The Church-Turing thesis and its variations are central to computation theory. Since, as an informal notion, the concept of effective calculability does not have a formal definition, the thesis, although it has near-universal acceptance, cannot be formally proven. The implications of this thesis is also of philosophical concern. Philosophers have interpreted the Church–Turing thesis as having implications for the philosophy of mind.
P vs NP problem
The P_vs_NP_problem is an unsolved problem in computer science and mathematics. It asks whether every problem whose solution can be verified in polynomial time (and so defined to belong to the NP class) can also be solved in polynomial time (and so defined to belong to the P class). Most computer scientists believe that P ≠ NP. Apart from the reason that after decades of studying these problems no one has been able to find a polynomial-time algorithm for any of more than 3000 important known NP-complete problems, philosophical reasons that concern its implications may have motivated this belief.
If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in "creative leaps", no fundamental gap between solving a problem and recognizing the solution once it's found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss.