The term proof theory, in German Beweistheorie, was introduced by Hilbert [1] in 1922 as the name of a subject which he had already described in a lecture [2] in 1917 as "an important, new field of research" where "we must make the concept of specific mathematical proof itself object of investigation, just as also the astronomer pays attention to his place of observation, the physicist must care about the theory of his instrument, and the philosopher criticizes reason itself".