> Should we even care for a mathematical proof that P=NP?
Should we care about the most important computational problem in the world today? I'd say yes.
> Suppose we'll be told that this is possible (i.e. P=NP): will this help up us to invent such algorithms? I don't see how, unless the proof will be by construction.
Why does the method of proof matter? If it is proved that P=NP, then it means most of the pressing computational problems today are solvable in polynomial time. So we can double our efforts in trying to find these solutions. If it is proven that P!=NP, then we can stop wasting our time or just work on solving subsets of these problems. The problem is that we don't know whether the problems can be solved in polynomial time.
Put it this way. Which scenario would you prefer.
Scenario 1: There MAY be a billion dollars hidden somewhere in Mount Denali.
Scenario 2: There IS a billion dollars hidden somewhere in Mount Denali.
Scenario 3: There ISN'T a billion dollars hidden somewhere in Mount Denali.
Currently, we are at scenario 1. We don't know whether our efforts are for naught. We don't know if we haven't found the billion dollars because it's in a place we haven't looked or if the billion dollars isn't even in the mountain.
Proving P=NP, would get us to scenario 2. We know there is a billion dollars there, but we just have to find it. Proving N!=NP would get us to scenario 3. We know the billion dollars isn't there so we don't have to bother wasting our time.
I'd say we'd be in a much better position if we were in scenario 2 or 3 than the current scenario 1 we are at right now.
Should we care about the most important computational problem in the world today? I'd say yes.
> Suppose we'll be told that this is possible (i.e. P=NP): will this help up us to invent such algorithms? I don't see how, unless the proof will be by construction.
Why does the method of proof matter? If it is proved that P=NP, then it means most of the pressing computational problems today are solvable in polynomial time. So we can double our efforts in trying to find these solutions. If it is proven that P!=NP, then we can stop wasting our time or just work on solving subsets of these problems. The problem is that we don't know whether the problems can be solved in polynomial time.
Put it this way. Which scenario would you prefer.
Scenario 1: There MAY be a billion dollars hidden somewhere in Mount Denali.
Scenario 2: There IS a billion dollars hidden somewhere in Mount Denali.
Scenario 3: There ISN'T a billion dollars hidden somewhere in Mount Denali.
Currently, we are at scenario 1. We don't know whether our efforts are for naught. We don't know if we haven't found the billion dollars because it's in a place we haven't looked or if the billion dollars isn't even in the mountain.
Proving P=NP, would get us to scenario 2. We know there is a billion dollars there, but we just have to find it. Proving N!=NP would get us to scenario 3. We know the billion dollars isn't there so we don't have to bother wasting our time.
I'd say we'd be in a much better position if we were in scenario 2 or 3 than the current scenario 1 we are at right now.