Ante rem Structuralism and the Myth of Identity Criteria
Siu, Ho Kin
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This thesis examines the connections between the motivations of ante rem structuralism and the problem of automorphism. Ante rem structuralists are led to the problem of automorphism because of their commitment to the thesis of structure-relative identity. Ladyman's and Button's solutions to the problem are both unsatisfactory. The problem can be solved only if ante rem structuralists drop the thesis of structure-relative identity. Besides blocking the problem of automorphism, there are further reasons why the thesis has to be dropped. (i) The purported metaphysical and epistemic purchase of adopting the thesis can be put into doubt. (ii) Primitive identity within a mathematical structure is more in line with ante rem structuralist's commitment to the faithfulness constraint and to the ontological priority of structure over positions. However, the cost of dropping the thesis is that ante rem structuralists cannot provide a satisfactory solution to Benacerraf's problem of multiple reductions of arithmetic.