Its not necessarily either. Each are factors that could have had a part or whole in the displacement of the ball bearing. There are reason(s) why the ball bearing was displaced, and with enough sophistication in the instruments of measure (whatever they may be) these reasons can be revealed and your question answered.
It neednt be any more complicated than that, and I dont see a reason to have a presumption of an underlying universal explanation.

What's interesting about the dome is that the ball's starting from rest, and, after a finite time, rolling in an arbitrary direction, is a valid solution to Newton's laws of motion. So, in that case, there would be no need for as much as an infinitesimal nudge in order to set the ball in motion. It could start moving (consistently with Newton's laws) in any direction at a moment in time when the net force being applied on it is was null. Newton's laws of motion also leave it open when it would start moving, after having been at rest for an arbitrary length of time.

I think quantum wave collapses are best viewed as events that lack a contrastive cause, just like this idealized Newtonian example exemplifies. But this idea of events that happen (in some sort of symmetry breaking way) without being caused to so happen also may be the best construal of events that are determined by microscopic (symmetry breaking) causes where those causes simply are irrelevant to the emergent macroscopic dynamics that we are interested in.

I like this better than your ball bearing:

Over 48 feet tall and 40 tons, the wind-carved rock balances precariously on a pedestal only 3 feet by 17 inches. (The color may have been manipulated in this photo)

It doesn't have to fall, but it probably will. Aside from human action, numerous independent factors could cause the rock to fall. Wind and earthquake come to mind, heating and cooling, particularly the freeze/thaw cycle--provided enough moisture was available at the right time in the right fissure. Very subtle influences, like the tidal pull of the moon, subliminal vibrations in the earth, shock waves from a meteor strike, lightning strikes, etc. could contribute to the fall.

Were we to carefully monitor wind, quakes, temperature, precipitation, the motion of the moon, and any other environmental factor we could measure, we could quite possibly not identify a single cause, because all these factors might be operating at the same time. Another thing, all the environmental forces operating on the rock are cumulative, and have been operating for a long time. (Don't we have to include the wind of millennia past as a causal factor?)

I cast my vote for "it fell" rather than "it was pushed".

If the net force acting on the ball bearing was zero, why would the ball move? On the other hand, can a ball sit on another ball of approximately the same size without moving? Or must it eventually move as a chance event? Which of Newton's laws is at play--a force pushing against it?

Its not necessarily either. ... There are reason(s) why the ball bearing was displaced, and with enough sophistication in the instruments of measure (whatever they may be) these reasons can be revealed and your question answered. — DingoJones

But here you seem to both say the right answer doesn't really matter, and yet also the right answer is that there will be some particular triggering cause that accounts for the "why". In the end, it will be a push that did it. And that is what counts most for your general view of the world.

In practical terms, you might be right. It seems we could always measure nature more closely and put our finger on some individual environmental disturbance as the guilty party.

But the essence of my question was metaphysical - what do we really want to believe about the truth of nature? So something is at stake. We ought to come down on one side or the other.

Could you imagine ceasing to care about the individual pushes and instead accepting that the generic impossibility of eliminating all disturbances is this deep truth?

I am guessing you would resist that alternative view strongly. The question becomes why? With what good justification?

If the net force acting on the ball bearing was zero, why would the ball move? On the other hand, can a ball sit on another ball without moving? Or must it eventually move as a chance event? — Bitter Crank

That's because of a specific queer mathematical property of the shape of the dome and how the system interacts with the vertical force of gravity. At the initial time when the ball gets rolling, its speed is zero. But the first derivative of its speed (its acceleration) also is zero. Hence, the force, at the instant in time when it is initially at rest, also is zero. This is why no force at all is required to set it in motion.

Another way to view it is to imagine the time reversal of the process where the ball is being sent rolling up the dome with just enough speed so that it will end up at rest at the apex, after a finite time. Thereafter -- and this is unmysterious -- it may remain at rest for an arbitrary period of time. If this is a valid solution to Newton's equations, then, so is the time reversal of this process where it remains at rest for some time and then "spontaneously" starts rolling (with an initial instantaneous null acceleration).

