Alors que l’applicabilité des mathématiques est devenue un sujet d’intérêt pour le débat philosophique récent, celui-ci n’a pas encore clairement mis l’accent sur les questions épistémiques posées par l’intervention des mathématiques dans les sciences et dans la vie quotidienne. Ces questions peuvent être formulées comme suit : comment pouvons-nous connaître la vérité d’une proposition scientifique, ou plus généralement d’une proposition sur certaines caractéristiques du monde naturel, lorsque cette proposition comprend des éléments mathématiques? Quelle sorte de justification avons-nous pour les parties mathématiques de notre connaissance empirique? Cette thèse de doctorat a un double objectif : d’une part, offrir une systématisation critique du débat philosophique en cours sur l’applicabilité. D’autre part, clarifier le problème épistémique posé par l’applicabilité des mathématiques et le séparer des problèmes métaphysiques corrélés. La première partie du travail est consacrée à la formulation des questions propres à une enquête épistémologique sur les mathématiques appliquées, ainsi qu’à la présentation de l’analyse classique de l’applicabilité offerte par Steiner. Dans la partie II, la présentation du débat récent sur l’applicabilité est organisée autour d’une distinction entre les approches qui considèrent les mathématiques pures et appliquées sur le même niveau épistémique, et celles qui distinguent le niveau de mathématiques pures du niveau des applications. Les positions étudiées sont, respectivement, les points de vue fregéen et néo-fregéen, et ce que l’on considère comme des points de vue ‘structurels’, à savoir le structuralisme mathématique (à la fois ante rem et éliminatif), la position de Field et la théorie de la mesure. La partie III introduit le débat sur l’indispensabilité des mathématiques dans les sciences, pour montrer comment les différentes formulations des arguments d’indispensabilité et les critiques qui leur sont adressées renouvellent l’attention sur les questions philosophiques liées à l’applicabilité, et clarifient la séparation entre les questions épistémiques sur les mathématiques pures (par exemple, le problème de l’accès) et les questions épistémiques sur les applications (par exemple, la justification des parties mathématiques de notre connaissance scientifique). La position de Christopher Pincock, qui théorise un traitement épistémique distinct pour les mathématiques pures et appliquées, est spécifiquement analysée. Enfin, la dernière partie en conclut ce que peuvent être les caractéristiques d’une théorie épistémologique adéquate pour les mathématiques pures et pour les mathématiques appliquées, et présentent plusieurs problématiques connexes et cruciales pour de futures recherches. / While the applicability of mathematics has become a topic of great interest in recent philosophical debate, the debate has not yet clearly focused on the fundamental epistemic questions that arise from the use of mathematics in science and in daily life. These questions can be basicallystated as follows: how can we affirm to know the truth of a scientific statement, or more generally that of any statement about a feature of the natural world, when that statement includes some elements of mathematics? What kind of justification do we have for the mathematical portions of our empirical knowledge? My PhD dissertation has a twofold purpose: on the one hand, it offers a critical systematization of the on-going philosophical debate on applicability. On the other hand, the epistemic problem posed by the applicability of mathematics is clarified and separated from correlated metaphysical issues. The first part of the work is devoted to the definition of the specific epistemic questions and the presentation of the classic analysis of applicability problem(s) offered by Steiner. In Part II, the recent debate on applicability is organized around a distinction between those approaches that take pure and applied mathematics to be on the same epistemic level, and those that keep the level of pure mathematics separate from the level of application. The positions investigated are, respectively, Fregean and Neo-Fregean views for the one-stage side, and what I refer to as ‘structural’ views for the two-stageside, namely, mathematical structuralism (both ante rem and eliminative), Field’s account, and measurement theory. Part III takes into account the related debate on the indispensability of mathematics to science, showing how the different