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Dans cette thèse, je propose une lecture renouvelée de l’itinéraire philosophique d’Hilary Putnam concernant la problématique du réalisme. Mon propos consiste essentiellement à défendre l’idée selon laquelle il y aurait beaucoup plus de continuité, voir une certaine permanence, dans la manière dont Putnam a envisagé la question du réalisme tout au long de sa carrière. Pour arriver à une telle interprétation de son oeuvre, j’ai essentiellement suivi deux filons. D’abord, dans un ouvrage du début des années 2000, Ethics without Ontology (2004), Putnam établit un parallèle entre sa conception de l’objectivité en philosophie des mathématiques et en éthique. Le deuxième filon vient d’une remarque qu’il fait, dans l’introduction du premier volume de ses Philosophical Papers (1975), affirmant que la forme de réalisme qu’il présupposait dans ses travaux des années 1960-1970 était la même que celle qu’il défendait en philosophie des mathématiques et qu’il souhaitait défendre ultérieurement en éthique. En suivant le premier filon, il est possible de mieux cerner la conception générale que se fait Putnam de l’objectivité, mais pour comprendre en quel sens une telle conception de l’objectivité n’est pas propre aux mathématiques, mais constitue en réalité une conception générale de l’objectivité, il faut suivre le second filon, selon lequel Putnam aurait endossé, durant les années 1960-1970, le même type de réalisme en philosophie des sciences et en éthique qu’en philosophie des mathématiques. Suivant cette voie, on se rend compte qu’il existe une similarité structurelle très forte entre le premier réalisme de Putnam et son réalisme interne. Après avoir établi la parenté entre le premier et le second réalisme de Putnam, je montre, en m’inspirant de commentaires du philosophe ainsi qu’en comparant le discours du réalisme interne au discours de son réalisme actuel (le réalisme naturel du commun des mortels), que, contrairement à l’interprétation répandue, il existe une grande unité au sein de sa conception du réalisme depuis les années 1960 à nos jours. Je termine la thèse en montrant comment mon interprétation renouvelée de l’itinéraire philosophique de Putnam permet de jeter un certain éclairage sur la forme de réalisme que Putnam souhaite défendre en éthique. / In this dissertation I propose a new reading of the philosophical itinerary of Hilary Putnam on the matter of realism. In essence, my purpose is to argue that there is much more continuity than is normally understood, and even a degree of permanence, in the way in which Putnam has viewed the question of realism throughout his career. To arrive at this interpretation of Putnam I essentially followed two veins in his work. First, in a volume published in the early 2000s entitled Ethics without Ontology (2004), Putnam establishes a parallel between his conception of objectivity in the philosophy of mathematics and in ethics. The second vein comes from a comment he made in the introduction to the first volume of his Philosophical Papers (1975) to the effect that the kind of realism he presupposed in his work of the 1960s and 70s was the same that he upheld in the philosophy of mathematics and wished to argue for at a later date in ethics. Following the first vein makes it possible to better grasp Putnam’s general conception of objectivity, but in order to understand how such a conception of objectivity is not unique to mathematics but is instead a general conception of objectivity one must follow the second vein. There, in the 1960s and 70s, Putnam adopted the same kind of realism in the philosophy of science and in ethics as he had in the philosophy of mathematics. Following this path, one realises that there exists a very strong structural similarity between Putnam’s first realism and his internal realism. After establishing this connection between Putnam’s first and second realism, I draw on Putnam’s remarks and compare the internal realism discourse to his current realism (the natural realism of ordinary people) to demonstrate, contrary to the prevalent interpretation, that there has been a great deal of consistency in his conception of realism from the 1960s to the present day. I conclude the dissertation by demonstrating how my new interpretation of Putnam’s philosophical itinerary makes it possible to shed light on the kind of realism he wishes to champion in ethics.

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.

The goal of this thesis in finance is to combine the use of advanced mathematical methods with a return to foundational economic issues. In that perspective, I study generalized rational expectations and asset pricing in Chapter 2, and a converse comparison principle for backward stochastic differential equations with jumps in Chapter 3. Since the use of stochastic methods in finance is an interesting and complex issue in itself - if only to clarify the difference between the use of mathematical models in finance and in physics or biology - I also present a philosophical reflection on the interpretation of mathematical models in finance (Chapter 4). In Chapter 5, I conclude the thesis with an essay on the history and interpretation of mathematical probability - to be read while keeping in mind the fundamental role of mathematical probability in financial models.

This dissertation is centered around a set of apparently conflicting intuitions that we may have about mathematics. On the one hand, we are inclined to believe that the theorems of mathematics are true. Since many of these theorems are existence assertions, it seems that if we accept them as true, we also commit ourselves to the existence of mathematical objects. On the other hand, mathematical objects are usually thought of as abstract objects that are non-spatiotemporal and causally inert. This makes it difficult to understand how we can have knowledge of them and how they can have any relevance for our mathematical theories. I begin by characterizing a realist position in the philosophy of mathematics and discussing two of the most influential arguments for that kind of view. Next, after highlighting some of the difficulties that realism faces, I look at a few alternative approaches that attempt to account for our mathematical practice without making the assumption that there exist abstract mathematical entities. More specifically, I examine the fictionalist views developed by Hartry Field, Mark Balaguer, and Stephen Yablo, respectively. A common feature of these views is that they accept that mathematics interpreted at face value is committed to the existence of abstract objects. In order to avoid this commitment, they claim that mathematics, when taken at face value, is false. I argue that the fictionalist idea of mathematics as consisting of falsehoods is counter-intuitive and that we should aim for an account that can accommodate both the intuition that mathematics is true and the intuition that the causal inertness of abstract mathematical objects makes them irrelevant to mathematical practice and mathematical knowledge. The solution that I propose is based on Rudolf Carnap's distinction between an internal and an external perspective on existence. I argue that the most reasonable interpretation of the notions of mathematical truth and existence is that they are internal to mathematics and, hence, that mathematical truth cannot be used to draw the conclusion that mathematical objects exist in an external/ontological sense.

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The goal of this thesis in finance is to combine the use of advanced mathematical methods with a return to foundational economic issues. In that perspective, I study generalized rational expectations and asset pricing in Chapter 2, and a converse comparison principle for backward stochastic differential equations with jumps in Chapter 3. Since the use of stochastic methods in finance is an interesting and complex issue in itself - if only to clarify the difference between the use of mathematical models in finance and in physics or biology - I also present a philosophical reflection on the interpretation of mathematical models in finance (Chapter 4). In Chapter 5, I conclude the thesis with an essay on the history and interpretation of mathematical probability - to be read while keeping in mind the fundamental role of mathematical probability in financial models. / Doctorat en Sciences économiques et de gestion / info:eu-repo/semantics/nonPublished