Dealing with infinity, an inherently abstract concept which defies concrete representation, involves distinct cognitive difficulties. I first review the research on the development of children's understanding of the endlessness of numbers and of the infinite gap between a large finite set and an infinite set. Second, I consider the contradictions with which mathematicians and philosophers have to cope when attempting to manipulate infinity concepts in a coherent way. These difficulties are highlighted by analyzing one perplexing paradox. Finally, I discuss the intellect's puzzling ability to transcend the limits of our finitely based intuitions, as manifested in science and higher mathematics, and the need to thoroughly explore the emergence of that kind of human capability.
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References
1.
Asimov, I. (1989). The relativity of wrong. The Skeptical Inquirer, 14, 35-44.
2.
Benacerraf, P. (1970). Tasks, super-tasks, and the modern eleatics. In W.C. Salmon (Ed.), Zeno's paradoxes (pp. 103-129). Indianapolis, IN: Bobbs-Merrill.
3.
Black, M. (1970). Achilles and the tortoise. In W.C. Salmon (Ed.), Zeno's paradoxes (pp. 67-81). Indianapolis, IN: Bobbs-Merrill.
4.
Bolzano, B. (1950). Paradoxes of the infinite (D.H. Steel, Trans.). London: Routledge & Kegan Paul. (Original work published 1851.)
5.
Borasi, R. (1985). Errors in the enumeration of infinite sets. Focus on Learning Problems in Mathematics, 7, 77-89.
6.
Clary, E.G. , & Tesser, A. (1983). Reactions to unexpected events: The naive scientist. Personality and Social Psychology Bulletin, 9, 609-620.
7.
Delbruck, M. (1986). Mind from matter? An essay on evolutionary epistemology. Palo Alto: Blackwell.
8.
Elkind, D. , Koegler, R.R., & Go, E. (1964). Studies in perceptual development: II. Part-whole perception. Child Development, 35, 81-90.
9.
Erlich, H.S. (1978). Adolescent suicide: Maternal longing and cognitive development. The Psychoanalytic Study of the Child, 33, 261-277.
10.
Evans, D.W. (1983). Understanding zero and infinity in the early school years. Unpublished doctoral dissertation, University of Pennsylvania.
11.
Evans, D. , & Gelman, R. (1982). Understanding infinity: A beginning inquiry. Unpublished manuscript, University of Pennsylvania.
12.
Falk, R. (1990). Children's concepts of infinity [in Hebrew]. In K. Benyamini, M. Amir, A. Dolev, E. Cohen, & I.M. Schlesinger (Eds.), Theory and applications in psychology: Volume in memory of Israel Lieblich (pp. 212-228). Jerusalem: Magnes.
13.
Falk, R. , & Ben-Lavy, S. (1989). How big is an infinite set? Exploration of children's ideas. Actes de la 13eme conference internationale: Psychology of Mathematics Education (Vol. 1, pp. 252-259). Paris.
14.
Falk, R. , Gassner, D., Ben-Zoor, F., & Ben-Simon, K. (1986). How do children cope with the infinity of numbers?Proceedings of the Tenth International Conference for the Psychology of Mathematics Education (pp. 13-18). London.
15.
Fischbein, E. , Tirosh, D., & Hess, P. (1979). The intuition of infinity. Educational Studies in Mathematics, 10, 3-40.
16.
Flavell, J.H. (1963). The developmental psychology of Jean Piaget. New York: Van Nostrand.
17.
Gamow, G. (1961). One, two, three.. infinity: Facts and speculations of science (rev. ed.). New York: Viking.
18.
Gardner. M. (1982). Aha! Gotcha: Paradoxes to puzzle and delight. New York: Freeman.
19.
Gelman, R. (1980). What young children know about numbers. Educational Psychologist, 15, 54-68.
20.
Gelman, R. , & Gallistel, C.R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.
21.
Gibilisco, S. (1990). Reaching for infinity: Puzzles, paradoxes and brainteasers # 3. Blue Ridge Summit, PA: TAB Books.
22.
Gigerenzer, G. (1991a). From tools to theories: A heuristic of discovery in cognitive psychology. Psychological Review, 98, 254-267.
23.
Gigerenzer, G. (1991b). How to make cognitive illusions disappear: Beyond `heuristics and biases'. European Review of Social Psychology, 2, 83-115.
Hartnett, P. , Gelman, R., & Meck, E. (1987). Inducing further numerical principles given some knowledge of counting and addition. Paper presented at the bi-annual meeting of the Society for Research in Child Development, Baltimore, MD.
26.
Inhelder, B. , & Piaget, J. (1964). The early growth of logic in the child (E.A. Lunzer & D. Papert, Trans.). New York: Harper & Row. (Original work published 1959.)
27.