formulations of indispensability arguments and the criticisms led to renewed attention to the philosophical questions about applicability in the early 2000s, along with a clarification of the separation between epistemic questions about pure mathematics (e.g. the access problem) and epistemic questions about applications (e.g. the justification of the mathematical portions of scientific knowledge). The account offered by Christopher Pincock, which provides a separate epistemic treatment for pure and appliedmathematics, is specifically analyzed. Finally, in the last part of the work, we draw particular conclusions about what would be, following our analysis, the features of a suitable epistemological treatment of both pure and applied mathematics, while several connected issues are identified as crucial for further inquiry. / L’applicabilità della matematica è diventata un argomento di grande interesse per il dibattito filosofico recente, ma il dibattito non si è ancora focalizzato sulle fondamentali questioni epistemologiche poste dall’uso della matematica nella scienza e nella vita quotidiana. Queste domande possono essere formulate come segue: come possiamo dire di conoscere la verità di un asserto scientifico, o più in generale di qualsiasi asserto su alcune caratteristiche del mondo naturale, quando tale asserto include elementi matematici? Che tipo di giustificazione possiamo avere per le porzioni matematiche della nostra conoscenza empirica? La presente tesi di dottorato ha un duplice scopo: da un lato, si offre una presentazione sistematica deldibattito filosofico in corso sull’applicabilità. Dall’altro lato, il problema epistemico posto dall’applicabilità della matematica è chiarito e separato dai correlati problemi metafisici. La prima parte del lavoro è dedicata alla definizione delle specifiche domande epistemiche sulla matematica applicata; si presenta inoltre l’analisi classica dei problemi legati all’applicabilità offerta da Steiner. Nella seconda parte, la presentazione del dibattito recente sull’applicabilità è organizzata attorno ad una distinzione tra le posizioni che considerano matematica pura e applicata sullo stesso livello epistemico e quelle che mantengono il livello della matematica pura separato dal livello applicativo. Le posizioni indagate sono, rispettivamente, la posizionefregeana e neo-fregeana da un lato, e le posizioni che definiremo ‘strutturali’ dall’altro, ovvero lo strutturalismo matematico (sia ante rem che eliminativo), la posizione di Field e la teoria della misura. La terza parte del lavoro affronta il dibattito sull’indispensabilità della matematica nella scienza, mostrando come le diverse formulazioni degli argomenti di indispensabilità e le critiche ad esse rivolte contribuiscano a rinnovare l’interesse per le domande filosofiche sull’applicabilità, oltre che a chiarire la separazione tra domande epistemiche sulla matematica pura (ad esempio Il problema dell’accesso) e domande epistemiche sulle applicazioni (ad esempio la giustificazione delle porzioni matematiche della nostra conoscenza scientifica). La proposta teorica di Christopher Pincock, che tratta separatamente l’epistemologia di matematica pura e applicata, è analizzata in modo specifico. Nell’ultima parte del lavoro, si traggono alcuni conclusioni su quali potrebbero essere, in seguito allo studio svolto, le caratteristiche di un trattamento adeguato dell’epistemologia della matematica pura e applicata. Infine, alcuni ulteriori problemi connessi sono individuati come cruciali per indagini future.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-27T18:28:27Z No. of bitstreams: 1 TeseALN.pdf: 2172393 bytes, checksum: 0233779408d5d19f4ecc2c9d4de83bf0 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-04T17:28:56Z (GMT) No. of bitstreams: 1 TeseALN.pdf: 2172393 bytes, checksum: 0233779408d5d19f4ecc2c9d4de83bf0 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-04T17:29:02Z (GMT) No. of bitstreams: 1 TeseALN.pdf: 2172393 bytes, checksum: 0233779408d5d19f4ecc2c9d4de83bf0 (MD5) / Made available in DSpace on 2016-10-04T17:29:09Z (GMT). No. of bitstreams: 1 TeseALN.pdf: 2172393 bytes, checksum: 0233779408d5d19f4ecc2c9d4de83bf0 (MD5) Previous issue date: 2015-09-29 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / This thesis provides a reading and interpretation of Wittgenstein’s writings on mathematics at the beginning of his “middle period” (more precisely, at the “mathematical chapters” of Philosophische Bemerkungen), placing