Hilbert's formalism and game formalism differences and similarities

Question

I've recently encountered differences between Hilbert's formalism and game formalism.

They seem pretty much similar in my eyes. I wish to understand in what way does Hilbert’s formalism resemble game formalism? In what way does it differ?

Answer 1

Alan Weir distinguishes between game formalism and Hilbert's formalism. He describes Hilbert's formalism as follows:

The Hilbertian position differs [from game formalism] because it depends on a distinction within mathematical language between a finitary sector, whose sentences express contentful propositions, and an ideal, or infinitary sector. Where exactly Hilbert drew the distinction, or where it should be drawn, is a matter of debate. Crucially, though, Hilbert adopted an instrumentalistic attitude towards the ideal sector. The formulae of this language are, or are treated as if they are, uninterpreted, having the syntactic form of sentences to which we can apply formal rules of transformation and inference but no semantics. Nonetheless they are, or can be useful, if the ideal sector conservatively extends the finitary, that is if no proof from finitary premisses to a finitary conclusion which takes a detour through the infinitary language yields a conclusion we could not have reached, albeit perhaps (herein lies the utility) by a longer, more unwieldy proof.

By contrast he describes game formalism as follows:

Returning now to our non-Hilbertian focus, the earlier formalism which Frege attacked does not divide mathematics into the aforementioned dual categories of the finitary/contentful, and the infinitary/essentially meaningless but, on the contrary, treats all of mathematics in a unitary and homogeneous fashion.

Game formalism treats all of mathematics as a game with no commitment to "an ontology of objects or properties". Hilbert's formalism divides mathematics into finitary and infinitary sectors with only the infinitary sector remaining uninterpreted.

---

Weir, Alan, "Formalism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.), forthcoming URL = https://plato.stanford.edu/archives/fall2019/entries/formalism-mathematics/.

Answer 2

Weir's description as reported in Frank Hubeny's answer is insufficient because it does not mention the crucial distinction between the "finitary sector" and the "infinitary sector". Namely, Hilbert is usually understood in the following sense. The finitary sector occurs at the metamathematical level, whereas the infinitary sector is at the object-language level. In this sense, they are not really "sectors" into which mathematics is divided, because they occur on different levels.

I should note also that the idea of "game formalism" is a concept that exists mainly in Errett Bishop's attacks on classical mathematics, expressed notably in his Foundations of Constructive Analysis (1967). Other (though not all) intuitionists and constructivists have also penned such attacks. Notably, Arend Heyting did not; see also this.