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

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.