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I am looking forward to saying less trivial things here, when I can carve out some more time, but anyway…
I don’t think “$*$-autonomous” is a good example of the red herring principle, because I’m pretty sure Michael Barr didn’t mean “autonomous” in the compact closed sense. It may not be well known, but “autonomous” is also an old term for “(symmetric monoidal) closed” which never quite caught on, and I think at the time of writing his monograph $*$-Autonomous Categories, Barr was sort of hoping to resurrect that usage. A $*$-autonomous category is an autonomous category in that sense, one which is equipped with a $*$-operator $A^* = D^A$ for some specified object $D$, such that the unit $A \to A^{* *}$ is a natural isomorphism.
Note: this “complaint” is in lieu of a query box. :-)
The statement about noncommutative rings is actually just incorrect. Every book in algebraic geometry says at the very beginning, “We take as a convention for the rest of the book that all of our rings will be commutative and unital, with all modules unital as well” à la Bourbaki. When one speaks of “noncommutative rings”, this is simply because many books on ring theory stress the commutative aspects of ring theory. If one is among algebraic geometers, they may take “ring” to mean “commutative ring”, because these are the most important rings to them, but it is, as I said, a convention and not a definition.
Every book in algebraic geometry says at the very beginning, “We take as a convention for the rest of the book that all of our rings will be associative and unital”
(You probably meant to write “commutative” instead of “associative”, since everyone assumes multiplicative associativity when speaking of a ring.)
Yes yes, my mistake.
@Harry, what’s the difference between a definition and a universally adhered to convention? (-:
@Todd, I don’t see why Barr’s original intent matters, if nowadays “autonomous” always refers to having duals.
@Mike: Because if you were to get into an argument about the definition of a ring, even the most dyed in the wool commutative algebraist would take your side that rings aren’t commutative unless we said so in advance.
The real example of the red-herring principle is of course the inclusion of associativity in algebra.
@Mike #5: For the reason I just said: $*$-autonomous categories are autonomous (in the old-fashioned sense), hence doesn’t quite fit the definition given on the page.
I think what this means is that more amplification might be a good idea at red herring principle, all the more because “red herring” in the usual sense is largely about intent: usually it means something intended to divert, to throw one off track or off the spoor (there’s a folk etymology here), as for example a rhetorical tactic for this purpose, or a novelistic device. I periodically have to return to that page to remind myself what is actually meant, since it’s not my usual association, and I’m sure I’m not alone in that.
So my usual association would be (ironically!) off here, but “red herring” as described on that page still gives a whiff of being the result of ineptitude or poor choice of naming. The specific amplification I have in mind here is that a “red herring” might be the outcome of older meanings being lost and forgotten due to changing fashions, so that they merely appear inept or inappropriate now.
I added a few notes to red herring principle, recapitulating what I said in #7. Please feel free to comment (or tweak, if need be).
That’s funny; up until now my understanding of the idiom didn’t require that it be a deliberate attempt to divert attention, but only something to which attention might be mistakenly diverted for whatever reason. But apparently that’s wrong, or at least less standard. Thanks.
My vision of what the “red herring principle” refers to is anything that might confuse the reader, regardless of the definition that the author has in mind. So even if the author thinks that “ring” doesn’t imply commutativity by definition, if s/he always uses “ring” to mean “commutative ring,” then a careless reader may easily come to think that “ring” is defined to mean “commutative ring.”
Who came up with ‘red herring principle’ in this context?
FWIW, my understanding of ‘red herring’ is that it’s supposed to be a deliberate distraction. However, in the context of mystery novels (where it most often appears IME), this gets watered down a bit. A red herring is deliberate on the part of the author, but it may or may not be deliberate on the part of a character in the story. So it can be a red herring externally without being a red herring internally, even if it does divert attention internally. So then it’s easy to call anything that diverts attention a ‘red herring’.
I don’t know who originated it, and I don’t even remember where I first heard it. Anyone willing to take the credit/blame?
My vision of what the “red herring principle” refers to is anything that might confuse the reader, regardless of the definition that the author has in mind. So even if the author thinks that “ring” doesn’t imply commutativity by definition, if s/he always uses “ring” to mean “commutative ring,” then a careless reader may easily come to think that “ring” is defined to mean “commutative ring.”
I disagree that the situation is like that at all. Would you say that the notion of a compactly generated weakly hausdorff space is a red herring because people sometimes just call them spaces (or call the category of them Top)? The case with commutative rings is rather simpler.
The first meaning of “red herring” that comes to my mind is as a rhetorical or debating tactic. It is a considered a general and particularly common form of “fallacy of irrelevance” (see for instance here).
