Topics in the History and Philosophy of Mathematics

discussions and reflections

Abstract

This article presents and analyzes the discussions developed throughout the course Topics in the History and Philosophy of Mathematics, taken in the second semester of 2025 within the Doctoral Program in Mathematics Education at PUC-SP. The text aims to reflect on the historical, philosophical, and epistemological constitution of mathematical knowledge and its implications for Mathematics Education, articulating different theoretical frameworks and formative experiences. The analyses are primarily grounded in the works Fundamental Concepts of Mathematics by Bento de Jesus Caraça and Introduction to the Philosophy of Mathematics by Bertrand Russell, as well as in texts by Ubiratan D’Ambrosio and contemporary contributions discussed during the course. From a historical perspective, the article examines the construction of numerical fields—natural, rational, and irrational numbers—as responses to concrete problems of counting and measurement, highlighting the dynamic and non-linear character of mathematical development. The notion of generalization, as presented by Caraça, is analyzed as a driving force of scientific evolution, emphasizing the role of the negation of negation in overcoming conceptual limitations. Issues such as infinity, incommensurability, and the continuity of the number line are addressed as examples of ruptures that fostered significant theoretical advances.

Within the field of the Philosophy of Mathematics, the text emphasizes the contributions of Russell and the axiomatization proposed by Peano, discussing the concept of number as a class of classes and the importance of one-to-one correspondences for mathematical definition. These reflections are articulated with D’Ambrosio’s critique of traditional mathematics teaching, advocating the integration of History, Philosophy, and pedagogical practice as a means to humanize teaching and promote critical thinking. In addition, the article analyzes a didactic experience based on Galileo’s problem, highlighting the potential of the history of mathematics and experimentation to foster meaningful learning in Basic Education. It is concluded that the History and Philosophy of Mathematics constitute fundamental tools in teacher education, as they enable an understanding of mathematics as a cultural, historical, and social production, broadening the meaning of teaching and learning mathematics.