Natural and Formal Infinities |
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Authors: | David Tall |
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Institution: | (1) Mathematics Education Research Centre, Institute of Education, University of Warwick, Coventry, CV4 7AL |
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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 self-contradictory' (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 built-in 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 re-interpreted 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.
This revised version was published online in July 2006 with corrections to the Cover Date. |
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