system
Given the following post, respond as if you're commenting on Reddit:
human
Does dx "really mean" something in formal mathematics? or is it just a form of heurism for "rapid integration".

The book I am currently studying says that dx=dy for example is meaningless, but we still do it for fast calculation and for the sake of programming, but at the end of the day, it's still the same as doing the rigorous way, for example, manipulating so that you can apply quotient rule. The prior is way taxing however. Am I understanding it correctly? Also, I want to know your opinion on this kind of pedagogy; Is it harmful, all this "simple tricks"?
gpt
Yes. No. Maybe.

Yes: when Liebniz invented the notation, they represented infinitesimals.

No: when others like Cauchy developed real analysis without the need for infinitesimals, the dx was no longer meaningful as in Liebniz's notation, but the notation is still used for historical reasons. And as we know it's okay to manipulate in this way.

Maybe: more recently, dx makes a comeback as a "differential form" (a 1-form, in fact). But this is more advanced that the average calculus user needs to know.