There's a wonderful passage in Philippians 4:6-7 where Paul writes, "Do not be anxious about anything, but in everything, by prayer and petition, with thanksgiving, present your requests to God. And the peace of God, which transcends all understanding, will guard your hearts and your minds in Christ Jesus." My question is, why the thanksgiving? What does thanksgiving contribute to the process? Prayer and petition I understand; indeed, I don't see the difference between them in this context. To petition God is to pray. To present a request to God in prayer is to petition him. That's another problem I won't go into; meanwhile, I'm still asking about the thanksgiving part. Is it just that to ask God for certain things requires that one is in a right relationship with him, and this requires, among other things, a general attitude of thanksgiving? I guess that's possible but I don't think that's what Paul is saying here.
Let me present my answer to this question by first pointing out that we tend to think of "faith" in two ways. The first is discussed in Hebrews 11:1: "faith is being sure of what we hope for and certain of what we do not see." Being certain of what we do not see suggests that everything that one is not seeing at the present moment is believed on faith. I think we can extend this beyond sight as the author of Hebrews presumably did not mean to exclude things that we are immediately experiencing via some other sense or introspection. Even so, taken this way, faith is incredibly broad. It would mean that I have faith that the wall behind me is still there while I'm typing this. I have faith that the dinner I just had with my family actually happened. The only thing we would not call faith are those things that one is immediately experiencing at the present moment. Since the future is precisely such a thing, we are exhorted to have faith that God will keep his promises. This kind of faith is propositional in nature, it is intellectual assent, and it is not the faith by which we are saved. When people say they believe that God exists by faith they are employing this concept of faith. You have faith that a particular proposition is true or false. I have faith that the wall behind me is still there, even though it's not immediately present to my mind. This is problematic however since God sometimes makes his presence known to people directly, in such a way that we are immediately aware of him. According to this first definition, we do not have faith in God at those moments. It's no good saying, "Yes, maybe in those moments we don't have faith because we know God is there," because faith has traditionally been conceived of as a kind of knowledge. So this strikes me as a problem for the first type of faith.
The other kind of faith is also in Hebrews 11:1. This is not intellectual assent, it is essentially trust. Faith like this is not faith that God exists, but trust in him. So the statement "I have faith in God" can be taken in two ways: it can be taken as an assertion that the person believes that God exists, or it can be taken as an assertion that since God exists, the person trusts him. The presence of this second kind of faith in Hebrews 11:1 is not self-evident; it comes from asking the question, "Why should we be sure of what we hope for?" The reason is that we can trust God. It is based on what God's character is like, based on the Bible and on one's experience of him. Our experience of him does not necessarily refer to theophanies or the like; it refers to the experiences we have had in which God was faithful to us, where he showed himself to be trustworthy by bringing us through difficulties or blessing us with things we asked for.
And now we can see why we should present our requests to God "with thanksgiving." We are being told that, when we ask God for something specific, we should reflect on the times when God showed himself to be eminently trustworthy. This doesn't mean that he will do everything and anything we ask for. It means that we can trust him to carry us through whatever happens. We can trust that he has our best interests at heart. Jesus said it: "Which of you fathers, if your son asks for a fish, will give him a snake instead? Or if he asks for an egg, will give him a scorpion?" We may think our best interests entail a particular state of affairs and pray that God bring about that state of affairs. God has a longer view than we do, and in all likelihood knows a few things about what we're asking for that we don't know. But he also loves us and loves to bless us. Regardless, when we ask God for something, we should do it with thanksgiving, thanksgiving for how he has carried us through the fire in the past. The point being that this should impact our confidence in asking for God to give us something in particular. In fact it should do more than impact it, since Paul tells us what the result will be of presenting our requests before God with thanksgiving: "And the peace of God, which transcends all understanding, will guard your hearts and your minds in Christ Jesus."
That's all I need to say about the Philippians passage, but I have more to say about these two concepts of faith. They are closely related. When I was a kid I had faith that my mother wouldn't deliberately hurt me or allow me to come to harm. This is clearly the second kind of faith. But just as clearly, it can be formed into a proposition that I believed, since I just did so. So I can have the first kind of faith that the proposition "My mother won't hurt me" is true. I can also just trust my mother based on my experience of her as a loving, gentle woman who loved me dearly.
Here's another point: neither kind of faith is incompatible with evidence or proof. If I had been intellectually sophisticated enough, I could probably have provided plenty of evidence attesting to my mother's character and that hurting her child is incompatible with this character. I could even have constructed this evidence into an argument demonstrating it. Did I have evidence? Yes. Could I have proven it beyond any reasonable doubt? Yes. Did I have faith? Yes. The only point where faith would conflict with evidence is the point mentioned above regarding the first kind of faith and something immediately present to the mind. But this is because my belief in something I'm directly experiencing -- that I'm looking at a computer screen right now, or that I have a headache, or whatever -- isn't really a product of evidence. I don't weigh the evidence for or against something I'm directly experiencing. I don't formulate an argument like: "via introspection I discover that my head hurts; introspection about what one is directly experiencing at the present moment cannot be mistaken; therefore I believe that my head really does hurt." My belief is not based on evidence or argument at all. Of course, this brings us back to the claim that God can and does reveal himself directly to our minds. This would then imply that we do not require evidence or argument to believe in him either. As mentioned above, this would seem to conflict with the first type of faith. But perhaps we can comfort ourselves in those moments by settling for the second.