What's interesting about the dome is that the ball's starting from rest, and, after a finite time, rolling in an arbitrary direction, is a valid solution to Newton's laws of motion. — Pierre-Normand

How so? If the ball has mass, it has inertia. A push is required to set it moving.

Of course, it the ball is already "in motion", then that acceleration already exists. But that then becomes my alternative story of the physical impossibility of eliminating such accelerations. A physical mass has to have some kind of internal thermal jitter and so - even when placed perfectly at rest - it is going to just throw itself over the edge and roll.

So we can't rely on the formalisms if the formalisms simply leave out the crucial physical facts.

How so? If the ball has mass, it has inertia. A push is required to set it moving. — apokrisis

Not so, as I've already explained to Bitter. The shape of the dome is such that, as the ball is getting infinitesimally close to the apex, the second derivatives of its horizontal motion tends towards zero; and hence, also, the horizontal component of the force.

I cast my vote for "it fell" rather than "it was pushed". For one thing, all the environmental forces operating on the rock are cumulative, and have been operating for a long time. — Bitter Crank

Yes, but this would be a case of innumerable accidents as you say. The "push" advocates would still reply that there would be one last event that finally did the trick. So it could have been the unlucky tourist that leant on it, or that lightning bolt which was the straw that broke the back of the camel.

I want to focus on the most extreme example where absolutely anything would be enough to be that straw. And so we can't really blame some particular straw anymore.

Could you imagine ceasing to care about the individual pushes and instead accepting that the generic impossibility of eliminating all disturbances is this deep truth? — apokrisis

I cant think of a reason that it would be impossible.

I am guessing you would resist that alternative view strongly. The question becomes why? With what good justification? — apokrisis

You guessed wrong unfortunately. Sorry.
Until today I hadnt given it much thought. If it turns out to in fact be impossible, i cant imagine why I would strongly resist.

Not so, as I've already explained to Bitter. The shape of the dome is such that, as the ball is getting infinitesimally close to the apex, the second derivatives of its horizontal motion tends towards zero; and hence, also, the horizontal component of the force. — Pierre-Normand

But that relies on the ball starting on a slope, not on the flat. It is only infinitesimally close to the apex and so also infinitesimally inclined towards rolling down in some direction. The forces acting upon it are already sufficiently off-kilter.

So again, my essential point remains. The ball can't be placed with exact precision at the apex. Our modelling incorporates that infinitesimal "swerve" as something that can't be eliminated.

As a way of thinking about what causes the ball to start to roll, the answer becomes we couldn't prevent that because any placement on the apex had to involve infinitesimal error.

Another way to view it is to imagine the time reversal of the process where the ball is being sent rolling up the dome with just enough speed so that it will end up at rest at the apex, after a finite time. Thereafter -- and this is unmysterious -- it may remain at rest for an arbitrary period of time. If this is a valid solution to Newton's equations, then, so is the time reversal of this process where it remains at rest for some time and then "spontaneously" starts rolling (with an initial instantaneous null acceleration). — Pierre-Normand

But now if you time reverse the story, you still only can arrive infinitesimally close to the apex, not actually perched exactly on it. So if the ball seems at rest, that is a mistake. It is only ever decelerating and then beginning to accelerate again. Inertia and friction might slow that transition in the real world. It might get stuck a while. But in the model, which presumes frictionless action and an actual perfect balance of forces, with no infinitesimal errors regarding its location at the apex to break the symmetry in advance, traditional thinking would seek a triggering push. And it is that framing of the situation which is the OPs target.

I cant think of a reason that it would be impossible. — DingoJones

But that is the easy presumption that is under attack here. Most people probably do find no reason to even question the possibility of being able to eliminate every possible source of perturbation in some physical system.

The habit is to think of a world that is essentially clean and simple. A blank slate. A void. And then you start populating this world with its little pushes and pulls, its atomistic play of events.

