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.

For the Kronecker's views, Brouwer's intuitionism and much more real motivations for Hilbert's program than 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.