Hilbertian Structuralism and the Frege-Hilbert Controversy†

Abstract

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, indiscernibility, and physical computation

“…The indiscernibility objection is a powerful challenge to both kinds of structuralism, one that should be understood as threatening them with a potentially fatal reductio ad absurdum. However, the connection also offers a potential route to salvation for computational structuralism, by adapting a recent defence of mathematical structuralism also provided by Doherty (2019). In the next section we will introduce this defence, before finally applying it to computational structuralism in Sect.…”

“…While she is the original source of the indiscernibility objection to computational structuralism, Doherty (2019) has in fact provided a defence of mathematical structuralism against the same indiscernibility objection by drawing on the early mathematical structuralism of David Hilbert (1862-1943. Our hope is that Doherty's very own defence can actually be adapted and made applicable to the computational case, in order to resolve her own objection to computational structuralism.…”

“…It should also be noted that Hilbert's Principle can be read as indicating that Hilbert's eliminative approach to mathematical structures supports a more general eliminativism with regard to mathematical objects. 13 In what follows we will treat Hilbert as non-eliminative about both mathematical objects and structures, following Doherty (2019). However, we want to make clear that nothing much hangs on this question of whether Hilbert should be understood as an eliminativist.…”

“…In the case of mathematical structuralism, Doherty (2019) advocates a Hilbertian solution to the indiscernibility objection that amounts to rejecting PII and thereby accepting that there can be structurally indiscernible but numerically distinct mathematical objects (such as i and −i). For pure computational structuralists, the rejection of PII amounts to endorsing a particular view of the non-semantic individuation of basic components in a physical computing system.…”

On consistency and existence in mathematics

“…But even if we are willing to take the latter 67 In at least the case of catgeorical theories, it may be possible to acknowledge the current point while also assimilating Hilbert's position to the familiar taxonomy of contemporary structuralisms. Such an attempt is made by Doherty (2019) who suggests that the general strategy of Hilbert's replies to Frege can be understood as anticipating a form of non-eliminative structuralism similar to that of Shapiro (1997). (A similar interpretation is also developed by Isaacson 2011.)…”