 Hilbert’s meta-proof of consistency, by contrast, Reichenbach thinks is genuinely important. The consistency of any “interpreted” formalism can be judged empirically if there is empirical evidence for the claims made within the formalism, and Reichenbach argues that the proposition within the formalism that asserts that the interpreted formal system is consistent has an empirical probability (the only kind Reichenbach allows, of course) at least as high as that of any other claim in the language. But for mathematics, the language must be separated into the part concerned with rational numbers, which can be interpreted so that measurements give rational numbers as values, and the part concerned with real or complex numbers, which correspond to no empirical measurement. “Since all theorems of applied mathematics are deducible from the subsystem [of physical interpretation of the field of rational numbers]…it is only this subsystem which is verified by the interpretations.” (1978, vol. I, p. 423) So we cannot empirically confirm any claims that are properly about real or complex numbers, so we cannot empirically confirm their consistency. Hence we need Hilbert’s program of metamathematics, which, by arithmetizing the claims of mathematical languages, reduces claims of their consistency to claims in finite mathematics about the manipulations of symbols—a claim that can be confirmed empirically (see the entry on  Hilbert’s program).  We leave to the reader to diagnose how this separatist account fits with Reichenbach’s insistence that Bayes theorem suffices to estimate an empirical probability for theoretical claims that are not “directly” empirically confirmable. 