Many seem to believe that $P\ne NP$, but many also believe it to be very unlikely that this will ever be proven. Is there not some inconsistency to this? If you hold that such a proof is unlikely, then you should also believe that sound arguments for $P\ne NP$ are lacking. Or are there good arguments for $P\ne NP$ being unlikely, in a similar vein to say, the Riemann hypothesis holding for large numbers, or the very high lower bounds on the number of existing primes with a small distance apart viz. the Twin Prime conjecture?
People are skeptical because:
- No proof has come from an expert without having been rescinded shortly thereafter
- So much effort has been put into finding a proof, with no success, that it's assumed one will be either substantially complicated, or invent new mathematics for the proof
- The "proofs" that arise frequently fail to address hurdles which are known to exist. For example, many claim that 3SAT is not in P, while providing an argument that also applies to 2SAT.
To be clear, the skepticism is of the proofs, not of the result itself.
Beliefs are orthogonal to proofs. Belief may direct attempted solutions by researchers or rather their main interest but this does not prevent them from checking a proof anyway.
The problem with $P \ne NP$ that many standard ways of attempting a proof are already excluded as not sufficient to infer anything, see here for further details.
There is no inconsistency in gathered poll of suspicions and educated conjectures. Also the belief that something will not be proven is not insightful in any way, without a proof of unprovability.
The years of attempts, claims, and discarded methods render people skeptical.
Please look at the prior papers that tried to contribute something towards the resolution.
"Extraordinary claims require extraordinary evidence."
This quite accurately characterizes the skepticism.
A few reasons, some generic and some specific.
The generic reason is that this is a long-known famous problem which many smart people have tried to solve, and many smart people have gotten wrong. The odds that any one new proof is valid is extremely low based off this history.
In this specific case, there has been research on what proofs don't work. It has been shown that basically all known proof techniques for proving things in computer science cannot prove P!=NP.
Wikipedia covers this and points out how "Relativizing proofs" (proofs that work regardless of what oracles your TM has access to), "Natural proofs" (involving circuit lower bounds), and "arithmetization" are all either insufficient to distinguish P and NP (show them equal or different), or any such proof would be a ridiculously more powerful result.
In short, not only have many smart people been working at this a long time and failed, along the way they have proven entire families of proofs cannot be used to solve this problem. So when someone comes up with P!=NP, there is natural skepticism, followed by noticing that one of the many proofs about such proofs is violated, and then there is no longer a need to check the rest of the result.
People don't believe any "proofs" because of the perceived difficulty.
Let's say we meet aliens who are better at maths than humans. Their average school child is about as good at maths as our greatest mathematicians. Not a smart school child, but an average school child.
They have proved the Riemann Hypothesis, the Twin Prime Theorem and the first Hardy-Littlewood Conjecture, and Goldbach's Hypothesis. What do they think about proving that the Travelling Salesman problem can be solved in polynomial time? They will find it unlikely that anyone could solve this. What do they think about proving that the Travelling Salesman problem cannot be solved in polynomial time? I think they will find it even less likely that someone could find a proof.
That's just my opinion, but if someone says they have a proof for P = NP or P ≠ NP, I won't believe it.
PS. The Riemann Hypothesis is open for a longer time because it is a classical mathematical problem that made sense to mathematicians 100 years ago. P ≠ NP is computer science, something a lot newer, and AFAIK the whole notion of NP comes from the 1970's only. There has been progress on the Riemann Hypothesis (we can't prove "all zeroes yada yada" but at least "a large portion of all zeroes yada yada"), unlike P ≠ NP. It's one-dimensional. It's about the zeroes of one single function. P ≠ NP is about all possible algorithms to solve a problem.
The reason people are skeptical of proof attempts of P!=NP is the same reason that people are skeptical of proofs of any famous conjecture: false proofs are published every few months and shot down. Meanwhile, correct proofs of famous conjectures seem to have little trouble getting attention, in spite of this (see, for example, the Poincare conjecture or Fermat's Last Theorem), but these proofs often rely on deep knowledge of large-scale efforts by groups of mathematicians (such as Hamilton's Ricci flow for the poincare conjecture or the Taniyama–Shimura–Weil conjecture for Fermat's Last Theorem) even if the final steps were done by a single theorist.
P vs NP is a particularly thorny problem because all the "obvious" methods have not only failed to yield a proof, but been proven to be useless with strong theorems. First time would-be provers are very likely think they have stumbled on a proof but instead have fallen into one of these well-known traps. Remarkably, showing that a number of ways of proving P != NP cannot work are the main advances in the field. It's somewhat outrageous that we cannot even show that 3Sat is not decidable linear time, let alone outside of polynomial time!
I would argue that very few people believe it won't be proven ever however. Indeed, the statement P != NP is a such a basic roadblock in our understanding of computational complexity that it's hard not to think that it's true for a simple and elegant reason.
However, if one wants to be cynical, P != NP is equivalent to the statement that just because a proof is easy (i.e. short) doesn't mean that it's not very hard to find the proof (i.e. takes super-polynomial search time). Indeed most theories believe that there is no sub-exponential time algorithm for finding proofs suggesting that, given any one method of finding proofs (i.e. a mathematician thinking or a computer search), there are many theorems with simple short proofs which are extremely difficult to find (potentially millennia of search time). Whether P != NP is such a theorem is not known of course!
That said, someone could publish the proof tomorrow.