But this is true of every level of mathematics. See humorous proofs that 2=3 amounting to manipulations like 0a = 0b, cut the zeros, a=b. This is no fault of "n

But this is true of every level of mathematics. See humorous proofs that 2=3 amounting to manipulations like 0a = 0b, cut the zeros, a=b. This is no fault of "nonrigorous elementary algebra", it's a matter of remembering all notations are abbreviations and you have to know how to manipulate them. You can get away with dy and dx if you just think "I'm not writing integral signs because they're annoying" and remember the rules of working with integrals. This is how stochastic differential equations work -- they're not even differential equations because Brownian motion is not differentiable, they're just notation.