It's always kind of hard to reply to post dissections in a thoughtful manner. I find the practice tends to devolve into a back and forth of quips and one-liners. I'll try to distill your main points as I see them.
I'm sure we can avoid that fate

. I certainly appreciate the time you are taking with this, and am confident that we can maintain a civil and academic tone. I doubt anything of significance was well communicated with one line quips after all

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I have to disagree with the assertion that personal observation of self-awareness is somehow a non-scientific observation.
Ah! I think my definition of scientific is quite a lot stricter than yours, I probably should have noticed this sooner. We should agree on semantics. It's hard to discuss something if you can't nail down what you're discussing after all

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Specifically, I agree with the definition that scientific observations should be empirical, repeatable, and public.
I think one good description is found in, "Experimental Methods for the Behavioral and Biological Sciences", by Dr Jay Gould, under "Requirements for Scientific Observation", which can be found here:
http://books.google.com/books?id=FSOKh3tsk90C&pg=PA24
Of special note, he further explains the definition of public, "By public it is meant that observations must occur in a manner, place, and time such that it is possible for others to make them."
He also specifically addresses the question of introspection, "Subjective observations of what is occurring in one's own mind is referred to as introspection ... Introspection is not public, and hence is not considered scientific observation." This, according to Gould, is because:
- Other individuals cannot share in the experience
- Instead, others can only observe reports of the experiences and then draw inferences about the actual mental events.
- The problem with introspection is that the reports may not be true reflections of the experiences -- they might be false, inaccurate, or incomplete; and the terms used to describe the experiences might be imprecise or ambiguous.
There is more here, I recommend reading the entire section. Below are some other examples of authors who require these three key characteristics for scientific evidence:
http://media.wiley.com/product_data/excerpt/28/04713691/0471369128.pdf
http://books.google.com/books?id=eCVU0VRPJIYC&pg=RA1-PA179
http://books.google.com/books?id=u7JmdWRlrAcC&pg=PA6&lpg=PA6#PPA4,M1
Would you be willing to accept this definition? It's not key to the argument, we really could use any definition, but I think this definition would more closely match the common understanding. Most would probably say that as much as the overworked grad student may wish them to be, his own introspective musings do not constitute scientific evidence.
However, if it is important to you, I am willing to call these kinds of observations scientific -- just as long as we're clear on our terms.
No doubt it is a very tricky subject at the very least because I have no proof for the claim that I am self-aware that I could show to someone else. I do, however, have proof which I myself can observe. My mind is very much observable, but it is locked in a room into which only I can enter, metaphorically speaking.
So, we both agree that this is another method of observation -- scientific or not. These observations clearly do not come through our five senses. Do you agree that these kinds of observations might be useful, along with the five senses, for determining the nature of things?
If so, one of the first observations I make is that I am a self-aware, conscious mind, with the capability to make decisions, rather than only a collection of particles whose behavior is completely determined by physical laws and QM. This model explains the physical domain just as well as the mechanical view, but it more closely matches the observations I make about my own mind. Are you willing to consider this model, and if not, why not?
I believe this will change in time. Already the process has begun. Through brain imaging an observer can witness different moods, or even what kinds of thoughts a person is having. For example whether they are thinking about a mathematical problem or an emotional one. These are the beginning inroads into the verifiable observation of the mind. I see no fundamental obstacle preventing the eventual thorough explanation of the mind.
Strictly speaking, what the observer would actually witness is the physical behavior of the brain -- not the actual mental state of the participant. Just as the example with the rock, all the observer can really determine is the physical state of the system.
Consider this for example: Suppose at some future date, you are lying in the ultimate medical imaging machine, which can completely determine the physical state of your brain at infinite detail, in real time. You may surmise that there would be a correlation between the physical state of your brain, and the internal state of your mind -- and you might be right.
But, you have just made a key admission. You have admitted that despite that perfect physical knowledge that the machine gives you, there are other attributes with which this perfect physical knowledge can be correlated -- namely, your internal observations. This is key -- according to a mechanical definition of the universe, there IS nothing other than the physical state. There should be nothing to correlate to -- once you know the complete physical state, you should know everything -- game over. The fact that you are now taking this perfect physical knowledge and trying to correlate it with something means you admit that there are attributes which are not physical.
Any attribute, in a mechanical universe, is literally defined by a physical observation or physical state. Yet, self awareness is not defined in this way, because quite simply, it is not a physical attribute. Any correlation between brain state and self-awareness which scientists "discover" would have to be based on taking a participant's word for it, which is faulty at best - and certainly not based on only physical evidence.
Included in that would be the operative processes of logic. Consider that even now certain logic operations can be replicated via computer. Obviously the computer has no awareness of these operations, or anything for that matter, but on an effective basis it is the same.
Sure, I think I still have not made clear what I was trying to say by bringing up logic. I certainly agree that computers can perform logical operations. What I am saying is that we believe logical operations are valid or invalid not by observation through the senses, but internal observation.
