In this entry in the stanford encyclopedia of philosophy, it is stated that the theory of nominalist formalism deals with the metatheory problem of formalism as follows:
Commendably, Goodman and Quine do not shy away from the metatheory problem, the difficulty that syntax and metamathematics itself seems as ontologically rich and committed to abstract objects as arithmetic. On the contrary, they face it squarely and attempt to make do entirely with an ontology of concrete objects, finitely many such objects in fact. (However they do assume fairly powerful mereological principles, essentially universal composition: they assume that any fusion of objects, however scattered or diffuse, is also an object in good standing.)
With much ingenuity they try to develop a syntax which “will treat mathematical expressions as concrete objects” (ibid.)—as actual strings of physical marks—and give concrete surrogates for notions such as ‘formula’, ‘axiom’ and ‘proof’ as platonistically defined. However they do not address the issue of the application of mathematics, construed in this concrete, formalistic fashion.
To me, this passage seems to suggest that this version of formalism takes actual physical representation to be the objects of mathematical study. This passage in particular appears to confirm this hypothesis:
Secondly, what can Goodman and Quine say about a sentence such as
2^2^2^2^2^2 + 1 is prime ?
(That is—with ‘2^n’ representing ‘2 to the power n’—‘[2^(2^(2^(2^(2^2))))]+1 is prime’; cf. Tennant, 1997 p. 152.) They cannot deny the sentence exists, for there is the token before our very eyes. But there are strong grounds for thinking that no concrete proof or disproof will exist, for the only methods available may use up more time, space and material than any human could have at her disposal, perhaps than actually exists.
So, is it indeed true that nominalist formalism proposes that the formal element of mathematics exist as actual, physical objects? If so, how exactly does this solve the metatheory problem?