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I have heard the term "priority" used in regards to research ideas. For example, How does one determine priority, prominence, and impact with regard to books, rather than articles? asks about priority (and other things) in regards to books.

From the usage I have seen in this community, priority appears to be established when you submit a manuscript (e.g., to a publisher or arXiv). This seems very weird to me, so I am worried that I am missing something. The time at which an idea is submitted for publication can be years after the idea was had, so it seems priority is not giving credit to the first individual who had the idea. Since the review process is confidential and it can take years for an idea to get published, priority is not being given to the person who first disseminated the idea either.

What exactly is priority and why does it matter?

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The way I use this term, "priority" means "who had this idea first". Of course, everyone can come up and say "hey, I totally had this idea to determine graph isomorphism in quasipolynomial time in 1990", so it is difficult to validate a priority claim unless there is some written material to back it up. But intellectual priority is the former concept; publication dates are just an imperfect way to measure it.

In fields where research is routinely disseminated before formal publication (using arXiv, preprints, or conferences), such as mathematics, there are (in my opinion) fewer possibilities for issues such as "X had the idea one year before Y, but Y's paper was published earlier" or "Y was a peer reviewer and scooped X's idea".

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Part of the work in a scientific opus is to trace back the origin of the ideas the authors convey. This can be a difficult work, as many sources are barely available, and not always in a known language. Sometimes the original ideas were only eludated to, given away in private communications. It often takes armies of historians to recover the whole story. Think for instance about:

Generally, a date of submission and a date of publication are public information that may stamp a priority, at least a prior date. For clear concepts, it is already complicated, see for instance in this chirplet (a type of of harmonic analysis for radar signals) debate around the first pair of inventors (Mann and Haykin or Mihovilovic and Bracewell).

But in many cases, honesty and deep referencing are guides, as ideas are often unconsciously borrowed and blended. Even a sentence like:

If You Steal From One Author, It’s Plagiarism; If You Steal From Many, It’s Research

has a debated origin (quote investigator).

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