A celebrated problem in the theory of computation, really had fascinated many. A few thoughts over the problem is given below. This approach of solving the problem ofcourse has a flaw. The whole thread of discussion can be found in google-groups-cs-theory link.
Given any problem, humans tend to think over the problem and find the solution over a span of few days or months or even years. Here is a list of steps:
The thought process can be considered to be an NP complete problem (Though we might actually making some assumptions inherently).
Any problem is bound either to have or not to have a solution. (Here is where we make a critical assumption violating Godel’s incompleteness).
If that is the case, then, this assumption says P=NP has some solution. Given any problem, we tend to find the solution in the thought space or the logical space of our mind and move to the problem’s solution in different paths whichever is logical and takes us to the solution.
This is nothing but the traversal in the solution graph of the problem presented. Since we do not know before hand which path would lead us to the solution, we face with no better method other than trying out all possible methods. (This is done by the mind inherently by finding the best method of attack from the description of the problem).
Hence, thinking and arriving the solution of the presented problem is an NP complete one. Once the problem’s solution is formulated, its always easier to read through the solution and hence find its correct! (The ‘P’ part of thought process).
We’re trying to find the solution of the P=NP by using a process which is inherently NP complete. If P=NP turns out to be true, which means, given a problem, its solution can be recovered through some form of algorithm without actually trying out all methods, we could rewrite the solution for FLT and even generalized Catalan’s problem.
Here is where Godel’s incompleteness comes into help. What if the problem cannot be determined of its solvability? In this case we’ll not have the solution and hence P will not be equal to NP.
The actual flaw lies in the assumption and proof of the considering the Thought process as an NP problem. For a problem to be NP complete, it not only suffices that the problem shouldnt have a known algorithm and maps to an existing NP complete problem with regard to the existance of verifier. Rather, the existance of finitely bound witnesses should be proved.
In the case of thought process, the bound on the number of problems is not finite and this is where the actual flaw creeps in.
