1) When we say the ball is at rest, and let’s grant it can be, doesn’t that mean velocity, acceleration, jerk, etc are all 0? And thus it will never move? There’s a single solution governing the ball if we say it’s truly at rest.
2) We can’t determine when a space invader will suddenly appear to us, but that isn’t some fundamental indeterminism, that’s just limits to the speed of light.
Quantum mechanics (potentially) has a radically different indeterminism than these in some of the interpretations (Copenhagen, GRW), where even some FTL and infinitely precise oracle couldn’t predict. Its fundamental randomness (in some interpretations).
I'm not sure if this will make you feel any better, but there's an interesting mathematical corner case at work with Norton's Dome that's responsible for the breakdown in the intuition you're expressing in (1).
You could formalize this intuition as the statement that, if I'm trying to describe a function f(t) and I know (a) the value of f and its derivative at t=0 along with (b) a second-order differential equation that f has to satisfy, this should be enough to nail down the entire function.
A big theorem, which many people just call something like "existence and uniqueness of solutions of ordinary differential equations", says that in most ordinary situations this is indeed true, and basically for the reason you probably intuitively think: you can imagine using the differential equation to make tiny "updates" to the value of f to move a little bit forward in time, and take the limit as the size of your time increment goes to zero. (You can read more about it in this somewhat technical Wikipedia article: https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_t....)
But there is a condition on the theorem which limits its scope: the right side of the differential equation has to be something called "Lipschitz continuous". The vast majority of differential equations that appear in Newtonian physics satisfy this condition, but the equation you get in the Norton's Dome example doesn't, and this is what's responsible for the lack of uniqueness in the solution. It turns out that there are many different trajectories for the particle that satisfy both the initial condition and the differential equation.
What relevance does this have to the actual universe? Personally, I think very little; it's a fact about a model of physics, not a fact about the actual universe. There are all sorts of reasons why you can't literally build Norton's Dome: matter is not actually continuous because it's made of atoms, and classical physics isn't an exact model of the universe anyway. But it's interesting to see that a feature of Newtonian physics that we usually take for granted isn't actually always true.
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.
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.
In classical mechanics it’s quite common to say something is “at rest” and then instantaneously acted upon by a force, so just because something is at rest doesn’t mean it will never move. It usually just means dr/dt = 0 in a particular coordinate system.
In fact[1] “at rest” doesn’t even mean the thing isn’t moving, accelerating or even that the rate of acceleration isn’t increasing. It just means than an inertial frame exists from which you can conveniently assume that these things aren’t happening, and Newton’s first law tells us that indeed such a frame always exists.
I’m not sure I buy the example of the Norton’s Dome though. I get the technical argument from continuity but it seems weak to me. Lots of differential equations (eg the wave equation) have a trivial solution where u(x,t)=0 for all x and t - that doesn’t actually translate to us not knowing whether or not something will move in the real world. As you have said it will fundamentally always move or always not move given a particular set of initial conditions - we may just be lacking enough information to say which of those cases may hold, but that doesn’t make it non-determanistic - just it appears so to us.
[1] I got this from Kleppner and Kolenkow, which is an amazing book if you’re interested in Classical Mechanics
1) When we say the ball is at rest, and let’s grant it can be, doesn’t that mean velocity, acceleration, jerk, etc are all 0? And thus it will never move? There’s a single solution governing the ball if we say it’s truly at rest.
2) We can’t determine when a space invader will suddenly appear to us, but that isn’t some fundamental indeterminism, that’s just limits to the speed of light.
Quantum mechanics (potentially) has a radically different indeterminism than these in some of the interpretations (Copenhagen, GRW), where even some FTL and infinitely precise oracle couldn’t predict. Its fundamental randomness (in some interpretations).