que, muito embora não tenham relação com conteúdos específicos, expõem o alcance do método de produção da Matemática. / In this thesis we present a proposal to insert Gödel's incompleteness theorem in Mathematics Education undergraduate courses. The main research question guiding this investigation is: How can the senses and meanings of Gödel's incompleteness theorem be updated in Mathematics Education undergraduate courses? In answering the research question, we start by presenting the mathematical scenario from the time when the theorem emerged; this scenario proposed a negative response to the project of Formalism, which aimed to formalize all Mathematics based upon Peano’s arithmetic. We also describe Logicism and Intuitionism, focusing on reasons that prevented the completion of these two projects which, in similarly to Formalism, were sought to support mathematics under other bases of Logic and finitists constructs. Gödel's incompleteness theorem, which offers a negative answer to the issue of arithmetic consistency, was a problem for Mathematics at that time, as the Mathematical field was passing though the challenge of demonstrating its consistency by depending upon the success of Formalism and upon the Mathematics’ rationale grounded in formalists’ ideal. We present the proof of Gödel's theorem by focusing on its two different versions, both being accessible and appropriate to be explored in Mathematics Education undergraduate courses. In the first one, the reader will have a chance to follow the details of the proof as developed by Gödel in 1931. The intention here is to expose Gödel’ ideas used at the time, as well as to clarify understanding of the proof. In the second one, the reader will be familiarized with another proof that validates the incompleteness theorem, presenting it in its formal version. The intention here is to highlight Gödel’s numbering experience and the construction of undecidable sentence, and to present the formal construction of the decision method concept from a theory. We also present a brief discussion of Bourbaki’s proposal for Mathematics, highlighting Bourbaki’s group perspective which reveals how Gödel’s incompleteness theorem was important and welcome in science, and how the field has developed since its result. It seems to us that Bourbaki’s group assumes that the incompleteness theorem does not preclude Mathematics from continuing its activity. Thus, from Bourbaki’s perspective, Gödel’s incompleteness theorem only indicates the arising of undecidable propositions, which are inevitable, occurring even in the theory of natural numbers. We suggest updating the senses and the meanings of Gödel's incompleteness theorem in Mathematics Education undergraduate courses by aligning Gödel's theorem with secondary mathematics school curriculum. We also suggest including discussion of this theorem in different moments of the secondary mathematics school curriculum, in which students will have elements to build understanding of the two proofs as a final comprehensive project. This study contributes to the literature by setting light on the importance of working with results of Mathematical Logic such as Gödel's incompleteness theorem in secondary mathematics courses and teaching preparation. It calls the attention of the Mathematical Education community, since its consequences are directly related to the design of mathematics and how it is being taught at all grade levels. Although some of these mathematics contents may not be related specifically to the theorem, the understanding of the theorem shows the broad relevance of the method in making sense of Mathematics.

<p>String theory is a unified framework for general relativity and quantum mechanics, thus being a theory of quantum gravity. In this thesis we discuss various aspects of quantum gravity for particular systems, having in common the existence of horizons. The main motivation is that one major challenge in theoretical physics today is in trying to understanding how time dependent backgrounds, with its resulting horizons and space-like singularities, should be described in a controlled way. One such system of particular importance is our own universe.</p><p>We begin by discussing the information puzzle in de Sitter space and consequences thereof. A typical time-scale is encountered, which we interpreted as setting the thermalization time for the system. Then the question of closed time-like curves is discussed in the combined setting where we have a rotating black hole in a Gödel-like universe. This gives a unified picture of what previously was considered as independent systems. The last three projects concerns $c=1$ matrix models and their applications. First in relation to the RR-charged two dimensional type 0A black hole. We calculate the ground state energy on both sides of the duality and find a perfect agreement. Finally, we relate the 0A model at self-dual radius to the topological string on the conifold. We find that an intriguing factorization of the theory previously observed for the topological string is also present in the 0A matrix model.</p>

String theory is a unified framework for general relativity and quantum mechanics, thus being a theory of quantum gravity. In this thesis we discuss various aspects of quantum gravity for particular systems, having in common the existence of horizons. The main motivation is that one major challenge in theoretical physics today is in trying to understanding how time dependent backgrounds, with its resulting horizons and space-like singularities, should be described in a controlled way. One such system of particular importance is our own universe. We begin by discussing the information puzzle in de Sitter space and consequences thereof. A typical time-scale is encountered, which we interpreted as setting the thermalization time for the system. Then the question of closed time-like curves is discussed in the combined setting where we have a rotating black hole in a Gödel-like universe. This gives a unified picture of what previously was considered as independent systems. The last three projects concerns $c=1$ matrix models and their applications. First in relation to the RR-charged two dimensional type 0A black hole. We calculate the ground state energy on both sides of the duality and find a perfect agreement. Finally, we relate the 0A model at self-dual radius to the topological string on the conifold. We find that an intriguing factorization of the theory previously observed for the topological string is also present in the 0A matrix model.

