Submitted by Automa??o e Estat?stica (sst@bczm.ufrn.br) on 2016-05-30T23:34:35Z No. of bitstreams: 1 AndreaMariaFerreiraMoura_TESE.pdf: 2515776 bytes, checksum: 2a8045cdd1293bd9f2d83943858c8ae2 (MD5) / Approved for entry into archive by Arlan Eloi Leite Silva (eloihistoriador@yahoo.com.br) on 2016-06-02T20:57:12Z (GMT) No. of bitstreams: 1 AndreaMariaFerreiraMoura_TESE.pdf: 2515776 bytes, checksum: 2a8045cdd1293bd9f2d83943858c8ae2 (MD5) / Made available in DSpace on 2016-06-02T20:57:12Z (GMT). No. of bitstreams: 1 AndreaMariaFerreiraMoura_TESE.pdf: 2515776 bytes, checksum: 2a8045cdd1293bd9f2d83943858c8ae2 (MD5) Previous issue date: 2015-03-27 / A rica hist?ria em torno da legitima??o dos negativos como n?meros se apresenta como o fio condutor da presente pesquisa, que tem como objetivo principal compreender, a dura rejei??o inglesa aos negativos, ocorrida na segunda metade do s?culo XVIII e in?cio do s?culo XIX. Para tanto, optamos por realizar uma an?lise nas obras A Dissertation on the Use of the Negative Sign in Algebra, publicada em 1758, Tracts on the Resolution of Affected Algebra?ick Equations, publicada em 1800, de Francis Maseres, e Principles of Algebra, publicada em 1796, de William Frend, pois essas obras foram consideradas as bases te?ricas aos que resistiam a aceitar a exist?ncia dos negativos na Inglaterra do fim do s?culo XVIII. Al?m delas, consideramos relevante estudar tamb?m a primeira obra inglesa sobre Teoria dos N?meros, An Elementary Investigation of the Theory of Numbers, publicada em 1811, e que ? de autoria de Peter Barlow, um dos ?ltimos a assumir a postura rejeitadora na Inglaterra. A an?lise das obras dos dois primeiros nos mostrou que a rejei??o aos negativos se alicer?ava na impossibilidade de uma defini??o clara e rigorosa para esses n?meros nos moldes da concep??o de matem?tica vigente, que tinha a geometria como ?nica norma de rigor e verdade, j? que era a ?nica ?rea da matem?tica axiomatizada at? o in?cio do s?culo XIX. Sendo assim, a postura rejeitadora alerta para a necessidade de uma fundamenta??o matem?tica mais coerente com as teorias em desenvolvimento, e dessa forma contribui com a mudan?a na concep??o de matem?tica ocorrida no s?culo XIX. Por outro lado, nossa pesquisa revelou tamb?m que as ideias rejeitadoras acarretavam limita??es e restri??es na matem?tica em desenvolvimento da ?poca, dentre as quais destacamos as limita??es impostas ?s opera??es alg?bricas e a impossibilidade de um n?mero negativo ser raiz de uma equa??o. Quanto ? obra de Barlow, nossa investiga??o evidenciou que a postura rejeitadora do autor acarreta a necessidade de ado??o de estrat?gias para adequar a teoria por ele em desenvolvimento com as limita??es impostas pelo seu posicionamento rejeitador dos negativos. Com o intuito de enriquecer o conhecimento sobre o tema principal, a problem?tica em torno da legitima??o dos negativos, apresentamos no in?cio da pesquisa um apanhado hist?rico, focado no uso, aceita??o, rejei??o e concep??o dos n?meros negativos ao longo dos s?culos, iniciando pela Antiguidade e terminando no s?culo XIX, quando finalmente toda a pol?mica foi resolvida com a amplia??o do conceito de n?mero, a axiomatiza??o da ?lgebra e a ado??o da atual concep??o de matem?tica como ci?ncia formal. / The main purpose of this research is to review the rich history around the legitimization of negative numbers. It is important to highlight that although negatives were born in Ancient times, their full acceptance as numbers only happened in the XIX century. During this struggle, that lasted almost 2 thousand years, English mathematics, especially the one developed in the second half of the XVIII century and the early XIX century, played a key role as a scenario in which the rejection to negative numbers was fiercer. Therefore, the general purpose of this research is to understand the strong English rejection to negatives that occurred between 1750 and 1830. Consequently, we analyzed the works A Dissertation on the Use of the Negative Sign in Algebra, published in 1758 and Tracts on the Resolution of Affected Algebra?ick Equations, published in 1800 by Francis Maseres, and Principles of Algebra, published in 1796 by William Frend, as we consider these works as the theoretical basis for