Convergent succession and Cauchy’s succession: mathematical equivalence and phenomenological equivalence

Abstract

This research is framed by Phenomenology, Mathematical Advanced Thinking and Representation Systems; we study a phenomenological equivalence to be added to the well known mathematical equivalence between a convergent sequence and a Cauchy sequence.

We introduce two phenomena associated to each definition (the finite limit of a sequence, and a Cauchy sequence); by using several high school mathematics textbooks, we recognize, in Spanish high schools textbooks, two phenomena organized by the first definition and, in British high schools textbooks only one of these phenomena. (Spanish textbooks were randomly selected, but they are not a representative sample; British textbooks were selected by “ad hoc” criteria and do not should be considered as representing the corresponding population.) We compare phenomena organized by each definition; we establish analogies and differences among them; we introduce a criterion of equivalence between phenomena and a criterion of phenomenological equivalence between mathematically equivalent definitions; finally, we answer affirmatively about the phenomenological equivalence among both definitions. Phenomenological equivalence appears to be more complex than mathematical equivalence, since it involves two couples of phenomena: the first one is observed within an intuitive context while the other couple is observed within a formal context. This paper extends results presented in Claros (2010)