Since Quine and Goodman not only impose nominalistic strictures, but also finitism in their joint article (Quine and Goodman 1947, §2), the syntactic and proof-theoretic notions defined still fall short of the usual platonist counterparts. Even if any given sentence or proof is finite in length, the platonist would hold that there are sentences and proofs of any finite length, and thus sentences and proofs that are too long to have a concrete inscription in a given finite universe.  Moreover, there are infinitely many (and indeed uncountably many) truths of mathematics, but—in particular, in a finite universe—there will only ever be finitely many inscriptions of theorems. Even if the universe is in fact infinite, perhaps a theory of syntax and proof should not make itself hostage to this circumstance.