You believe "A and not A" is a logical contradiction simply because it does not match the hard wiring of your own mind. Similarly, if you accept "All As are Bs", and "C is an A", then you will accept "C is a B" -- again, because it matches the basic nature of your mind.
I was simply saying that basic logic, which really is a fundamental basis of all reason and scientific thought, is another valid observation we make by internal examination which does not come through the senses.
Your second question I think offers some fertile grounds for discussion; what evidence could be offered to support the existence of the supernatural scientifically? This of course varies widely depending on the theory presented. Let's take the topic theory of this thread. The theory goes, god exists. That really doesn't give us much to work with. First we'd have to define god. We've hit a roadblock already. Nobody agrees on what god would actually be. The best we could do is to pick one of the theories involving a god and seek to test that one. We'll take the christian mythos since most of us here are familiar with it. According to the christian theory god exists and that he has a set of properties.
1. He created and controls everything
2. He is all-knowing, all-powerful, and all-good
3. He can and does intervene in human affairs
4. He rewards and punishes souls for their deeds in life
For each one of these properties evidence would have to be put forth to justify their belief. Beginning with the first some kind of explanation would have to be offered describing how god would have created the universe and from that theory predictions would have to be made. Perhaps one might theorize that god would write his name on things. That seems sort of absurd, but what other predictions could be made based on the theory that god caused existence? I know of none. The second half of the assertion is a little easier to address. Control of something implies action. If it could be shown that something which used to work in a certain way, but now works in another way, with a degree of certainty greater than the probability that it is simply human error could be counted as evidence that some force is acting upon the universe. An example would be, fire used to burn but now it does not. It's not the humidity, it's not the wood, it's not our ignition. The universe has changed in an arbitrary way. That would be the beginning of evidence for a supernatural god. We should note here, such a thing has never been observed.
But, don't you think such historical evidence would be discounted? After all, the ancient Greeks and Romans used to believe in gods, and claimed that they interacted with people all the time. There are numerous stories about dragons, travels to the underworld, travels to heaven, etc, in the ancient texts. People for millenia believed that the world was flat, and there are stories about travels to the edge of the earth. Yet we do not believe that dragons really existed with people, and went extinct, or that people really traveled to the underworld, or to the edge of the earth.
If fire stopped burning, our descendants would quite likely view our fire burning as a curious series of mythical stories which we concocted based on our primitive view of the universe.
Furthermore, something would have to change for fire not to burn. Even if the fundamental laws of the universe were changed, it would have to manifest itself in a certain way. You say we should check that "It's not the humidity, it's not the wood, it's not our ignition". Yet, if a god chose to change the universe so fire did not burn, he/she would no doubt change one of these things. For example, perhaps the chemical composition of wood would change, or the atomic composition of carbon, hydrogen, and oxygen. Perhaps the amount of energy contained in these chemical bonds would change. Something would change -- and science would say, "aha! we knew this was not a supernatural event! Now, let us examine why these chemical bonds are not as strong as they used to be". There would indeed be a reason for that .. and on we would go. Perhaps, if we were really bright, we could follow it all the way down to find that a fundamental variable -- say Planck's constant, had changed. Would this be evidence for that god? Hardly.
Take this for example:
http://www.universetoday.com/2003/11/25/fundamental-force-of-nature-has-changed-over-time/
Quote: "Physicists from Northeastern University believe that a fundamental force of nature, the bond between electrons and protons, has been strengthening since the Big Bang."
Now, assuredly these physicists have all become theists, No?
Just as the sphere can always, self consistently explain himself as a two dimensional circle on a plane, we will always be able to describe the purely physical in terms of the purely physical, and what is left is simply "changing variables" or "randomness". It is impossible to prove the third dimension by looking only at the first two.
Let me note here that going straight for a description of evidence for God is a bit like modeling a heavy metal before trying hydrogen, but we can go with it. The point I was trying to make is that using public, repeatable, objective, physical evidence, one could only prove truths that are physical, public, and repeatable. Thus, any example of where science could jump outside these bounds would suffice.
The second listed property would be extremely difficult to ascertain. In order to determine if a being was all-knowing, we would first have to be able to communicate with them. No common theory of god supposes that you can seriously have a conversation with it. Further, in order to judge all-knowing-ness we ourselves would have to know everything! We could formulate no test without possessing such. All-powerful could be tested to the degree that our imaginations can formulate actions. All-good is practically an oxy-moron. I suggest that the very notion could not be.
I could speak to how we could formulate theories about God, but I think we'd be getting ahead of our selves. I'm looking for scientific evidence that could prove the supernatural. I think you'd agree none of these ideas are scientific.
