Zooming in on infinitesimal 1-.9.. in a post-triumvirate era

TL;DR

This paper explores alternative formalizations of infinity and infinitesimals, analyzing the meaning of recurring decimal notation and its implications for mathematical evaluation and cognition.

Contribution

It offers a proceptual analysis of the ellipsis in .999..., discusses infinitesimal-enriched systems, and examines debates on infinitesimals' conceptual foundations.

Findings

Infinitesimal-enriched systems include quantities with infinite 9s in decimal expansion.

The ambiguity of the ellipsis impacts the evaluation of .999... as 1.

Cognitive concepts influence the understanding of infinitesimals and limits.

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. Infinitesimal-enriched number systems accomodate quantities in the half-open 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 non-standard 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.