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My answer for Naming in math: from red herrings to very long names has been deleted. According to Why and how are some answers deleted?, the only two relevant reasons are:

  • commentary on the question or other answers
  • not even a partial answer to the actual question

Still, I don't see how they apply to this answer. The question asks:

So my preferred options are to choose a derived name or a new name. Derived names are [...]. But new names can be [...]. My question is: Which do you think is the best option?

And my answer addresses that (well, technically not the literal question). It explains why the best strategy may be to ignore the original meaning of the name and find a way to create new meaning on it. It also provide some methods to coin a new name. I don't see how it's just a commentary or not even a partial answer.

In this comment, Stanley Yao Xiao said:

The answer almost entirely misses the point of the question, in my opinion. A specific critique is the claim that "Fourier, Banach, or Peter-Weyl are meaningless", which is something that almost no mathematician would agree with.

But I don't understand this point. Before and after this statement are the actually explanations. This is just a minor example, so if it's really wrong, you can simply ignore it. For me this critique misses my point.

However, I'm aware that I may have the illusion of transparency. This is a human tendency and I cannot avoid it. But I need feedback to know where I'm unclear.

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    $\begingroup$ This may be for the deleters to answer. But I don't think the reasons given in that Help post were meant to be comprehensive. My own guess is that the deleters felt that your answer was largely opinion-based, not obviously rooted in experience with writing mathematics papers, and indeed arguably gave bad advice for writing mathematics -- to the point it would be better to remove it than leave it there. I myself strongly disagreed with much of it, inasmuch as it purports to do with writing mathematics. $\endgroup$ – Todd Trimble Feb 3 at 5:00
  • $\begingroup$ Thanks for the feedback. Ironically I consider my other answer as bad, and I actually prefer to have it deleted. I indeed don't have experience in writing math paper, but I do read some math (up to representation theory), and I think it goes well with my experience. It will be great if you can explain why you strongly disagree with it. $\endgroup$ – Ooker Feb 3 at 7:52
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    $\begingroup$ In particular, I disagreed with everything you said about "forest". That was a case where nothing needs to be done: it's an evocative term already, and widely understood; for that reason your suggestions seem uninformed and way off the mark (and the OP didn't say he had a problem with "forest", did he?). I disagree with what you said about "abelian": it's so commonly understood and entrenched that I know no one but you who has any problem with it. I really dislike the "sabio" suggestion. It's "deja vu", not "de javu". I think the suggestion to use a portmanteau engine is probably a bad one. $\endgroup$ – Todd Trimble Feb 3 at 12:20
  • $\begingroup$ Well, the OP has problem with red herring names, and I guess "forest" is a red herring name, no? In your comment on my other answer you say that it's not even an analogy, suggesting that you dislike it. I find your statement "it's an evocative term already, and widely understood" contradicts to everything I know in that question. The "sabio" or "abelian", again, is just an example, and your dislike and my dislike on them are just personal preferences I think. I'd like to know why my main point is wrong: that it may be better to create new meaning on the established name. $\endgroup$ – Ooker Feb 3 at 12:49
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    $\begingroup$ Because mathematics is a community effort, and it takes a substantial period of time to overcome dialogue like "Sorry, what is a sabio group?" "oh, it's just an abelian group". There should be seriously good justification before undergoing such a project. A related point is already made by the top-voted answer in that thread! $\endgroup$ – Mike Miller Feb 3 at 13:50
  • $\begingroup$ @MikeMiller That would be a good point. It seems like although I see my answers (1) align well with my experience during the time I learn math/physics, and (2) base on cognitive psychology and linguistics, they are more on daily basis, not on math basis. This makes sense to me. The tax to coin new word in daily basis is much lower than in math, and I don't think the theories on those fields base on math papers anyway. So do you think that the answer is not wrong per se, it's just not suitable in math? Anyway, I think my other answer aligns with the top one, especially the challenge section. $\endgroup$ – Ooker Feb 3 at 14:25
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    $\begingroup$ "Forest" is most definitely not a red herring, in my understanding of how "red herring" is used here: ncatlab.org/nlab/show/red+herring+principle $\endgroup$ – Todd Trimble Feb 3 at 14:48
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    $\begingroup$ Ooker, I think your last comment is on the right track: however good your advice might be for other areas, I don't think much of it applies well in math. For example, the advice to invent a fictional character (which you even emphasize in a footnote: "Please do this"). For heaven's sake, please don't do this!! Maybe (maybe) John H. Conway could get away with this, because (1) he's a genius, as everyone knows, and because (2) he has a long established record of linguistic whimsy, and people will indulge him. But for those just starting a mathematical career, IMO it sounds like terrible advice. $\endgroup$ – Todd Trimble Feb 3 at 18:21
  • $\begingroup$ @ToddTrimble thanks. I've rewritten it in the answer below, can you check it. $\endgroup$ – Ooker Feb 4 at 8:52
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OK, I get your point. I did visit the red herring page several times, but apparently I missed a lot of subtleties. It turns out that red herring names in mathematics are not simply wrong analogies, and coining new term may add more problems than fixing it like normal. I now understand and agree with the deletion.

