Mathematical proof is the easiest, you either get the right answer or not.
Physics proof says you make an assumption or a series of assumptions about matter then you set up experiments to see if matter or particles behave the way you think they do. If they do, you have your proof, if they don't its back to the drawing board, but the kink here is that more complicated theories as they evolve may prove things in overlapping or even discrete areas of physics, and so the new theory is considered correct because it proves 2 or more ideas in areas of physics.
Legal proof is the subject of a myriad TV shows and is supposed to involve all parties (even the defense, theoretically) searching for the truth of an occurrence so that if a person caused something to happen deliberately, or carelessly, they can be held accountable. Short of eyewitness testimony by the Mormon Tabernacle Choir, its almost always a judgement call. OJSimpson is the poster child for the difficulty of legal proof.
Now we get to where things are complicated because faith and belief come into the issue. How about biological proof, to wit, evolution. Evolution, even as coarsely stated by Darwin and Alfred Russell Wallace who came up with the same idea at nearly the same time, explains how animals change through time. It explains how a single finch species could evolve into several finch species over an extended period of time on the Galapagos Islands, and there have been endless refinements since then. There is a very good sequence of fossils indicating quite clearly that a terrestrial bear-like animal adapted to a more watery environment and eventually became permanently aquatic while still breathing air, hence all the species of whales. There are sequences like that for a number of species, but they are a minority because the fossilization process is so complicated and unlikely. And we haven't even touched on the fact that this contradicts a strict extremely literal reading of Genesis.
Science and Religion have different purposes, different goals, and different kinds of proof and occupy different realms in human understanding. Religion is a societal distillate of philosophy and ethics and history and cosmology and perhaps even biology which speaks mostly to how man should treat man. Science in the case of paleontology, the study of the fossil record, seeks to tie data points on a timeline and complexity grid together with the simplest explanation not involving supernatural involvement. This derives from a 13th Century English Churchman (Catholic) named William of Occam and goes by the name of Occam's Razor.
The fact that dinosaurs go unmentioned in the Bible does not mean they didn't exist, or that man existed almost as long as the earth existed and must have been contemporaneous with those unmentioned dinosaurs. The Judaeo-Christian Religion speaks to the societal experience of a relatively small group of Semites in the Middle East, and what rules they needed and wished to put into place to help them make sense of their lives and to give their lives meaning. The dietary laws of the Old Testament tended to keep the population healthy, the moral codes therein tended to keep the populace cohesive and respectful of one another, and even respectful of some outsiders. It tended to promote survival and a sense of ethical goodness not seen in some of the other Kingships that they outlived. And all of that comes through to us as Jews and Christians today. It is a backbone which even the least religious of us inherits and benfits enormously from. It underlies a great deal of the law that society enacts, and it certainly is the basis of our legal system, of judgement based on the best evidence, and the best advocacy, evaluated by a segment of the community.
It is unfair and simply wrong to expect religion and the Bible to literally explain everything, but it gives superb guidance on how to confront new issues, it is a philosophy of how to live. It is a book which distills perhaps 4000 years more or less of a people's experience and teaches us how to learn, how to adapt to new circumstances without losing our humanity and how to adapt without losing contact with the past. Having thought the issue over since Ian McKellan's pronouncement, and my agreement with Mark in regard to that, I have re-thought my position. Philosophy is not fiction, nor is it fact, it is a means of looking at the world, and at people and occurrences and making sense out of them of learning how to treat people as ends in themselves, not as instruments to an end (Immanuel Kant's Categorical Imperative).
I wouldn't want to live in a world without Science, but if forced to choose, I could always try to invent Science. I would never wish to live in a world without Religion the things that derive from religion. The respect for one another that the best parts of religion engenders, or the legal processes that have grown out of it.
Rob