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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
Article

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 

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References

  1. Bolzano, B.: 1950, Paradoxes of the infinite (translated from the German of the posthumous edition by Fr. Prihonsky and furnished with a historical introduction by Donald A. Steele), Routledge & Paul, London.Google Scholar
  2. Dubinsky, E. and McDonald, M.A.: 2001, ‘APOS: A constructivist theory of learning in undergraduate mathematics education research’, in Derek Holton, et al. (eds.), The Teaching and Learning of Mathematics at University Level: An ICMI Study, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 273–280.Google Scholar
  3. Falk, R.: 1994, ‘Infinity: A cognitive challenge’, Theory and Psychology 4(1), 35– 60.Google Scholar
  4. Fischbein, E.: 1987, Intuition in Science and Mathematics, Reidel Publishing, Kluwer Academic Publishers, Dordrecht.Google Scholar
  5. Fischbein, E.: 2001, ‘Tacit models of infinity’, Educational Studies in Mathematics 48(2/3), 309–329.CrossRefGoogle Scholar
  6. Galilei, G.: 1974, Galileo: Two new sciences (translated with introduction and notes by Stillman Drake), University of Wisconsin Press, Madison, WI.Google Scholar
  7. Hauchart, C. and Rouche, N.: 1987, Apprivoiser l’infini: Un enseignement des d'ebuts de l’analyse, CIACO, Louvain.Google Scholar
  8. Kleiner, I.: 2001, ‘History of the infinitely small and the infinitely large in calculus’, Educational Studies in Mathematics 48(2/3), 137–174.CrossRefGoogle Scholar
  9. Knobloch, E.: 1999, ‘Galileo and Leibniz: Different approaches to infinity’, Archive for History of Exact Sciences 54, 87–99.CrossRefGoogle Scholar
  10. Lakoff, G. and Núñez, R.: 2000, Where Mathematics Comes From, Basic Books, New York.Google Scholar
  11. Mamona-Downs, J.: 2001, ‘Letting the intuitive bear upon the formal: A didactical approach for the understanding of the limit of a sequence”, Educational Studies in Mathematics 48(2/3), 259–288.CrossRefGoogle Scholar
  12. Monaghan, J.: 2001, ‘Young people’s ideas of infinity’, Educational Studies in Mathematics 48(2/3), 239–257.CrossRefGoogle Scholar
  13. Moore, A.W.: 1995, ‘A brief history of infinity’, Scientific American 272(4), 112–116.Google Scholar
  14. Moore, A.W.: 1999, The Infinite, 2nd ed., Routledge & Paul, London.Google Scholar
  15. Núñez Errázuriz, R.: 1993, En deça de transfini, '{E}ditions Universitaires, Fribourg.Google Scholar
  16. Piaget, J.: 1970, Genetic Epistemology, Columbia University Press, New York and London.Google Scholar
  17. Poincaré, H.: 1963, Mathematics and science: Last Essays (translated by John W. Bolduc), Dover, New York.Google Scholar
  18. Rabinovitch, N.L.: 1970, ‘Rabbi Hasdai Crescas (1340–1410) on numerical infinities’, Isis 61(2), 224–230.CrossRefGoogle Scholar
  19. Richman, F.: 1999, ‘Is $.999... = 1?’, Mathematics Magazine 72(5), 396–400.Google Scholar
  20. Rucker, R.: 1982, Infinity and the Mind: The Science and Philosophy of the Infinite, Birkhauser, Boston.Google Scholar
  21. Sierksma, G. and Sierksma, W.: 1999, ‘The great leap to the infinitely small. Johann Bernoulli: Mathematician and philosopher’, Annals of Science 56, 433–449.CrossRefGoogle Scholar
  22. Sierpi‘{n}ska, A.: 1987, ‘Humanities students and epistemological obstacles related to limits’, Educational Studies in Mathematics 18(4), 371–397.CrossRefGoogle Scholar
  23. Stavy, R. and Tirosh, D.: 2000, How Students (mis-)Understand Science and Mathematics, Teachers College Press, New York.Google Scholar
  24. Tall, D. and Schwarzenberger, R.L.E.: 1978, ‘Conflicts in the learning of real numbers and limits’, Mathematics Teaching 82, 44–49.Google Scholar
  25. Tirosh, D.: 1999, ‘Finite and infinite sets: Definitions and intuitions’, International Journal of Mathematical Education in Science & Technology 30(3), 341–349.Google Scholar
  26. Tsamir, P.: 1999, ‘The transition from comparison of finite to the comparison of infinite sets: Teaching prospective teachers”, Educational Studies in Mathematics 38, 209– 234.CrossRefGoogle Scholar
  27. Weller, K., Clark, J., Dubinsky, E., Loch, S., McDonald, M. and Merkovsky, R.: 2003, ‘Student performance and attitudes in courses based on APOS Theory and the ACE Teaching Cycle’, in A. Selden, E. Dubinsky, G. Harel, and F. Hitt (eds.), Research in Collegiate Mathematics Education V, American Mathematical Society, Providence, pp. 97–131.Google Scholar

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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