Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos-Based Analysis: Part 1

Abstract

This paper applies APOS Theory to suggest a new explanation of how people might think about the concept of infinity. We propose cognitive explanations, and in some cases resolutions, of various dichotomies, paradoxes, and mathematical problems involving the concept of infinity. These explanations are expressed in terms of the mental mechanisms of interiorization and encapsulation. Our purpose for providing a cognitive perspective is that issues involving the infinite have been and continue to be a source of interest, of controversy, and of student difficulty. We provide a cognitive analysis of these issues as a contribution to the discussion. In this paper, Part 1, we focus on dichotomies and paradoxes and, in Part 2, we will discuss the notion of an infinite process and certain mathematical issues related to the concept of infinity.KEY WORDS: actual and potential infinity, APOS Theory, classical paradoxes of the infinite, encapsulation, history of mathematics, human conceptions of the infinite, large finite sets

8th Grade Students’ Processes of the Construction of the Concept of Slope

“…Bahsedilen bu zihinsel mekanizmalar içselleştirme (interiorization) ve kapsülleme (encapsulation) iken, bilişsel yapılar ise eylem (action), süreç (process), nesne (object) ve şema (schema)'dır (Dubinsky, Weller, Mcdonald ve Brown, 2005). Bu bilişsel yapılar aynı zamanda kavramsal öğrenme aşamaları olarak da ele alınmaktadır.…”

Procesos iterativos infinitos y objetos trascendentes: un modelo de construcción del infinito matemático desde la teoría APOE

“…El infinito ha sido una noción particularmente estudiada desde la perspectiva de la teoría APOE (Roa-Fuentes y Oktaç, 2014;Dubinsky, Weller & Arnon, 2013;Brown, McDonald & Weller, 2010;Stenger et al, 2008;Dubinsky, Weller, McDonald & Brown, 2005a, 2005bWeller et al, 2004). Estos estudios han contribuido no sólo a la investigación en Matemática Educativa, sino también a la teoría misma.…”

“…Estas características particulares que se han propuesto a partir del análisis de individuos pensando en diferentes situaciones ha llevado a la definición de dos constructos: los procesos iterativos infinitos y los objetos trascendentes (Stenger et al, 2008). En los primeros análisis teóricos referentes al infinito Dubinsky et al, (2005a) hacen una distinción inicial entre el infinito potencial y actual:…”

Mathematics under the Microscope

“…APOS theory describes how Actions become interiorized into Processes and then encapsulated as mental Objects, which take their place in more sophisticated cognitive Schemas. See [90,91] for a detailed discussion of APOS as applied to the concept of infinity, and David Tall for a review of APOS. 3 It is worth to remember that "sums" of infinite series and limits of sequences can be meaningfully defined in many ways, depending on context.…”