  Most philosophers of mathematics writing on Hilbert have read him as an instrumentalist (including  Kitcher 1976,  Resnik 1980,  Giaquinto 1983,  Sieg 1990,  and in particular  Detlefsen 1986)  in that they read Hilbert’s explanation that the ideal propositions “have no meaning in  themselves”  (Hilbert, 1926,  381) as claiming that classical mathematics is a mere instrument, and that statements of transfinite mathematics have no truth value. To the extent that this is accurate, it must be understood as a methodological instrumentalism: A successful execution of the proof-theoretic program would show that one could pretend as if mathematics was meaningless. The analogy with physics is therefore not: transfinite propositions have no meaning just as propositions involving theoretical terms have no meaning, but: transfinite propositions require no direct intuitive meaning just as one does not have to directly see electrons in order to theorize about them.  Hallett (1990),  taking into account the 19th century mathematical background from which Hilbert came as well as published and unpublished sources from Hilbert’s entire career (in particular  Hilbert 1992,  the most extensive discussion of the method of ideal elements) comes to the following conclusion: