The Library
Natural and formal infinities
Tools
Tall, David. (2001) Natural and formal infinities. Educational Studies in Mathematics, Vol.48 (No.2 ). pp. 199238. ISSN 00131954

PDF
WRAP_Tall_dot2001pesminfinity.pdf  Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (268Kb) 
Official URL: http://dx.doi.org/10.1023/A:1016000710038
Abstract
Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conception as natural infinities.Research has shown that individuals' natural conceptions of infinity are `labile and selfcontradictory' (Fischbein et al.,1979, p. 31). The formal approach to mathematics in the twentieth century attempted to rationalize these inconsistencies by selecting a finite list of specific properties (or axioms) from which the conception of a formal infinity is built by formal deduction. By beginning with different properties of finite numbers, such as counting,ordering or arithmetic,different formal systems may be developed. Counting and ordering lead to cardinal and ordinal number theory and the properties of arithmetic lead to ordered fields that may contain infinite and infinitesimalquantities. Cardinal and ordinal numbers can be added and multiplied but not divided or subtracted. The operations of cardinals are commutative, but the operations of ordinals are not. Meanwhile an ordered field has a full system of arithmetic in which the reciprocals of infinite elements are infinitesimals. Thus, while natural concepts of infinity may contain builtin contradictions, there are several different kinds of formal infinity, each with its own coherent properties, yet each system having properties that differ from the others. The construction of both natural and formal infinities are products of human thought and so may be considered in terms of embodied cognition' (Lakoff and Nunez,2000). The viewpoint forwarded here, however, is that formal deduction focuses as far as possible on formal logic in preference to perceptual imagery, developing a network of formal properties that do not depend on specific embodiments. Indeed, I shall show that formal theory can lead to structure theorems, whose formal properties may then be reinterpreted as a more subtle form of embodied imagery. Not only can natural embodied theory inspire theorems to be proved formally, but formal theory can also feed back into human embodiment, now subtly enhanced by the underlying network of formal relationships.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Social Sciences > Institute of Education 
Library of Congress Subject Headings (LCSH):  Infinity, Mathematical ability 
Journal or Publication Title:  Educational Studies in Mathematics 
Publisher:  Springer Netherlands 
ISSN:  00131954 
Date:  November 2001 
Volume:  Vol.48 
Number:  No.2  
Page Range:  pp. 199238 
Identification Number:  10.1023/A:1016000710038 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  Artigue, M., 1991: ‘Analysis’. In D. O. Tall (Ed.), Advanced Mathematical Thinking, (pp.166–198), Dordrecht: Kluwer. Berkeley, G., 1951: ‘The Analyst or A discourse Addressed to an Infidel Mathematician Wherein it is examined whether the object, principles, and inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than religious Mysteries and points of Faith’. In A. A. Luce (Ed.), The Works of George Berkeley Vol 4, London: Thomas Nelson. Bishop, E., 1967: Foundations of Constructive Analysis, McGrawHill. Bishop, E., 1977: Review of ‘Elementary Calculus’ by H. J. Keisler, Bulletin of the American Mathematical Society, 83, 2, 205–208. Cantor, G., 1895: Beiträge zur Begründung der transfiniten Mengelehre, (English trans.: Contributions to the Founding of the Theory of Transfinite numbers, 1915). Cohen, P. J., 1966: Set Theory and the Continuum Hypothesis, New York: Benjamin. Cornu, B., 1981: ‘Apprentissage de la notion de limite: modèles spontanés et modèles propres,’ Actes due Cinquième Colloque du Groupe International P.M.E., Grenoble 322–326. Cornu, B., 1991: ‘Limits’. In D. O. Tall (Ed.), Advanced Mathematical Thinking, (pp.153–166), Dordrecht: Kluwer. Duffin, J. M. & Simpson, A. P., 1993: ‘Natural, conflicting and alien,’ Journal of Mathematical Behaviour, 12, 4, 313–328. Fischbein, E., Tirosh, D., & Hess, P. 1979: ‘The intuition of infinity’, Educational Studies in Mathematics, 10, 3–40. Frid, S., 1994: ‘Three approaches to undergraduate calculus instruction: Their nature and potential impact on students’ language use and sources of conviction.’ In E. Dubinsky, J. Kaput & A. Schoenfeld (eds.), Research in Collegiate Mathematics Education I, Providence, RI: AMS. Gödel, K., 1931: ‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme,’ Monatshefte für Mathematik und Physik, vol. 38. Gray, E. M., & Tall, D. O., 1994: ‘Duality, ambiguity and flexibility: A proceptual view of simple arithmetic,’ Journal of Research in Mathematics Education, 25 2, 115–141. Henle, J. & Kleinberg, E., 1979: Infinitesimal Calculus, MIT Press, 1979. Hilbert, D., 1900: The Problems of Mathematics. Address to the International Congress of Mathematicians, Paris. Translated into English by M. W. Newson, for the Bulletin of the American Mathematical Society, (1902), 8, 437–479. Houdé, O, Zago, L., Mellet, E., Moutier, S. Pineau, A., Mazoyer, B, TzourioMazoyer, N., 2000: ‘Shifting from the perceptual brain to the logical brain: the neural impact of cognitive inhibition training’, Journal of Cognitive Neuroscience, 12 5, 721–728. Keisler, H. J., 1976: Elementary Calculus, Prindle, Weber & Schmidt. Kennedy, H. C. (ed.), 1973: Selected works of Giuseppe Peano (1858–1932) translated from the Italian. London: Allen and Unwin. Kleiner, I., in press: ‘The Infinitely Small and the Infinitely Large in Calculus’. Educational Studies in Mathematics. (This volume.) Lakoff, G., 1987: Women, Fire and Dangerous Things, Chicago: University of Chicago Press. Lakoff, G. and Johnson, M., 1999: Philosophy in the Flesh. New York: Basic Books. Lakoff, G. & Nunez, R., 2000: Where Mathematics Comes From. New York: Basic Books. Lindstrøm, T., 1988: ‘An Invitation to Nonstandard Analysis.’ In N. Cutland (ed.), Nonstandard Analysis and its Applications, (pp. 1–105). Cambridge: Cambridge University Press. Peano, G., 1908: Formulario Mathematico. Torino: Fratelli Bocca. Pinto, M. M. F., 1998: Students’ Understanding of Real Analysis. Unpublished PhD Thesis, Warwick University. Pinto, M., M. F. & Tall, D. O., 1999: ‘Student constructions of formal theory: giving and extracting meaning,’ Proceedings of the Twenty third International Conference for the Psychology of Mathematics Education, Haifa, Israel, 2, 41–48. Pinto, M., M. F. & Tall, D. O., 2001: ‘Following students’ development in a traditional university classroom.’ In Marja van den HeuvelPanhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education 4, 57–64. Utrecht, The Netherlands. Robert, A., 1988: Nonstandard Analysis, New York: Wiley. Robinson, A., 1966: Nonstandard Analysis. North Holland. Stewart, I. N. & Tall, D. O., 1977: Foundations of Mathematics. Oxford: O.U.P. Sullivan, K., 1976: ‘The teaching of elementary calculus: an approach using infinitesimals,’ American Mathematical Monthly 83, 370–375. Tall D.O., 1980a: ‘The notion of infinite measuring number and its relevance in the intuition of infinity’, Educational Studies in Mathematics, 11, 271–284. Tall, D. O., 1980b: ‘Looking at graphs through infinitesimal microscopes, windows and telescopes’, Math. Gazette, 64, 22–49. Tall, D. O. & Vinner, S., 1981: ‘Concept image and concept definition in mathematics, with special reference to limits and continuity’, Educational Studies in Mathematics, 12, 151–169. Tall, D. O., 1982: ‘Elementary axioms and pictures for infinitesimal calculus’, Bulletin of the IMA, 18, 43–48. Tall, D. O., 1992: ‘The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity, and Proof’. In D. A .Grouws (ed.) Handbook of Research on Mathematics Teaching and Learning, (pp. 495–511). New York: Macmillan. Tall, D. O., Gray, E. M., Ali, M. B., Crowley, L. R. F., DeMarois, P., McGowen, M. A., Pitta, D., Pinto, M. M. F., Thomas, M. O. J. & Yusof, Y. B. M., 2001:. ‘Symbols and the Bifurcation between Procedural and Conceptual Thinking’, Canadian Journal of Science, Mathematics and Technology Education 1, 81–104. 
URI:  http://wrap.warwick.ac.uk/id/eprint/476 
Actions (login required)
View Item 