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The two main criticisms against the philosophical position on mathematics which is called game formalism (see here for details) are,

The first is the question of applicability: if mathematics is just a calculus in which we shuffle uninterpreted symbols (or symbols whose interpretation is a matter of no importance), then why has it been applied so successfully—and in so many ways, to so many different things—ordinary physical objects, sub-atomic objects, fields, properties, and indeed from one part of mathematics to another (we can count the number of dimensions in a pure geometric space)?

and,

The problem this raises for the formalist is this: the metatheory is itself a substantial piece of mathematics, ostensibly committed to an infinite realm of objects which are not, on the face of it, concrete. Tokens of the expressions of the object language game calculus may be finite—ink marks and the like; but since there are infinitely many expressions, theorems and proofs, these themselves must be taken to be abstract types. At best, the formalist can achieve no more than a reduction in commitment from the transfinite realms of some mathematical theories, such as set theory, to the countably infinite, but still presumably abstract, realm of arithmetic, wherein the syntax and proof theory of standard countable languages such as those of standard set theory, can, as Gödel showed, be modelled.

However, in the article, the author also writes that,

Not that these are the only objections to formalism, but they are two fundamental ones.

However, I can't find any objection to Hilbert's version of Formalism (apart from the question of applicability) and this made me wonder two things,

  • What are the other objections to game formalism?

  • What are the objections to Hilbert's version of Formalism other than the two mentioned above?

  • But the "philosophical" position of Hilbert was not: "mathematics is just a calculus in which we shuffle uninterpreted symbols". The matheatical study of a mathemtical theory (called meta-math) must treat it as an uninterpreted calculus whose "formal" properties must be investigated: consistency and completeness make little sense for e.g the game of chess. – Mauro ALLEGRANZA Dec 16 '17 at 9:46
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  • @MauroALLEGRANZA: I am sorry that I misinterpreted your comment. You are right in saying that for Hilbert "mathematics is [not] just a calculus in which we shuffle uninterpreted symbols". Then there is a different problem for a Hilbertian formalist because for a Hilbertian formalist every formal system is equally "meaningful" and hence he needs to answer the question: Why for some formal system we can interpret them but for some we can't (at least that doesn't seem to be the case)? – user 170039 Dec 17 '17 at 3:53
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    Very useful The Frege-Hilbert Controversy: in a nutshell, Hilbert has the "informal" notion of modern soundness theorem: if a system is consistent, then it is satisfiable (i.e. has a model). – Mauro ALLEGRANZA Dec 17 '17 at 8:45
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    A good book that link Formalism to history of math is Craig Smorynski, Adventures in Formalism, College Publications (2012). – Mauro ALLEGRANZA Dec 18 '17 at 8:31

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