Zooming in on infinitesimal 1.9.. in a posttriumvirate era
Abstract
The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis"..." in the real formula .999... = 1. Infinitesimalenriched number systems accomodate quantities in the halfopen interval [0,1) whose extended decimal expansion starts with an unlimited number of repeated digits 9. Do such quantities pose a challenge to the unital evaluation of the symbol ".999..."? We present some nonstandard thoughts on the ambiguity of the ellipsis, in the context of the cognitive concept of generic limit of B. Cornu and D. Tall. We analyze the vigorous debates among mathematicians concerning the idea of infinitesimals.
 Publication:

arXiv eprints
 Pub Date:
 March 2010
 arXiv:
 arXiv:1003.1501
 Bibcode:
 2010arXiv1003.1501K
 Keywords:

 Mathematics  History and Overview;
 97A20;
 97C30
 EPrint:
 18 pages, to appear in Educational Studies in Mathematics