[pt] O objetivo dessa tese é expor a base filosófica por trás das propostas de Wittgenstein para a filosofia da matemática. Procuramos mostrar que há um núcleo semântico das quais essas propostas são derivadas: a idéia de que o significado de uma sentença deveria ser tomado como sendo suas condições de verdade. Procuramos acompanhar esse insight desde seu aparecimento em Frege, depois no Tractatus até a última fase do pensamento de Wittgenstein. Em nossos dois últimos capítulos discutimos as mudanças que essa abordagem sofreu nessa última fase do pensamento do filósofo em conexão com a idéia de triangularização da semântica subjacente. Tratamos em com algum nível de detalhamento as interpretações propostas por Wittgenstein de alguns tipos de proposições matemáticas elementares, especialmente proposições aritméticas. / [en] This thesis deals with the philosophical basis behind Wittgenstein s proposals in the philosophy of mathematics. We try to show that there is a semantic core from which these proposals are derived: the idea that the meaning of a sentence should be taken to be its truth conditions. We sort out this insight from its inception in Frege through the Tractatus all the way to the last phase of Wittgenstein s thought. In our last two chapters we discuss the changes this approach went through in this last phase of the philosopher s thought in relation to the idea of triangularization of the underlying semantics. We then deal in some detail with Wittgenstein s proposed interpretation of some types of elementary mathematical propositions, especially arithmetical propositions.

[pt] A partir das reflexões de Frege em seus Grundlagen der Arithmetik, destaca-se como o fio condutor da presente dissertação o problema que se refere à determinação da natureza numérica. A análise que Frege dedica a este problema almeja caracterizar a noção de número com o auxílio da noção de objeto lógico, e tal aproximação receberá um intenso ataque teórico por parte de Paul Benacerraf. Este ataque teórico, por sua vez, será auxiliado pela sugestão de que, na medida em que o interesse dos matemáticos (enquanto matemáticos) permanece em um âmbito estruturalista, uma pesquisa filosófica sobre a matemática deveria, em princípio, preservar semelhante aspecto. Esta argumentação de Benacerraf, aliada aos trabalhos anteriores do grupo Bourbaki, impulsionou as mais diversas pesquisas no âmbito estruturalista, constituindo-se os trabalhos de Stewart Shapiro como um de seus mais interessantes desenvolvimentos. São assim apresentadas as bases ontológicas e epistemológicas que sustentam tal teoria – intitulada como estruturalismo ante rem – para, por fim, se delinear um histórico de tais concepções. Sublinham-se, então, alguns debates ocorridos ao final do século XIX e início do século XX, com a intenção de se iluminar as heranças conceituais de Shapiro e alguns possíveis pressupostos de Benacerraf, oferecendo-se alguma notoriedade às figuras do matemático alemão Richard Dedekind e do grupo francês Nicholas Bourbaki. / [en] Frege’s reflections on the nature of numbers, presented in his Grundlagen der Arithmetik (1884), provide the guiding theme for this dissertation. His characterization of numbers as logical objects was sharply criticized by Paul Benacerraf (1965), who suggested that since mathematicians (as mathematicians) work within a structural framework, philosophical research on mathematics should preserve this structural aspect of mathematics. Benacerraf’s arguments, as well as earlier works by the Bourbaki group, gave raise to several structuralist research projects, one of the most interesting of which is developed by Stewart Shapiro. We present the ontological and epistemological basis of Shapiro’s socalled ante rem structuralism, as well as some of the debates from the late 19th and early 20th centuries which throw some light on the conceptual presuppositions of Shapiro and Benacerraf. In this connection we emphasize the work of the German mathematician Richard Dedekind and of the French group Nicholas Bourbaki.