Mes travaux de recherche portent sur la théorie de la démonstration, le lambda-calcul et l'informatique théorique, dans la ligne de la correspondance de Curry-Howard entre les preuves et les programmes.<br /><br />Dans ma thèse de doctorat, j'ai étudié les opérateurs de mise en mémoire pour les types de données. Ces notions, qui sont introduites par Krivine, permettent de programmer en appel par valeur tout en utilisant la stratégie de la réduction de tête pour exécuter les $\lambda$-termes. Pour cette étude, j'ai introduit avec David une extension du $\lambda$-calcul avec substitutions explicites appelée $\lambda$-calcul dirigé. Nous en avons déduit une nouvelle caractérisation des termes de mise en mémoire et obtenu des nombreux résultats très fins à leur sujet. En ce qui concerne le typage des opérateurs de mise en mémoire, Krivine a trouvé une formule du second ordre, utilisant la non-non traduction de Gödel de la logique classique dans la logique intuitionniste, qui caractérise ces opérateurs. Je me suis attaché à diverses généralisations du résultat de Krivine pour les types à quantificateur positif dans des extensions de la logique des prédicats du second ordre.<br /><br />J'ai poursuivi, après ma thèse, une activité de recherche sur l'extension de la correspondance de Curry-Howard à la logique classique, au moyen des instructions de contrôle. J'ai étudié des problèmes liés aux types de données dans deux de ces systèmes : le $\lambda \mu$-calcul de Parigot et le $\lambda C$-calcul de Krivine. J'ai donné des algorithmes très simples permettant de calculer la valeur d'un entier classique dans ces deux systèmes. J'ai également caractérisé les termes dont le type est l'une des règles de l'absurde. J'ai étendu le système de Parigot pour en obtenir une version non déterministe mais où les entiers se réduisent toujours en entiers de Church. Curieusement, ce système permet de programmer la fonction ``ou parallèle''.<br /><br />Je me suis intéressé aux systèmes numériques qui servent à représenter les entiers naturels au sein du $\lambda$-calcul. J'ai montré que pour un tel système, la possession d'un successeur, d'un prédécesseur et d'un test à zéro sont des propriétés indépendantes, puis qu'un système ayant ces trois fonctions possède toujours un opérateur de mise en mémoire. Dans un cadre typé, j'ai apporté une réponse négative à une conjecture de Tronci qui énonçait une réciproque du résultat précédent.<br /><br />La notion de mise en mémoire ne s'applique qu'à des types de données. Une définition syntaxique a été donné par Böhm et Berarducci, et Krivine a proposé une définition sémantique de ces types. J'ai obtenu avec Farkh des résultats reliant la syntaxe et la sémantique des types de données. Nous avons proposé également des définitions des types entrée et des types sortie pour lesquelles nous avons montré diverses propriétés syntaxiques et sémantiques.<br /><br />J'ai réussi à combiner la logique intuitionniste et la logique classique en une logique mixte. Dans cette logique, on distingue deux genres de variables du second ordre, suivant que l'on peut, ou non, leur appliquer le raisonnement par l'absurde. Ce cadre m'a permi de donner le type le plus général pour les opérateurs de mise en mémoire. Vu le rôle important que cette logique semble devoir jouer dans la théorie de ces opérateurs, j'en ai mené avec A. Nour une étude théorique approfondie. Le système de logique mixte propositionnelle auquelle nous avons abouti évoque les sytèmes $LC$ de Girard et $LK^{tq}$ de Danos, Joinet et Schellinx.<br /><br />Je me suis intéressé avec David à l'équivalence induite par l'égalité entre les arbres de Böhm infiniment $\eta$-expansés. Avec Raffalli, je me suis également intéressé à la sémantique de la logique du second ordre.