According to mathematical structuralism, mathematics is the science of pattern and structure.1 A structure, as the notion is understood in contemporary mathematics, is a set (or possibly a proper class) with distinguished relations (which may be operations). Examples are orderings, rings, groups, fields, lattices, Boolean algebras, trees, etc.

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To introduce a central motivation for structuralism, consider the representation of the system N of natural numbers within set theory. Suppose we define 0 as Ø. Then we may define the successor of x to be x ∪ {x} or to be {x}. If we then take the smallest set containing 0 and closed under successor, then both definitions – and indeed countless others – yield isomorphic ω -sequences. But none of these representations seems any more basic or privileged than any other. Questions of the form ‘is 2 an element of...