Online distance mathematics education in Brazil: research, practice and policy

In this article, we address online distance mathematics education research and practice in Brazil, which are relative newcomers to the educational scene. We present the national context of education in Brazil, highlighting the organization of the educational system, and also a summary of national legislation on distance education and an overview of digital inclusion in the country. We outline the potential and relevance of distance education for the Brazilian educational system and show how it could intervene in the system. With respect to research and practice in online mathematics education, we present support for research, examples of studies and highlight different aspects being addressed, including its essential components. In addition, we discuss the synergy between distance education and teacher education, and mathematics distance education and modeling, as well as other initiatives in the national scenario.

Husserl's two notions of completeness: husserl and hilbert on completeness and imaginary elements in mathematics

In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures he gave in Göttingen in 1901 and other related texts of the same period, a problem that had occupied Husserl since the beginning of 1890, when he was planning a never published sequel to Philosophie der Arithmetik (1891). In order to solve the problem of imaginary entities Husserl introduced, independently of Hilbert, two notions of completeness (definiteness in Husserl's terminology) for a formal axiomatic system. I present and discuss these notions here, establishing also parallels between Husserl's and Hilbert's notions of completeness. © 2000 Kluwer Academic Publishers.