The principal focus of this research is a comprehensive defence of the theory of strict finitism as a foundation for mathematics. I have three broad aims in the thesis; firstly, to offer as complete and developed account of the theory of strict finitism as it has been described and discussed in the literature. I detail the commitments and claims of the theory, and discuss the best ways in which to present the theory. Secondly, I consider the main objections to strict finitism, in particular a number of claims that have been made to the effect that strict finitism is, as it stands, incoherent. Many of these claims I reject, but one, which focuses on the problematic notion of vagueness to which the strict finites seems committed, I suggest, calls for some revision or further development of the strict finitist’s position. The third part of this thesis is therefore concerned with such development, and I discuss various options for strict finitism, ranging from the development of a trivalent semantic, to a rejection of the commitment to vagueness in the first instance.

According to Plato, we live in a substitute world. The things we see around us are shadows of reality, imperfect imitations of perfect originals. Beyond the world of the senses, there is another, changeless world, more real and more beautiful than our own. But how can we get at this world, or attain knowledge of it, when our senses are unreliable and the perfect philosophical method remains out of reach? In the Divided Line passage of the Republic, Plato is clear that mathematics has a role to play, but the debate about the exact nature of that role remains unresolved. My reading of the Divided Line might provide the answer. I propose that the ‘mathematical’ passages of the Meno and Phaedo contain evidence that we can use to construct the method by which Plato means us to ascend to knowledge of the Forms. In this dissertation, I shall set out my reading of Plato’s Divided Line, and show how Plato’s use of mathematics in the Meno and Phaedo supports this view. The mathematical method, adapted to philosophy, is a central part of the Line’s ‘way up’ to the definitions of Forms that pure philosophy requires. I shall argue that this method is not, as some scholars think, the geometric method of analysis and synthesis, but apagōgē, or reduction. On this reading, mathematics is pivotal on our journey into the world of the Forms.

My thesis is an exposition and defence of Aristotle's philosophy of mathematics. The first part of my thesis is an exposition of Aristotle's cryptic and challenging view on mathematics and is based on remarks scattered all over the corpus aristotelicum. The thesis' central focus is on Aristotle's view on numbers rather than on geometrical figures. In particular, number is understood as a countable plurality and is always a number of something. I show that as a consequence the related concept of counting is based on units. In the second part of my thesis, I verify Aristotle's view on number by applying it to his account of time. Time presents itself as a perfect test case for this project because Aristotle defines time as a kind of number but also considers it as a continuum. Since numbers and continuous things are mutually exclusive this observation seems to lead to an apparent contradiction. I show why a contradiction does not arise when we understand Aristotle properly. In the third part, I argue that the ontological status of mathematical objects, dubbed as materially [hulekos, ÍlekÀc] by Aristotle, can only be defended as an alternative to Platonism if mathematical objects exist potentially enmattered in physical objects. In the fourth part, I compare Aristotle's and Plato's views on how we obtain knowledge of mathematical objects. The fifth part is an extension of my comparison between Aristotle's and Plato's epistemological views to their respective ontological views regarding mathematics. In the last part of my thesis I bring Frege's view on numbers into play and engage with Plato, Aristotle and Frege equally while exploring their ontological commitments to mathematical objects. Specifically, I argue that Frege should not be mistaken for a historical Platonist and that we find surprisingly many similarities between Frege and Aristotle. After having acknowledged commonalities between Aristotle and Frege, I turn to the most significant differences in their views. Finally, I defend Aristotle's abstractionism in mathematics against Frege's counting block argument. This whole project sheds more light on Aristotle's view on mathematical objects and explains why it remains an attractive view in the philosophy of mathematics.

