  The analogy with physics is striking, and one can find similar ideas in Hilbert’s own writing—perhaps Hilbert was influenced in this by Weyl. Although Hilbert’s first proposals focused exclusively on consistency, there is a noticeable development in Hilbert’s thinking in the direction of a general reductivist project of a sort quite common in the philosophy of science at the time (as was pointed out by  Giaquinto 1983).  In the second half of the 1920s, Hilbert replaced the consistency program with a conservativity program: Formalized mathematics was to be considered by analogy with theoretical physics. The ultimate justification for the theoretical part lies in its conservativity over “real” mathematics: whenever theoretical, “ideal” mathematics proves a “real” proposition, that proposition is also intuitively true. This justifies the use of transfinite mathematics: it is not only internally consistent, but it proves only true intuitive propositions (and indeed all, since a formalization of intuitive mathematics is part of the formalization of all mathematics).