Kasner, E. , & Newman, J. (1949). Mathematics and the imagination. London: Bell and Sons.
28.
Knop, L.E. (1985). Cantor's disappearing table. The College Mathematics Journal, 16, 398-399.
29.
Lau, R.R. , & Russell, D. (1980). Attributions in the sports pages. Journal of Personality and Social Psychology, 39, 29-38.
30.
Love, W.P. (1989). Infinity: The twilight zone of mathematics. Mathematics Teacher, 82, 284-292.
31.
Maor, E. (1987). To infinity and beyond: A cultural history of the infinite. Boston, MA: Birkhauser.
Mazursky, D. , & Ofir, C. (1990). 'I could never have expected it to happen': The reversal of the hindsight bias. Organizational Behavior and Human Decision Processes, 46, 20-33.
34.
Moreno, L.E.A. , & Waldegg, G. (1991). The conceptual evolution of actual mathematical infinity. Educational Studies in Mathematics, 22, 211-231.
35.
Paik, M.K. (1983). Empty or infinitely full?Mathematics Magazine, 56, 221-223.
36.
Peter, R. (1976). Playing with infinity: Mathematical explorations and excursions (Z.P. Dienes, Trans.). New York: Dover. (Original work published 1957.)
37.
Piaget, J. (1929). The child's conception of the world (J. & A. Tomlinson, Trans.). London: Routledge & Kegan Paul.
38.
Piaget, J. (1952). The child's conception of number (C. Gattegno & F.M. Hodgson, Trans.). London: Routledge & Kegan Paul. (Original work published 1941.)
39.
Piaget, J. , & Inhelder, B. (1967). The child's conception of space (F.J. Langdon & J.L. Lunzer, Trans.). New York: Norton.(Original work published 1948.)
40.
Popper, K. (1976). Unended quest: An intellectual autobiography. Glasgow: Fontana/Collins.
41.
Pyszczynski, T.A. , & Greenberg, J. (1981). Role of disconfirmed expectancies in the instigation of attributional processing. Journal of Personality and Social Psychology, 40, 31-38.
42.
Rosenblum, T. , & Pinker, S.A. (1983). Word magic revisited: Monolingual and bilingual children's understanding of the word-object relationship. Child Development, 54, 773-780.
43.
Ross, S. (1976). A first course in probability. New York: Macmillan.
44.
Rucker, R. (1982). Infinity and the mind: The science and philosophy of the infinite. London: Paladin.
45.
Russell, B. (1963). Mysticism and logic and other essays. London: Allen & Unwin.
46.
Russell, B. (1970). The problem of infinity considered historically. In W.C. Salmon (Ed.), Zeno's paradoxes (pp. 45-58). Indianapolis, IN: Bobbs-Merrill.
Shmukler, A. (1980). Introduction to set theory [in Hebrew]. Jerusalem: Academon.
49.
Skemp, R.R. (1971). The psychology of learning mathematics. Harmondsworth: Penguin.
50.
Smullyan, R.M. (1985). Satan, Cantor, and infinity. The College Mathematics Journal, 16, 118-121.
51.
Tall, D. (1980). The notion of infinite measuring number and its relevance in the intuition of infinity. Educational Studies in Mathematics, 11, 271-284.
52.
Thomson, J. (1970a). Comments on Professor Benacerrafs paper. In W.C. Salmon (Ed.), Zeno's paradoxes (pp. 130-138). Indianapolis, IN: Bobbs-Merrill.
53.
Thomson, J. (1970b). Tasks and super-tasks. In W.C. Salmon (Ed.), Zeno's paradoxes (pp. 89-102). Indianapolis, IN: Bobbs-Merrill.
54.
Tirosh, D. (1985). The intuition of infinity and its relevance for mathematical education. Unpublished doctoral dissertation [in Hebrew with English abstract], Tel Aviv University.
55.
Vilenkin, N.Ya. (1968). Stories about sets (Scripta Technica, Trans.). New York: Academic Press. (Original work published 1965.)
56.
Vollmer, G. (1984). Mesocosm and objective knowledge. In F.M. Wuketits (Ed.), Concepts and approaches in evolutionary epistemology (pp. 69-121). Dordrecht: Reidel.
57.
von Glasersfeld, E. (1987). Subitizing-the role of figural patterns in the development of numerical concepts. In E. von Glasersfeld (Ed.), The construction of knowledge: Contributions to conceptual semantics (pp. 279-306). Seaside, CA: Intersystems Publications.
58.
Vygotsky, L.S. (1962). Thought and language (E. Hanfmann & G. Vakar, Ed. and Trans.). Cambridge, MA: MIT Press. (Original work published 1934.)
59.
Wilkinson, A. (1976). Counting strategies and semantic analysis as applied to class inclusion. Cognitive Psychology, 8, 64-85.