these writings in the context of two crises. The first, internal to his thought, consists of inconsistencies regarding what the Tractatus prescribed as the result of the application of logic and the effective logical analysis of certain domains of reality, which characterized, in Wittgenstein’s view, a crisis in the foundations of logic. On the other hand, controversies about the foundations of mathematics were intensified throughout the 1920s, and debates between three schools who attempted to impose their way not only of conceiving mathematics, but also of doing it, became increasingly frequent. This crisis, also called Grundlagenkrise der Mathematik, is an important historical and conceptual background for these early writings immediately after Wittgenstein’s return to philosophy in 1929. If in the Tractatus Wittgenstein had positioned himself only with regard to Frege’s and Russell’s logicism, in these writtings he tries in his own way to contrast his thought with the prevailing trends of his time: the intuicionism of Brouwer and Weyl, Hilbert’s formalism and, finally, Ramsey’s renewed logicism. This thesis develops, in its concluding Chapter, a reflection on Wittgenstein’s posture with respect to these three classical schools and with respect to the problems faced by them. / A tese fornece uma leitura e interpretação dos escritos de Wittgenstein sobre a matemática no início do seu “período intermediário” (mais precisamente, nos “capítulos matemáticos” das Philosophische Bemerkungen), situando estes escritos no contexto de duas crises. A primeira, interna ao pensamento do autor, diz respeito a inconsistências referentes `aquilo que o Tractatus prescrevera como resultado da aplicação da lógica e a análise lógica efetiva de certos domínios do real, o que configurava, aos olhos de Wittgenstein, uma crise nos fundamentos da lógica. Por outro lado, controvérsias acerca dos fundamentos da matemática se acirraram ao longo da década de 1920, e debates entre três escolas que buscavam impor o seu modo não apenas de conceber a matemática, mas também de fazê-la tornavam-se cada vez mais frequentes. Essa crise, que recebera o codinome de Grundlagenkrise der Mathematik, constitui um importante pano de fundo histórico-conceitual para estes primeiros escritos de Wittgenstein após seu retorno `a filosofia em 1929. Se, no Tractatus, Wittgenstein se posiciona apenas em relação ao logicismo de Frege e Russell, nestes escritos ele procura, a seu modo, contrapor seu pensamento em relação às tendências dominantes de sua época: o intuicionismo de Brouwer e Weyl, o formalismo de Hilbert e, por fim, o logicismo renovado de Ramsey. A tese desenvolve, em seu Capítulo conclusivo, uma reflexão sobre a postura de Wittgenstein ante estas três escolas clássicas e ante os problemas por elas enfrentados

In this thesis, I aim to motivate a particular philosophy of mathematics characterised by the following three claims. First, mathematical sentences are generally speaking false because mathematical objects do not exist. Second, people typically use mathematical sentences to communicate content the truth of which does not require mathematical objects to exist. Finally, in using mathematical language in this way, speakers are not doing anything out of the ordinary: they are performing straightforward assertions. In Part I, I argue that the role played by mathematics in our scientific explanations is a purely expressive one, merely allowing us to say more than we otherwise would be able to about, or yielding a greater understanding of, the physical world. Mathematical objects to not need to exist for mathematical language to play this role. This proposal puts a normative constraint on our use of mathematical language: we ought to use mathematically presented theories to express belief only in the consequences they have for non-mathematical things. In Part II, I will argue that what the normative proposal recommends is in fact what people generally do in both pure and applied mathematical contexts. I motivate this claim by showing that it is predicted by our best general means of analysing natural language. I provide a semantic theory of applied arithmetical sentences and show that they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which shows that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show the hermeneutic fictionalist position that emerges is preferable to any alternative which interprets mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects.

