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

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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 ( 2013) 
Library of Congress Subject Headings (LCSH):  Infinity, Mathematical ability 
Journal or Publication Title:  Educational Studies in Mathematics 
Publisher:  Springer Netherlands 
ISSN:  00131954 
Official 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, 
URI:  http://wrap.warwick.ac.uk/id/eprint/476 
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