Are faith and proof mutually exclusive? Must something be unprovable to permit faith in it? Is it possible to have faith in something which is certain provable fact? If something is proven beyond any doubt can you still have faith in it? How?
The much used ode to faith in the NT book of Hebrews tells us a little bit about what faith is. Here are the first couple verses:


  Hebrews 11:1-3 (ESV)
  Now faith is the assurance of things hoped for, the conviction of things not seen. For by it the people of old received their commendation. By faith we understand that the universe was created by the word of God, so that what is seen was not made out of things that are visible.


The passage goes on to give many examples, but this bit helps us scope out a definition of faith that is somewhere between the two extremes you suggest in your question. First of all we see that there is assurance. True faith is not blind, it is based on evidence and knowledge. It is a kind of trust. Trust between two parties is established in past encounters but looks forward to the future knowing what to expect from the other party. Secondly it is a conviction. This doesn't have to be without any doubt (in fact many of the people listed as examples in the passage experienced doubt) but it does have to come to a conclusion and really believe/trust.

Faith is built on evidence and knowledge as well as a trust that enables us to be sure the parts we don't fully understand will work out.

Faith and proof are not mutually exclusive -- faith projects evidence of past events into the future.