Abstract
In the chapter, it is argued that a transparent precedent to the inferentialist doctrine of Robert Brandom can be traced back to so-called axiomatism, particularly in the form advocated by Hilbert and, implicitly, by Frege. The dialectical role that axiomatism has played in the history of mathematics provides an important exegetical tool to demonstrate the validity of the grounding principles of inferentialist philosophy including, surprisingly, the social perspective on knowledge.
Accordingly, the chhapter interprets the occurrence of the phenomenon of Gödel incompleteness theorems within Hilbert's symbolic program as a split of mathematical self-consciousness into two consciousnesses-known in mathematical logic under the names of "truth" and "proof"-to be interpreted as players in the game of giving and asking for reasons. In this, Lorenzen's transformation of Hilbert's purely symbolic project of operative mathematics and logic into Lorenzen's and Lorenz's dialogical logic is of particular interest, as are Lorenzen's metamathematical concepts of semi- and full-formalism.