Thanks to Odder, all good info. Yes, we believe that Lorentz invariance holds to about 1 part in 10^17, and even then, we're (probably) measuring the limits of the experimental setup; our technological abilities. That is to say, Einstein's theories of relativity (both special and general), have never been violated, to the best of humans' ability to measure such a thing. There is, in fact, a theoretical particle that moves backwards in time, called a tachyon, but if they do exist, there is no known interaction with us, or any baryonic matter, for that.. err, matter. Here's a classic joke among particle physicists: I'm afraid I can't do that, Dave. This is in the realm of known physics. You are correct in thinking that a grand unifying theory of physics may require reformulating everything that we know in a completely novel set of axioms. String theory is one attempt at this, though it doesn't make testable predictions that would reveal its superiority over existing theories. And I'm personally under the impression that there will never be another singular person (another "Einstein") capable of bestowing us with a grand unifying theory; it will probably require teams of people working iteratively over several generations. Things have just gotten too damn complicated. But I hope I'm wrong. It's healthy to muse on things like this, though, I don't want to discourage that. I can guarantee that I would look like Hubski's ultimate fool if I were start bumbling on about economics, poetry, politics, or anything that isn't physics. Apparently, to become an educated physicist these days, it requires hardly knowing anything else, at least until you get the PhD. And I've already cried about it, my tear ducts are drier than the Sahara, at this point.
If you are an educated physicist please turn OFF everything you have learned just for a minute, and just let your mind wander a bit into the unknown.
Your "2001: A Space Odyssey" reference made me smile, I like it. And that is our cage. That little "can't" ... Speaking in psychological sense (nothing to do with physics) we know that we "do not know" but, we do not know how to look for that unknown or usually we are very bad at it. Most of the time way how we look for the things can be compared with brutal force method in computing. Psychologically if we say that something is impossible, we are shutting down our brains to look for solutions, we are saying this is it, this is all we can ever hope for. Every time we say it is "impossible", "it will never work", "there is no point" those cage bars become a bit bigger and stronger. Do not get me wrong, I just like more when people intelligently try and failed thousand times trying to do something new, than when they give up at the beginning or just hold dearly to one standard already known thing. I'm afraid I can't do that, Dave. This is in the realm of known physics.