David Hilbert wanted all mathematics to be proved by logical steps. Kurt Gödel showed that no axiomatic system could be complete and consistent at the same time:
On Monday, September 8, 1930, Hilbert opened the annual meeting of the Society of German Scientists and Physicians in Königsberg with a famous discourse called “Logic and the knowledge of nature.” He ended with these words:
“For the mathematician there is no Ignorabimus, and, in my opinion, not at all for natural science either…
“The true reason why [no-one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem. In contrast to the foolish Ignorabimus, our credo avers: We must know, We shall know.”
In one of those ironies of history, during the three days prior to the conference opened by Hilbert’s speech, a joint conference called Epistemology of the Exact Sciences also took place in Königsberg. On Saturday, September 6, in a twenty-minute talk, Kurt Gödel (1906–1978) presented his incompleteness theorems. On Sunday 7, at the roundtable closing the event, Gödel announced that it was possible to give examples of mathematical propositions that could not be proven in a formal mathematical axiomatic system even though they were true.
The result was shattering. Gödel showed the limitations of any formal axiomatic system in modeling basic arithmetic. He showed that no axiomatic system could be complete and consistent at the same time.Daniel Andrés Díaz Pachón, “Faith is the most fundamental of the mathematical tools” at Mind Matters News
So it’s not a question of faith vs. reason but faith so we can have reason.