Orientador: Maria Aparecida Viggiani Bicudo / Resumo: A continuidade é um tema que sempre trouxe desafios aos filósofos e matemáticos, desde a Grécia antiga com os paradoxos sobre o movimento, que persistem até os dias atuais, quando nos encontramos discutindo a continuidade da consciência e do tempo. Com o advento da tecnologia digital uma perspectiva se abre nessa discussão, pois modelos matemáticos contínuos estão sendo aplicados a problemas numéricos computacionais, que são caracteristicamente discretos. Essa possível discretização do contínuo mostra-se de modo claro, levando-nos a investigar a presença do contínuo ao se produzir Matemática junto ao computador. Investigaremos esse assunto realizando um movimento característico do pensar filosófico, tomando o tema como uma lectio, entendida como momento de discussão sobre textos numa dimensão argumentativa filosófica sobre nossa interrogação de pesquisa, ou seja, nossa quaestio. Nossa compreensão dos textos é exposta de maneira articulada e dividida em três seções, que versam sobre os estudos realizados no âmbito das Ciências Matemática, Filosofia e Computação. Finalizaremos trazendo uma meta-compreensão dos estudos realizados, tomando como centro articulador da reflexão a interrogação formulada. Nosso objetivo com esse exercício filosófico é compreender o fenômeno “contínuo-discreto” na região de inquérito das Ciências Ocidentais e sua presença na computação / Abstract: Continuity has been a challenging topic to philosophers and mathematicians, since the ancient Greece, with paradoxes of movement, until present days when continuity of consciousness and of time are discussed. With the advent of digital technology another perspective has brought into the discussion, because continuous mathematical models are being applied in numerical computational problems, which are characteristically discrete. This possibility of continuos’ discretization is drawing our attention. Therefore, this research aims to understand the presentification of continuous when we are producing Mathematics with computers. We will investigate this subject via a philosophical approach. This thesis is constituted as a lectio, understood as a moment of discussion about texts in a philosophical argumentative dimension about our research question, that is, our quaestio. Our understanding of the texts is articulated and divided into three sections, which deal with the studies carried out in Mathematics, Philosophy and Computer Science. Our goal with this philosophical exercise is to explore the "continuous-discrete" phenomenon under Western Sciences influence and its presence in computation. / Doutor

As computers become a more prevalent commodity in mathematical research and mathematical proof, the question of whether or not a computer assisted proof can be considered a mathematical proof has become an ongoing topic of discussion in the mathematics community. The use of the computer in mathematical research leads to several implications about mathematics in the present day including the notion that mathematical proof can be based on empirical evidence, and that some mathematical conclusions can be achieved a posteriori instead of a priori, as most mathematicians have done before. While some mathematicians are open to the idea of a computer-assisted proof, others are skeptical and would feel more comfortable if presented with a more traditional proof, as it is more surveyable. A surveyable proof enables mathematicians to see the validity of a proof, which is paramount for mathematical growth, and offer critique. In my thesis, I will present the role that the mathematical proof plays within the mathematical community, and thereby conclude that because of the dynamics of the mathematical community and the constant activity of proving, the risks that are associated with a mistake that stems from a computer-assisted proof can be caught by the scrupulous activity of peer review in the mathematics community. Eventually, as the following generations of mathematicians become more trained in using computers and in computer programming, they will be able to better use computers in producing evidence, and in turn, other mathematicians will be able to both understand and trust the resultant proof. Therefore, it remains that whether or not a proof was achieved by a priori or a posteriori, the validity of a proof will be determined by the correct logic behind it, as well as its ability to convince the members of the mathematical community—not on whether the result was reached a priori with a traditional proof, or a posteriori with a computer-assisted proof.

In broadest terms, this thesis is concerned to answer the question of whether the view that arithmetic is analytic can be maintained consistently. Lest there be much suspense, I will conclude that it can. Those who disagree claim that accounts which defend the analyticity of arithmetic are either unable to give a satisfactory account of the foundations of mathematics due to the incompleteness theorems, or, if steps are taken to mitigate incompleteness, then the view loses the ability to account for the applicability of mathematics in the sciences. I will show that this criticism is not successful against every view whereby arithmetic is analytic by showing that the brand of "conventionalism" about mathematics that Rudolf Carnap advocated in the 1930s, especially in Logical Syntax of Language, does not suffer from these difficulties. There, Carnap develops an account of logic and mathematics that ensures the analyticity of both. It is based on his famous "Principle of Tolerance", and so the major focus of this thesis will to defend this principle from certain criticisms that have arisen in the 80 years since the book was published. I claim that these criticisms all share certain misunderstandings of the principle, and, because my diagnosis of the critiques is that they misunderstand Carnap, the defense I will give is of a primarily historical and exegetical nature. Again speaking broadly, the defense will be split into two parts: one primarily historical and the other argumentative. The historical section concerns the development of Carnap's views on logic and mathematics, from their beginnings in Frege's lectures up through the publication of Logical Syntax. Though this material is well-trod ground, it is necessary background for the second part. In part two we shift gears, and leave aside the historical development of Carnap's views to examine a certain family of critiques of it. We focus on the version due to Kurt Gödel, but also explore four others found in the literature. In the final chapter, I develop a reading of Carnap's Principle - the `wide' reading. It is one whereby there are no antecedent constraints on the construction of linguistic frameworks. I argue that this reading of the principle resolves the purported problems. Though this thesis is not a vindication of Carnap's view of logic and mathematics tout court, it does show that the view has more plausibility than is commonly thought.

