The are some parts of math in which you encounter easily new structures, obtained by modifying or generalizing existing ones. Recent examples can be tropical geometry, or the theory around the field with one element. If one works in those areas, one cannot avoid the problem of naming new objects.

When working with such a "new" notion, more general than an existing one, you have different options to name it. Either the red herring option, like group without inverses, a brand new name, like monoid, a derived name, like semigroup (which is actually a group without inverses and without an identity element), or no name at all, so a very long name, i.e. the category $M$ of sets with an associative binary operation. Or even you can also decide to use the old name with a new meaning. (The examples I wrote don't pretend to have a historical justification).

Although the red herring construction is used everywhere in math, I feel that it is not a good practice. To use the old name with a different meaning can be the origin of a lot of errors. And the option of not giving any name at all is like you elude your responsibility, so if someone needs to use it they will have to put a name to it (maybe your name?).

So my preferred options are to choose a derived name or a new name. Derived names are quite common: e.g. quasicoherent, semiring, pseudoprime, prescheme (which is an old term), and they contain some information which is useful, but sometimes they are ugly, and it could seem you don't really want to take a decision: you just write quasi/semi/pseudo/pre in front of the name. But new names can be difficult to invent, to sell and to justify: if you decide to give the name jungle to a proposed prototype of tropical variety, because it sounds to you that in the tropics are plenty of jungles, it is a loose justification and probably will have no future (unless you are Grothendieck).

My question is: Which do you think is the best option?

In fact, the situation can be worse in some cases: what happens if some name has already been used but you don't agree with the choice? Is it adequate to modify it, or can it be seen as some sort of offense?

I could put some very concrete examples, even papers where they introduce red herrings, new meanings for old names, new names and no-names for some objects, all in the same paper. But my point is not to criticize what others did but to decide what to do.

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    $\begingroup$ Naming in mathematics is definitely a problem. Reuse and overuse of weak metaphors like 'normal', 'regular', people's names like 'boolean', 'abelian', 'Grothendieck', frivolous names like 'Tropical'. Just needing names is an (unavoidable) problem in math (half of math is remembering what the name is supposed to mean). So any ways of making that easier are welcome. I vote for 'derived' because entirely new names tend to be undecipherable/difficult to attach to the concepts. $\endgroup$ – Mitch Aug 15 '18 at 13:51
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    $\begingroup$ Best is to find a name that’s at least partially descriptive of the phenomenon. But this can be surpassingly difficult. $\endgroup$ – Lubin Aug 15 '18 at 15:25
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    $\begingroup$ I'm not sure I've seen jungle, but there are forests, more precisely: "A forest is an undirected graph without cycles (a disjoint union of unrooted trees), or a directed graph formed as a disjoint union of rooted trees." Graph theorists seem quite adept at creating new names, see en.wikipedia.org/wiki/Glossary_of_graph_theory_terms $\endgroup$ – Joe Silverman Aug 15 '18 at 15:58
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    $\begingroup$ primarily opinion based... $\endgroup$ – Gerald Edgar Aug 15 '18 at 17:22
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Let me mention as a counterpoint that there is less need for new terminology than one might expect. Mathematical exposition is often more successful and clearer without new terminology, and one should consider whether one needs any new terminology at all.

It seems to be a typical beginner's mistake to name everything in sight, introducing all kinds of fancy names and cluttering one's writing with unnecessary terminology and jargon. To be sure, this naming process is easy, as well as fun; one feels like Adam or Eve in the garden. I've succumbed to the attraction of it myself. But now I view this more negatively, for it imposes a kind of tax on the reader. One opens the article and finds a theorem stated there:

Theorem. Every big-topped parade is heartily divisible.

The jargon prevents it from having an immediate meaning, even for an expert in the subject, and one must hunt down the definitions of the various terms. I am sure that many mathematicians have had this experience. Articles are almost never read front to back, and so the definitions of new terms are often missed. The question of whether the article will be read at all is often settled by browsing through it and seeing if the theorems are interesting. The jargon tax is a tax many readers are not willing to pay — when one can't find meaningful mathematics easily enough, then one simply looks elsewhere, and one article (perhaps your article!) is dropped in favor of another.

