If you look at the P = NP problem, the results are universally almost dismal. This is because people want it to be true or false, and a number of people run around in circles with this notion in their heads. Occasionally, some bits of wisdom filter down to them - but one does not really know whether it has just stuck in their heads, or it is some bit of wisdom which has filtered down. One needs to solve the problem 1st, which is no good, because the bits of wisdom were supposed to be a help.
(The same could be said for me, because of course the one thing that crazy people are sure on is that there are other crazy people. It is part of the gag. (http://www.win.tue.nl/~gwoegi/P-versus-NP.htm is the best place for a running gag, maintained by someone who has stomach is much greater than my own for critiquing through the details http://www.win.tue.nl/~gwoegi/) (In fact another example of recursion, cranks know how to use the Internet as well))
For example one person latched on to Gödel's proof - and proceeded to, as a professor once said, miscite a conclusion even as he used it. It is with this problem that I am wrestling with - it raises the specter of cranks with every turn. I do not want to have archiveX make twin blunders of rising the cranks, and making the ordinary mathematician think I am a crank - which is the most logical assumption in the world to make. Wrong in this case, but there are so many other counterexamples that a person could logically not miss the finer point.
This point is that P = NP is neither true nor false, but is instead indeterminable - but in a special way. For the person who is physically contained, he does not even have access to the real mathematical universe - because he is constrained by - at least – Plank's Constant. In other words he is a realist. But let us take “God” - in a metaphysical way, not believing if he exists or not - who has access to the idealist view of the universe. The realist has points of matter which are described by the plank constant, the realist has points which have no limit in size. These are 2 distinct points which do not actually describe the other. Unfortunately, I am not the person to find the way to do this - it is for someone else of greater intelligence to do that. But what I am capable of doing is joining a circle - that is the realist point of view - to a tangent, which is the idealist pov - and describe the difference between the 2 points. The realist point is physical and we have a good idea about its diameter. And electron contains a Higgs equivalent, but light does not. So it is in the penumbras of this that want see a difference between 2 physical points. But in the idealist world, unconstrained with any physical worldview, point is exactly 0 in all dimensions but time. It is, so we cannot draw this we can imagine. This is why Cohen, among others, wanted to destroy the realist point of view - because it made paradoxes. One of which is the NP problem.
Let us start with an example of this problem – the Subset sum problem. It is one of the problems which makes P = NP it seem intractable. But effectively, it shows that we have a problem with the problem and not the solution. The subset problem is this: given a set are there any which total 0? it is easy to solve - because one has all of the information. But it is hard to compute. This leads many people to decide that P =NP cannot be solved. But this is incorrect, because all paradoxes have a way of being described such that they do not have a solution. However, just because a problem has a hardest possible solution which cannot be resolved, does not mean that there is not an easier solution which can be resolved. And I will show that P = NP is such a dilemma ( or actually trilemma).The solution is in "Set Theory and the continuum hypothesis”. If Cohen had another partner, he might have seen the way out of the dilemma that he found him self in. but as I have mentioned before, the 3 members who could help him were Turing - who was jailed for being homosexual, this in a time where homosexuality was regarded as deviant – Gödel, who was a recluse, and only communicated with him in small bytes - and Nash - who in this case tripped over a related problem, which Turing had actually solved in part, and was for much of the time, paranoid. Thus he did not have a sounding board, and in a plea did not see the solution which was right in front of him. I feel sure that if a sounding board had presented itself, Cohen would have seen the solution, given that the computer was already present.
The Ur-cite is on 117-8 of Cohen's text, where he shows that anything can be forced. But there are a few problems - some which are intentionally not thought out correctly, and others of which do not see that recursion is necessary. But the other problem is that everything cannot be resolved. Even God's algorithm ( Rubik's cube fame) cannot work for everything. It is thus anything but not everything which is actually the recursion factor - one needs to, in computer science terms - go down a level to solve the paradox. For example in our subset problem, one needs to understand that the subset is lower down a Gödel solvability problem - and then pop up to solve the subset, where the numbers have meaning. So in other words - one drops down to a lower set of solution, 1 takes the number to be solved as the undecidability number, and then dumps the numbers to find which one will be undecidable, then one pops up - and remember that being an undecidability problem means that 2 is all that one needs to solve, one does not need to know what the number is, just are the 2 numbers equal. The problem then is that the algorithm which searches, destroys part of the information it needs to solve. In quantum logic this is not a problem, but P =NP is not quantum in nature.
In SQL terms, this is one level higher than one needs to solve the problem - but the information is actually available. So the actual problem, is that the undecidable version throws out much of the information.
Gödel "On formally undecidable propositions of Principia Mathematica and related systems"
Cohen "Set theory and the Continuum Hypothesis"