Three varieties of mathematical structuralism (Q2765252)

This paper compares three varieties of structuralism that have come to the fore in recent philosophy of mathematics. These are (i) set-theoretic structuralism (STS), which defines mathematical structures in familiar model-theoretic ways, (ii) the `sui generis' structuralism (SGS) of \textit{Stewart Shapiro} (as in his book: Philosophy of mathematics: structure and ontology. New York: Oxf. Univ. Press (1997; Zbl 0897.00004)), which considers structures to be a special kind of object implicitly defined in a second-order theory, and (iii) the author's own modal-structuralism (MS) (as in his: Mathematics without numbers. Oxford: Clarendon Press (1989; Zbl 0688.03001)), which avoids literal quantification over structures, and instead appeals to the second-order logical possibility of a domain and its relations. The author argues that while all three agree about ordinary mathematics, they diverge with regard to some extraordinary mathematics. He presents a number of problems that STS faces but SGS and MS avoid, and then some other problems that SGS faces that MS avoids. To the charge that the modal primitives of MS are themselves problematic, the author replies that both STS and SGS are implicitly committed to similar sorts of modality.