Four conceptions of infinity

Abstract

The paper is the result of extensive research carried out among Czech students and focuses on a conception of infinity. The questionnaire survey was taken by 861 students ranging from grades 7 to 13. The aim of the research was to describe the development of students’ conceptions of infinity. These conceptions are built on the intuitive phenomenon of the horizon. We monitor the proportional representation of these conceptions in four combinations of views (into the distance and in depth) and contexts (arithmetical and geometrical). It can safely be maintained that the development of the proportional representation of the earliest conception comprehension of the concept of infinity, the so-called natural infinity, is not concurrent with the students’ age. The development of the proportional representation of the conception of actual infinity is non-decreasing, at least in the view into the distance, in both contexts with respect to age. In general, it can be stated that the proportional representation of individual conceptions of infinity is strongly dependent on both context and view.

KEYWORDS:

• Conception of infinity
• natural infinity
• potential infinity
• omega-epsilon position
• actual infinity
• horizon

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Interesting is its comparison with the arithmetic of extremely large numbers with which ancient Indian mathematics worked and which was followed by an early idea of infinity. The Jains (sixth century)

had an interest in very large numbers and infinity and they classified all numbers into three groups, namely enumerable, innumerable, and infinite. […] The Jains were the first to discard the idea that all infinities are the same, an idea held in Europe until Georg Cantor’s work in the late nineteenth century delineated the countable infinity from uncountable infinities. (Nataraj & Thomas, 2009, p. 100)

2 So far we have used the term omega position in our research (Cihlář et al., 2013, 2015) but we have decided to change it with reference not only to infinitely large (omega as the ordinal number of the set of natural numbers, which is their extension), but also to infinitely small (i.e. epsilon as an infinitely small quantity).

3 The Czech Republic school system: Primary education lasts for a period of nine years and is divided into two stages: i.e. a 5-year stage (from the age of 6 until the age of 11) and a 4-year stage (from the age of 11 until the age of 15). Some of the primary schools focus on the teaching of gifted children, i.e. selective language schools, eight-year secondary grammar schools and schools with extended instruction in mathematics. Secondary education comprises three main types of schools: secondary general schools (grammar schools), secondary technical and business schools and secondary vocational schools. Schools of this type will be called high schools. Grammar schools prepare students for their further studies at institutions of higher education. The above mentioned secondary technical schools and numerous 4-year courses at secondary vocational schools prepare students for a wide range of professions, as well as for further studies at institutions of higher education. The 2-year and 3-year courses at Czech vocational schools prepare students for a great variety of professional activities.

4 Non-increasing development, or non-decreasing development, respectively, is identified whereby there can be interlaid a non-increasing function, or a non-decreasing function respectively, into all of the 95% confidence intervals.

 

Additional information

Funding

This work was supported by the Czech Science Foundation under Grant [number 406/07/1026].