[meroes]

Wow thank you. You’re right there’s likely some hidden assumptions that I had taken for granted that a unique solution is relying on when solving DE’s. I will have to read up to make this more clear mathematically, but at least mathematically I think you’ve answered my concern about 1). Now whether things break down when we model a potentially discrete world with continuous math, that’s for another day like you said. And what it means for something “at rest” to start moving if all its position derivatives are 0. But those might be more philosophical.

[nicf]

You're welcome! There's actually one more point that I thought of after sending that reply, which since you mentioned it again I should maybe flag.

Totally apart from physics, it may seem intuitively plausible that if you have a function f and (a) all f's derivatives exist everywhere, and (b) f(0), f'(0), f''(0), etc. are all zero, then f must be the zero function. This is actually also not true! For a counterexample, you can look at this article on bump functions: https://en.wikipedia.org/wiki/Bump_function.