Text in Afrikaans, abstract in Afrikaans and English / Hierdie studie is daarop gemik om te bepaal tot welke mate wiskundeleerlinge op skool en onderwysstudente in wiskunde, onderrig in logika ontvang as agtergrond vir strenge bewysvoering. Die formele aspek van wiskunde op hoerskool en tersiere vlak is besonder belangrik. Leerlinge en studente kom onvermydelik met hipotetiese argumente in aanraking. Hulle leer ook om die kontrapositief te gebruik in bewysvoering. Hulle maak onder andere gebruik van bewyse uit die ongerymde. Verder word nodige en voldoende voorwaardes met stellings en hulle omgekeerdes in verband gebring. Dit is dus duidelik dat 'n studie van logika reeds op hoerskool nodig is om aanvaarbare wiskunde te beoefen. Om seker te maak dat aanvaarbare wiskunde beoefen word, is dit nodig om te let op die gebrek aan beheer in die ontwikkeling van 'n taal, waar woorde meer as een betekenis het. 'n Kunsmatige taal moet gebruik word om interpretasies van uitdrukkings eenduidig te maak. In so 'n kunsmatige taal word die moontlikheid van foutiewe redenering uitgeskakel. Die eersteordepredikaatlogika, is so 'n taal, wat ryk genoeg is om die wiskunde te akkommodeer. Binne die konteks van hierdie kunsmatige taal, kan wiskundige toeriee geformaliseer word. Verskillende bewystegnieke uit die eersteordepredikaatlogika word geidentifiseer, gekategoriseer en op 'n redelik eenvoudige wyse verduidelik. Uit 'n ontleding van die wiskundesillabusse van die Departement van Onderwys, en 'n onderwysersopleidingsinstansie, volg dit dat leerlinge en studente hierdie bewystegnieke moet gebruik. Volgens hierdie sillabusse moet die leerlinge en studente vertroud wees met logiese argumente. Uit die gevolgtrekkings waartoe gekom word, blyk dit dat die leerlinge en studente se agtergrond in logika geheel en al gebrekkig en ontoereikend is. Dit het tot gevolg dat hulle nie 'n volledige begrip oor bewysvoering het nie, en 'n gebrekkige insig ontwikkel oor wat wiskunde presies behels. Die aanbevelings om hierdie ernstige leemtes in die onderrig van wiskunde aan te spreek, asook verdere navorsingsprojekte word in die laaste hoofstuk verwoord. / The aim of this study is to determine to which extent pupils taking Mathematics at school level and student teachers of Mathematics receive instruction in logic as a grounding for rigorous proof. The formal aspect of Mathematics at secondary school and tertiary levels is extremely important. It is inevitable that pupils and students become involved with hypothetical arguments. They also learn to use the contrapositive in proof. They use, among others, proofs by contradiction. Futhermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practice Mathematics satisfactorily. To ensure that acceptable Mathematics is practised, it is necessary to take cognizance of the lack of control over language development, where words can have more than one meaning. For this reason an artificial language must be used so that interpretations can have one meaning. Faulty interpretations are ruled out in such an artificial language. A language which is rich enough to accommodate Mathematics is the first-order predicate logic. Mathematical theories can be formalised within the context of this artificial language. Different techniques of proof from the first-order logic are identified, categorized and explained in fairly simple terms. An analysis of Mathematics syllabuses of the Department of Education and an institution for teacher training has indicated that pupils should use these techniques of proof. According to these syllabuses pupils should be familiar with logical arguments. The conclusion which is reached, gives evidence that pupils' and students' background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what Mathematics exactly entails. Recommendations to bridge these serious problems in the instruction of Mathematics, as well as further research projects are discussed in the final chapter. / Curriculum and Institutional Studies / D. Phil. (Wiskundeonderwys)

Orientador: Enéias Forlin / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas / Made available in DSpace on 2018-08-16T15:26:39Z (GMT). No. of bitstreams: 1 Nolasco_FabioMascarenhas_M.pdf: 1121338 bytes, checksum: 679280a1fe00df842faf6eedecac94db (MD5) Previous issue date: 2010 / Resumo: Tentou-se nesta dissertação apresentar um ponto de vista - fragmentário e inacabado como necessariamente deve ser - acerca do percurso histórico do imbricamento entre matemática e filosofia que marcou intensamente o séc. XVII, imbricamento este assinalado pelos termos more geometrico, mathesis universalis, algebra, dentre outros. Isto com o intuito distante de abordar, posteriormente, a questão mais genérica de situar tal imbricamento específico do séc. XVII perante outras conformações da relação entre filosofia e matemática, notadamente, as que presentemente mais nos interessam, a do