In lectures given in 1922 at Göttingen, Hilbert launched his Beweistheorie — proof theory. The aim was to take the proofs of mathematics and make them themselves the object of investigation, to establish that no contradiction could ever be derived within the system.
As he put it, we move to a higher level of contemplation, from which the axioms, formulae and proofs of the mathematical theory are themselves the objects of investigation.
Not: are these results correct. But: is this way of reaching results sound at all.