My question is:
Do we need to search MSE (or blogs, math forums, ...) to make sure someone hasn't already proven a result when writing a paper?
What if we are already aware of a them (so no need for searching)?
Is not citing such a post in these two cases considered plagiarism?
As I understand, the common practice is to check standard reviewed reputable publication venues (journals, conferences, maybe arXiv) and also with experts in the area to make sure a result is not already published nor a well-known folklore result. No one is going to search all over the internet and check every post that Google returns and citing other resources is very uncommon. I think checking MatheOverflow can be considered similar to the later (checking with experts) (also see this discussion on MO but that doesn't seem to apply to a site like MSE. I am not going to cite a discussion with some random person on the street (not a professional mathematician) who claimed to have a solution or an idea for a solution for a problem (which is not passed peer-review process and I might not want even want to spend time understanding or checking the correctness of the solution).
What are the accepted practice for checking originality of a result?
What is expected from authors regarding this before making a paper submission?
Some clarification since there seems to be a misinterpretation of the question about being academic honesty. The question is not about posts that
- you are aware of,
- contain a complete rigorousness solution (not just ideas), and
- you are confident the solution is correct.
In my experience I think the accepted practice is searching the peer-reviewed literature in your field. I'm not saying results published on websites/MSE, etc. aren't valid, they just aren't part of the expected search. Anyways, I don't think peer-reviewed journals would react well to web citations.