JJ Sylvester e a comunidade formada através do desenvolvimento da teoria dos invariantes

Abstract

This thesis focuses on the contexts that accompanied the development of the work of James Joseph Sylvester (1814- 1897) in a period of consolidation of British research in mathematics. This period begins in 1837, the year in which the first specialized journal in the field was founded, Cambridge Mathematical Journal (CMJ), and continues until the emergence of the London Mathematical Society (LMS) in 1865. We investigated the mathematician’s performance through the following aspects: the key practices and concepts developed by him throughout the period investigated; the relationship these practices had with publications by other British authors, as well as the way they were received on the continent; in addition to the developments that contributed to transforming our research character into one of the pillars of the formation of what we call British Community of Invariants. The selection of the corpus of texts occurred after a prior analysis of Sylvester’s work, where we identified six fundamental themes: Elimination Theory, Locus of Algebraic Curves and Surfaces, Contact Problems, Linear Transformations, Polynomial Reduction and Rotation Movements. The organization of the articles was carried out using the text network methodology, which were established based on different criteria that allowed us to elucidate the aspects mentioned. The networks of interactions showed us how British authors related to each other through their interests in topics associated with Theory of Invariants; thematic networks reveal how key concepts were articulated between publications; and the dissemination/reception networks highlighted the way in which British practices were disseminated on the European continent and brought credibility to the United Kingdom’s style of mathematical knowledge production in the mid-19th century. From these organizations, it was possible to highlight the geometric vision shared in practices on invariants and understand the role of each of the authors, who were related to Sylvester’s work, in the formation of the British Community of Invariants . In this context, we reveal the influence of the work of Julius Plücker (1801- 1868), in addition to developments that brought credibility to the mathematician, compared to other centers, as in the case of the Law of Inertia for quadratic forms and its definitive contribution to the problem of nature of the roots of a polynomial of degree two, which is associated with the Secular Equation problem. We also highlight the diffusion of Theory of Invariants in textbooks that dealt with the topic in a didactic way, such as in the modern algebra books published by George Salmon (1819- 1904) in four editions until 1885. We understand that these analyzes represent more a contribution to investigations into the history of mathematics in the 19th century.