I don’t know who coined “red herring principle” (maybe John knows), but I don’t mind its use so long as the use is clarified.
My understanding of the origins of the phrase is that it is very definitely deliberate: it was a false trail laid to put the hunt off the scent. Whether or not that’s an urban legend, I have no idea!
Whether or not that’s an urban legend, I have no idea!
Or, maybe it’s a red herring… (Sorry, couldn’t resist.)
Seriously, here is an excellent discussion of the origin of the term, at least in the figurative sense.
I don’t need any more convincing that “red herring” usually refers to something deliberate; as soon as Todd mentioned it, I went and looked it up and discovered that my previous understanding was wrong.
I feel differently about “space” than about “ring.” Restricting to compactly generated spaces is a “niceness” assumption; we don’t really change what we’re studying noticeably, but we just get a somewhat nicer category to work in. By contrast, the theories of commutative rings and of noncommutative rings are radically different, and there is a huge literature on noncommutative rings as well as on commutative ones.
However, when you say you study “ring theory”, it means that you generally study noncommutative rings, while if you say that you study “commutative algebra”, you study commutative rings.
Okay, I concede; I created a new “non-examples” section at red herring principle to discuss this question. Feel free to modify.
My understanding of the origins of the phrase is that it is very definitely deliberate: it was a false trail laid to put the hunt off the scent. Whether or not that’s an urban legend, I have no idea!
It is probably true that this is the origin of the phrase; certainly this is how the metaphorical meaning first appeared in print.
But there is no evidence that this was ever actually done, with a real kipper and real hounds. It was always just a metaphor.
Edit: As Ian’s link says. But my version is shorter. (^_^)
Todd wrote:
I don’t know who coined “red herring principle” (maybe John knows)…
I don’t know, but Todd’s right that I’m the kind of guy who likes wasting time on such questions…
From Wikipedia:
In a literal sense, there is no such fish species as a “red herring”; rather it refers to a particularly strong kipper, meaning a fish—typically a herring but not always—that has been strongly cured in brine and/or heavily smoked. This process makes the fish particularly pungent smelling and turns its flesh red (and makes it very noticeable, notably for the idiom).[2] This term, in its literal sense as a type of kipper, can be dated to the late Middle Ages, as quoted here c1400 Femina (Trin-C B.14.40) 27: “He eteþ no ffyssh But heryng red.” Samuel Pepys used it in his diary entry of 28 February 1660 “Up in the morning, and had some red herrings to our breakfast, while my boot-heel was a-mending, by the same token the boy left the hole as big as it was before.”[3]
The idiomatic sense of “red herring” has, until very recently, been thought to originate from a supposed technique of training young scent hounds.[2] There are variations of the story, but according to one version, the pungent red herring would be dragged along a trail until a puppy learned to follow the scent[4]. Later, when the dog was being trained to follow the faint odour of a fox or a badger, the trainer would drag a red herring (whose strong scent confuses the animal) perpendicular to the animal’s trail to confuse the dog.[5] The dog would eventually learn to follow the original scent rather than the stronger scent. An alternate etymology points to escaping convicts who would use the pungent fish to throw off hounds in pursuit.[6]
And then, after these exciting distractions, the article admits:
In reality, the technique was probably never used to train hounds or help desperate criminals. The idiom probably originates from an article published 14 February 1807 by journalist William Cobbett in the polemical Weekly Political Register.[7] In a critique of the English press, which had mistakenly reported Napoleon’s defeat, Cobbett recounted that he had once used a red herring to deflect hounds in pursuit of a hare, adding “It was a mere transitory effect of the political red-herring; for, on the Saturday, the scent became as cold as a stone.”[7] As British etymologist Michael Quinion says, “This story, and [Cobbett’s] extended repetition of it in 1833, was enough to get the figurative sense of red herring into the minds of his readers, unfortunately also with the false idea that it came from some real practice of huntsmen.”[7]
Yes, but what’s the origin of the term in our sense?
Before I came to the lab I did not hear of white and red herring principle but of weak horse versus just a horse. Weak horse in mathematics is neither horse nor weak as some old reference explained to me... :) Now when I know the discussion about white and red, I can not remember which one was white and which one was red.
I’ve not heard of a white herring before (at least, not in the context of red herrings!). I think the general rule is that given a red foo, sometimes it is not a foo (e.g. infinitary Lawvere theory - which isn’t a Lawvere theory). Then we have to talk about white foos as distinct from red ones (but actually as defined all foos are white, so the adjective is redundant).
