[politician]

Please excuse my imposition, as I am a humble programmer who spends his days adding and subtracting 1, and not a mathematician.

Yet, this discussion of the confusion and potential confusion of misinterpreting notation strikes me as something that has long (well, in the sense of programming) been solved in my area with type systems and syntax highlighting.

Do mathematicians not have these tools?

[Sharlin]

There is no syntax highlighting on the blackboard.

Math notation is not designed. It has haphazardly evolved over centuries. It is not rigorous even though math itself is (attempts to be) rigorous, it is a language as imperfect as its users. But it does its job well enough.

[ABeeSea]

They do! It’s called abstract algebra and it’s very similar to type theory in a lot of ways.

But to get to the rigorous mathematician definition of manipulating dx and dy, it requires a large amount of the machinery from abstract algebra that’s hard to quickly absorb or explain.