Indistinguishable elements and mathematical structuralism

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According to mathematical structuralism, mathematical objects are defined by their positions in mathematical structures. Structures are under stood in the standard sense, as domains over which certain privileged relations are explicitly defined and that may contain identified constant elements and functions. In talking about structures we use a language (for the purposes of this article, and for model theory in general, a first-order language) whose signature contains symbols for the relevant relations, constant elements, and functions. The theory of a structure is the set of sentences true in that structure. To take a familiar example, we specify the structure X = (N, x, +, s, <, 0) in a way that makes clear its signature and hence the language 3?x that we use to talk about it. The theory of this structure, Th (X) = [cp I cp is a sentence of 2?$ and X \= (p\, is complete arithmetic. One objection to mathematical structuralism is that there seem to be facts about the domain of a structure that cannot be 'stated' in terms of the relations, elements, and functions available within the structure. That is, there are facts about elements of the domain that do not seem to be reducible to facts about positions in the domain. Thus John Burgess, commenting on the complex field C = (C, x, +, 0,1) in relation to the ver sion of structuralism proposed in Shapiro (1997): We have two roots to the equation z2 + 1 = 0, which are additive inverses of each other, so that if we call them / and / we have / = ?i and / = -/. But the two are not distinguished from each other by any algebraic properties, since there is a symmetry or automorphism of the field of complex numbers, which is to say an isomorphism with itself, which switches / and /'. On Shapiro's view the two are distinct, although there seems to be nothing to distinguish them. (Burgess 1999: 288)