Eric Snyder

According to non-eliminative structuralism, the referents of numerical singular terms, such as the numeral ‘two’ or ‘the number two’, are numbers, construed as positions within the natural number structure. However, a potential problem comes in the form of sentences like ‘{∅, {∅}} is the number two among the von Neumann ordinals’. If this is an identity statement, then its truth would seemingly require identifying the second position of the natural number structure with a particular set, thus giving rise to a version of Benacerraf’s famous Identification Problem. In response, Stewart Shapiro (1997) draws an analogy to expressions like ‘the Vice President’, which are ambiguous between denoting an office-holder (e.g. Kamala Harris) or an office (the office of the Vice Presidency). Similarly, Shapiro suggests that in ordinary arithmetic contexts, such as ‘Two is less than three’, we view positions as analogous to office-holders, while in other contexts, we view them instead as analogous to offices occupied by entities playing the role of numbers, e.g. {∅, {∅}}. However, this suggestion faces two serious challenges. First, what exactly is the nature of this purported ambiguity, and what empirical evidence, if any, is there for it? Second, even if we grant the ambiguity, we appear to get a revenge version of the Identification Problem anyway: just permute the positions within the natural number structure. The purpose of this talk is to defend Shapiro’s ambiguity thesis, by supplying the empirical support required, and explaining how, when appropriately understood, the semantics assumed does not give rise to a revenge form of the Identification Problem.

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