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Belief: P != NP

True? Maybe.

Justification: Experimental evidence

Basically the justification for the belief is that despite lots of research nobody has managed to discover an efficient solution for any NP-complete problem. If the belief turns out to be true, is this knowledge according to JTB? That is, is it a Gettier case?

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    This is not justified in the relevant sense to be knowledge, the relevant standard of justification in mathematics is higher (rigorous proof). Nor is this a Gettier case, where the standard is facially met, albeit for wrong reasons. If one's justification was instead a seeming proof with a subtle unnoticed flaw - that would be a Gettier case.
    – Conifold
    May 25 at 10:26
  • This is a major unsolved problem in computer science and also one of the seven Millennium Prize Problems referenced here, so it's the same as other unproven math conjectures like [Riemann hypothesis][2] for which despite lots of research nobody has managed to discover an solution (proof) yet. In CS theory or math enumerative justification doesn't count as a universal proof, so you can claim you may have some particular JTB like knowledge in certain cases of these conjectures but no knowledge yet as for its universal conclusion... [2]: https: May 25 at 15:14
  • I would say yes. This is similar to lottery cases which are also considered JTB without knowledge. Given a fair and large lottery you can have a JTB that your ticket lost, based on probability, but you cannot know that it lost just based on probability -- for that you need to get information about the actual result.
    – Eliran
    May 25 at 22:24
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Note that we know this well enough that entire industries depend upon it, and they are considered safe. RSA Public Key Cryptography is an international standard that backs SSL and other internet security. We do security audits of institutions dependent upon that technology and find them secure.

We may know it better than we know how to build safe bridges. So, do we not know how to build safe bridges?

The components of this definition seem categorical, but they are all open to degrees of interpretation.

Examples like this do not present a challenge to the definition itself, so much as they raise the 'sorites' problem implicit in everything with a degree. How certain is certain? How justified is justified? If I add one strong hint toward reliability at some point does that change conjecture into knowledge?

In math, clearly not. In industry, probably so, though that point is not consistent or clear. You know subjectively when you are past it, but you cannot observe yourself passing it.

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