But I say why isn't the inverse of that a closer match to observer reality? Why don't we start with a world already chock full of pushes and pulls, then see if we can imagine subtracting them all completely away.

Quantum mechanics tells us we can't in fact achieve a void. There are always going to be infinitesimal or virtual fluctuations.

So in fact we have a well-motivated reason for taking the opposite view - the one that presumes the impossibility of suppressing all physical disturbances. And so - as a metaphysics - that would flip the usual comfortable view on its head.

Where before there was no reason to think an absence of fluctuation was impossible, now there is no reason to think it might be possible. Hence the idea of a triggering cause loses its previously fundamental-seeming metaphysical status. The interesting condition is the one where such causes have become so suppressed that all their particularity has been lost and there is only now the generic concept of "the inevitability of spontaneous outcomes". Perturbation itself becomes a primal feature of "the void".

But now if you time reverse the story, you still only can arrive infinitesimally close to the apex, not actually perched exactly on it. — apokrisis

No so. What you are saying would be true for any number of smooth convex domes, including spherical domes. But the particular shape being discussed in the paper that you linked to in the OP, referred to as "the dome" and defined radially as the surface with height h = -(2/3g) r^(3/2), where g is the vertical acceleration of gravity, is such that the equation of motion for a small spherical mass being perched at rest at the apex at t = 0 admits of two different sorts solutions. The first solution describes the mass remaining at rest at all times. The second class of solutions have the mass moving away from the apex with radial positon r(t) = (1/144) (t – T)^4, where T is an arbitrary time and t <= T; and r(t) = 0 at any time t before T.

So, in the time reversal scenario, when the ball is sent sliding up towards the apex with just the right speed, it doesn't slow down asymptotically as a function of time. It slows down to rest in a finite time and then (consistently with Newton's laws) remains at rest for an arbitrary amount of time at the apex before sliding back down in the same, or another arbitrary, radial direction.

If "the dome" didn't have this very specific shape, then, the equation of motion of the sphere instantaneously at rest at the apex, at a time, would (in most cases) be deterministic. It would necessarily remain at rest at all times. In the case of "the dome", indeterministic outcomes are consistent with Newton's laws of motion even in the idealized case where there is no initial disturbance at all away form an initial state of instantaneous rest! That's what makes "the dome" (Norton's) such an interesting shape.

But that is the easy presumption that is under attack here. Most people probably do find no reason to even question the possibility of being able to eliminate every possible source of perturbation in some physical system. — apokrisis

It is certainly not a presumption on my part. I in fact cannot think of a reason why it would be impossible.
You seem like you want me to accept simething that you feel is possible as something that is true. It takes more than something to just be possible in order for me to accept that it is true. This doesnt seem like a particularly high or difficult standard.
Im open minded to all possibilities, but I need a good reason to accept the possibility is actually true and I told you that I cannot think of a good reason to accept that it is impossible to eliminate all disturbances. Do you have one?

I'm not sure I understand your point. If the particle is perfectly balanced on top of the dome, then there it shall remain until some net force moves it. You didn't specify what that net force would be, or how it could be interpreted as a fall versus a push. The object is in an unstable position thermodynamically speaking, because any movement will cause the potential energy to drop, at least if we only consider movements along the dome as opposed to lifting it off the dome. However, that has nothing to do with the nature of the net force that would come along and cause the particle to move from an equilibrium position.

I told you that I cannot think of a good reason to accept that it is impossible to eliminate all disturbances. Do you have one? — DingoJones

Quantum mechanics.

I'm not sure I understand your point. If the particle is perfectly balanced on top of the dome, then there it shall remain until some net force moves it. — LD Saunders

That's fairly intuitive, right? Yet, the mathematical analysis of the case contradicts this intuition. Look at the equation of motion of the ball bearing. At any time when it is at the apex, both the net force being applied on it, and its acceleration, are zero, as they should be, or else this equation would not be a solution to Newton's laws of motion. And yet, also, the ball bearing is radially being displaced away from the apex a finite distance after only a finite time. It needs not be the case that, as our intuition about causality seems to demand, that the cause (viz., the net force) begins operating before the effect (viz., the acceleration) starts being manifested. In this idealized Newtonian case, they both come into existence together (i.e. at any time t > T) without there being some other physical cause that accounts for the moment in time T when the ball starts moving away from the apex (with only an initial null acceleration at t=T, and at all t <= T).