For this reason, I find it desirable, when possible, for one to state one's theorems in a manner that can be readily understood with ordinary terminology, even if some new-fangled jargon would make it slightly shorter or would perfectly express some extremely abstract connection.

In time, of course, new objects and ideas find an established usage, and there will be a need for new names. My comment is merely a caution against over-exuberant naming.

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    $\begingroup$ A similar point can be raised with respect to unnecessary notation, used to express relations that can be easily expressed in ordinary language. $\endgroup$ – Joel David Hamkins Aug 15 '18 at 16:21
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    $\begingroup$ I thought Kerfluski had found a big-topped parade that was only semi-heartily divisible. $\endgroup$ – Gerry Myerson Aug 18 '18 at 0:39
  • $\begingroup$ Indeed! Making up names is not the same as actually doing mathematics, after all... :) $\endgroup$ – paul garrett Nov 17 '18 at 19:01

Let me address the question "what happens if some name it has already been used but you don't agree with the choice?", by giving a recent example from (mathematical) physics. The 2012 experiment that discovered a "Majorana fermion" in a superconductor attracted much attention because it would be a realisation of a non-Abelian anyon. The name was a red herring, because a fermion by definition has Abelian statistics.

In a discussion on Wikipedia it was argued we could keep the name, for the same reason that "We should not rename the "jellyfish" Wikipedia article into "Cnidaria" just because jellyfish are not fish." But the oxymoron of a "non-Abelian fermion" was sufficiently unpleasant that the name has been banned in favor of "Majorana zero-mode" --- less pretty, without the neat pointer to the Majorana-Fermi duo, but more accurate. Referees played a decisive role in pushing this change through, these days you just can't get a paper published on "non-Abelian fermionic statistics".

So yes, I do think it is appropriate to avoid red herrings in the nomenclature, and if they exist, to modify them, preferrably by a minor change in the name. The change in this recent example, from "fermion" to "zero-mode", was major. A minor change that I recall from longer ago is from "quantum chaos" to quantum chaology.

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    $\begingroup$ The name "quantum chaology" makes me feel kind of sick $\endgroup$ – Qfwfq Aug 15 '18 at 21:40
  • $\begingroup$ Is "chaology" good Greek? Not "chaosology"? $\endgroup$ – Todd Trimble Aug 23 '18 at 12:24
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    $\begingroup$ @ToddTrimble "chaology" looks more like English Greek to me than "chaosology", which looks like a formation within English, taking the suffix to be "-ology". The reason why I say English Greek is that in Greek, words such as analogy are αναλογια, which would be analogia, so the final y is of purely English origin. Chaology comes from "chaos" and "logos" in an analogous way to how topology comes from "topos" and "logos" and genealogy comes from "genea" and "logos". Taking "-ology" as the suffix instead of "-logy" came later. $\endgroup$ – Robert Furber Aug 23 '18 at 21:29
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    $\begingroup$ @ToddTrimble However, I am against this kind of classical language snobbery, and think we need less of it, not more. Nobody insists that the plural in English of yogurt is yogurtlar (the word comes from Turkish), and it is unreasonable to expect people to understand the detailed grammatical processes of every language a word derives from etymologically. $\endgroup$ – Robert Furber Aug 23 '18 at 21:33
  • $\begingroup$ @RobertFurber Thanks for your response! $\endgroup$ – Todd Trimble Aug 23 '18 at 22:20

For me, I think naming is not as important as understanding what it is. We all understand Fourier transform, Banach space, Peter-Weyl theorem, Pauli matrix. If we rename just one of them, what would we call?

As someone who used to work as a translator and is very inquiring about how an idea forms and evolves, I have some chances to observe how a same concept is translated to different terms, and how a same object is renamed several times in different contexts. Perhaps this practice is exotic in academical mathematics, and I admit that I have zero experience in graduate research, but cognitively I don't think our brains are that different.

