Conway's notion of real number and the principle of complementarity, some contributions to the development of epistemological reference models

Abstract

The objective of this article is to highlight the potential of Conway's theory compared to the classical concept of number with a view to contributing to the development of Epistemological Reference Models for teaching Differential and Integral Calculus. The search for a single answer to the epistemological question “What is a number?” has mobilized Mathematics epistemologists for centuries, considered essential for the foundation of this concept. John Horton Conway, an English mathematician from Princeton University, dedicated himself to researching this issue and resulted in a theory presented in the 1970s. In this article we bring elements about this theory highlighting its contributions to the evolution of the foundation of the concept of number. Conway's definition of number meets the complementarity of the intensional and extensional aspects of this concept, bringing advantages to Mathematics teaching. Scientific investigations and results of teaching practices in the field of teaching have encouraged questions about the importance of the role that the concept of real numbers has for learning Calculus and Real Analysis. Add to this question, and for Mathematics in general, and for the formation of analytical thinking, and for mathematical thinking? The reflections carried out in this article aim to raise epistemological and cognitive aspects about the classical construction of number, seeking to have an impact on current epistemology.