Some examples of reality that are not explainable by the scientific method (and these are pretty standard responses to an old question)-
Perhaps my point was made poorly. I asked for things not explainable by "natural processes". I probably should have said something like: "things that can only be explained by a supernatural intelligence." Or perhaps for which the best explanation is a supernatural intelligence.
1) Logical and mathematical proofs cannot be proven science, since science must assume mathematical proofs are true. You cannot use science to then prove it so.
In a way you can, because you can test logical and mathematical proofs and they remain consistently true. But that's not my point. How do these serve as evidence for God? Bear in mind that God could not have created these laws, since he himself is bound by them. Unless you have formulated a new answer to "can God create a rock so big even he can lift it".
2) Metaphysical reality - that is, there are external minds that exist outside one's one, or that past was not created when a person born.
If you wish to use this as evidence, you need to first provide evidence for this.
3) Ethical beliefs - that murder is wrong, or that love exists cannot be proven with science. Any value system in society cannot be proven with science. How do you show by the scientific method that the Nazi scientists in Germany were evil?
Science can demonstrate that people feel that murder is wrong, and that people experience something they call love. But again, not my point. These once again are not evidence for the supernatural. Ethics result from evolution (groups that didn't murder each other survived better), the ability to reason, and societal pressures.
4) Aesthetic judgements - a pretty girl, a delicious steak, a cool beer watching football are good things to many men, but prove it using the scientific method
5) Science itself has many unprovable assumptions - for example, in special relativity the speed of light is constant. Or unknown possibilities - in a given experiment, light will act as a wave or as a particle. (Also see Schrodinger's Cat). In fact physics has stated unequivocally that one cannot observe without changing the experiment in some manner.
Again, this comes down to miscommunication between what exactly we were talking about. Neither serves as evidence for the supernatural.
What kind of things is evidence that is beyond scientific evidence?
The distinction I was making was largely semantic. I would say that any of the things you mentioned, if used to support or cast doubt on a theory, would constitute scientific evidence.
Finally you state that God does NOT intervene in the natural world. There is a ton evidence of the testimonial type that would disagree with you and state for a fact that He has. If you have millions of people state that God intervenes in their life in a real and physical manner, are you simply going to discount this evidence?
Can they present evidence that this intervention occurred? I asked for examples, and you provided a few that didn't really fit what I was looking for, although I asked the question poorly, so this could merely be the result of a mistake.
As for testimonials, so what? They are often contradictory, referencing different gods. And they likely merely reflect misunderstood events. I had some experiences early in my life that I interpreted as divine, but looking back, I realized that they were just coincidences or inventions of my own mind that I interpreted as divine. Never were the laws of physics violated or anything demonstrably miraculous. I do not discount testimonial evidence. I believe those people experienced things. I do not believe that they accurately interpreted those experiences.
And finally, God did intervene in the most direct manner possible in incarnation of Jesus Christ.
The whole point here is that I am asking for evidence that this actually occurred.
1) The physical evidence is that the universe had a beginning in the first place. The logic from this one bit evidence will show that God exists. (read - circumstantial evidence)
This is evidence that the universe exists. It either always existed in some form or another, or came to exist. If it came to exist, it either came from nothing, (things come from nothing all the time, look up virtual particles) or something. If something, that something could very well have been something natural that we do not yet understand properly. There are theories about how the universe came to be and what happened "before" the big bang.
And if God cause it, then what caused God? If God needs no cause, then why not simply a natural phenomenon that also needs no cause? The universe's existence alone in no way points to an intelligent agent.
2) You only need to talk to any Christian to see how God makes a difference.
You could also talk to any Buddhist or Muslim to hear similar things. Or talk to an atheist who feels newly liberated from religion. "Because it makes me happy" is in no way evidence of a fact about reality.