I see now what you mean. It was a mistake to link to that paper as Norton’s dome is a special case. Do you buy his story? I don’t get where his sudden spontaneous motion could come from. And why it is a solution permitted by his particular curve. Need to read more yet. This helped - https://blog.gruffdavies.com/tag/the-dome/

But anyway, I just meant to talk about the standard example as an illustration of spontaneous symmetry breaking.

Do you buy his story? — apokrisis

Not really, because it assumes metaphysical realism: the idea that there might conceivably be an external God's eye view of the world that amounts to a complete description of it, including its laws and causal relations. I think Kant has shown this not to a be possible account of our (or of any conceivable) empirical world. But this case remains an instructive mathematical possibility. I'll say a bit more later on.

My question was about a poised situation and how we think about its symmetry breaking. It seems a problem that we need a first cause to actually pop up out of nowhere and set the ball in motion. But that problem can be removed by imagining a world where nothing is ever absolutely at rest. The second metaphysical picture is less conventional, but better fits the facts, I would suggest.

Quantum mechanics — apokrisis

I have never heard that QM shows us the impossibility of eliminating all disturbances, I was under the impression that QM is still struggling to lock down exactly whats happening at the level it focuses in.
Ok, so it sounds like you already knew the answer to your question. At the QM level the ball doesnt get pushed, at the level we most interact with it does.

I would say you are missing the point. I am discussing how we might view the metaphysics of accidents - spontaneous or random events.

On one view there are no accidents as every event will turn out to have some particular cause. Look close enough and you will find the nudge that actually did the trick of toppling the perfectly balanced ball bearing or pencil. So the accident becomes in fact micro determined. It only looks an accident while we are ignorant of the fine detail.

I want to contrast that usual view with its opposite. The story can be turned around by laying the stress on the other facts. In the end, it was the impossibility of eliminating all sources of environmental disturbance that was the cause of the toppling. Yes, there was some individual nudge that did it. But a nudge of some kind was also absolutely inevitable. We thus have no good reason to point a finger at some particular nudge as if it were significant in its own right. It was nothing special. If it had failed to act, the ball bearing would have still fallen just as surely because an unlimited number of other nudges were there to step in and do the same job.

So the odds of the accident happening would be 100% from that point of view. We could say that the ball bearing simply has the propensity to fall. It doesn’t need a particular push. It is generically set up to respond to a perturbation. Identifying some individual nudge as the actual culprit adds no real information to an account of the causality.

Clearly this line of reasoning then takes you into the interesting metaphysical questions about how the Big Bang could happen out of “quantum nothingness”, or what causes an unstable particle to decay.

You are just choosing to look at it generically, there is still a fact of the matter as to why the ball rolled. You say it “adds no real information”, but it certainly does. It would add the information of what was actually happening. This is not changed by your infinite other possible nudges. Sure, they were potential nudges, but not the ones that happened.
It is simply not accurate to say the ball has a propensity to fall, or that our view of things bottoms out at some generic level and we cant or could never be specific. You are just choosing to be less accurate, choosing to not get too specific.
So you want this discussion to be unmoored from logic, sense and causality. Thats fine, its some kind of thought experiment, but as you pointed out I am missing the point.
In what way is it useful? You mentioned something from nothing and unstables particles...but we have ways of determing those things, those are questions physicists answer with precisely things like quantum mechanics and mathmatics. Why is it that you think these tools are insufficient?
What is the advantage of operating from the basis you are suggesting?

