On the Adequacy and Substantiality of the Structuralist Thesis
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Babeș-Bolyai University / Cluj University Press
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The idea that positions in structures have no mathematically significant non-fundamental features is a constitutive trait of non-eliminative structuralism; it underpins the restricted structuralist thesis that all fundamental properties are structural. So, a seemingly straightforward strategy to uphold the eligibility of non-eliminative structuralism is to prove a formal rendition of the thesis. However, the soundness of the strategy depends on two key aspects: the thesis has to be substantial, and materially adequate. The substantiality of the thesis is predicated on the non-synonymy of fundamental and structural properties. The adequacy is predicated on the synonymy between the formal definition of fundamental properties and the intuitive content of the notion. Two remarkable abstractionists accounts claim to have proven a formal, non-trivial, consistent version of the thesis. The first one, developed Linnebo and Pettigrew, arguably fails to satisfactorily accomplish this goal. However, the more formally sophisticated second one, developed by Schiemer and Wigglesworth, succeeds. This will be focus of the paper. I am going to argue that, precisely because it proves a non-trivial formal version of the thesis, their account of fundamental properties fails to be adequate. More precisely, I will show that the formal specifications of the fundamental properties needed to ensure the substantiality and soundness of the proof undergenerate and overgenerate structural properties. In the end, it seems that there is a trade-off between substantiality and adequacy. The arguments will inform some pessimistic conclusions about the overall strategy of establishing the eligibility of non-eliminative structuralism by means of such a proof of the structuralist thesis.