The OP presupposes an utterly impossible entity. It would have ended rather abruptly had it's author noticed this fatal flaw. — creativesoul

Yeah, it was my mistake to link to that paper for sure. I should have just left it at the gif I was taking.

But still, Norton's dome is also its own interesting debate. I'm just saying don't keep mixing the two things up.

As apokrisis has said, the ball effectively vibrates, as its internal molecules move about (Unless the experiment takes place at absolute zero), so it 'pushes' itself, if nothing else does so first. No need even for QM, just Brownian motion is enough to explain it. — Pattern-chaser

Reflecting on things further, we do seem to wind up with the issue that there always needs to be some kind of probabilistic tunnelling process for there to be an actual causal issue. The symmetry of the initial conditions is already broken in the sense that there is both a probabilistic process - some kind of quantum or even classical jitter - and the barrier preventing it breaking through until a threshold is accidentally breached.

So the ball bearing perched on the dome is going to be able to survive any nudge until there is one strong enough to overcome the frictional forces that would oppose the ball moving. Instead of the accelerative nudge tipping the ball bearing, that energy would instead start to heat up the environment via friction.

So again, if we really zero in on the full physics of the thought experiment, we find ourselves being pointed down the path to a fuller thermodynamical conception of this little causal universe. Not by accident, we look to be headed towards Feynman's Brownian ratchet and how that imposes an ultimate physical cut-off on determinism.

The general principle that follows is that we need to view first causes - the spontaneous breakings of symmetries - not as something hot happening to something cold (as in the bump that pushes the ball bearing), but instead as something cold happening to something hot - the fall, as in a generic fall in temperature that suddenly allows a ratchet and pawl to quit hopping about and start turning mechanically in a single deterministic direction.

That explains particle decay. In a hot environment, the particle isn't even stable. It is already melted. But if the environment is cooled, a particle can form. It will lock up a degree of internal instability that has some lingering propensity of overcoming the thermal decay barrier represented by a now cold world. By quantum uncertainty, that barrier will be spontaneously crossed because the particle fluctuated into a higher energy state that wasn't forbidden to it.

So if we want laws of nature that are generic enough to capture the causality of a Big Bang cosmos, this is the direction our causal thinking has to head in.

Classical causality is about something hot happening to something cold. But we need to flip that model on its head. The deeper causal story is about something cold happening to something hot. It is the context that tells the story, not the event. As the temperature drops generically, then localised heat can start to become the new big thing. But only after the temperature has dropped generically.

So the context has to be changed in order for the ball not to fall? eliminating each classical cause is the sisyphean task? Like attempting to end cancer while still having cell division?
Edit: corrected mistake.

The symmetry of the initial conditions is already broken in the sense that there is both a probabilistic process and the barrier preventing it breaking through until a threshold is accidentally breached. — apokrisis

I didn't quite complete this thought. As I am trying to make plain, my own interest here would be in the question of cosmic creation - the causality of the Big Bang as an example of spontaneous symmetry breaking. So the difficulty becomes getting beyond a story - like tunnelling - which already presumes a causally broken situation. We have to get the bit just beyond where there are either the trapped propensity, or the barrier that is trapping it.

Tunnelling is good for explaining why there might be delays in events happening - like particle decays. And the decays - being statistically random - seem good evidence that we are glimpsing a quantumly indeterministic realm beyond. With quantum tunnelling, we can see flashes of the fundamental uncertainty breaking through.

However, the primal story would seem to have to go beyond a trapped propensity and the threshold holding it back for "a time".

So the context has to be changed in order for the ball not to fall? eliminating each classical cause is the sisyphean task? — JupiterJess

I would instead say that to arrive at the classical situation, you would have to keep adding classical constraints. My argument is that indeterminism can never actually be eliminated. It can only be contextually regulated.

So we arrive at classicality as a terminus - the result of adding enough complex restrictions to produce an apparent causal simplicity. We have to remove the jitter, the friction, the heat - all that messy thermodynamic stuff - to arrive at one round object perched motionlessly on top of another round object with now no other object in sight to disturb that ridiculously unstable situation.

Maximum instability is presented as absolute stability. And then somehow this is the causal model of the world that most people want to defend.

But still, Norton's dome is also its own interesting debate. I'm just saying don't keep mixing the two things up. — apokrisis

I understand that you intended to raise issues for causality that are more general than those that arise from the peculiar features of Norton's dome. But I also think the specific issues raised by Norton with respect to this peculiar case are relevant to some features of diachronic/synchronic emergence, the arrow of time, and the metaphysics of causation. Those features intersect with the broader questions you are interested in. Maybe I'll come to discussing some of them in due course. Meanwhile, I apologize for the temporary side-tracking.