As a way of thinking about what causes the ball to start to roll, the answer becomes we couldn't prevent that because any placement on the apex had to involve infinitesimal error. — apokrisis

So the act of placement is really a push, because placement cannot be precise. And, if the act of placement could be precise enough, or the surface flat enough, then a push would be needed. Therefore it's always a push.

the answer becomes we couldn't prevent that because any placement on the apex had to involve infinitesimal error. — apokrisis

That is my view. The ball fell because when it was released by whatever was holding it on the apex, its centre of mass was not exactly above the point of contact with the dome, so it started falling.

One can foresee an objection that says 'But what if the CoM was exactly above the point of contact when released?' The response to this is:

1. The probability of that being the case is zero, as the horizontal coordinates of the CoM would have to match two exact real numbers.

2. In addition, the CoM would have to have both horizontal components of its velocity exactly zero upon release. In practice no object can have an exactly stationary CoM, because of the same probability argument.

That is my view. The ball fell because when it was released by whatever was holding it on the apex, its centre of mass was not exactly above the point of contact with the dome, so it started falling. — andrewk

Have you looked at the paper linked to in the OP, though? The case has been specially contrived such that even if the ball is placed exactly at the apex, and with no initial velocity at all, then, consistently with Newton's laws of motion, it could either remain stationary or start falling towards an arbitrary radial direction with a distance from the apex: r(t) = (1/144) (t – T)^4 (where T is the time when it would spontaneously start moving in the absence of any net force at that time).

This would be missing the point of the thought experiment. We can take it for granted the ball bearing is actually at rest on the apex in perfectly balanced fashion and then just focus on the event that would be needed to topple it.

I agree that in practice this precision would be impossible, but that isn’t the point being made. The point is about how we like to assign causality to particular triggering events, but if a triggering event is almost sure to happen, then the particular loses its hallowed explanatory status. The cause can be treated as completely generic.

Making the generic cause to be about the impossibility of placing a ball with arbitrary accuracy on an apex is both another way of saying the same thing, but not quite as strong a version as focusing on the impossibility of eliminating triggering fluctuations.

Making the generic cause to be about the impossibility of placing a ball with arbitrary accuracy on an apex is both another way of saying the same thing, but not quite as strong a version as focusing on the impossibility of eliminating triggering fluctuations. — apokrisis

I rather like the idea of a generic cause of the symmetry breaking mechanism. The generic cause, in this case, is the (practically inobservable) fluctuation background that serves as the explanation of the emergence of the indeterministic law that governs the (practically) observable events of symmetry breaking. There being such a generic cause doesn't entail that there is a law on account of which a contrastive explanation can be given as to why the ball fell in one rather than another direction, or fell immediately rather than at a slightly later time. There may be no such law, and no such contrastive cause. (There may however be a emergent law specifying the half-life of a ball's staying poised before starting to fall).

(There may be a emergent law specifying the half-life of a ball's staying poised before starting to fall). — Pierre-Normand

Yeah. I think this is the next interesting case to dig into.

Another slant on the OP would be the more standard example of a phase transition where a system is in a state of correlations of infinite length. It is right on the cusp of a global symmetry breaking and any perturbation at all will push it across the threshold. At that point you can throw away the need to identify the triggering disturbance. It was always going to happen somewhere.

All this is about how to view spontaneous symmetry breakings. The formalism usually reduce the description of the physical system to its perfectly poised symmetry, leaving the issue of a triggering cause - the source of the spontaneous change - outside the model. This leads to some rather arbitrary metaphysical conclusions by those only prepared to consider what the formalism is prepared to cover.

So that is the issue. How can we talk about the lighting of the blue touchpaper, the first cause, in a way that it is part of the model and not some ad hoc extra? If fluctuations are treated as generic, then that would answer the question. The causal problem is flipped as what would now become surprising is if some critical instability could be prevented from breaking.

In short, why does existence exist? It no longer requires a particular triggering cause that broke a prior quiescent nothingness. Now the generic issue is how could wild fluctuations ever get suppressed? The causal story becomes about the physical mechanism that could limit the possibility of fluctuations to the point where stable order finally starts to reign.