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[en] MATHEMATICS AND KNOWLEDGE IN THE PLATO S REPUBLIC / [pt] MATEMÁTICA E CONHECIMENTO NA REPÚBLICA DE PLATÃOALEXANDRE JORDAO BAPTISTA 18 June 2007 (has links)
[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.
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L'infini en poids, nombre et mesure : la comparaison des incomparables dans l'œuvre de Blaise Pascal / Infinity in weight, number and measure : the comparison of incomparables in the works of Blaise PascalFigueiredo Nobre Cortese, João 30 October 2017 (has links)
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
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Tessitura sobre discursos acerca de Resolução de Problemas e seus pressupostos filosóficos em Educação Matemática: cosi è, se vi pare / Tessiture on discourses about Problem Solving and their philosophical presuppositions in Mathematical Education: cosi è, se vi pareLeal Junior, Luiz Carlos [UNESP] 10 September 2018 (has links)
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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.
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Mathematical thinking: From cacophony to consensusArgyle, Sean Francis 09 August 2012 (has links)
No description available.
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"Presences of the infinite" : J.M. Coetzee and mathematicsJohnston, Peter January 2013 (has links)
This thesis articulates the resonances between J.M. Coetzee's lifelong engagement with mathematics and his practice as a novelist, critic, and poet. Though the critical discourse surrounding Coetzee's literary work continues to flourish, and though the basic details of his background in mathematics are now widely acknowledged, his inheritance from that background has not yet been the subject of a comprehensive and mathematically- literate account. In providing such an account, I propose that these two strands of his intellectual trajectory not only developed in parallel, but together engendered several of the characteristic qualities of his finest work. The structure of the thesis is essentially thematic, but is also broadly chronological. Chapter 1 focuses on Coetzee's poetry, charting the increasing involvement of mathematical concepts and methods in his practice and poetics between 1958 and 1979. Chapter 2 situates his master's thesis alongside archival materials from the early stages of his academic career, and thus traces the development of his philosophical interest in the migration of quantificatory metaphors into other conceptual domains. Concentrating on his doctoral thesis and a series of contemporaneous reviews, essays, and lecture notes, Chapter 3 details the calculated ambivalence with which he therein articulates, adopts, and challenges various statistical methods designed to disclose objective truth. Chapter 4 explores the thematisation of several mathematical concepts in Dusklands and In the Heart of the Country. Chapter Five considers Waiting for the Barbarians and Foe in the context provided by Coetzee's interest in the attempts of Isaac Newton to bridge the gap between natural language and the supposedly transparent language of mathematics. Finally, Chapter 6 locates in Elizabeth Costello and Diary of a Bad Year a cognitive approach to the use of mathematical concepts in ethics, politics, and aesthetics, and, by analogy, a central aspect of the challenge Coetzee's late fiction poses to the contemporary literary landscape.
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Proof, rigour and informality : a virtue account of mathematical knowledgeTanswell, Fenner Stanley January 2017 (has links)
This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour.
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Le statut des mathématiques en France au XVIe siècle : le cas d'Oronce Fine / The status of mathematics in France in the sixteenth Century : The case of Oronce FineAxworthy, Angela 05 December 2011 (has links)
Cette thèse se propose de déterminer les apports d’Oronce Fine (1494-1555) à la philosophie des mathématiques de la Renaissance. En tant que premier titulaire de la première chaire royale de mathématiques, ce mathématicien a joué un rôle important dans la revalorisation de l’enseignement des mathématiques dans la France du XVIe siècle. Dans cette mesure, sa conception des mathématiques permet de montrer l’évolution du statut épistémologique et institutionnel de ces disciplines dans le milieu académique parisien de cette période. Parmi les thèmes abordés par Fine dans sa définition du statut des mathématiques, nous avons choisi d’étudier, dans une première partie, la nature des objets du mathématicien, le statut épistémologique de l’astronomie, la nature des procédures démonstratives et des principes des mathématiques, ainsi que la fonction du quadrivium dans le processus éducatif. Dans une seconde partie, notre analyse de la pensée de Fine porte sur le statut des mathématiques pratiques et des disciplines subalternes des mathématiques, à savoir la perspective et la géométrie, ainsi que sur le profit qui peut être obtenu de l’apprentissage du quadrivium. / The aim of this study is to determine the contributions of Oronce Fine (1494-1555) to Renaissance philosophy of mathematics. As first Royal lecturer in mathematics, Fine played a major part in the reassertion of the value of mathematical teaching in sixteenth-century France. Thus, his thought concerning mathematics allows to set forth the evolution of the epistemological and institutional status of these sciences within the parisian academic context of the period. Among the questions tackled by Fine in his definition of the status of mathematics, we consider, in a first part, the ontological status of mathematical things, the epistemological status of astronomy, the nature of mathematical demonstrations and principles, as well as the function of the quadrivium in the educative process. In a second part, our analysis of Fine’s conception on mathematics deals with the status of practical mathematics and of the sciences which are subalternated to mathematics, that is optics and geography, concluding with the definition of the profit which may be obtained from learning mathematics.
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The foundations of linguistics : mathematics, models, and structuresNefdt, Ryan Mark January 2016 (has links)
The philosophy of linguistics is a rich philosophical domain which encompasses various disciplines. One of the aims of this thesis is to unite theoretical linguistics, the philosophy of language, the philosophy of science (particularly mathematics and modelling) and the ontology of language. Each part of the research presented here targets separate but related goals with the unified aim of bringing greater clarity to the foundations of linguistics from a philosophical perspective. Part I is devoted to the methodology of linguistics in terms of scientific modelling. I argue against both the Conceptualist and Platonist (as well as Pluralist) interpretations of linguistic theory by means of three grades of mathematical involvement for linguistic grammars. Part II explores the specific models of syntactic and semantics by an analogy with the harder sciences. In Part III, I develop a novel account of linguistic ontology and in the process comment on the type-token distinction, the role and connection with mathematics and the nature of linguistic objects. In this research, I offer a structural realist interpretation of linguistic methodology with a nuanced structuralist picture for its ontology. This proposal is informed by historical and current work in theoretical linguistics as well as philosophical views on ontology, scientific modelling and mathematics.