Below is my draft to rewrite it. Hope to have your feedback (and your proofing).


Take the red herring name noncommutative geometry as an example. Its correct form should be not necessarily commutative geometry (or possibly non-commutative geometry), but the adverbs are trimmed in a rush to explain an idea in which this concept is not the main focus. Because the main focus is not the concept itself, the concept is explained fleetingly, and this shortened form stabilizes in the circle.

In the insiders' minds, those subtleties have been implemented into the words, making them a variation of the original meaning. In other words, the word noncommutative transforms into a polysemy. It now has two senses:

  • The dominated sense: truly noncommutative. This sense doesn't have subtlety (or we can say that it has the subtlety truly)
  • The newfound sense: not necessarily commutative, possibly non-commutative

The problem is that one can only get this newfound sense after a period of learning. This is not the same in daily basis, in which the new subtleties can easily be explained after a small talk. So we have the dilemma:

  • Linguistically, this is a natural process and unavoidable
  • Cognitively, the required energy to deal with this linguistic process is not natural at all

This applies well to my experience when learning math. For example, here is the formal definition of "irreducible representation" and my translation in the margin:

  • Formal definition: A representation U(G) on V is irreducible if there is no non-trivial invariant subspace V with respect to U(G). (Wu-Ki Tung, Definition 3.5)
  • My translation: When a representation on a space is reduced to the point that only that space and {0} are its only two subspaces that can hold their vectors from being pulled out, then we have an irreducible representation.

I do the translation by nesting definitions of each term in the definition, then rewriting it. I think this is the way to restore the true intuition of irreducible representations before it is conceptualized. Notice how 4 times negation is used in the formal definition (ir-, no, non-, in-), flipping the meaning back and forth. Logically, they can be canceled out, but perhaps linguistically the 4 negation version is more rigorous than the none one? I think, the formality of logical connectives (and, or, not, if, iff) and quantifiers (∀,∃) significantly trims down the subtleties that are necessary to understand formal definitions correctly: impossible, necessary, unavoidable, indispensable, can't exist without, together with, there is no way, never again, etc.

I think textbook authors may do a better service to their students if they start with this informal definition before getting to the formal one. Likewise, paper authors should emphasize the implicit assumptions in the term before using it, in case it is read by outsiders.

Having said that, even when both sides are aware of this, the execution to transfer the subtleties may face these obstacles:

  • By definition, subtlety or tacit knowledge is extremely hard to recall. Even when the author is aware that they need to make it explicitly, they may not know where to start
  • Even if it is explicitly explained, the readers may overlook it, because they are overwhelmed with the information and just want to skim the paper. This makes them even more overwhelming

My advice to deal with this:

  • The author needs to take every single note about their struggles when they learn/invent the concept, because after it becomes tacit knowledge, their notes are the only way to help them explain it to others
  • The student may need to equip the knowledge about mathematic polysemy so that (1) they have a strong motivation to carefully read the section explaining the subtleties, and (2) the author doesn't have to include a lecture about linguistics in a math book
  • The author should make the section about subtleties more overt and standing out, so that it will not be skipped when the paper is read F-shapedly

I cannot emphasize the last bullet enough. Maybe they should be put in a separate box with yellow background and the title "Attention!". Maybe there should be a popup message with a checkbox "I have read and agreed with all the subtleties" before the continue button is clickable. Making sure everyone to be in a same page is important.

