Red herring names are names not either red or a herring. That is, it is a misleading name in math. 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 the speaker knows that the listener has this sense in their mind already. 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 conversations, in which the new subtleties can easily be explained after a quick explanation. 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”:
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)
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
- 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 mathematical 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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