those who refused to accept the existence of negative numbers in England by the end of the XVIII century. Besides these two works, we also consider relevant to study the first English work on the Theory of Numbers, An Elementary Investigation of the Theory of Numbers, published in 1811 by Peter Barlow, one of the last ones to adopt a position against negatives in England. Seeking to understand the English rejection to negatives, we set the following specific goals: to identify the theoretical basis of the rejecting stream, analyze the main implications that this rejection posture provoked in the mathematics under development by the end of the XVIII century and study how the rejection theory affected the production of the first English work on the Theory of Numbers. As a result of the analysis developed on Maseres? and Frend?s works, we concluded that the rejection to negatives was based on the impossibility to clearly and strictly define these numbers within the mathematical concepts that existed in those days. We also noticed that the rejection defended by English mathematicians, on the one hand alerted on the need for a more coherent mathematical substantiation with regards to the theories that were being developed and on the other hand, rejection ideas provoked limitations and restrictions in the mathematics under development by the end of the XVIII century. Among these limitations we highlight those imposed to algebraic operations and the impossibility of having a negative number as an equation root. As for Barlow?s work, research showed that the author?s position against negatives implied the need to adopt strategies to adapt his theory to the limitations imposed by his rejection to negatives. Seeking to improve our knowledge on this issue, we introduce a historical review focused on the use, acceptance, rejection and conception of negative numbers since the early days until the XIX century, when finally the whole controversy was solved with the widening of the concept of number, the axiomatization of algebra and the adoption of modern mathematics as a formal science.

Made available in DSpace on 2014-12-17T15:04:50Z (GMT). No. of bitstreams: 1 MartaFA.pdf: 422236 bytes, checksum: c082c32935e784fcbc231f73fd326d64 (MD5) Previous issue date: 2008-06-26 / The present study seeks to present a historico-epistemological analysis of the development of the mathematical concept of negative number. In order to do so, we analyzed the different forms and conditions of the construction of mathematical knowledge in different mathematical communities and, thus, identified the characteristics in the establishment of this concept. By understanding the historically constructed barriers, especially, the ones having ontologicas significant, that made the concept of negative number incompatible with that of natural number, thereby hindering the development of the concept of negative, we were able to sketch the reasons for the rejection of negative numbers by the English author Peter Barlow (1776 -1862) in his An Elementary Investigation of the Theory of Numbers, published in 1811. We also show the continuity of his difficulties with the treatment of negative numbers in the middle of the nineteenth century / O presente estudo visa apresentar um incurso hist?rico-epistemol?gico no desenvolvimento do conceito matem?tico de n?mero negativo. Para tanto, inter-relacionamos as diferentes formas e condi??es de constru??o do conhecimento matem?tico e, assim, identificando as conseq?entes caracter?sticas no estabelecimento do referido conceito, em diferentes comunidades matem?ticas e momentos hist?ricos. Com isso, diante de um entendimento das barreiras constru?das historicamente perante o conceito de n?mero negativo, especialmente, as de origem ontol?gicas, que incompatibilizava o conceito de n?mero negativo com o conceito de n?mero natural, impedindo o desenvolvimento do conceito de n?mero negativo. Assim, esbo?amos as raz?es de rejei??o aos n?meros negativos, como ra?zes de equa??es, pelo ingl?s Peter Barlow (1776 1862) em An Elementary Investigation of the Theory of Numbers, publicada em 1811. N?s tamb?m diagnosticamos a continuidade das dificuldades com o tratamento com os n?meros negativos, j? em pleno transcurso do s?culo XIX