For the Kronecker's views, Brouwer's intuitionism and much more real motivations for Hilbert's program that the usual tales about paradoxes as main reasons of crisis (they are only a pretext and the reasons go back to 19th century reincarnation of mathematics) see for example
Jeremy Avigad and Erich Reck "Clarifying the nature of the infinite: the development of metamathematics and proof theory".
You may also look at many other important historical works referenced there.