Non-eliminative Structuralism, Fregean Abstraction, and Non-rigid Structures

Wigglesworth, John

doi:10.1007/s10670-018-0096-3

Cited by 4 publications

(1 citation statement)

References 12 publications

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“…The indiscernibility objection against noneliminativist structuralism has recently been the focus of much attention, essentially arguing that these accounts are committed to the structural identity of patently nonidentical mathematical objects. Versions of this objection can be found in the likes of Burgess (1999), Keränen (2001), Parsons (2004), MacBride (2006), Linnebo andPettigrew (2014), andWigglesworth (2018). We will briefly review the objection before applying an analogous version of it to computational structuralism.…”

Section: The Indiscernibility Objection To Mathematical Structuralismmentioning

confidence: 99%

Structuralism, indiscernibility, and physical computation

Doherty¹,

Dewhurst

2022

Synthese

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Structuralism about mathematical objects and structuralist accounts of physical computation both face indeterminacy objections. For the former, the problem arises for cases such as the complex roots i and $$-i$$ - i , for which a (non-trivial) automorphism can be defined, thus establishing the structural identity of these importantly distinct mathematical objects (see e.g. Keränen in Philos Math 3:308–330, 2001). In the case of the latter, the problem arises for logical duals such as AND and OR, which have invertible structural profiles (see e.g. Shagrir in Mind 110(438):369–400, 2001). This makes their physical implementations indeterminate, in the sense that their structural profiles alone cannot establish whether a given physical component is an AND-gate or an OR-gate. Doherty (PhilPapers, https://philpapers.org/rec/DOHCI-3, 2021) has recently shown both problems to be analogous, and has argued that computational structuralism is threatened with the absurd conclusion that computational digits might be indiscernible, such that, if structural properties are all that we have to go on, the binary digit 0 must be treated as identical to the binary digit 1 (rendering pure structuralism absurd). However, we think that a solution to the indiscernibility problem for mathematical structuralists, drawing on the work of David Hilbert, can be adapted for the analogous problem in the computational case, thereby rescuing the structuralist approach to physical computation.

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