Orientador: Marcos Vieira Teixeira / Resumo: Realizamos um histórico da introdução das “matemáticas mistas”, e identificamos que o filósofo Francis Bacon (1561-1626) é creditado por meio das publicações “Proficience and Advancement of Learnings” em 1605 e “De Dignitate et Augmentis Scientiarum” em 1623. Entretanto, para responder questões pertinentes da Filosofia Natural, essa classificação matemática gerou um conflito entre as ciências matemáticas e a metafísica. Desse modo, as ciências físico-matemáticas surgem como uma tentativa de utilizar a matemática para abordar tópicos relacionados as causas naturais do mundo real. No ano de 1751, Jean le Rond D'Alembert (1717-1783), realizou uma nova classificação dos conhecimentos humanos, divulgada na obra Discours Préliminaire, que foi o texto de abertura da primeira edição da Encyclopédie, editada em conjunto com Denis Diderot (1713-1784). No que se refere a matemática, foi mantida como uma ramificação da metafísica, mas essa nova classificação a dividiu em: “matemática pura”, “matemática mista” e “ciências físico-matemáticas”. Porém, no decorrer do século XVIII, estimulado principalmente pelas críticas de Kant (1724-1804) em relação ao conhecimento puro, se deu início a uma discussão quanto a metafísica, e consequentemente refletiu nas ciências que estavam subordinadas a esse ramo do conhecimento. Desse modo, o século XIX culminou no desuso da expressão “matemáticas mistas”, contudo, veremos que essas ciências forneceram elementos para fomentar o aparecimento das “matemáti... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: Our aim is to realize a history of the introduction of the "mixed mathematics", and we find that the philosopher Francis Bacon (1561-1626) is credited through the publications "Proficiency and Advancement of Learnings" in 1605 and "De Dignitate et Augmentis Scientiarum" in 1623. However, to answer pertinent questions of Natural Philosophy, this mathematical classification generated a conflict between the mathematical sciences and metaphysics. In this way, the physico-mathematics sciences appeared as an attempt to use mathematics to address topics related to the natural causes of the real world. In 1751, Jean le Rond D'Alembert (1717-1783), made a new classification of human knowledge, published in the Discours Préliminaire, which was the opening text of the first edition of the Encyclopedie, edited together with Denis Diderot 1713-1784). As far as mathematics is concerned, was maintained as a branch of metaphysics, but this new classification divided it into "pure mathematics", "mixed mathematics" and "physico-mathematics sciences". But in the course of the eighteenth century, stimulated mainly by Kant (1724-1804) and his critiques of pure knowledge, a discussion of metaphysics began, and consequently reflected in the sciences which were subordinate to this branch of knowledge. The 19th century culminated in the disuse of the expression "mixed mathematics," however, these sciences provided elements to foster the emergence of "applied mathematics" as well as contributed to the... (Complete abstract click electronic access below) / Doutor

[pt] A proximidade entre matemática e filosofia em Platão é algo historicamente estabelecido e que pode ser constatado desde o primeiro contato com a sua obra e com as linhas gerais de seu pensamento. Nesse sentido, encontramos em alguns dos seus principais Diálogos, particularmente em A República, concepções sobre a natureza da matemática relacionadas, sobretudo, à metodologia matemática. Na República Platão aborda criticamente aspectos referentes ao método e ao status epistemológico das disciplinas matemáticas em dois momentos. O primeiro no Livro VI, na célebre passagem da Linha Dividida (509d - 511e), e o segundo no Livro VII, por ocasião da descrição do programa de estudos preparatórios à dialética (521c-534e) e, em ambos, considerando-se o que Platão diz em outras oportunidades, o teor da crítica platônica surpreende. Na Linha, as disciplinas matemáticas são descritas como formas de conhecimento intermediárias entre a opinião e a dialética, a única a merecer o título de ciência legítima. No Livro VII para ilustrar a distinção entre o conhecimento alcançado pelas disciplinas matemáticas, de um lado, e pela dialética, de outro, é dito que apesar de apreender alguma coisa da essência o matemático estaria para o dialético como aquele que dorme e sonha está para aquele que está acordado e vivendo a realidade (533b - 534e). O objetivo desse trabalho, portanto, é investigar por que Platão considera as matemáticas ciências intermediárias e qual a noção de conhecimento que serve de critério para essa classificação. / [en] The