I am aware that this seems a very stupid (or too obvious to state) question. However, I am confused at some point.
We can show that P $=$ NP if and only if we can design an algorithm that solves any given instance of problem in NP in polynomial time.
However, I do not understand how on earth can we prove that P$\neq$NP. Please excuse me for the following similitude as it might be so irrelevant, but telling someone to prove if P is not equal to NP appears to me like telling someone to prove that God does not exists.
There is a set of problems, those are unable to be solved by a Non-deterministic Finite Automata (NFA) with polynomial number of states regardless of the current technology (I know this is a sloppy definition). In addition, we have a considerably large set of algorithms that makes some crucial problems (shortest path, minimum spanning tree, and even sum of integers $1 + 2 + \dots + n$) polynomial-time problems.
My question in short: If I believe that P $=$ NP, you would say "then show your algorithm that solves an NP problem in polynomial time!". Suppose that I believe P$\neq$NP. Then what would you exactly ask? What would you want me to show?
The answer is clearly "your proof". However, what kind of proof shows that an algorithm cannot exist? (in this case, a polynomial time algorithm for an NP problem)