The central point about ball throwing is the argument that it involves implicitly approximating the solutions to differential equations.

That's a fact, eh?

So, when a mathematician is approximating solutions to differential equations, their brain is functioning the same as if they were catching a ball?

To catch a ball in midair, requires the same hand eye coordination as moving a drumstick to hit a drum at the right time.

But what I'm talking about, is not how the hand moves to catch the ball or hit the drum or find the right fret.

To catch a ball, the hand moves to catch a ball. The ball is the input, the catch the result.

What I'm talking about happens before hand-eye coordination ever begins.

To play music, you are mentally throwing multiple balls all the time, and then catching them at the right time. The part of music that engages in the brain is not catching the ball, it is mentally placing the ball on a discrete grid.

Again, if you disagree, I'll take your word for it.