proximity between mathematics and philosophy in Plato is something historically acknowledged and that can be verified from the first contact with his work and with the general lines of his thought. Thus, one can find in some of his main Dialogues, particularly in the Republic, conceptions on the nature of mathematics mainly related to the mathematical methodology. In the Republic Plato approaches critically aspects regarding the method and the epistemological status of the mathematical disciplines in two moments. The first in Book VI, in the famous fragment of the Divided Line (509d - 511e), and the second in Book VII, while describing the program of preparatory studies to dialectics (521c-534e) and, in both cases, considering what Plato says in other fragments, the character of Plato s criticism surprises. In the Line, the disciplines of mathematics are described as a way of knowledge in-between opinion and dialectics, the last being the only one entitled to be considered a legitimate science. In Book VII, in order to show the distinction between the knowledge reached by mathematical disciplines, on one side, and the dialectics, on another, it is stated that despite learning some of the essence, the mathematician is for the dialectical as one who sleeps and dreams is for those who are awake and living reality itself (533b 534e). Therefore, the aim of this work is to investigate why Plato considers the disciplines of mathematics in-between sciences and what notion of knowledge was used as the criteria for that classification.

Ce travail montre l'unité de l'œuvre de Pascal dans ce qui concerne la « comparabilité des incomparables » : la comparaison, langagière ou mathématique, qui se fait entre des choses qui ne pourraient pas en principe être rapprochées. Il s'agit de faire une approche historique et linguistique pour poser des questions philosophiques par rapport à la comparaison, notamment sur le rôle de principe que l'infini y joue selon Pascal. Nous identifions la comparaison des incomparables sous trois formes.La première partie de ce travail est consacrée à formuler une forme rhétorique d'analogie que nous nommons l'« analogie de disproportion » (nous inspirant de Secretan 1998). Si l'analogie est généralement dite faire une comparaison entre deux rapports, chacun desquels existe entre des choses homogènes, l'analogie de disproportion permet en revanche de montrer une ressemblance entre des rapports d'hétérogénéité, entre des disproportions ou entre des distances infinies: deux choses sont aussi différentes entre elles que deux autres. Pascal étant un auteur qui souligne surtout les disproportions, nous montrons qu'il compare ces disproportions, notamment pour délimiter à l'homme ce qu'il ne peut pas connaître parfaitement.La deuxième partie analyse la pratique mathématique de Pascal « en poids, nombre et mesure » : il s'agit de montrer que dans la méthode des indivisibles des Lettres de A. Dettonville, dans le Traité du triangle arithmétique et dans la comparaison du courbe et du droit, toujours l'infini (ou plutôt l'indéfini) intervient comme un facteur qui permet la comparabilité de ce qui semblait être incomparable. La troisième partie fait une discussion proprement philosophique sur l'infiniment petit et l'infiniment grand, prenant en compte la pratique mathématique de Pascal analysée dans la deuxième partie. Il est question de discuter sur la nature des « indivisibles », des « différences » et des « distances infinies ». Nous proposons que l'« infini » dans la pratique mathématique de Pascal relève plutôt de l'« indéfini », reliant cela à une distinction entre le sens absolu et le sens relatif des mots. Une exception dans la pratique mathématique de Pascal est la géométrie projective, où il faut accepter des éléments à distance infinie. La « rencontre » des deux infinis, finalement, permet de montrer la réciprocité de l'infini de grandeur et de l'infini de petitesse. Une discussion est faite à ce propos, reliant la proportion inverse entre les deux infinis à la grandeur et la petitesse de l'homme et au caractère paradoxal de certaines vérités selon Pascal, lesquelles sont résolues dans la personne du Christ. On conclut que Pascal propose non pas une connaissance directe de l'infini, mais plutôt une approche à la relation que l'homme, être fini, possède avec l'infini / This thesis shows the unity of Pascal's work in what concerns the "comparability of incomparables'': the comparison, either in mathematics our natural language, between things which could not in principle be brought together. The approach is both a historical and a linguistic one, and it aims to recovery some important questions regarding the philosophical