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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
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.
Falk, R.: 1994, ‘Infinity: A cognitive challenge’, Theory and Psychology 4(1), 35– 60.
Fischbein, E.: 1987, Intuition in Science and Mathematics, Reidel Publishing, Kluwer Academic Publishers, Dordrecht.
Fischbein, E.: 2001, ‘Tacit models of infinity’, Educational Studies in Mathematics 48(2/3), 309–329.
Galilei, G.: 1974, Galileo: Two new sciences (translated with introduction and notes by Stillman Drake), University of Wisconsin Press, Madison, WI.
Hauchart, C. and Rouche, N.: 1987, Apprivoiser l’infini: Un enseignement des d'ebuts de l’analyse, CIACO, Louvain.
Kleiner, I.: 2001, ‘History of the infinitely small and the infinitely large in calculus’, Educational Studies in Mathematics 48(2/3), 137–174.
Knobloch, E.: 1999, ‘Galileo and Leibniz: Different approaches to infinity’, Archive for History of Exact Sciences 54, 87–99.
Lakoff, G. and Núñez, R.: 2000, Where Mathematics Comes From, Basic Books, New York.
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.
Monaghan, J.: 2001, ‘Young people’s ideas of infinity’, Educational Studies in Mathematics 48(2/3), 239–257.
Moore, A.W.: 1995, ‘A brief history of infinity’, Scientific American 272(4), 112–116.
Moore, A.W.: 1999, The Infinite, 2nd ed., Routledge & Paul, London.
Núñez Errázuriz, R.: 1993, En deça de transfini, '{E}ditions Universitaires, Fribourg.
Piaget, J.: 1970, Genetic Epistemology, Columbia University Press, New York and London.
Poincaré, H.: 1963, Mathematics and science: Last Essays (translated by John W. Bolduc), Dover, New York.
Rabinovitch, N.L.: 1970, ‘Rabbi Hasdai Crescas (1340–1410) on numerical infinities’, Isis 61(2), 224–230.
Richman, F.: 1999, ‘Is $.999... = 1?’, Mathematics Magazine 72(5), 396–400.
Rucker, R.: 1982, Infinity and the Mind: The Science and Philosophy of the Infinite, Birkhauser, Boston.
Sierksma, G. and Sierksma, W.: 1999, ‘The great leap to the infinitely small. Johann Bernoulli: Mathematician and philosopher’, Annals of Science 56, 433–449.
Sierpi‘{n}ska, A.: 1987, ‘Humanities students and epistemological obstacles related to limits’, Educational Studies in Mathematics 18(4), 371–397.
Stavy, R. and Tirosh, D.: 2000, How Students (mis-)Understand Science and Mathematics, Teachers College Press, New York.
Tall, D. and Schwarzenberger, R.L.E.: 1978, ‘Conflicts in the learning of real numbers and limits’, Mathematics Teaching 82, 44–49.
Tirosh, D.: 1999, ‘Finite and infinite sets: Definitions and intuitions’, International Journal of Mathematical Education in Science & Technology 30(3), 341–349.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dubinsky, E., Weller, K., Mcdonald, M.A. et al. Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos-Based Analysis: Part 1. Educ Stud Math 58, 335–359 (2005). https://doi.org/10.1007/s10649-005-2531-z
Issue Date:
DOI: https://doi.org/10.1007/s10649-005-2531-z