Läroboken styr till stor del vilket innehåll som behandlas i matematikundervisningen. Med detta i åtanke har fem svenska läroboksserier har utsatts för en latent och manifest innehållsanalys av hur de presenterar de aritmetiska räknelagarna och räknereglerna. I studien framkommer både kvantitativ och kvalitativ data. Den kvantitativa datan indikerar att få läroboksserier tar upp associativa och distributiva lagen explicit. Den kvalitativa datan pekar på att räknelagarna ofta beskrivs i andra sammanhang. Flera exempel i läroböckerna gör generaliseringar som riskerar leda till begränsad förståelse för räknelagarna och räknereglerna. / In mathematics education textbooks to a large extent determine what is offered for students to be learnt. With this in mind, in this study, five Swedish textbooks series is reviewed in a latent and manifest content analysis approach where both quantitative and qualitative data is presented. The result of the quantitative data indicate that only a few textbooks series mentions the associative and distributive law in explicit manners. The result of the qualitative data shows that the basic laws of arithmetic is often described in other contexts. Many examples in the textbooks makes generalizations that may lead to limited understanding of the basic laws and rules of arithmetic.

Frege wanted to show that arithmetical truths are logical by proving them from purely logical basic laws. But how do we know that these basic laws are logical? Frege uses generality and undeniability to make a prima facie case for logicality—if a truth is general and undeniable, then it's likely logical. I argue that Frege could, did, and had to make a deeper case for why we're right in recognizing his basic laws as logical. Implicit in his work is a view of logical laws as epistemically analytic—his arguments for his basic laws serve to elicit a reflective awareness of the fact that understanding them is sufficient for recognizing them to be true. This view both fits with Frege's comments concerning the connection between logic, truth, and normativity, and serves to explain why and in what sense he took logic to be general and conceptually undeniable. In my view, semantics must play a distinctive role in any rational reconstruction of Frege's case for logicality—the aforementioned “reflective awareness” must be an explicit appreciation of how the truth of formulas expressing Frege's laws follows quickly from his stipulations governing terms which figure in those formulas. Opposing this view is the elucidatory interpretation of Thomas Ricketts, Warren Goldfarb, and Joan Weiner, which holds that Frege's arguments for his basic laws can't be taken at face value, and must serve the merely elucidatory purpose of easing us into the language. Another reading is the correctness interpretation of Richard Heck and Jason Stanley, which holds that Frege's primary purpose in his arguments is justifying the claim that Frege's axioms, qua formulas, are true. I argue against both of these interpretations, and in doing so clarify the role and limits of semantics in Frege's enterprise.

Made available in DSpace on 2016-04-26T20:27:26Z (GMT). No. of bitstreams: 1 Jose Claudio Pavao Santana.pdf: 859518 bytes, checksum: 34a608dbe3d0c18fb57e8d614a6a6ef7 (MD5) Previous issue date: 2008-06-06 / The present work deals with the formation of the constitutionalism in the American continent. It argues the daily payconstitutionalism and the nature of the Basic Laws of the Maranhão as contribution to the constitutionalism / O presente trabalho trata da formação do constitucionalismo no Continente Americano. Discute o préconstitucionalismo e a natureza das Leis Fundamentais do Maranhão como contribuição ao constitucionalismo

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-26T14:58:18Z No. of bitstreams: 1 DissAGFC.pdf: 1978531 bytes, checksum: e57d2335b2038eca2d8a6468869e05fa (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-27T19:36:51Z (GMT) No. of bitstreams: 1 DissAGFC.pdf: 1978531 bytes, checksum: e57d2335b2038eca2d8a6468869e05fa (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-27T19:36:56Z (GMT) No. of bitstreams: 1 DissAGFC.pdf: 1978531 bytes, checksum: e57d2335b2038eca2d8a6468869e05fa (MD5) / Made available in DSpace on 2016-09-27T19:37:02Z (GMT). No. of bitstreams: 1 DissAGFC.pdf: 1978531 bytes, checksum: e57d2335b2038eca2d8a6468869e05fa (MD5) Previous issue date: 2015-07-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Since the end of the nineteenth century and early twentieth century, some scholars such as Frege, Russell, Dedekind, Wittgenstein, among others, started to seek the foundations of the mathematics. Specifically, Frege developed studies in order to build the arithmetic foundation based on classical logic, i. e., using logic, he intended to build a system capable of formalizing mathematical definitions and proof methods. These works resulted in the publication of The Foundations of the Arithmetic in 1884 and subsequently in 1893 and 1902, The Basic Laws of Arithmetic. However, Frege’s attempt to reduce arithmetic to logic was inadequate due to a paradox discovered by Bertrand Russell in 1902. The aim of this research was to reconstruct mathematically and logically the Russell paradox in its original formulation in the Frege’s The Basic Laws of Arithmetic. This study had as primary bibliography Frege’s works and as secondary bibliography, works of his commentators, as well as the correspondence between Frege and Russell. This research provides a logical, philosophical and mathematical formation for the educator who is in contact with this event that covers both the areas that is disciplinary today. It is a fertile moment in the history of philosophy of mathematics and logic, configured as a watershed for mathematical theories since it enabled Gödel's incompleteness theorems and non-classical logics to be formulated, and also has repercussions in contemporary philosophy and which is of unquestionable value for the teacher formation. / A partir do fim do século XIX e início do século XX, alguns estudiosos, como Frege, Russell, Dedekind, Wittgenstein, dentre outros, buscaram alcançar os fundamentos últimos para a Matemática. Especificamente, Frege desenvolveu trabalhos a fim de fundamentar a Aritmética tendo como base a Lógica Clássica, ou seja, utilizando a lógica ele pretendia construir um sistema capaz de formalizar definições matemáticas e métodos de prova. Esses trabalhos culminaram na publicação de Os Fundamentos da Aritmética em 1884 e, posteriormente, em 1893 e 1902, em As Leis Básicas da Aritmética. No entanto, a tentativa proposta por Frege de reduzir a Aritmética à Lógica se mostrou inadequada, devido a um paradoxo na teoria apontado por Bertrand Russell em 1902. Assim sendo, o estudo aqui proposto tem como objetivo reconstruir lógicomatematicamente o paradoxo de Russell em sua formulação original nas Leis Básicas da Aritmética de Frege. Para realização deste estudo, o presente trabalho fundamentou-se numa pesquisa bibliográfica englobando, como bibliografia primária, as obras de Frege e, como bibliografia secundária, as obras de seus comentadores, assim como a correspondência entre Frege e Russell. A pesquisa proporciona uma formação lógica, filosófica e matemática para o educador que percorre este evento de fronteira entre as áreas que se configuram disciplinares na atualidade. Este é um momento fecundo na história e filosofia da Matemática e da Lógica, configurando-se como um divisor de águas para as teorias matemáticas, já que abre espaço para os Teoremas da Incompletude de Gödel e as lógicas não clássicas, possuindo também desdobramentos na Filosofia Contemporânea e que é de inquestionável valor nessa formação.