The identity problem for realist structuralism

Philosophia Mathematica 9 (3):308--30 (2001)
According to realist structuralism, mathematical objects are places in abstract structures. We argue that in spite of its many attractions, realist structuralism must be rejected. For, first, mathematical structures typically contain intra-structurally indiscernible places. Second, any account of place-identity available to the realist structuralist entails that intra-structurally indiscernible places are identical. Since for her mathematical singular terms denote places in structures, she would have to say, for example, that 1 = –1 in the group (Z, +). We call this the identity problem and conclude that nominalism is presently the safest route for the structuralist.
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