<p>This paper studies the philosopher Ludwig Wittgenstein during his years (1920–26) as an elementary school teacher in remote Niederösterreich, Austria. The paper gives a survey of his life, and also a brief account of three of his main works: Tractatus Logico-Philosophicus, Philosophical Investigations and Remarks on the Foundations of Mathematics. Attention is given to his alphabetical word list, Wörterbuch für Volksschulen, published for educational use in elementary schools. The study is focused on Wittgenstein’s educational practise, and establishes a connection between his experience as a teacher and his late philosophy.</p>

This paper studies the philosopher Ludwig Wittgenstein during his years (1920–26) as an elementary school teacher in remote Niederösterreich, Austria. The paper gives a survey of his life, and also a brief account of three of his main works: Tractatus Logico-Philosophicus, Philosophical Investigations and Remarks on the Foundations of Mathematics. Attention is given to his alphabetical word list, Wörterbuch für Volksschulen, published for educational use in elementary schools. The study is focused on Wittgenstein’s educational practise, and establishes a connection between his experience as a teacher and his late philosophy.

Current reform in mathematics teaching and learning is rooted in a changing vision of school mathematics, one that includes constructivist learning, student-centered pedagogy, and the use of worthwhile tasks (National Council of Teachers of Mathematics, 1989, 1991, 2000). This changing vision not only challenges teachers’ beliefs about mathematics instruction but their philosophies of mathematics as well (Dossey, 1992). This study investigates the processes that four teachers’ go through as they implement a new task-based mathematics curriculum while exploring their personal philosophies of mathematics. The participants were part of a graduate-level course that examined, through the writings of Davis and Hersh (1981), Lakatos (1976), Polya (1945/1973), and others, a humanist/fallibilist philosophy of mathematics. These participants shared, through reflective writings and interviews, their struggles to, first, define mathematics and its purpose in society and in schools, and second, to reconcile their views of mathematics with their instructional practices. The study took place as the participants, two classroom teachers and two instructional coaches, engaged in the initial implementation of a reform mathematics curriculum, a reform based in social constructivist learning theories. Using narrative analysis, this study focuses on the unique mathematical stories of four experienced educators. Each of the participants initially expressed a traditional, a priori view of mathematics, seeing mathematics as right/wrong, black/white, a subject outside of human construction. The participants’ expressed views of mathematics changed as they attempted to align their personal philosophies of mathematics with their (changing) classroom practices. They shared their personal struggles to redefine themselves as mathematics teachers through a process of experimentation, reflection, and adaptation. This process was echoed in their changing philosophies of mathematics. These participants came to see mathematics as fluid and a human construct; they also came to see their philosophies of mathematics as fluid and ever-changing, a process more than a product.

Traditionally, mathematics and metaphor have been thought of as disparate: the former rigorous, objective, universal, eternal, and fundamental; the latter imprecise, derivative, nearly - if not patently - false, and therefore of merely aesthetic value, at best. A growing amount of contemporary scholarship argues that both of these characterizations are flawed. This dissertation shows that there are important connexions between mathematics and metaphor that benefit our understanding of both. A historically structured overview of traditional theories of metaphor reveals it to be a notion that is complicated, controversial, and inadequately understood; this motivates a non-traditional approach. Paradigmatically shifting the locus of metaphor from the linguistic to the conceptual - as George Lakoff, Mark Johnson, and many other contemporary metaphor scholars do - overcomes problems plaguing traditional theories and promisingly advances our understanding of both metaphor and of concepts. It is argued that conceptual metaphor plays a key role in explaining how mathematics is grounded, and simultaneously provides a mechanism for reconciling and integrating the strengths of traditional theories of mathematics usually understood as mutually incompatible. Conversely, it is shown that metaphor can be usefully and consistently understood in terms of mathematics. However, instead of developing a rigorous mathematical model of metaphor, the unorthodox approach of applying mathematical concepts metaphorically is defended.

