Alas, I’ve seen a lot of experienced people who don’t really see the difference between “function $f$” and “function $f(x)$”. I would even say it’s a bit frightening.

“$\dot{x} = f(x)$” seems weird whatever meaning one puts into $x$. This $x$ could be a function, but traditional math notation then implies that $f(x)$ is a single vector from a vector field $f$, picked in point $x$. But if $x$ is point, then type-checking for the LHS fails: derivative of a point is meaningless (zero at best).

The notation “$\dot{x} = f \circ x$” (or “$D(x) = f \circ x$”) looks more consistent from the geometrical point of view, becuse it emphasises that $x\colon \mathbb{R} \to \mathbb{R}^n$ is a curve in the space $\mathbb{R}^n$, and both $f \circ x$ and $\dot{x}$ are curves in the corresponding tangent space $T \mathbb{R}^n$.

Hence, I could just rewrite the ODE in the form . No quantifiers required. It is not a big change, but at least it type-checks.

However I have some criticism. I think that even the “moderate” use of quantifiers could be almost as harmful as the lousy notation.

You could (and did) get rid of by introducing the abstract in place of . When you have a statement about functions you don’t have to care about their parameters at all. And that is good, because this is one of those abstractions that make math powerful.

And I think that the quantifier is unnecessary too. Without it, you’d got the equation. It is not a definition, and it is not an assignment. It is just another mathematical object, like a function, or a vector, or a vector field, and it is valuable by itself.

ODE is not a statement that “something exists”. There could be only a trivial solution to ODE, or no solution at all. The absence of solutions doesn’t necessarily make this particular ODE uninteresting.