Cool Skill,
Metaphysical: Beyond physical.
Metalogical: Beyond logic.
..............................................
I made the word metalogical up to be the term for something that is beyond logic like metaphysical means beyond physical.
There happens to be a science that is also called metalogics which has nothing to do with this.
There is an issue of semantics going on here.
"Beyond physical" is not exactly the best way to describe metaphysical. In fact, it's more about things underlying the physical rather than being 'beyond' it. Onefinity has already mentioned the origin of the word. Aristotle wrote a series of books. Some were grouped together under the term, "The Physics" and the books following these were grouped together under the term, "Metaphysics". This grouping was, I believe, performed not by Aristotle himself, but by later scholars.
Anyway, the Metaphysics were divided into three sections: Ontology, Theology, and Universal Science.
I believe that by modern definitions, theology has moved into it's own field and metaphysics now consists of ontology and universal science which are closely related in my mind. In a nutshell, they are concerned with properties of the physical. The redness of the apple. The roughness of sandpaper. Etc...
Physics are about physical objects. Metaphysics are about the physics of physical objects. Meta-metaphysics are about the metaphysics of the physics of physical objects. And so on.
Logic is the same way.
Metalogic is about logic. Meta-metalogic is about the metalogic of logic.
As Turdurcken has mentioned, Godel, Escher, Bach is a definite must read on this subject.
As to the question:
The question is about whether or not there exists concepts that cannot be conceived by logic, but can be conceived by using more sophisticated intelligence.
The answer is... absolutely.
Logic suffers from problems that are unavoidable. A system based solely on formal logic is subject to the principle exposed in Godel's Incompleteness Theorum and in Turing's Halting Problem. Or. More commonly and intuitively understood by most, the Blue Screen of Death.
Paradox is the common mechanism where this is demonstrated as has been mentioned several times already. Let's examine a couple of paradoxes.
"This sentence is false."
We have a paradox. If the sentence is false, then it must be true. But if it's true then it must be false. Paradox. But. A useless paradox with no solution. There is no answer that can be intuitively known except that the statement is nonsense and should not be taken seriously. (Which, I suppose
is an example of intuitive reasoning. And which is also beyond the capability of a machine without explicit programming detailing exactly this circumstance.)
Anyway. Let's consider another paradox which, on the surface, appears to be the same but has quite different results.
"This well-formed formula is unprovable in the system."
The 'well-formed' part of the sentence stipulates that it cannot consist of the same paradox of the former type. It cannot be self-contradictory such as "A equals not A." And note also that true or false is not used, rather the term 'unprovable'.
Ok. So. We have a sentence that exists and we are evaluating it within the rules of a formal system. It can either be true or false. But which?
Let's consider that it's false. If it's false, then that means that it
is provable within the system. Which means that it's true. Which means that it is unprovable. This is a blatant self-contradiction.
So. We're left with this:
a.) The statement is true in a consistent system but unprovable.
b.) The statement is false in an inconsistent system.
So we're left with a question as to the state of the system. Consistent or inconsistent? Which is interesting, because the question can be posed as a sentence. "This system is consistent." Which is basically the same as "This well-formed sentence is unprovable within this system." And we're back at square one, but at a
deeper level.
We've gone from logic. To metalogic. And deeper? Lies metametalogic. And metametametalogic.
This is a 'strange loop' and can go infinitely deep. To a machine, a creature ruled by a formal and consistent system, there is no escape except for preprogrammed escape clauses programmed with this
exact loop in mind. The ensuing state of the machine without the escape route would be the Blue Screen of Death. Infinite recursion.
But, a man, a creature ruled by an informal and inconsistent system (although it is capable of containing pockets of formality and consistency) is able to stop the loop anytime he wishes. He is able to informally deduce a satisfactory answer or he can determine that an answer is unreachable.
A man can
guess.
A machine can't.
Now. As to the preprogrammed escape clause.
This leads into Godels theorum. For any formal system, there is a formula that can't be proven in that system. This is its Godel Formula. To try to prove this formula would lead to this infinite recursion and halting problems. But. The designer of that system can figure out what the Godel Formula for the system is and design it so that the Formula exists within the system as an Axiom. But. In so doing, he's changed the system. And so now we're dealing with a new formal system with a Godel Formula of it's own. And this too slips into an infinite recursion.
Infinities everywhere.
When you find infinity, you know that you're bashing your head against the system itself. For instance, our number system is unable to deal with division by zero. This is not necessarily because division by zero is impossible, but rather it's an artifact of the system. The same goes for things like dividing 10 by 3. It only leads to an infinitely repeating decimal because of the design of the system.