In this dissertation on Hilary Putnam's philosophy, I investigate his development regarding meaning and necessity, in particular mathematical necessity. Putnam has been a leading American philosopher since the end of the 1950s, becoming famous in the 1960s within the school of analytic philosophy, associated in particular with the philosophy of science and the philosophy of language. Under the influence of W.V. Quine, Putnam challenged the logical positivism/empiricism that had become strong in America after World War II, with influential exponents such as Rudolf Carnap and Hans Reichenbach. Putnam agreed with Quine that there are no absolute a priori truths. In particular, he was critical of the notion of truth by convention. Instead he developed a notion of relative a priori truth, that is, a notion of necessary truth with respect to a body of knowledge, or a conceptual scheme. Putnam's position on necessity has developed over the years and has always been connected to his important contributions to the philosophy of meaning. I study Hilary Putnam's development through an early phase of scientific realism, a middle phase of internal realism, and his later position of a natural or commonsense realism. I challenge some of Putnam’s ideas on mathematical necessity, although I have largely defended his views against some other contemporary major philosophers; for instance, I defend his conceptual relativism, his conceptual pluralism, as well as his analysis of the realism/anti-realism debate.

Orientador: Marcelo de Carvalho Borba / Banca: Maria Aparecida Viggiani Bicudo / Banca: Nilson José Machado / Banca: Siobhan Victoria Healy / Banca: Marcus Vinicius Maltempi / Resumo: Os processos perceptivos que fundamentam a experiência humana podem nos parecer absolutamente naturais. Devido a isso, costumamos não tematizá-los. Este trabalho é um esforço de evidenciação da percepção na educação matemática, mais especificamente na geometria dinâmica e no ensino de funções com auxílio de representações dinâmicas. Percepção é entendida em uma concepção fenomenológica. Sustenta-se que: (a) é da natureza humana certa capacidade de perceber comportamentos de dependências entre eventos do mundo físico, isto é, existe um sentido de percepção de dependência; (b) as representações dinâmicas de funções, como os Dynagraphs (conhecidos na literatura) e os funcionetes (propostos no trabalho), são depreendidas pelo sentido de percepção de dependência; (c) o emprego de representações dinâmicas no auxílio ao ensino de funções abre novos sentidos para funções matemáticas, conceitos, propriedades e teoremas correlatos, justificando o interesse em sua aplicação; além disso, os sentidos abertos são perceptivos e, por isso, diretos, imediatos e evidentes (conforme a fundamentação fenomenológica); (d) existe um sentido de percepção de restrições ou impedimentos; (e) na resolução interativa (geometria dinâmica) de sistemas de restrições geométricas, o sentido de percepção de restrições apresenta, ao trazer perceptivamente as restrições para primeiro plano, as construções geométricas como uma combinação de restrições. No desenvolvimento dessas ideias: apresentam-se os funcionetes planos e sua aplicação na construção de uma abordagem pedagógica para o conceito (da álgebra linear) transformação linear, que é um tipo de função; abordam-se os tópicos: autovetores de um operador linear, propriedade de linearidade e núcleo de uma transformação linear, inclusive o teorema do núcleo e da imagem... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The perceptive processes that provide the basis for human experience can seem absolutely natural. Therefore we do not have the habit of focusing on them as the object of study. The present theoretical study aimed to make perception evident in the context of mathematics education, specifically dynamic geometry and teaching of functions using dynamic representations. Perception is understood as a phenomenological conception. It is maintained that: (a) it is of human nature to be able to perceive dependent behaviors among events in the physical world, i.e. a sense of perception of dependence; (b) dynamic representations of functions, such as Dynagraphs (known in the literature) and "funcionetes" (proposed here), are ascertained through the sense of perception of dependence; (c) the use of dynamic representations to aid in the teaching of functions opens up new senses for mathematical functions, concepts, properties and correlated theorems, justifying interest in its application; in addition, these newly-opened senses are perceptive in nature, and therefore direct, immediate and evident (according to foundations of phenomenology); (d) there exists a sense of perception of constraints or impediments; (e) in the interactive resolution (dynamic geometry) of constraint systems for geometric domain, the sense of perception of constraints presents geometric constructions as a combination of constraints as it perceptively brings the constraints to the foreground. The concept of "funcionetes planos" is presented and their use proposed as part of an approach for teaching the concept (from linear algebra) of linear transformation, which is a type of function. Topics addressed include: eigenvectors of a linear operator, the property of linearity and nucleus of a linear transformation, including the theorem of nucleus and of image, presented in a perceptive sense... (Complete abstract click electronic access below) / Doutor

No description available.

