I agree with most of what you said, and the missing dollar question is a great example, but I would say that (from my non-mathematical perspective) there is an important difference. I know you were talking about words as they relate to expressing mathematical concepts, but I'm going to take you slightly out of context and disagree with the idea that language and math essentially work the same. Math requires the logic and internal consistency in order to communicate anything of value, while in language this is not necessarily the case. For example, metaphors do not always communicate ideas logically, and in fact we value them because of that. "A fish out of water" is a phrase that we know to mean someone who is struggling because they are not in their accustomed environment (work environment, social environment, whatever). Yet logically, a fish out of water is dead, even though that's not at all what we're saying.

You could argue that commonly accepted metaphors are part of a language's "logic", that is to say, everyone knows that they don't mean what they are logically saying, therefore it's still logical in a certain sense. But lots of metaphors work without being as well-known as "a fish out of water". They work because our minds are able to discard the logical answer as incorrect and instead work out the illogical inferences. You could say, "He slept with an angry frog in his throat" and most people would be able to work out that this is a metaphor for snoring. You could say, "Life's the fattest turkey you've ever seen and it's always Thanksgiving morning," and this could mean either that the world is going to eat you alive or that life is great and should be seized and enjoyed to the fullest every day. It's both illogical and ambiguous, yet it still works.

Perhaps there are higher forms of math where information can be conveyed even though the strict logic is false and it's unclear exactly what is meant? Like I said, I'm not a mathematician.

But let's take it to extremes. In math you couldn't write 1+4=27 and have it mean anything, because it goes against the logical rules as we understand them (unless there's some sort of messed up negative-fractional base where that string makes sense). But you can screw with the logical rules of language and still convey information (even if it's not the best way to go about it), for example by sentencing the reverses, chewzing alternut methuds uv konveighing infermashun, wronging out opposites, jubmling the lettres in yuor sentneces, and just plain upping the down inside the where with a fiddle (scandalous!). You can make up words that still convey information either through familiarity, through conjointification (simplexity itself) or through context, but I feel fanstagingly presitive that it's not so easy in math. In language the reader can get a different meaning out of a sentence than the writer intended and everything's still kosher, but in math it means what it says.

At the end of the day (there's another one of those metaphors), I think that math is better suited for what it does than words are. Even though language can be bent to serve math's purpose, it is fundamentally too slippery to be trusted with the job.

Okay, whew, that was totally unnecessary on my part, but dammit, it was fun.