I don't understand why anyone would think this is even the slightest bit weird. The situation described is dynamically unstable. The object remaining at rest forever is only possible in the Platonic ideal: zero friction, infinitely rigid materials, no thermal motion, no external perturbations. As soon as the state diverges from the Platonic ideal in any way, positive feedback will amplify that divergence. So the prediction for the Platonic ideal is exactly what one would expect: the object might stay still, or it might start to move at any time without any apparent cause.
Of course, this situation can never actually be observed because even if you could somehow construct it (and good luck with that), you couldn't actually look at it because merely shining a light on the object would give it a nudge.
The whole point is that Newtonian mechanics doesn't uniquely predict the motion of the ideal ball on that ideal shape. The ball could stay there forever, but it could also start moving down along the shape at any point in time - both are valid possibilities in the idealized model. This is the unintuitive part.
The ball could stay there forever, but it could also start moving down along the shape at any point in time
Only if the fourth derivative spontaneously changes from zero to nonzero. It doesn't seem any more surprising than the conditions f(0)=f'(0)=f''(0)=0 not uniquely determining f(x) for all x.
The condition imposed by the construction of the problem and the laws of motion is that f''(t) = sqrt(t), and that f''(t) = 0 => F(t) = 0. The function given as an example in the article, f(t) = {(1/144) (t-T)^4, t >= T | 0, t < T}, obeys both laws, just as much as f(t) = 0 does.
I'm not sure what the fourth derivative has to do with this argument.
Well, yeah, but my point is that it shouldn't be surprising. If you think about it, if it is even possible to bring a particle to rest for a finite time in the Platonic ideal then that plus time reversal necessarily entails non-determinism. So non-determinism should be no more surprising than the fact that it is actually possible to bring a particle to rest for a finite time.
I think the only reason this example surprises people is that everyone just assumes that bringing a particle to rest is possible/easy without really thinking through what this would actually require in the Platonic ideal. It's actually very challenging to stop things from moving without friction.
> then that plus time reversal necessarily entails non-determinism.
No. Generally speaking in the "Platonic ideal", we assume that if we reversed time, the particle would move from rest in a deterministic way, that there's only one solution for its movement. The surprising part here is that there are multiple valid solutions. Which takes a very-specially designed "dome" to demonstrate -- which is very surprising indeed.
> It's actually very challenging to stop things from moving without friction.
No it's not. All it takes is a billiard ball in motion hitting another billiard ball at rest. The first billiard ball will now have stopped moving. Which is kind of the canonical example of how we expect things to be deterministically reversible.
Yes, you're right, and this occurred to me after I posted but while I was in the shower so I couldn't correct it :-) I should have said "if it is possible to bring all of the constituent particles in a system to rest for a non-zero time..."
You're just reframing the article as though it's obvious and then claiming that it is obvious. The problem is - the article already does just that.
Further, you have changed your stance from one of "of course it moves from some noise in the environment" to one more closely resembling the article's main points.
You basically aren't making sense. Your initial abuse of "platonic ideals" is the sort of thing that gives philosophical arguments a bad reputation.
This is just one example of non-uniqueness condition. There are first-order differential equations that don't have unique solution for a given initial condition.
Maybe I'm just stupid but I found it very surprising that such a shape exists (i.e. stopping a ball at the top in finite time) even in a platonic ideal.
Sure, but more in the “I knocked the plank off the desk by accident while measuring it, by hitting it really hard with the ruler” sense, and less in the quantum physics is weird sense.
Consider how many pop-sci singularitans think their god in a box could predict the future through mere calculation. Many people do have a naive clockwork-universe model of causality.
Of course, this situation can never actually be observed because even if you could somehow construct it (and good luck with that), you couldn't actually look at it because merely shining a light on the object would give it a nudge.