Hilbert and the internal logic of mathematics

Hilbert's programme is shown to have been inspired in part by what we can call Kronecker's programme in the foundations of an arithmetic theory of algebraic quantities.While finitism stays within the bounds of intuitive finite arithmetic, metamathematics goes beyond in the hope of recovering classical logic. The leap into the transfinite proved to be hazardous, not only from the perspective of Gödel's results, but also from a Kroneckerian point of view.Hilbert's rare admission of a Kroneckerian influence does not constitute the basis of such a reconstruction; it is rather Kronecker's mathematical practice which is seen as a forerunner of Hilbert's endeavour in the foundations of mathematics.I am indebted to an anonymous referee for many helpful critical remarks.