Tall, David. (1980) The notion of infinite measuring number and its relevance in the intuition of infinity. Educational Studies in Mathematics, Vol.11 (No.3). pp. 271-284. ISSN 0013-1954

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Abstract

In this paper a concept of infinity is described which extrapolates themeasuring properties of number rather thancounting aspects (which lead to cardinal number theory). Infinite measuring numbers are part of a coherent number system extending the real numbers, including both infinitely large and infinitely small quantities. A suitable extension is the superreal number system described here; an alternative extension is the hyperreal number field used in non-standard analysis which is also mentioned. Various theorems are proved in careful detail to illustrate that certain properties of infinity which might be considered false in a cardinal sense are true in a measuring sense. Thus cardinal infinity is now only one of a choice of possible extensions of the number concept to the infinite case. It is therefore inappropriate to judge the correctness of intuitions of infinity within a cardinal framework alone, especially those intuitions which relate to measurement rather than one-one correspondence. The same comments apply in general to the analysis of naive intuitions within an extrapolated formal framework.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Social Sciences > Institute of Education
Library of Congress Subject Headings (LCSH):	Infinity, Mathematics
Journal or Publication Title:	Educational Studies in Mathematics
Publisher:	Springer Netherlands
ISSN:	0013-1954
Date:	1980
Volume:	Vol.11
Number:	No.3
Page Range:	pp. 271-284
Identification Number:	10.1007/BF00697740
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	[1] Fischbein, E.: 1978, ‘Intuition and mathematical education’, Osnabrücker Schriften zür Mathematik, 1, 148-176. [2] Fischbein, E., Tirosh, D. and Hess, P.: 1979, ‘The intuition of infinity’, Educational Studies in Mathematics 10, 3 40. [3] Keisler, H. J.: 1976, Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston. [41 Robinson, A.: 1966, Non-standard Analysis, North Holland, Amsterdam. [S] Stewart, I. N. and Tall, D. O.: 1977, Foundations of Mathematics, Oxford University Press, Oxford. [6] Tall, D. O.: 1980, ‘Infinitesimals constructed algebraically and interpreted geometrically’, Mathematical Education for Teachers (to appear). [7] Tall, D. O.: 1980, ‘Looking at graphs through infinitesimal microscopes, windows and telescopes’, Mathematical Gazette (to appear).
URI:	http://wrap.warwick.ac.uk/id/eprint/506

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