Semantics and the Ontology of Number

Abstract: What are the meanings of number words, such as 'four', and how might they shed light on the ontological question of whether numbers exist? This short monograph attempts to answer these questions by first outlining a recent debate between two popular kinds of philosophical analyses concerning the semantic function of number words (substantivalism and adjectivalism ), their motivations, and their difficulties, and secondly by developing a comprehensive, polymorphic semantics for number words which overcomes those difficulties. Ultimately, I argue for three major claims: i) Number words can take on a wide variety of different but related semantic functions, and thus meanings; ii) explaining how these various meanings are systematically related requires recognizing that they share an element in common, namely a number; and iii) in light of this, independent of whether number words function referentially or non-referentially as a matter of their lexical meaning, no uses of number words, including non-referential uses, would be true if numbers did not exist. Put differently, making sense of the polysemy of number number words provides a powerful, and novel, argument for realism. 

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