Anyway. Besides GEB, another interesting read on this subject is
John Lucas Against Mechanism. (Here's his
home page.
And, in Sciforums, another thread on basically the same subject was started long ago even though it sort of petered out without a satisfactory conclusion:
Humanity as the Principia Mathematica-Inspired by Godel.
Now. For some biological discussion.
There is a disorder (which I forget the name and would have to go digging deep in a stack of magazines and books to find the exact reference) in which portions of the brain which provide emotional stimulus are damaged and the afflicted people have no emotional predilection for a specific outcome. What they do, then, is build tally lists in their minds. Lists of pros and cons to any particular action. Some actions are, of course, easily decided upon. The pros far outweigh the cons or vice versa and the decision is made. But, there are many decisions that are not so clear-cut. And in these situations the people
freeze. They can't choose. They can't
act. They can only sit there and build tally lists that grow and grow and grow.
I imagine that there is also a form of stack overflow error in this condition. When the list grows too long, earlier entries are lost and repeated. In this way, the list can keep growing indefinitely.
We also enter into a loop within a loop here. The decision to stop adding to the tally list and come to a decision could also be subject to its own tally list and its own decision on when to stop adding to the tally list and come to a decision which would have its own tally list...
We
cannot exist in a world of pure logic.
From Durant:
"....we pretend to be constucting edifices of impartial thought, when actually we are selecting only such acts and agreements as will give dignity to some personal or patriotic wish" --The Mansions of Philosophy.
Now for some quotes:
Superliminal said:
There are savants that calculate, compose, etc. in an apparently intuitive way (they say they are not thinking through what they do - it just sort of happens) that completely defy our current understanding of brain function.
In this instance, I'd think that a better term would be nonlinguistically than intuitively. Autists are unable to explain how they do what they do. But, what they do is generally not conceived to be illogical. Just not typical. Individual autists are unique in their disorder, but many of them (that are somewhat able to convey what they do) think visually. Thinking in pictures.
Many autistic tricks depend upon rote memorization.
In fact, what is the functional difference between intuition and logic? These are two recognized ways of knowing. Is intuition simply "logic" working in the brain on a subconscious, automatic level?
I'd say, yes.
Logic is dependant upon its symbology. There is a part of the mind that speaks. And there are far more parts of the mind that don't. Logic is worked out in stages through the medium of language (whatever language). Intuition occurs on a deeper level and is presented to the speaking mind as a feat accompli. It is then the task of the speaking mind to explain to itself why the 'irrational' intuition is rational.
Sometimes it succeeds. Sometimes it fails.
Intuition is invaluable to logic in that it points the way. One follows hunches. Sometimes the hunches play out. Sometimes they don't. But without the hunch, then you need some type of system to delineate the order of investigation. Alphabetical. Or whatever.
Nasor said:
Infinity is a purely logical concept. There’s pretty much no other way to arrive at the idea than through mathematical logic, since it’s not really something that you can stumble upon in your every-day life.
Exactly. Infinity is often a consequence of the shortcomings of purely logical systems. We then use intuition to lead our explorations to explain the inconsistency.
Black holes, for instance.
Ecclesiastes said:
hey what about quantum entanglement?
noone's commenting on that
Quantum entanglement defies intuition. It is dependent upon logic. It is logically derived even though its conclusions can be termed illogical by intuitive processes.
Watcher said:
I absolutely agree that rationalism, the cornerstone of Western thinking, is very limiting.
Yeah. It's so limiting that our western society is far inferior to eastern society. I mean. Man on the moon? Pshaw...
Gendanken said:
Your emotions cry out to you and your mind stumbles in trying to explain it.
The mind stumbling trying to explain? That's what the mind excels at. And from such rationalizations, filtered through rigorous discipline, leads to the wonders in which we live today.
Or you can just go with the rationalizations with no rigorous treatment which leads to ESP carnivals.
Anything past logic isn't. Call it "emotional"......or is that too humble for you?
No entirely emotional, but emotion is surely a leading part to it. There are more hidden systems in the brain than emotion though. Someone's mentioned blindsight. This is an excellent example of an alternative system other than emotion.
However, I would call emotion and logic sister levels in that they both work with the products of the sensory cortexes. They apply labels and meanings to the products of the senses.
Ok. Long enough. My eyes are swimming. There were many other things said in here worthy of commenting on, but maybe later.