On Ambiguity: The case of fraction, its meanings and roles

TL;DR

This paper explores ambiguity in mathematical language, especially regarding fractions, proposing precise terminology and a categorization approach to clarify their meanings and roles.

Contribution

It introduces new terms to distinguish different aspects of fractions and argues that 'fraction' functions as a category rather than a single mathematical concept.

Findings

Proposed new terminology for fractional concepts

Resolved ambiguity in elementary arithmetic discourse

Compared structuralist and analytical views of number systems

Abstract

We contemplate the notion of ambiguity in mathematical discourse. We consider a general method of resolving ambiguity and semantic options for sustaining a resolution. The general discussion is applied to the case of `fraction' which is ill-defined and ambiguous in the literature of elementary arithmetic. In order to clarify the use of `fraction' we introduce several new terms to designate some of its possible meanings. For example, to distinguish structural aspects we use `fracterm', to distinguish purely numerical aspects `fracvalue' and, to distinguish purely textual aspects `fracsign' and `fracsign occurence'. These interpretations can resolve ambiguity, and we discuss the resolution by using such precise notions in fragments of arithmetical discourse. We propose that fraction does not qualify as a mathematical concept but that the term functions as a collective for several concepts, which we simply call a `category'. This analysis of fraction leads us to consider the notion of number in relation to fracvalue. We introduce a way of specifying number systems, and compare the analytical concepts with those of structuralism.