This is a description of an emblem, a vivid and possibly surreal image, with various associated meanings. I think that deliberately crafting emblems for oneself is possibly useful (I have written about the project in a post called "Deliberately creating mental-entities; chunks"), but certainly fun.
The idea is, when you are trying to do something, and you get stuck, you can switch your attention from pursuing the original goal using the methods that got you stuck, and instead view the obstacle as a thing in itself. You can exhibit curiosity about it: what qualities does it have, does it show up in other contexts, how does it relate to other things (other obstacles? other things in general?), and that this switch is often a good move.
(This is perhaps an example of Kegan's subject-object shifts; I am not too sure.)
Sometimes when I try to remember what goes in a particular spot in a particular diagram, I can't remember it. So for example, for a while I couldn't remember what pair of letters were located at the 7th and 8th position of the QuickScript alphabet. By noun-ing my blank sensation, I can still rehearse around it, with self talk like: "okay, this one I know, okay, this one I know, oh there's THAT familiar blank, I wonder what's behind it, then there's this one that I know". I also had a blank at the 17th through 20th positions, but by noun-ing it, I can distinguish the one blank from the other.
I didn't give the blanks explicit names - maybe "that blank at 7th and 8th" or "that blank at 17th through 20th" - but I think it would be reasonable to. Suppose I decided to name the first blank "Balthazar" and the second "Frank".
When I eventually either figured out or looked up which letters are "behind" that blank, I can link the now-familiar entity, the named blank, to those letters, and so where previously my thought process might have stopped at that blank, I can now navigate through the named blank, it is now part of the indexing structure of my memory of those letters. So the associations as I navigate through my memory are something like "7th letter -> first row -> second half -> Balthazar -> the letter "thaw".
An example of a blank from the domain of Mathematics: Decartes's systematic method in geometry recommends drawing a diagram of the situation, labeling the known parts (he used letters like A, B, C, for these) and the unknown parts (he used, and IIUC established the mathematical convention, letters like x, y, z for these). If you think of this labeling of unknown parts as transforming "I don't know" (or possibly "consulting my intuitions, I feel blank about what that might be") into "I do know what that is, it's x, and I merely don't know all its qualities yet", this is an example of reifying a blank.
Sometimes when I try to build something that is possibly novel by extrapolating from some plausible starting point or points, I seem not to be able to, and not just because I haven't thought of the answer yet, but because there's something like a rumple in a rug that isn't going to go away, leading me to believe there's something like a contradiction or disharmony in my starting points, that I will never be able to extrapolate those starting points in that direction.
Examples of clashes in Mathematics
An example of a clash from mathematics: You might look at the formulas for area of a circle, area of an ellipse, perimeter of an ellipse, and think "What is the formula for the perimeter of an ellipse?" and experience a clash - there is no neat formula for the perimeter of an ellipse. See Standup Maths "Why is there no equation for the perimeter of an ellipse‽"
Clashes happen over and over again in mathematics. Another example of a clash from mathematics is in Conway and Coxeter friezes. A Conway and Coxeter frieze is a kindof doodle consisting of a grid of numbers, where every little square of numbers a, b, c, d fulfill a condition "a * b - c * d = 1". (Mathematicians think this condition is very natural.) If you start with a row of 1s and then a row that alternates 1, 2, 1, 2, then you can complete the doodle easily (with a final row of 1s), using the local rule to figure out what the remaining pattern has to be. But not all repeating sequences of numbers can be completed neatly like that. Conway and Coxeter figured out a characterization of precisely which repeating sequences work nicely, and which don't.
A third example from math: forbidden minors in graph theory. A systematic examination of a particular graph, such as the Petersen graph, might naturally lead to a proof that that particular graph is non-planar. But if you could change your focus to "the obstacles to planarity" as an entity in themselves, then you might find (might have found) Kuratowski's theorem, that there are essentially two obstacles to planarity, K5 (the complete graph on five vertices) and K3,3 (complete bipartite graph on six vertices).
Yet more examples of clashes in other domains
An example of a clash in programming is from Jackson Structured Programming (and this domain is where I took term "clash" from). JSP is a methodology for writing code, which has a set number of steps, and some of the steps are not guaranteed to work. As you're working through them, you need to be alert to the possibility that you might not be able to succeed smoothly and naturally, and notice that and back up rather than force your way forward. Another example of a clash in programming is called the object relational impedence mismatch. The term "impedence mismatch" is an idea from power electronics, which is being used metaphorically here.
An example of a clash from power electronics is in Bond Graphs. Bond Graphs are a methodology for creating models of power trains, that uses diagrams shaped like molecules. An early step in Bond Graph modeling is to orient each edge by drawing a little stroke at one end of the edge or the other. In Bond Graph terminology, the orientation of the edge is called "causality", and if you experience difficulties in annotating the graph with causal strokes, the Bond Graph methodology recommends going backwards and modeling the system differently.
What would a clash in linguistics / phonology look like? I can imagine a kind of clash if you had dimensions (such as the two dimensions of the vowel quadrilateral) that were working for you in classifying and organizing vowel phonemes, and then you found that you were unable to extend this scheme to also fit consonant phonemes. If you could exhibit curiosity about this clash, then you might be able to discover or characterize a distinction between vowels and consonants, or another pair of related concepts, such as "pitchy" versus "noisy" sounds.
Backtracking search and conflict-driven clause learning
Conflict-driven clause learning is an algorithm (or a technique) in constraint propagation. If you are playing or writing an algorithm for a puzzle like Sudoku or Kakuro, and you know the idea of backtracking, you might at first view the "rules" of the game as fixed - for example, one rule for each of the 9 columns, 9 rows, and 9 blocks in Sudoku, or one rule for each of the "clues" in Kakuro. However, the idea of conflict-driven clause learning is that if you do a few assumptions, and from them derive a contradiction, then you can add a new rule, which will head off that particular contradiction. V
The point is that instead of exploring the tree as if you were an idiot, and ramming into the same obstacle at multiple different leaves of the tree, you try to look into the qualities of the obstacle, certainly enough that you could recognize if if you rammed yourself into it again (in an analogous portion of the tree), but possibly enough that you can change your search procedure.
Learning while searching like this is something that the Soar cognitive architecture (Laird, Newell, Rosenbloom, 1983) can do. Impasse-driven subgoaling and chunking is one of the characteristic qualities of Soar among other theories.
Lock-and-key structures in Zelda-like dungeon design
There is an idea from game design (or specifically, Zelda-like dungeon design), that you should arrange for the player to encounter an obstacle, such as a row of bushes, large rocks, or a river before they encounter the "key" - a tool, such as a bush-cutting sword, rock-exploding bombs, or fish-leather shorts (that allow the player to swim), that they will eventually use to overcome that obstacle. (See Joris Dorman's "Mission and Space" and "Unexplored" work, Tom Coxon's "Metazelda" dungeon generator and "Lenna's Inception", Mark Brown's "Boss Keys" videos about Zelda dungeon design).
I visualize this emblem as an enormous closed and locked door in the sky, surreal in its height and imposingness, directly blocking a beautiful tree-lined path, with a bottomless crevasse, also blocking the same path, in front of the door. (The word "crevasse" reminds me of "impasse" - rhymes are silly, but associations are associations.) You can't continue, you can't jump, even if you could jump there's not enough space to stand on the other side, and even if you could, you wouldn't be able to get to where the handle or keyhole would be because it's too high, and there isn't even a handle or a keyhole, and anyway you don't have a key.