Maybe we can put the horse vs weak horse version on red herring principle…an interesting non-anglophone perspective I think.
I added a new kind of example: not ‘homogeneous polynomial’ (which is not a red herring) but ‘homogeneous polynomial of degree $n$’ (which is a red herring in precisely one degenerate case).
I added “multivalued function” and “planar ternary ring” (page not written yet) as examples.
@Mike - I’m curious about your sudden interest in planar ternary rings. Something you’re formalising? I has a small wonder about the internalisation/constructivisation of the definition, and it seems like similar issues that hold for fields hold here.
No, actually just that I’m teaching projective geometry next semester. (-: I’ll probably actually stop at division rings rather than getting to the planar ternary ones, but I like to get deeper into a subject myself than what I’m going to tell my students, so that I have some … er … perspective.
[rimshot]
I’m teaching projective geometry next semester
How about a subproject of presenting it in HoTT? I was thinking the other day about ’the’ in the context of Euclidean geometry, and whether things are helped by typing. There’s a conversation in Lewis Carroll’s ’Euclid and his modern rivals’
§ 6. The Principle of Superposition.
Min. The next subject is the principle of ’superposition’. You use it twice only (in Props. 4 and 8) in the First Book: but the modern fancy is to use it on all possible occasions. The Syllabus indicates (to use the words of the Committee) ’the free use of this principle as desirable in many cases where Euclid’ prefers to keep it out of sight.’
Euc. Give me an instance of this modern method.
Min. It is proposed to prove I. 5 by taking up the isosceles triangle, turning it over, and then laying it down again upon itself.
Euc. Surely that has too much of the Irish Bull about it, and reminds one a little too vividly of the man who walked down his own throat, to deserve a place in a strictly philosophical treatise?
Min. I suppose its defenders would say that it is conceived to leave a trace of itself behind, and that the reversed triangle is laid down upon the trace so left.
Euc. That is, in fact, the same thing as conceiving that there are two coincident triangles, and that one of them is taken up, turned over, and laid down upon the other. And what does their subsequent coincidence prove ? Merely this: that the right-hand angle of the first is equal to the left-hand angle of the second, and vice versa. To make the proof complete, it is necessary to point out that, owing to the original coincidence of the triangles, this same ’ left-hand angle of the second’ is also equal to the left-hand, angle of the first: and then, and not till then, we may conclude that the base-angles of the first triangle are equal. This is the full argument, strictly drawn out. The Modern books on Geometry often attain their much-vaunted brevity by the dangerous process of omitting links in the chain; and some of the new proofs, which at first sight seem to be shorter than mine, are really longer when fully stated. In this particular case I think you will allow that I had good reason for not adopting the method of superposition?
Min. You had indeed.
Euc. Mind, I do not object to that proof, if appended to mine as an alternative. It will do very well for more advanced students. But, for beginners, I think it much clearer to have two non-isosceles triangles to deal with.
One issue seems to be about taking a triangle as both an ordered set of three line segments and as the 2-dimensional patch contained within such an ordered set. Different types.
Of course, I always have HoTT in the back of my mind nowadays, but it doesn’t usually seem relevant, aside from the fact that “of course” all of mathematics is naturally typed.
I think nowadays one generally regards superposition as not justified by Euclid’s axioms. I think Hilbert replaced it by taking SAS as an axiom, since making it precise requires some “higher-order” assumptions about what sorts of automorphisms exist (leading eventually, as you probably know, to an Erlangen sort of viewpoint).
I see III.6 here. Hilbert sees triangles as ’broken lines’.
What I’m after is a nice case from ordinary life where a type $X$ has some nontrivial isotopy, so that one shouldn’t really say ’The $X$’ as it’s non-contractible.
Anyway, this is worse than a red herring suitable for this thread, as it’s not even misleadingly about red herrings.
Not sure you’re still interested, Mike, but apparently there is an equivalence of groupoids between that of planar ternary rings (PTRs) and projective planes with a chosen quadrangle. See this 15-year-old(!) website for a mention of full-faithfulness of the functor $PTR\to ProjPlane$, and the fact every projective plane gives rise to a PTR give essential surjectivity. Possibly this article could help address the case of the categories of such things.
If I get time to sort things out, I may put it on the (yet nonexistent) page planar ternary ring.
Interesting, thanks!
David Roberts: vaguely related to this topic, I sent a private email to an address I have for you, but I’m not sure it’s an up-to-date address. Could you check when you have a moment?
Got it, but I was asleep!
And, continuing #32, it means that if the automorphism group of the plane acts transitively on the set of quadrangles, the PTR coordinatising it is unique.
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