The Negative Case

We do not know for sure if P equals NP because we can't find any polynomial-time solutions to NP-complete problems. Additionally, we have been unable to prove whether P does not equal NP. We suspect P does not equal NP because it has been so difficult to prove that P = NP.

That said, it's actually more complicated to prove the negative case. To prove the positive case, that P = NP, we simply need to solve an NP-complete problem like TSP in polynomial time. In order to prove the negative case, that P != NP, we would need to exhaustively prove that there's no possible way to solve TSP in polynomial time. That's a lot trickier.