 Hilbert’s proposed project (sometimes taken to be the starting point for formalism in the philosophy of mathematics) was to replace talk of these infinitary entities themselves with talk of the finitely long sentences that we ordinarily interpret as being about the entities. His goal was to axiomatize the theories of these infinite objects, and then to prove, using finitary means of syntactic reasoning about the language, that these theories are consistent. While this idea doesn’t deny the existence of the infinite objects, it suggests a methodological approach of only literally accepting the finitary objects, whether strokes standing for the integers or sentences. (See the entry on  Hilbert’s program.)