A math formula is one of the objectively simplest ways of expressing something. But it's not easily understood unless you know the math well, thus violating Gruber's rule. Feynman was honest about this, because magnetism can be explained very simply, but not in any familiar way to things most people already know about.
At least as good as math (and capturing the objective complexity better I think) is a working program. "What I can't program, I don't understand."
When people ask for a simple explanation, they usually expect it to be easy for them too, because we all want simple and easy at the same time, even though only one of those is objective. If your reply to "how do magnets work?" is to start by writing down Maxwell's equations, you're gonna get crap for it, but someone who uses the fake rubber band analogy will be well received. But who really understands magnetism better?
But I think that's the deeper point of the "if you can't explain it, you don't understand it": Sure, you can write down the equations, but then what exactly have you done? You wrote down some string of characters and showed that it can be derived from some other strings of characters by applying a series of arbitrary-looking rules. By itself, that doesn't tell anything. It's only when you manage the connect formula to the part of reality that it tries to describe - and connect that part to the rest of reality - that it gains any sense.
Trying to explain a concept in "simple terms" forces you to view it in terms of its connection to other, well-known phenomena.
> You wrote down some string of characters and showed that it can be derived from some other strings of characters by applying a series of arbitrary-looking rules.
That's what it looks like if you only look at the formula. (And why I'd say that formulas are not an "objectively simple way of expressing something*)
If you can somehow keep in mind what the formula is supposed to represent, the operations on characters will let you find some insight about that thing and will stop standing only for itself.
The point is that the term "simple" is inherently subjective - there are plenty of concepts that are simple and intuitive once learned, but are hard to learn.
Also, mathematical formulas are essentially a different language. If people don't speak it, then it's a moot point.
I'd say we do it quite well, just not in in higher brain parts. We find it hard to integrate 'intuition' with the later 'reasoning' parts, instead, it just biases the logic.
Well an equation is certainly the more precise way of describing something, but not really the simplest. My most vivid memory of this is my Automata class; the instructor led with the equations before describing in plain language. Take push down automata as an example [1]. The equations are found in the formal definition, but the informal definition is much clearer.
An equation is compressed, which helps give it its precision, and can create an illusion of simplicity. Just like "a witch did it" is compressed, but if you expand it all of the complexity is inside "witch" and "it", and as you expand "it" you might arrive at some true simplifications in the untwisting sense like physical laws that make you wonder about the need for "witch"... But even if you expand the math it's still all together one of the simplest ways to describe something. That's why I like programs more, there's less implicit compression and it's easier to expand things you aren't familiar with. And there's the formalization of Kolmogorov complexity you can use for things.
As you say, informal explanations are great for understanding and bringing clarity, a lot of the time they're just used to help expand the unfamiliar math, but also a lot of the time they trade off precision, conciseness, or accuracy. At worst they complicate things with twists and relations in the informal explanation that don't exist in the real deal. Sometimes that might be helpful in the same way technically unrelated mnemonics are but it's important to point them out.
The conflation of simple, easy, concise, precise, clear, intuitive.. is the root of the issue when discussing what makes a good explanation.
At least as good as math (and capturing the objective complexity better I think) is a working program. "What I can't program, I don't understand."
When people ask for a simple explanation, they usually expect it to be easy for them too, because we all want simple and easy at the same time, even though only one of those is objective. If your reply to "how do magnets work?" is to start by writing down Maxwell's equations, you're gonna get crap for it, but someone who uses the fake rubber band analogy will be well received. But who really understands magnetism better?