This paper reveals David Hilbert’s position in the philosophy of mathematics, circa 1900, to be a form of non-eliminative structuralism, predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the Frege-Hilbert controversy in virtue of this early structuralist approach. To demonstrate that this historical position deserves contemporary attention I show that Hilbertian structuralism avoids a recent wave of objections against non-eliminative structuralists to the effect that they cannot distinguish between structurally identical but importantly distinct mathematical objects, such as the complex roots of $-1$.

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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