If you are interested in the techniques to reveal subtleties, check out my article: Making concrete analogies and big pictures. You can also check out a basic course or a survey research on polysemy in cognitive linguistics.

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  • $\begingroup$ Sure, providing the intuition first before the formal definition can be great. But care needs to be taken to ensure that it remains correct and meaningful. Your "translation" is neither. You just introduce new terms that have not been defined (reducing a representation and "holding vectors from being pulled out"), neither of which easily translate to anything I could make precise while keeping the definition correct. Also, ending with "then we have..." is really not a good way to end a definition in mathematics. $\endgroup$ – Tobias Kildetoft Feb 4 at 10:43
  • $\begingroup$ This, again, is just an example and not my main point. A discussion of this is available in Math Edu: Why are proofs written in flowery language incomprehensible?. Can you read the question, the answer, and my comment in that answer? $\endgroup$ – Ooker Feb 4 at 12:52
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    $\begingroup$ I mean, maybe all this is worth thinking about if you're a textbook author, writing for students still struggling with the peculiar ways in which mathematicians use language, but the OP's question was more in view of writing for other mathematicians who are familiar with the peculiarities, and would probably resent such elaborations or over-explanations in a paper. The OP's question is also about naming new concepts, a problem which comes up hardly at all in writing textbooks. Therefore much of this still reads to me as off-topic. $\endgroup$ – Todd Trimble Feb 4 at 12:54
  • $\begingroup$ @ToddTrimble you are correct that my experience is limited to textbook. Anyway, what do you think about the first part of it (before I talk about my experience)? I think this is still useful in this question. In fact I think it can be a synthesis for my both answers. As for the naming new concepts, what do you think about the part in the deleted one? So far I still haven't receive much critiques about it. $\endgroup$ – Ooker Feb 4 at 13:05
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    $\begingroup$ The answer by Benoît Kloeckner in that Math Edu thread is excellent, and I think you should take it to heart, but the discussion is about something other than what the OP referred to in this discussion is asking about, and so it's a bit of a distraction. I'm disinclined to enter a long discussion about all of this, but I do wonder why you want to weigh in so much on a topic which you admit you have little experience with. MO and MO meta are not there for you (or other non-professional mathematicians) to hone your thinking about your own attempted answers. $\endgroup$ – Todd Trimble Feb 4 at 13:13
  • $\begingroup$ @ToddTrimble Perhaps it is because I have a strong interest in this topic, and I wasn't aware of the difference between my struggle to learn math and the struggle of professional mathematicians to produce math. I can only fully know it when I become a mathematician myself, but since this is not the case, this unawareness still follows me, making me wonder what I am missing. I know you don't want a long discussion, but I wonder why the first part isn't interesting to the OP, or why knowledge on other fields isn't interesting for mathematicians writing for mathematicians. $\endgroup$ – Ooker Feb 4 at 14:58
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    $\begingroup$ I think some of the downvotes here and elsewhere are arising because of an impression that you are very determined to offer suggestions on what should be done while speaking from a relatively inexperienced position as a researcher in mathematics or a writer in mathematics or a teacher of mathematics $\endgroup$ – Yemon Choi Feb 4 at 21:00
  • $\begingroup$ @YemonChoi Do you think that if I cut the advice part and only focus on the linguistic/cognitive aspects of it, it will be more welcomed? Maybe this is the reason why I feel you are passing my point and I am passing your point. Perhaps it is because the OP asks for solution that makes me want to give advice. Todd said that it's good, it's just not suitable in math. $\endgroup$ – Ooker Feb 5 at 6:12
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    $\begingroup$ Why would it be welcomed if it is not suitable for math? This is a forum for mathematicians and that was a thread about mathematics. $\endgroup$ – Mike Miller Feb 5 at 16:10
  • $\begingroup$ @MikeMiller but it is about the advice part only. The first part is not advice and about math. I am asking about it only $\endgroup$ – Ooker Feb 5 at 16:19

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