Basically, I'm saying, that if there is no piece of scientific evidence even in theory which would prove to you that the non-physical exists, then the lack of such evidence shows nothing, and the question is not scientific. Any description of a scientific test which, if successful, would prove the existance of something that is non-physical or supernatural will do -- I think coming up with scientific tests to prove the attributes of God is a bit of a tall order

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The third property is the kicker here, because it largely justifies any interest in a god, and consequently is the most accessible to science. Testing this would be fairly simple. All one would have to do is record a miracle. Not just good luck, a real miracle. A man walking on water without any assistance or bringing the dead back to life without the aid of any applicable scientific knowledge. In order for this to be evidence of the christian theory of a god some connection would have to be found with the christian god. For example are these acts only performable by those who meet certain criteria in line with Christianity, be they priests or prophets or simply good people by the metrics of Christianity. Once again, I must point out, no such record has ever been made.
Try to break this down into specific experiments. Let's take the example of walking on water. Now, for this to be real scientific evidence, it must be public, repeatable, etc. Let's say we observe that all people who claim to be Christians can walk on water, repeatably. We do this in the lab. Now, it has to happen in a certain way -- let's say the water viscosity increases enough so that it can support their weight momentarily.
We would then isolate for false claims -- let's try people who for the purposes of the test make the claim to be Christian, but aren't. Suppose we find that it does not work for them. We then will isolate for other attributes. Perhaps a particular brain state is common among the people who are Christians. Surely a belief like this would generate a different brain state in certain ways. Now we're getting somewhere.
For now, we will label this a theory -- this physical brain state causes the increase in water viscosity beneath the feet. There will be further investigations, to break the causal chain down further, but it has already become science, not requiring the supernatural.
This is the same thing we do will all observations. Think about it. Why do you not consider it a miracle when two magnets repel each other at a distance? I am sure most children are surprised by it the first time they see it. It certainly seems "spooky" to me. Yet, because we are familiar with it, and have worked hard to develop detailed descriptions of how it works (by "how" here I mean only the manner, not the cause), we label it "science".
If magnets only worked half of the time would it be considered supernatural? No, probably not, we'd just try to figure out what causes it not to work sometimes, or if we couldn't do that, we'd mark it up as a random effect, like we do with QM.
Or, take QM for instance. Here we have the definition of "spooky" behavior. We've got random causation, particles behaving like waves sometimes and like particles other times. Particles going from one point to another without passing any point in between. Yet, this is not considered supernatural.
Finally there's the topic of immortal souls. What is it that a soul does again, exactly? What does it affect? I don't think any serious answer to those questions have ever been put forth. One would have a difficult time scientifically verifying souls for the very fact that the definition of a soul is so empty. In other words, there is no evidence to support the belief in a soul. It's not required to explain anything.
To me, mind=soul, and the mind can be observed directly, by introspection. But again, I really didn't intend to ask for a description of possible evidence for all ideas connected to the traditional Judeo-Christian idea of God.
One example of a scientific experiment which would prove any non-physical or supernatural entity is all I'm looking for.
I've indulged your question here, and in the end it must be said such an approach is the absolute wrong way to seek knowledge. One does not produce a theory and then seek evidence for it. The correct method is to observe phenomena and then to propose a theory explaining it.
I certainly agree, and I fear you have indulged me too much, although I certainly appreciate your thoughts on this. I absolutely agree that we shouldn't jump ahead to theories before we've examined the underlying evidence.
What I was trying to say is that physical, objective, repeatable, public scientific evidence (and here I am using Dr. Gould's definition) cannot prove or disprove the non-physical, just as I cannot prove the third dimension by observing only two. If we accept this idea, then it follows that the question of the existence of the non-physical can only be answered by evidence not gathered using these criteria.
There is no phenomena for which god is a theorized explanation. For these reasons do I and many others reject the idea that a god exists.
I disagree, of course, I think that there are observations for which God is a theorized explanation. But, before we get there, I'd really like to focus on the idea of the non-physical mind. We must agree that the non-physical exists before we can discuss the nature of the non-physical universe.
To do otherwise would be something like discussing models of the orbit of the electron among two parties, one of whom does not believe electrons exist.
For each and every theory it must be evaluated on it's own merits. There is no single argument against the supernatural, but one for each claim. That's a rather exhausting approach to take to supernatural phenomena anyway. I'd much prefer simply treating it like any other theory; present the evidence. It's been said to death but the onus is on the claimer. If you believe that some supernatural phenomena is occurring, then prove it.
Ah, I agree! I think the mind is the first and most obvious example of a supernatural entity. A purely mechanical model, and a supernatural model of the mind are equally predictive of observations by the senses. But, when we consider the evidence of self-observation, it's quite obvious we perceive ourselves to be self-aware, conscious, decision making minds. Thus, I would say the latter model has the advantage, because while both equally match the evidence taken through the five senses, the latter much more closely matches observations taken in that, "room into which only I can enter", as you would put it, using observations which you may call scientific, and I might not, but I think we both would consider valid.
I find it hard to address your other suggestion that there were kinds of observation which are not scientific. Perhaps my imagination is failing me here. If you could give an example of a phenomena which you believe is described and explained without scientific methods, then perhaps I will be able to respond.
I was mainly referring to observations about our own minds. The mixup is only because we have been using different definitions of "scientific". I think that will be cleared up one way or another by the next post.
Regards