Denotators are purely mathematical structures which do not specify “what is the case” and what is not. This chapter deals with the existence problem of music-related objects in contrast to mathematical ‘fiction’. This amounts to loading mathematical constructs with an additional semiotic signification process in order to express “which denotators are the case”. These existence specifications instantiate an interface between mental potentiality and historical actuality. It reveals two fundamentally different existentialities, termed “textual” and “paratextual” signification, respectively. The former involves predicates defined by classical extension over specific denotators, whereas the latter transcends pure extensionality and thus points into domains of open semiosis. Both, textual and paratextual predicates are essential enrichments of mathematical constructs: The platonic ontology is thereby supplemented by a differentiation which cannot be reduced to pure “mentality”. The variety of textual predicates follows certain construction rules of logical and geometric nature and is founded in a triply typed set of “atomic” predicates of (a) mathematical, (b) musical, and (c) deictic types.
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