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felt like archiving a quote by Paul Taylor somewhere, it is now at folklore.
Besides being funny, it is actually a useful comment for the newbie, and so I linked to it from category theory.
Thanks, and thanks to Paul as well.
I have added a more general (and more boring) paragraph. But I really shouldn’t get further into that now…
I notice that Paul says “… in category theory”. Does the rest of his quote actually apply to category theory any more than to any other branch of mathematics?
Probably in principle, but in detail I found his quote quite reminiscent of category theory. It raised vivid memories in me of long discussions on the category theory mailing list about who said the word “triple” first over which coffee break at which meeting in 196x.
Other fields have other ways of stating their folklore theorems. In algeraic topology they will say instead “Hopkins knows it”, in string theory they say “the second superstring revolution has provided immense evidence that…”. ;-)
“they will get shot down in flames”
I wonder how true that is. There is some McGill University thesis, Butler’s theorems, where a guy named Butler proved a whole bunch of folklore theorems. Or so that is my rough understanding; I also seem to remember hearing about it in somewhat laudatory terms.
Something that often goes in hand with folklore theorems is when those things or even simple well known facts can only be found in exercises left to the reader to complete or even determine if true. Sure doing an exercise is a good way to learn material but if you really don’t want to fully understand some paper but just check some fact it can be frustrating.
I recall that Tom references some exercises about codensity.
If we list examples and anecdotes now:
the last folk theorem that really struck me was the equivalence between dependent type theory and locally cartesian closed categories. When I sat down a few months back intended to look up the details and write them down on the $n$Lab, it eventually turned out that all the standard references were either incomplete (most of them) or flawed (some which tried to be complete) — and that state of affairs was only rectified last December! (Dec 2011, that is). (The details are in the comments here).
added this example:
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
I’d like to add a link somewhere to Clément Aubert’s “Categories for Me, and You?” (assuming it’s not already been added and I missed it). Would the ’folklore’ page be an appropriate place?
Sure.
Ah, but there ought to be some comment to go with this. Otherwise it looks now like this is the reference on folklore. ?!
I thought you meant to add an example, along the lines of “A discussion fo folklore in category theory is in …”
It looks like the paper may be intended as a reference that quotes Taylor’s comment given above in the entry. (The link that accompanies that comment now goes to a Page Not Found, although apparently it was a comment on Andrej Bauer’s blog.)
The paper itself doesn’t thrill me. Which folklore results does it purport to publish for the first time?
re #14 Yes, I didn’t think it a model exposition either.
for no particular reason, I added a line on the popular adjective: “well-known”.
The (common) 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.
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