Skip to main content
SpringerLink
Log in

The notion of infinite measuring number and its relevance in the intuition of infinity

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fischbein, E.: 1978, ‘Intuition and mathematical education’,Osnabrücker Schriften zur Mathematik 1, 148–176.

    Google Scholar 

  2. Fischbein, E., Tirosh, D. and Hess, P.: 1979. ‘The intuition of infinity’Educational Studies in Mathematics 10, 3–40.

    Google Scholar 

  3. Keisler, H. J.: 1976,Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston.

    Google Scholar 

  4. Robinson, A.: 1966,Non-standard Analysis, North Holland, Amsterdam.

    Google Scholar 

  5. Stewart, I. N. and Tall, D. O.: 1977,Foundations of Mathematics, Oxford University Press, Oxford.

    Google Scholar 

  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).

Download references

Author information

Authors and Affiliations

  1. Mathematics Education Research Centre, Warwick University, USA

    David Tall

Authors
  1. David Tall
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tall, D. The notion of infinite measuring number and its relevance in the intuition of infinity. Educ Stud Math 11, 271–284 (1980). https://doi.org/10.1007/BF00697740

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00697740

Keywords

Search

Navigation