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When answering a question on Physics, should answers that make reference to non-peer reviewed material that is found on, for example, arXiv, be discouraged, or even removed?

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No. A non-peer reviewed reference is better than no reference. The reader can judge the legitimacy of a reference him/herself. In general, Phys.SE encourages answerers to back up claims with references.

(Concerning self-citations, see this meta post. For link-only answers, see this meta post.)

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    $\begingroup$ I agree, and I would add that when an answer cites a source, we should interpret it as though the poster of the answer is vouching for the correctness of the cited material. In other words, a citation should not be thought of as "[source] says so, therefore it is so" (that'd be appeal to authority), but rather as "I say so, and [source] also says so, and I'm saying [source] is trustworthy". It's the assertion of correctness that is key, and peer review is only minimally correlated with correctness (or lack of peer review with incorrectness). $\endgroup$ – David Z May 11 '16 at 8:40
  • $\begingroup$ In which case, what is the point of having a peer review system? $\endgroup$ – jim May 12 '16 at 8:06
  • $\begingroup$ @jim The formal review that journals do before accepting a paper for publishing mostly checks that the offering is complete enough to allow replication, uses the language of the discipline, and would represent an interesting contribution if true. But that's only the beginning of "peer review" in the broader sense, which takes place as the community reads, comments on, perhaps refutes, replicates (or fails to achieve the same), and either does or does not make use of the results/conclusions of the paper. In that sense peer review isn't a step finished when the paper sees print but a process. $\endgroup$ – dmckee Sep 20 '16 at 1:06

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