Made available in DSpace on 2014-06-11T19:31:43Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-11-03Bitstream added on 2014-06-13T20:02:50Z : No. of bitstreams: 1 figueiredo_oa_dr_rcla.pdf: 599343 bytes, checksum: 2107a2036f27ecde0dcfdbff1f4a2bf3 (MD5) / Os processos perceptivos que fundamentam a experiência humana podem nos parecer absolutamente naturais. Devido a isso, costumamos não tematizá-los. Este trabalho é um esforço de evidenciação da percepção na educação matemática, mais especificamente na geometria dinâmica e no ensino de funções com auxílio de representações dinâmicas. Percepção é entendida em uma concepção fenomenológica. Sustenta-se que: (a) é da natureza humana certa capacidade de perceber comportamentos de dependências entre eventos do mundo físico, isto é, existe um sentido de percepção de dependência; (b) as representações dinâmicas de funções, como os Dynagraphs (conhecidos na literatura) e os funcionetes (propostos no trabalho), são depreendidas pelo sentido de percepção de dependência; (c) o emprego de representações dinâmicas no auxílio ao ensino de funções abre novos sentidos para funções matemáticas, conceitos, propriedades e teoremas correlatos, justificando o interesse em sua aplicação; além disso, os sentidos abertos são perceptivos e, por isso, diretos, imediatos e evidentes (conforme a fundamentação fenomenológica); (d) existe um sentido de percepção de restrições ou impedimentos; (e) na resolução interativa (geometria dinâmica) de sistemas de restrições geométricas, o sentido de percepção de restrições apresenta, ao trazer perceptivamente as restrições para primeiro plano, as construções geométricas como uma combinação de restrições. No desenvolvimento dessas ideias: apresentam-se os funcionetes planos e sua aplicação na construção de uma abordagem pedagógica para o conceito (da álgebra linear) transformação linear, que é um tipo de função; abordam-se os tópicos: autovetores de um operador linear, propriedade de linearidade e núcleo de uma transformação linear, inclusive o teorema do núcleo e da imagem... / The perceptive processes that provide the basis for human experience can seem absolutely natural. Therefore we do not have the habit of focusing on them as the object of study. The present theoretical study aimed to make perception evident in the context of mathematics education, specifically dynamic geometry and teaching of functions using dynamic representations. Perception is understood as a phenomenological conception. It is maintained that: (a) it is of human nature to be able to perceive dependent behaviors among events in the physical world, i.e. a sense of perception of dependence; (b) dynamic representations of functions, such as Dynagraphs (known in the literature) and “funcionetes” (proposed here), are ascertained through the sense of perception of dependence; (c) the use of dynamic representations to aid in the teaching of functions opens up new senses for mathematical functions, concepts, properties and correlated theorems, justifying interest in its application; in addition, these newly-opened senses are perceptive in nature, and therefore direct, immediate and evident (according to foundations of phenomenology); (d) there exists a sense of perception of constraints or impediments; (e) in the interactive resolution (dynamic geometry) of constraint systems for geometric domain, the sense of perception of constraints presents geometric constructions as a combination of constraints as it perceptively brings the constraints to the foreground. The concept of “funcionetes planos” is presented and their use proposed as part of an approach for teaching the concept (from linear algebra) of linear transformation, which is a type of function. Topics addressed include: eigenvectors of a linear operator, the property of linearity and nucleus of a linear transformation, including the theorem of nucleus and of image, presented in a perceptive sense... (Complete abstract click electronic access below)