  Hilbert’s formalism was thus quite sophisticated: It avoided two crucial objections: (1) If the formulas of the system are meaningless, how can derivability in the system generate any kind of belief? (2) Why accept the system of \(\PA\) and not any other consistent system? Both objections are familiar from Frege; both questions are (in part) answered by a conservativity proof for real statements. For (2), furthermore, Hilbert has a naturalistic criterion of acceptance: we are constrained in the choice of systems by considerations of simplicity, fecundity, uniformity, and by what mathematicians actually do; Weyl would add that the ultimate test of a theory would be its usefulness in physics.