nature of comparisons, more specifically, the role of the infinite in Pascal's thought. The comparison of incomparables may be identified in three different formsIn the first part, we formulate a rhetorical form of analogy that we call an "analogy of disproportion'' (inspired by Secretan 1998). If the analogy is generally said to make a comparison between two relations, each of which exists between homogeneous things, the analogy of disproportion, on the other hand, shows a resemblance between relations of heterogeneity, between disproportions or between infinite distances: two things may be as different from each other as any two other things. Even if disproportions are a central theme to Pascal, he did not shy away of comparing such disproportions -- in particular to delimit what man cannot know perfectly.The second part analyzes the mathematical practice of Pascal "in weight, number and measure'': it is necessary to show that in the method of indivisibles of the Lettres de A. Dettonville, in the Traité du Triangle Arithmétique and in the comparison of the curved and the straight lines, always the infinite (or rather the indefinite) intervenes as a factor that allows the comparability of what would seem to be incomparable. The third part makes a philosophical discussion on the infinitely small and the infinitely large, taking into account Pascal's mathematical practice, which was analyzed in the second part. We discuss the nature of "indivisibles'', "differences'' and "infinite distances''. We suggest that the "infinite'' in Pascal's mathematical practice is rather an "indefinite'', linking it to a distinction between the absolute and the relative meaning of words. An exception in Pascal's mathematical practice is his projective geometry, where it is necessary to accept elements at an infinite distance. The "encounter'' of the two infinites makes it possible to show the reciprocity of the infinity of greatness and the infinity of smallness. Finally, we analyze the inverse proportionality between the two infinites with regard to the greatness and the wretchedness of man and to the paradoxical nature of certain truths according to Pascal, which are concealed in the person of the Christ. The conclusion is that Pascal arrives not at a direct knowledge of the infinite, but to an approach to the relation that man, a finite being, has with the infinite

Submitted by Luiz Carlos Leal Junior (jhcleal@gmail.com) on 2018-09-20T02:45:01Z No. of bitstreams: 1 Tese-merged.pdf: 3471711 bytes, checksum: fe5869f22f40e81b431c89e95dda7c88 (MD5) / Rejected by Adriana Aparecida Puerta null (dripuerta@rc.unesp.br), reason: Prezado Luiz Carlos, O documento enviado para a coleção Campus Unesp Rio Claro foi recusado pelo(s) seguinte(s) motivo(s): - Capa: Falta o ano de defesa (Página desconfigurada. Ano foi para a página seguinte). - Página de rosto: está sem Cidade e sem Ano de defesa ao final da página - Folha de aprovação está incompleta (Norma ABNT NBR 14724). Faltam os dados obrigatórios de: natureza (tese); nome da instituição à qual o trabalho é apresentado, indicando o título pretendido (mestre, doutor, bacharel, especialista etc). E também constar a informação "APROVADO" (deve ser solicitada à Seção de Pós-Graduação e deve ser inserida após a ficha catalográfica). - Agradecimentos: A Portaria nº 206, de 04/09/2018 Dispõe sobre obrigatoriedade de citação da CAPES nos agradecimentos da seguinte forma: "Art. 3º Deverão ser usadas as seguintes expressões, no idioma do trabalho: "O presente trabalho foi realizado com apoio da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Código de Financiamento 001" Agradecemos a compreensão e aguardamos o envio do novo arquivo. Atenciosamente, Biblioteca Campus Rio Claro Repositório Institucional UNESP https://repositorio.unesp.br on 2018-09-20T17:38:32Z (GMT) / Submitted by Luiz Carlos Leal Junior (jhcleal@gmail.com) on 2018-09-21T02:26:15Z No. of bitstreams: 1 Tese.pdf: 3603529 bytes, checksum: 9817094ff93cb248e58c8c43adc1d2ce (MD5) / Approved for entry into archive by Adriana Aparecida Puerta null (dripuerta@rc.unesp.br) on 2018-09-21T13:10:45Z (GMT) No. of bitstreams: 1 lealjunior_lc_dr_rcla.pdf: 3603529 bytes, checksum: 9817094ff93cb248e58c8c43adc1d2ce (MD5) / Made available in DSpace on 2018-09-21T13:10:45Z (GMT). No. of bitstreams: 1 lealjunior_lc_dr_rcla.pdf: 3603529 bytes, checksum: 9817094ff93cb248e58c8c43adc1d2ce (MD5) Previous issue date: 2018-09-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Práticas de Resolução de Problemas são problematizadas nesta pesquisa na forma de uma tessitura. Elas estão articuladas em torno de um estudo analítico acerca do tema Resolução de Problemas e