Educational Studies in Mathematics

, Volume 58, Issue 3, pp 335–359 | Cite as

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

  • Ed DubinskyEmail author
  • Kirk Weller
  • Michael A. Mcdonald
  • Anne Brown


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 


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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Ed Dubinsky
    • 1
    Email author
  • Kirk Weller
    • 2
  • Michael A. Mcdonald
    • 3
  • Anne Brown
    • 1
  1. 1.Kent State UniversityHermonU.S.A.
  2. 2.University of North TexasDentonU.S.A.
  3. 3.Occidental CollegeLos AngelesU.S.A.

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