idealismo alemão do início do séc. XIX e a dos princípios do que veio a se chamar de filosofia analítica, do início do séc. XX. Para tanto, desta forma, buscamos apresentar dois momentos desse imbricamento especificamente seiscentista entre matemática e filosofia que, em nossa opinião, abrangem uma vastíssima gama de temas sintomáticos do período: (i) a conformação cartesiana entre matemática e filosofia, dita inauguradora da modernidade, tipificada pelo conceito de intuicionismo, e (ii) a conformação leibniziana, tipificada pelo conceito de formalismo. Julgamos que a apresentação do percurso apenas tocando estes dois pólos do movimento, mesmo que incompleta por não tratar de temas da filosofia de Hobbes, Pascal, Spinoza, Locke, dentre outros, apresenta, todavia, os contornos gerais do desenvolvimento buscado: como se o momento da crítica de Leibniz a Descartes, e a re-formulação leibniziana do dito imbricamento, pudesse ser tomado como uma caracterização exemplar do percurso do imbricamento de matemática e filosofia do séc. XVII / Abstract: Our effort in this dissertation was to present a point of view - fragmentary and unachieved as it ought to be - concerning the historical development of the conjunction between Mathematics and Philosophy which has so characteristically marked the XVIIth century; a conjunction denoted by the terms more geometrico, mathesis universalis, algebra, to name but a few. And this, due to the further goal of approaching, afterwards, the more generic question of situating such a conjunction, specific of the XVIIth century, in regards to other conformations of the relation between Philosophy and Mathematics, namely the ones which are of our present interest, that of the German Idealism of the early XIXth century, and that of the first moments of what came to be called as Analytic Philosophy, of the first years of the XXth century. So as to approach the more basic part of this goal, we have thus tried to present two moments of this specifical XVIIth century conjunction between Mathematics and Philosophy which, in our opinion, touch a very large array of symptomatic questions of the period: (i) the cartesian conformation between mathematics and philosophy, said to be the inauguration of modernity, typified by the concept of intuitionism, and (ii) the leibnizian conformation, typified by the concept of formalism. We claim that such an exposition, which only touches the two poles of the movement, even if incomplete for not treating themes from Hobbes, Pascal, Spinoza and Locke's Philosophy, in among others, nevertheless presents the general outlines of the development we are trying to enlighten: as if the moment of Leibniz' criticism to Descartes, and the leibnizian re-conformation of the said conjunction, could be taken as an exemplary characterization of the development of the relation between mathematics and philosophy of the XVIIth century / Mestrado / Historia da Filosofia Moderna / Mestre em Filosofia

Social theorists have long argued that literacy is one of the principal causes and hallmark features of complex society. However, the relationship between literacy and social complexity remains poorly understood because the relevant data have not been assembled in a way that would allow competing hypotheses to be adjudicated. The project set out in this thesis provides a novel account of the multiple origins of literate behaviour around the globe, the principal mechanisms of its cultural transmission, and its relationship with the cultural evolution of large-group human cooperation and complex forms of socio-political organisation. A multi-method large-scale cross-cultural approach provided the data necessary to achieve these objectives. Evidence from the societies within which literate behaviour first emerged, and from a representative sample of ethnographically-attested societies worldwide (n=74), indicates that literate behaviour emerged through the routinization of rituals and pre-literate sign systems, eventually spreading more widely through classical religions. Cross-cultural evidence also suggests that literacy assumed a wide variety of forms and socio-political functions, particularly in large, complex groups, extending evolved psychological mechanisms for cooperation, which include reciprocity, reputation formation and maintenance systems, social norms and norm enforcement systems, and group identification. Finally, the results of a cross-cultural historical survey of first-generation states (n=10) reveal that simple models assuming single cause-and-effect relationships between literacy and complex forms of socio-political organisation must be rejected. Instead, literacy and first-generation state-level polities appear to have interacted in a complex positive feedback loop. This thesis contributes to the wider goal of transforming social and cultural anthropology into a cumulative and rapid-discovery science.