de seus pressupostos filosóficos. Considerando-se a falta de explicitação e objetivação destes pressupostos teórico-filosóficos que amparam práticas em Resolução de Problemas, objetiva-se realizar um estudo analítico acerca dos discursos que permeiam, engendram, potencializam e põem em funcionamento práticas, teorias, teorizações e outros discursos sobre a Resolução de Problemas tanto no cenário nacional quanto internacional. Para tanto, procedemos a análise do discurso, pautada pela arqueogenealogia de Michel Foucault enquanto uma caixa de ferramentas, para compor uma análise com o corpus desta pesquisa, que consiste em entrevistas, questionários, artigos, livros, teses, dissertações e demais materiais acadêmicos. Uma questão diretriz a ser trabalhada nessa tessitura é: Como e quais pressupostos filosóficos operam, tessem ou põem em funcionamento discursos presentes nas pesquisas em Resolução de Problemas? Bem como seus desdobramentos sobre práticas discursivas relacionadas ao tema. Desse modo, observamos que há momentos, movimentos, práticas e discursos que possuem uma fundamentação teórica bastante consistente com os pressupostos filosóficos que lhes dão suporte. Por outro lado, há aqueles que não têm preocupações críveis com a teoria, residindo na práxis enquanto eixo estruturador de suas práticas em Resolução de Problemas. Há situações em que a Resolução de Problemas aproxima-se da égide de uma metodologia, enquanto que, em outro panorama, ela pode ser concebida como algo mais amplo e complexo, que visa dar conta de campos, elementos e conceitos problemáticos como: sujeito, objeto (matemático), sociedade, Educação Matemática, fazer e/ou ter ciência, valores, conhecimento (matemático), pedagogia, didática, enfim, uma gama de assuntos podem ser trabalhados sob essa perspectiva, aproximando-a de uma Filosofia da Educação Matemática. Isso permite inferir de alguma forma que, para entender-se a Resolução de Problemas, com seus princípios, bases e propostas de pesquisa educativa e educacional, faz-se extremamente necessário entender-se seus pressupostos teóricos, pois são eles que lhe darão o tom de algo restrito ou amplo, uma metodologia ou uma filosofia. Contudo, tal concepção será sempre local e regional, sendo ela validada e legitimada pela comunidade que a pratica. / Problem Solving practices are problematized in this research in the form of a “tessiture” . They are articulated around an analytical study on the subject of Problem Solving and its philosophical tenants. Considering the lack of conceptual understanding of the theoretical - philosophical presuppositions that bear on practices in Problem Solving, we aim to carry out an analytical study of the discourses th at permeate, engender and potentiate elements as: practices , theories, theorizations and other discourses on Problem Solving both on the national and international scene , besides running them . In order to do so, we proceeded to Michel Foucault’s discourse analysis based on archaeogenealogy, as a tool box, to compose an analysis with the corpus of this research, which consists of interviews, questionnaires, articles, books, theses, dissertations and other academic materials. A guiding question to be addressed in this “ tessiture” is: How and which philosophical presuppositions work or running discourses present in the researches in Problem Solving? As well as its implications on discursive practices related to the theme. Thus, we observe that there are moments, movements, practices and discourses that have a theoretical foundation very consistent with the philosophical tenants that support them. On the other hand, there are those who do not have credible concerns with theory, residing in praxis as the structuring axis of their practices in Problem Solving. There are situations in which Problem Solving fits into the aegis of a methodology . In another scenario , it can be conceived as something broader and more complex, which aims to deal with problematic fields, elements and concepts such as subject, mathematical objects, pedagogy, mathematical knowledge, society, didactics, finally, a range of subjects can be worked from this perspective, approaching it of a Philosophy of Mathematics Education. This allows us to infer, in some way, that in order to understand Problem Solving with its principles, bases and proposals for research, educational and educative practice , it is extremely necessary to understand its theoretical presuppositions, because that will give the tone of something restricted or broad, a methodology or a philosophy. However, such a conception will always be local and regional, being validated and legitimized by the community that practices it.