When does a perfect name come? When you are in a rush. When you are in a rush, you don't have time to think about correctness, you just want to make it quick to solve another important, urgent problem. Your brain will cut all unnecessary information about the object, leaving just enough bit so you can jump to conclusion, or in this case, a name. Those unnecessary bits may be essential to for the concept to form at the first place, but unfortunately, don't really relate to the surrounding concepts in the sentence. The surviving bits that constitute the new name are the ones closer to the surroundings. So even when the original name is short and accurate, it will be replaced by a new name that fits the context.¹

In research, you are expected to be accurate, and you have plenty of time to learn it, but the principle is the same. My advice is to try using the concept to study many more other concepts, and see how your mind reacts with it when you are in a rush. You can also discuss those other concepts with your colleagues, and see how they complete this sentence when you are stuck: "you mean the ______?"

Of course in the realm of cognitive science and linguistic, sometimes you have to accept a sticking bad name. But lucky for us, this is also the realm of math, and unlike jellyfish or pineapple, non-abelian fermion or group without inverses aren't that imaginative, so they will always invoke an unpleasant feeling when reading it. Hopefully one day we can go around them, and let them rest in peace.

Here is a challenge, inspired from the small-world network: try explaining a topic by introducing only 6 intermediate terms. This will force you to twist what you already know about it, so that you can view it in a different perspective. You have to be bold to cut off the details that rock your soul, but by then the big picture will emerge. Only after seeing the big picture that naming can become a piece of cake.

I have an article for this, you can check it out: Making concrete analogies and big pictures.

¹ This is my own theory, but is inspired from Gentner's Structure Mapping Theory
Related: Hyphens after the prefixes “non-” and “anti-” in mathematics

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    $\begingroup$ I don't have an unpleasant feeling reading "non-abelian fermion" or "group without inverses" because they are unimaginative. The latter I find unpleasant because I wouldn't know whether the author meant "monoid" or "semigroup", and it sounds like he made up a term because he doesn't know standard terms. The former I would find less than congenial mostly because I'm not fluent in physics, but also because it doesn't sound like a useful concept without specifying the group (and only now do I see Carlo Beenakker's more erudite response to a phrase like this). $\endgroup$ – Todd Trimble Aug 23 '18 at 12:16
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    $\begingroup$ I'm also very skeptical about the bit of pop psychology that says you find the perfect term while in a rush. Mainly I feel that bad terminology arises when mathematicians don't think through carefully the likely consequences of their poor word choices. $\endgroup$ – Todd Trimble Aug 23 '18 at 12:20
  • $\begingroup$ @ToddTrimble In regard of your pop psychology bit, I just want to have a view from cognitive science - psychology - linguistics, rather than experience in math only. I know I stretch myself too thin, and none of the field I have strong understanding on. But I think the idea is legit for *analogical terms, like "jungle" or "representation", and I should have stressed that in my post. OP is unsatisfied with "jungle" because it is in accordant with "tropical", not because it actually captures the essence of the concept, and in a rush you won't feel "jungle" legit anymore. $\endgroup$ – Ooker Aug 23 '18 at 13:09
  • $\begingroup$ If there are cognitive science studies that show that well-chosen terms are found when one is in a rush, then that would be interesting (and I would be happy to edit my comment). The term "tropical" is in my opinion not a good choice (chosen for somewhat whimsical and unevocative reasons), but I think it proves my point about authors choosing a spur-of-the-moment terminology, not considering the long run. $\endgroup$ – Todd Trimble Aug 23 '18 at 13:48
  • $\begingroup$ @ToddTrimble yeah this is only my theory after all, based on anecdotal evidence, and I do see that's not very persuasive at the moment (read: I'm in a rush :P). Regarding the "tropical" that doesn't really capture the essence and fit the long run, I call that "weak analogy" in my article. Regarding the spur-of-the-moment vs well-chosen term, they both come when one is in a rush, but the latter requires the author has extensively studied it in various perspectives. Only then that they can speak up its name reflexively in a rush. $\endgroup$ – Ooker Aug 23 '18 at 14:29

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