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Frege, Hilbert, and StructuralismBurke, Mark January 2015 (has links)
The central question of this thesis is: what is mathematics about? The answer arrived at by the thesis is an unsettling and unsatisfying one. By examining two of the most promising contemporary accounts of the nature of mathematics, I conclude that neither is as yet capable of giving us a conclusive answer to our question. The conclusion is arrived at by a combination of historical and conceptual analysis. It begins with the historical fact that, since the middle of the nineteenth century, mathematics has undergone a radical transformation. This transformation occurred in most branches of mathematics, but was perhaps most apparent in geometry. Earlier images of geometry understood it as the science of space. In the wake of the emergence of multiple distinct geometries and the realization that non-Euclidean geometries might lay claim to the description of physical space, the old picture of Euclidean geometry as the sole correct description of physical space was no longer tenable. The first chapter of the dissertation provides an historical account of some of the forces which led to the destabilization of the traditional picture of geometry. The second chapter examines the debate between Gottlob Frege and David Hilbert regarding the nature of geometry and axiomatics, ending with an argument suggesting that Hilbert’s views are ultimately unsatisfying. The third chapter continues to probe the work of Frege and, again, finds his explanations of the nature of mathematics troublingly unsatisfying. The end result of the first three chapters is that the Frege-Hilbert debate leaves us with an impasse: the traditional understanding of mathematics cannot hold, but neither can the two most promising modern accounts. The fourth and final chapter of the thesis investigates mathematical structuralism—a more recent development in the philosophy of mathematics—in order to see whether it can move us beyond the impasse of the Frege-Hilbert debate. Ultimately, it is argued that the contemporary debate between ‘assertoric’ structuralists and ‘algebraic’ structuralists recapitulates a form of the Frege-Hilbert impasse. The ultimate claim of the thesis, then, is that neither of the two most promising contemporary accounts can offer us a satisfying philosophical answer to the question ‘what is mathematics about?’.
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L’application des mathématiques aux phénomènes naturels chez LeibnizElawani, Jeffrey 08 1900 (has links)
Ce mémoire porte sur la réponse leibnizienne à la question de l’utilité des
mathématiques pour la connaissance de la nature, c’est-à-dire, en l’occurrence, pour la
connaissance des phénomènes corporels et de leurs relations. Dans le premier chapitre,
nous nous intéressons à la façon dont les notions abstraites mathématiques entrent dans la
connaissance la plus immédiate des choses. à travers le mode par lequel nous apparaît
l’individualité des phénomènes. Après avoir fourni des éclaircissements métaphysiques sur
la conception leibnizienne de l’individuation, nous nous plongeons dans l’étude de la
position spatiale à la lumière de l’analyse géométrique leibnizienne. Ce dernier prédicat
fournit une manière de déterminer les individus qui ne sont pas bien distingués par nous au
moyen de leurs qualités réelles. Considérés sous le seul angle de leur individuation spatiale,
les phénomènes ont un caractère idéal et indéterminé qui les rend immédiatement
susceptibles d’un traitement mathématique. Dans le second chapitre, nous nous intéressons
à la question de savoir pourquoi les explications physiques qui font usage des
mathématiques sont pour Leibniz préférables épistémologiquement. Nous nous tournons
en conséquence vers ses raisons d’adhérer à la philosophie mécanique, qui contient une
composante mathématique essentielle, afin d’étudier celle qui tient à la plus grande
intelligibilité du mécanisme. Nous tentons de montrer que la composante mathématique du
mécanisme contribue à cette intelligibilité parce que les mathématiques proposent une
mode de raisonnement valide et expressément adapté à la situation épistémologique des
esprits finis. Ce mode produit des raisonnements nécessaires aux moyens de notions
incomplètes. Il suscite également la découverte de nouvelles vérités en offrant à
l’imagination un support sensible, contrôlable et évident. / This thesis explores Leibniz’s solution to the problem of how mathematics are
useful to our understanding of the world, i.e., to our understanding of corporeal phenomena
and their relations. In the first chapter, it focuses on how abstract mathematical notions
enter in our most immediate understanding of the world. Here, the aim is connecting the
pervasiveness of mathematics to the peculiar way by which the individuality of phenomena
manifests itself to us. After some metaphysical remarks on Leibniz’s conception of
individuation, we study spatial position in the light of the new leibnizian geometrical
analysis : Analysis Situs. Spatial position provides us with a way to further distinguish
between individual phenomena whose qualities relevant to their real individuation remain
ignored. In the sole light of spatial individuation, phenomena are ideal and indeterminate.
This situation renders them susceptible to mathematical treatment without further
elaboration. In the second chapter, we turn our attention to the question of why
mathematical methods in philosophy of nature are epistemologically superior in Leibniz’s
eyes. We explore Leibniz’s reason to espouse a mechanical philosophy which comprise
indispensable mathematical notions. Leibniz believes that mechanical philosophy is the
most intelligible explanation of nature and we mean to assess how mathematics enter this
picture. We try to show that the mathematical aspects of mechanical philosophy make it
more intelligible by virtue of mathematics’ peculiar mode of reasoning. This mode of
reasoning is valid as well as most suited for our finite minds. It provides necessary
arguments through incomplete notions. It also encourages the discovery by assisting the
imagination with controlled and sensible support that makes knowledge more evident.
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