<p dir="ltr">The concept of infinity plays a pivotal role in mathematics, yet its precise definition remains elusive. This conceptual ambiguity has given rise to several puzzles in contemporary philosophy of mathematics. In response, this dissertation embarks on a rational reconstruction of Hegels concept of infinity and applies it to resolve two groups of mathematical puzzles, including challenges in applied mathematics, especially the application of differential calculus, and the conceptual ground of set theory, especially Cantors paradox.</p><p dir="ltr">The exploration begins with a historical survey of the concept of infinity in philosophy. It becomes evident that a prevailing interpretation characterizes infinity as the unlimited. In addition, this unlimitedness has taken various forms, including endlessness (Aristotle), all-inclusiveness (Spinoza), and self-sufficiency (Kant).</p><p dir="ltr">The heart of the dissertation lies in reconstructing Hegels concept of genuine infinity. Hegel argues that the unlimited as the negation of the limit entails either the completely indeterminate or another limited entity, neither of which is genuinely infinite. Instead, Hegel points out that genuine infinity is the self-relation of a limited entity. By self-relation, Hegel means that the limited entity alters into another limited entity that is isomorphic to the original one.</p><p dir="ltr">Subsequently, Hegel’s concept of genuine infinity can be translated into a mathematical framework as the intrinsic alteration of quantum (roughly speaking, quantum is Hegel’s term for the variable), which is captured by the corresponding relation among quanta. It is argued that this relation serves as the necessary condition for three mathematical entities traditionally considered infinite: arbitrarily large (small) numbers, infinite sets, and endless sequences. Thus, for Hegel, this intrinsic relation among quanta constitutes the essence of mathematical infinity.</p><p dir="ltr">Hegels concept of mathematical infinity can help us resolve difficulties within contemporary mathematics. First, it addresses the question of why infinite mathematical structures can be applied to describe and predict seemingly finite physical phenomena. The application of mathematics is usually explained by the similarity between mathematical structures and empirical systems, but the lack of apparent empirical counterpart leads one to doubt the application of infinite mathematical structures. Hegels concept of mathematical infinity directs us to focus on the structural similarity between infinite mathematical structures and empirical systems, specifically between the intrinsic alteration of quantum and the change of physical properties with time. With this structural similarity, the application of mathematics can be explained. Second, the dissertation investigates the conceptual ground of set theory, especially the relationship between a set and its members. Hegels analysis of genuine infinity provides a twofold clarification: (1) members of set must be a unit first, which entails that the set of all sets (the Universe) is not a set; (2) members of a set are simultaneously distinct (due to their independent logical content) yet indistinguishable (due to their common structure as a unit). Clarification 1 resolves Cantors paradox as it excludes the Universe; clarification 2 explains arithmetic operations.</p>

Made available in DSpace on 2016-06-02T20:02:50Z (GMT). No. of bitstreams: 1 4511.pdf: 2657557 bytes, checksum: 0bd43e75994792068eae8c02c5edee1a (MD5) Previous issue date: 2012-03-30 / Financiadora de Estudos e Projetos / The main goal of this investigation is to analyze students&#8223; speeches and writings of students in the 9th year (Ensino Fundamental) about the concept of area , from guiding educational activities (Moura, 1996), involving the contents of notable areas of polygons:rectangles, triangles, , parallelogram, trapezoid triangle and losanges, including, the composition and the decomposition of plane figures. The activities constitute a puzzle. They were elaborated and developed by the researcher in the classroom and in the context of a marble yard.The investigation is qualitative and it can be characterized as a case study. It was conducted by the researcher the whole time, since he is the class teacher. It means, the teacher not only observed the class but also took notes of the movement that happened in the class. The question guiding the study is: what do students from the 9th year (Ensino Fundamenta) say and write about the area concept while they really live those guiding educational activities, inside the class and also in a marble yard context? The analysis of speech and writing were made by observing the categories related to the contents. As a result, the researcher tried to gather information, identify and comprehend, from the speeches and writings, what the students learned by showing it, as well as their difficulties or still, the relations that they make between what happens inside class and every day, giving emphasis on the use of formulas for calculating the areas and the application of these ones in the marble yard. It was also produced: guiding educational activities about the polygons area concepts.It is noteworthy, that this investigation summarizes theoretical and methodological the movements that happened inside the class and also in the marble yard. / O objetivo desta investigação é identificar e compreender o processo de apropriação e construção do conceito de área, por estudantes do 9º ano do Ensino Fundamental, analisando suas falas e suas escritas, a partir de atividades orientadoras de ensino (Moura, 1996) que envolvem os conteúdos de áreas dos polígonos notáveis: Retângulo, Triângulo, Paralelogramo, Trapézio e Losango, incluindo-se aí, a composição e a decomposição de figuras planas. As atividades se constituem por quebracabeças. Foram elaboradas pelo pesquisador , desenvolvidas na sala de aula e no contexto de uma Marmoraria. A investigação é qualitativa e pode ser caracterizada como estudo de caso. Foi conduzida pelo pesquisador em todos os momentos, uma vez que este é o professor da sala, ou seja, o professor não se limitou apenas a observar e a anotar o movimento ocorrido na sala de aula. A questão que norteia o estudo é: o que estudantes do 9º ano do Ensino Fundamental falam e escrevem sobre o conceito de área enquanto vivenciam atividades orientadoras de ensino, tanto na sala de aula, quanto no contexto de uma marmoraria? A análise das falas e das escritas foi feita mediante categorias de análise relacionadas aos conteúdos envolvidos. Como resultado do trabalho, procurou-se reunir informações, identificar e compreender, a partir das escritas e das falas, o que os estudantes evidenciam que aprenderam, bem como suas dificuldades ou ainda as relações que fazem entre o que ocorre na sala de aula e no cotidiano, destacando-se o uso das fórmulas para o cálculo de áreas e aplicação destas na marmoraria. Produziram-se ainda, atividades orientadoras de ensino sobre os conceitos de área de polígonos. Ressalta-se que esta investigação sintetiza teórica e metodologicamente os movimentos ocorridos tanto na sala de aula, quanto na marmoraria.

The Theses deals with applications of mathematical knowledge in teaching biology and it is divided into four chapters. Each chapter is dedicated to another application: 1. Application of conditional probability in medical diagnostics, 2. Application of exponential function in population ecology, 3. Application of logic functions in mathematical modelation of neuron and 4. Aplication of binomial theorem and binomial distribution in genetics. Each application contains solved problems, a worksheet for students and a solution for each worksheet. Two application (1. and 2.) have been tested in teaching and as an assessment of my lessons students filled questionnaires. Results of these questionnaires are processed in the end of these chapters. This Thesis can be used in teaching or self-studying.