Made available in DSpace on 2016-04-27T16:58:16Z (GMT). No. of bitstreams: 1 Fernando Eduardo de Souza.pdf: 10643917 bytes, checksum: e615f7376e8504d0923f2af03f15cfd4 (MD5) Previous issue date: 2007-05-11 / Secretaria da Educação do Estado de São Paulo / It discipline Differential and Integral Calculus appear of the resume of some courses of the area of Exacts or Human Sciences, such as Engineering, Physics, Chemistry, Computer Sciences, Administration and others. Its teaching if has supported many times in one practical "traditional" methodological based in: definitions, theorems, properties, examples and exercises. Innumerable research demonstrates that this methodology has resulted in a very high index of retention. The present work search to analyze the relations between the conceptions on the concept of Integral disclosed for professors, as well as its ways to analyze the production of the pupils, of form to get practical indications on the educative ones of these professionals. To deal of this objective, we use two methodological instruments, a questionnaire and interviews. The first one, objectifying to produce information for the interview, was applied the thirty pupils of two particular universities of São Paulo, having approached different aspects of the concept of Integral, as: definitions, representations, techniques of integration and applications. We analyze the answers produced to the light of the theoretical reference of Concept Image and Concept Definition of Tall & Vinner. The interviews had been carried through with three professors of a particular university of São Paulo, based in the methodological proposals of Bogdan & Binklen and Gaskell, G. on interviews in group and had been analyzed in accordance with the ideas of Paul Ernest concerning the Philosophy Absolutist and Fallibilist Philosophy of the Mathematics. We understand that the conceptions supported for the professors if approach more to the view absolutist of the mathematics, therefore in the majority of the productions analyzed, all seem to accept that this science is the domain of the absolute truths and that the knowledge in mathematics consists of descriptions of the mathematical beings, of the relations between them and of the logical structure that supports them. However, the professors interviewed manifest the possibility of that the mathematical knowledge be it fallible or opened the critical and corrections / A disciplina Cálculo Diferencial e Integral consta do currículo de vários cursos da área de Ciências Exatas ou Humanas, tais como Engenharia, Física, Química, Ciências da Computação, Administração e outros. Seu ensino tem se apoiado muitas vezes numa prática metodológica tradicional baseada em: definições, teoremas, propriedades, exemplos e exercícios. Inúmeras pesquisas demonstram que esta metodologia tem redundado em um índice muito alto de retenção. O presente trabalho busca analisar as relações entre as concepções sobre o conceito de Integral revelada por professores, bem como suas maneiras de analisarem a produção dos alunos, de forma a obtermos indícios sobre as práticas educativas desses profissionais. Para atender a esse objetivo, utilizamos dois instrumentos metodológicos, um questionário e entrevistas. O primeiro, objetivando produzir dados para as entrevista, foi aplicado a trinta alunos de duas universidades particulares de São Paulo, abordando diferentes aspectos do conceito de Integral, como: definições, representações, técnicas de integração e aplicações. Analisamos as respostas produzidas à luz do referencial teórico de Conceito Imagem e Conceito Definição de Tall & Vinner. As entrevistas foram realizadas com três professores de uma universidade particular de São Paulo, baseadas nas propostas metodológicas de Bogdan & Binklen e de Gaskell, G. sobre entrevistas em grupo e foram analisadas de acordo com as idéias de Paul Ernest acerca da Filosofia Absolutista e Filosofia Falibilista da Matemática. Notamos que as concepções sustentadas pelos professores se aproximam mais à visão absolutista da matemática, pois na maioria das produções analisadas, todos parecem aceitar que essa ciência é o domínio das verdades absolutas e que o conhecimento em matemática consiste em descrições dos entes matemáticos, das relações entre eles e da estrutura lógica que os sustenta. No entanto, os professores entrevistados manifestam a possibilidade de que o conhecimento matemático seja falível ou esteja aberto a críticas e correções

