In mathematics (also in mathematical physics and neighbouring disciplines) a theorem is sometimes said to be folklore when the community feels that it has been around and generally accepted as true for a long time, without however a proof of it having been submitted and published in the usual manner.
For instance the cobordism hypothesis – which is an intuitively evident statement, whose formalization and proof, however, is notoriously subtle – is referred to as “folklore” in Stolz 14, p. vi.
The sociology of folklore theorems can be subtle. In semi-formalized areas such as theoretical physics folklore convictions, correct ones and incorrect ones, have seriously impeded progress.
The (popular) claim that some statement is “well-known” – without, however, there being a reference for it – may signify folklore: If the truth of a statement really is well-known then it must be easy to give a definite reference for it. If it feels “well-known” but just doesn’t have a definite reference, then it’s folklore.
Paul Taylor on folklore in category theory (Taylor, blog comment Sept 2012):
folklore, $[\ldots]$ is a technical term for a method of publication in category theory. It means that someone sketched it on the back of an envelope, mimeographed it (whatever that means) and showed it to three people in a seminar in Chicago in 1973, except that the only evidence that we have of these events is a comment that was overheard in another seminar at Columbia in 1976. Nevertheless, if some younger person is so presumptuous as to write out a proper proof and attempt to publish it, they will get shot down in flames.
Clark Barwick on folklore in homotopy theory/algebraic topology (Barwick 17, Item 3 ):
We have no good culture of problems and conjectures. The people at the top of our field do not, as a rule, issue problems or programs of conjectures that shape our subject for years to come. In fact, in many cases, they simply announce results with only an outline of a proof – and never generate a complete proof. Then, when others work to develop proofs, they are not said to have “solved a problem of So-and-So”, rather, they have “completed the write-up of So-and-So’s proof” or “given a proof of So-and-So’s theorem”. The ossification of a caste system – in which one group has the general ideas and vision while another toils to realize that vision – is no way for a subject to flourish. Other subjects have high-status visionaries who are no sketchier in details than those in homotopy theory, but whose unproved insights are nevertheless known as conjectures, problems, and programs.
Paul Taylor’s comment on folklore is quoted in
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