This post started life as a rather long comment in response to a comment that my favourite Poor Unhappy Soul wrote to this post. But I'm not sure how many of you read the comments, and I'd actually quite like some input on this, because my ramblings may not make a lot of sense.
The whole point about faith is that you believe something even though you have no proof. I don't BELIEVE IN the cup of tea on the table beside me - I have objective proof that it's there, and I would be stupid to deny its existence, as would anyone else who was in the room with me.
You, on the other hand, have no way of knowing whether I have a cup of tea beside me or not. It would be reasonable for you to assert whether or not you believe that I have a cup of tea here. You could then share your belief with another person, and they could also choose whether or not to believe you.
There is an objective reality there, but you would have no proof that your belief was true. In fact, you could be wrong about it and never know - perhaps there is no cup on the table beside me, or even no table, or perhaps there is a cup, but it contains coffee rather than tea.
I could be wrong in my belief that there is a God - I have no way of objectively proving that I'm right. If I did, that would be knowledge and not belief. As it is, I KNOW what the Church teaches, and I BELIEVE it to be true. I can't prove it to you, and any attempt to prove it would contain the sort of false logic you see in my arithmetical example.
The fact that you wouldn't be able to prove to a third party whether or not I have a cup of tea on the table beside me (because you can't see me, don't know who I am, don't know where I live, and so have no way of objectively verifying my assertion) doesn't alter the fundamental and objective truth. Either there is a cup of tea here or there isn't.
And either there is a God or there isn't. I believe there is. Drac believes there isn't. Only one of us can be right. And neither of us can genuinely prove it one way or the other. The absence of a rock solid proof that is accessible to both of us does not disprove the existence of God.