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

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 ar...

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Bibliographic Details
Published in:Educational studies in mathematics 2005-03, Vol.58 (3), p.335-359
Main Authors: Dubinsky, Ed, Weller, Kirk, McDonald, Michael A., Brown, Anne
Format: Article
Language:English
Subjects:
Analysis
Cognitive Processes
Encapsulation
Human information processing
Infinite
Infinite sets
Infinity
Logic
Logical Thinking
Mathematical Concepts
Mathematical functions
Mathematical objects
Mathematical sets
Mathematics
Mathematics Education
Natural numbers
Paradox
Paradoxes
Polarity
Study and teaching
Tennis balls
Word Problems (Mathematics)
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Summary: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.
ISSN:0013-1954
1573-0816
DOI:10.1007/s10649-005-2531-z