I have a weakness for using an over-complicated analogy because I think it's cool, regardless of whether it will actually help. Come to think of it, I think many people have this weakness.
In mathematics, you often have a graph like "y=x2 except at x=2 where y=-57.2 instead of 4". Everyone can see that even though that's what it LOOKS like, what it really OUGHT to be is y=x2 EVERYWHERE. I think most people's intuition would say that that is what the graph "really" is, and that the best way of learning anything about it is to consider y=x2 and then consider the odd-point-out separately if necessary. And on this occasion, oddly enough, their intuition would be exactly right. And mathematicians are the same, except they (a) have a special word for it and (b) typically don't even MENTION the possibility, just assume it's taken care of, except when pedantically enumerating the things that MIGHT go wrong.
Contrariwise, if you have a graph like "y=1/x" you'll notice that it's perfectly smooth and nice when x is GREATER than 0, and perfectly smooth and nice when x is LESS than 0, but that the two halves don't match up: one gets bigger and bigger as it gets closer to 0, and the other gets more and more negative. So whatever value y is given at x=0, it doesn't mater, the graph is still in at least two separate parts, and can't be fixed up, nohow. In fact, you'll notice that "y=1/x" doesn't define the value at x=0 at all (because it would be 1/0, which is undefined in normal arithmetic), and to be correct you'd have to specify a value (or that that point wasn't defined), but actually whatever answer you give, it doesn't matter.
The first is a removable singularity. The second is a non-removable singularity. (A related term is "essential singularity", but that only applies if the ends of the graph are even more incompatible than the second case. And all are much more relevant in complex analysis than real analysis).
The comparison to plot holes in movies
I've previously used this comparison to whether or not a plot hole in a movie is important. Basically, I see a difference between (a) a flaw in a movie which could be fixed by changing one single isolated point in the movie to something else which would serve the same purpose to the rest of the movie, but not be so stupid. And (b) a flaw in a movie which connects to multiple points, and couldn't be removed without damaging all the good parts.
For instance, Pulp Fiction begins when Wallace sends the two gangsters to collect a briefcase, and then lots of artistic violence occurs for the rest of the film. But the briefcase is only mentioned that once, right at the beginning. Most people forget it. It doesn't matter. It's a classic macguffin (except being unobtrusive enough that people don't spend the whole film asking what it's for, as many macguffins do). If instead of the briefcase, there had been some evidently stupid thing Wallace sent the gangsters to retrieve, it would have been annoying in itself, but the rest of the film would have worked equally well.
In the Matrix, we are told sleeping humans are used as "batteries" for machines. If they say "massively parallel organic computers", the absurdity is massively reduced while changing almost nothing else in the film.
However, other films have flaws which (however big or small) are woven throughout them. If, unlike war of the worlds, the aliens do MULTIPLE things for no known reason, it saps the tension, because you can't just accept the premise and move on, you have to keep saying "oh look, they're in a dire situation. I bet the aliens will do something ELSE inexplicable about now".
The comparison to deliberate ambiguity in movies
However, it just now occurs to me a similar comparison can be used for ambiguity.
In Blade Runner, it is hinted that the main character may be an illegal android. However, whether he is or not, all the emotional decisions he makes during the film still happened and are still important. Thus, the film works equally well if he is revealed to be human, or revealed to be not human, or is left ambiguous. Most people on some level want to know "is he or isn't he" but in many ways that's almost beside the point.
However, in other films, a character goes through lots of emotional development, and then is revealed to possibly have been acting it all all the way through. In this case, the truth DOES make a difference, because one way all the emotional stuff you believed in really happened in the film, and the other way lots of different stuff happened (often, someone PRETENDING to do important and good things for other reasons or no reason and cynically manipulating the people around them).
Maybe people OUGHT to be able to hold those two contradictory feelings in their head at once, but they typically CAN'T. With the exception of films where the uncertainty of the main character is explicitly the main point, I think stories are better served by NOT being ambiguous.
In fact, in many cases, if you develop a long story with a character, and then say "oh, actually, they were a spy in disguise" people will simply not want to believe it, and go on believing the simpler version you told them earlier whether it's "correct" or not.
In mathematics, you often have a graph like "y=x2 except at x=2 where y=-57.2 instead of 4". Everyone can see that even though that's what it LOOKS like, what it really OUGHT to be is y=x2 EVERYWHERE. I think most people's intuition would say that that is what the graph "really" is, and that the best way of learning anything about it is to consider y=x2 and then consider the odd-point-out separately if necessary. And on this occasion, oddly enough, their intuition would be exactly right. And mathematicians are the same, except they (a) have a special word for it and (b) typically don't even MENTION the possibility, just assume it's taken care of, except when pedantically enumerating the things that MIGHT go wrong.
Contrariwise, if you have a graph like "y=1/x" you'll notice that it's perfectly smooth and nice when x is GREATER than 0, and perfectly smooth and nice when x is LESS than 0, but that the two halves don't match up: one gets bigger and bigger as it gets closer to 0, and the other gets more and more negative. So whatever value y is given at x=0, it doesn't mater, the graph is still in at least two separate parts, and can't be fixed up, nohow. In fact, you'll notice that "y=1/x" doesn't define the value at x=0 at all (because it would be 1/0, which is undefined in normal arithmetic), and to be correct you'd have to specify a value (or that that point wasn't defined), but actually whatever answer you give, it doesn't matter.
The first is a removable singularity. The second is a non-removable singularity. (A related term is "essential singularity", but that only applies if the ends of the graph are even more incompatible than the second case. And all are much more relevant in complex analysis than real analysis).
The comparison to plot holes in movies
I've previously used this comparison to whether or not a plot hole in a movie is important. Basically, I see a difference between (a) a flaw in a movie which could be fixed by changing one single isolated point in the movie to something else which would serve the same purpose to the rest of the movie, but not be so stupid. And (b) a flaw in a movie which connects to multiple points, and couldn't be removed without damaging all the good parts.
For instance, Pulp Fiction begins when Wallace sends the two gangsters to collect a briefcase, and then lots of artistic violence occurs for the rest of the film. But the briefcase is only mentioned that once, right at the beginning. Most people forget it. It doesn't matter. It's a classic macguffin (except being unobtrusive enough that people don't spend the whole film asking what it's for, as many macguffins do). If instead of the briefcase, there had been some evidently stupid thing Wallace sent the gangsters to retrieve, it would have been annoying in itself, but the rest of the film would have worked equally well.
In the Matrix, we are told sleeping humans are used as "batteries" for machines. If they say "massively parallel organic computers", the absurdity is massively reduced while changing almost nothing else in the film.
However, other films have flaws which (however big or small) are woven throughout them. If, unlike war of the worlds, the aliens do MULTIPLE things for no known reason, it saps the tension, because you can't just accept the premise and move on, you have to keep saying "oh look, they're in a dire situation. I bet the aliens will do something ELSE inexplicable about now".
The comparison to deliberate ambiguity in movies
However, it just now occurs to me a similar comparison can be used for ambiguity.
In Blade Runner, it is hinted that the main character may be an illegal android. However, whether he is or not, all the emotional decisions he makes during the film still happened and are still important. Thus, the film works equally well if he is revealed to be human, or revealed to be not human, or is left ambiguous. Most people on some level want to know "is he or isn't he" but in many ways that's almost beside the point.
However, in other films, a character goes through lots of emotional development, and then is revealed to possibly have been acting it all all the way through. In this case, the truth DOES make a difference, because one way all the emotional stuff you believed in really happened in the film, and the other way lots of different stuff happened (often, someone PRETENDING to do important and good things for other reasons or no reason and cynically manipulating the people around them).
Maybe people OUGHT to be able to hold those two contradictory feelings in their head at once, but they typically CAN'T. With the exception of films where the uncertainty of the main character is explicitly the main point, I think stories are better served by NOT being ambiguous.
In fact, in many cases, if you develop a long story with a character, and then say "oh, actually, they were a spy in disguise" people will simply not want to believe it, and go on believing the simpler version you told them earlier whether it's "correct" or not.