Made available in DSpace on 2016-08-17T13:54:14Z (GMT). No. of bitstreams: 1 RAIMUNDO SANTOS DE CASTRO.pdf: 868577 bytes, checksum: 1cb44fa7dd80f6f4b2afebd98e83b94e (MD5) Previous issue date: 2009-05-15 / In this study, it is analyzed the conceptions of Mathematics sustained by students of the last period of the course of Degree in Mathematics of the Federal Center of Technological Education of Maranhão, thirst São Luis, and their implications for the pedagogic practice of the future mathematical educator. For his accomplishment of this research was ruled in the qualitative approach, once this has for concern basic to bring the present reality the study object in a dynamic social reality, intextualizando relationships, interactions and implications proceeding of that. For so much, it was made use of the semi-structured interview while methodological procedure. The historical constitution of the teacher's of Mathematics formation is characterized in Brazil, trying to identify the conception of present Mathematics in their phases. It is discussed the Mathematical Education, the Philosophy of the Mathematical Education and the foundations of the school Mathematics, his teaching and her learning. Identifies to the light of the Philosophy of the Mathematics, of the Philosophy of the Mathematical Education, of the History of the Mathematics and of the Mathematical Education, the conceptions of Mathematics sustained by the conclusive students of the Course of Degree in Mathematics of CEFET-MA. Finally, search to discuss and to analyze the possible implications of the conceptions of Mathematics for the educational practice of the future mathematical educator. The reflection about the teacher's conceptions and on the current social practices of such conceptions it can point us the roads for the search of improvements of the teaching that will bring impacts the learning of the and in the Mathematics. In the perceptions of the totality of the subject of the research, some exist points that converge for a Mathematics of specific sense, difficult, disentailed of the external world to her. This sends us to the understanding that there is an absolutist inclination in their conceptions regarding the theme in subject. However, a change perspective appears while horizon. Being, therefore, possible to affirm that the conceptions concerning the Mathematics, his teaching and her learning, sustained by the students come in transition period of an absolutist conception for one that takes into account the produced mathematical knowledge while to know human and with significant applicability in the social contexts out of the school. / Neste estudo, analisam-se as concepções de Matemática sustentadas por estudantes do último período do curso de Licenciatura em Matemática do Centro Federal de Educação Tecnológica do Maranhão, sede São Luís, e suas implicações para a prática pedagógica do futuro educador matemático. A tecitura desta pesquisa pautou-se na abordagem qualitativa, uma vez que esta tem por preocupação básica contextualizar o objeto de estudo numa realidade social dinâmica, intertextualizando relações, interações e implicações advindas daquela. Para tanto, fez-se uso da entrevista semiestruturada enquanto procedimento metodológico. Caracteriza-se a constituição histórica da formação de professores de Matemática no Brasil, tentando identificar a concepção de Matemática presente em suas fases. Discute-se a Educação Matemática, a Filosofia da Educação Matemática e os fundamentos da Matemática escolar, o seu ensino e sua aprendizagem. Identifica-se, à luz da Filosofia da Matemática, da Filosofia da Educação Matemática, da História da Matemática e da Educação Matemática, as concepções de Matemática sustentadas pelos estudantes concludentes do Curso de Licenciatura em Matemática do CEFET-MA. Por fim, busca-se discutir e analisar as possíveis implicações das concepções de Matemática para a prática docente do futuro educador matemático. A reflexão sobre as concepções dos professores e sobre as práticas sociais decorrentes de tais concepções poderá nos apontar os caminhos para a busca de melhorias do ensino que impactará a aprendizagem da e na Matemática. Nas percepções da totalidade dos sujeitos da pesquisa, existem alguns pontos que convergem para uma Matemática de sentido específico, laboral, desvinculada do mundo exterior a ela. Isto nos remete ao entendimento de que há um viés absolutista em suas concepções a respeito do tema em questão. Contudo, uma perspectiva de mudança surge enquanto horizonte. Sendo, portanto, possível afirmar que as concepções acerca da Matemática, seu ensino e sua aprendizagem, sustentadas pelos estudantes apresentam-se em fase de transição de uma concepção absolutista para uma que leva em consideração o conhecimento matemático produzido como saber humano e com aplicabilidade significativa nos contextos sociais fora da escola.