For this thought too I would very much appreciate comments. — andrewk

I'll comment later since I'm taking a pause to read Norton's paper.

Hmmm... Sounds eerily similar to Zeno. — creativesoul

There is indeed an analogy to be made with Zeno's dichotomy paradox. When classical mechanics is being portrayed as a picture of the way the world is, in itself, at a fundamental material level, this picture is usually accompanied by a Humean conception of event-event causation (displacing the traditional Aristotelian picture of powerful substance-causation). Furthermore, 'events' are being identified with the 'states' of systems at a instantaneous moment in time. (The state of a system consists in the specification of the positons, momenta and angular momenta of all the particles and rigid masses comprising it). So, on that view, the (event-)cause of an (event-)effect are conceived as two instantaneous states of a system such that the later can be derived from the former in accordance with the dynamical laws of evolution of the system.

So, on that view, the cause of the state of motion (and position) of the ball at a moment in time can be identified with its state of motion at an earlier time. In the case of Norton's dome, if the ball has begun moving exactly at time Ti = 0, and is moving at a determinate positive speed at time T > Ti, then it was already moving at a determinate (and smaller) positive speed at time T2 = T/2. Its state of motion at that earlier time can thus be viewed as the cause of its state of motion at T. And likewise for its state of motion at time T3 = T/4, which can be viewed as the cause of its state of motion at T2. As long as the ball is in motion, there is an earlier cause (indeed, infinitely many causes) of its current state of motion. But those ordered causal chains don't extend in the past beyond Ti = 0. They don't even reach this initial time. So, there is no initial cause of this temporally bounded infinite sequence of events, even though all the events occurring after Ti have a sufficient cause.

Apokrisis: But, who's claiming that the point object is absolutely at rest? Who could say such a thing? All we can say is that the point object is at rest, with respect tp the dome below it. If the point object and dome were in your kitchen, for example, then we know they are moving, because the Earth is spinning through space and orbiting the Sun. So, you are claiming that the particle can move relative to the dome without any forces being applied to either the dome or the particle? Wouldn't that violate the 2nd law of thermodynamics?

I think the discussion moved on, but I wouldn't trust using the higher derivatives of radial displacement here to mathematically isolate a cause for the ball rolling. The radial displacement and higher derivatives are constant over direction - thus regardless of the indeterminism of the initial parameter, the direction the ball falls is also of the same character; undetermined by the model but consistent with its equations.

IE, so even if we specified a starting time for the ball rolling, that's still an incomplete description - we need a start time and a direction.

It is an inertial frame. And I’m not claiming that there is no accelerating force. I argue that the necessary force ought to be considered generic rather than particular. The environment did it. Accidents happen because they can’t be suppressed.

IE, so even if we specified a starting time for the ball rolling, that's still an incomplete description - we need a start time and a direction. — fdrake

The differential equation that constrains the equation of motion, and, in this case, that has been set up to ensure that Newton's second law is obeyed at all times, admits of a multiplicity of solutions. So, it's true that leaving out the direction of the motion that is beginning at the initial time T, such that this initial time is the only one (or the last one) when the particle is at rest, underspecifies the equation of motion. But it doesn't underspecify the "state" of the system at the initial time. Newton's laws of motion are supposed to govern the evolution of material systems on the basis of specifications merely of their "states" at a time, where those states are being fully characterized by the positions and momenta of the material constituents of the system. (The higher order time derivatives of the momenta are irrelevant to the determination of the "state" of a mechanical system, as far as Newton's laws are concerned). So, the fact that the initial state, in conjunction with specification of the forces, and the laws, underspecifies the equation of motion (and hence, also, the future direction of motion), precisely is what makes this system indeterministic (as constrained only by Newton's laws).

It is an inertial frame. And I’m not claiming that there is no accelerating force. I argue that the necessary force ought to be considered generic rather than particular. The environment did it. Accidents happen because they can’t be suppressed. — apokrisis

I am in broad agreement with this. I've finished reading Norton's paper, now. It's very good even though the whole discussion presupposes a broadly Humean conception of causation, and of the laws of nature, that is inimical to me. Nevertheless, if this presupposition is granted (as it can be for the sake of the discussion of the structure of idealized physical theories), Norton offers very good replies to the main attempt by critics to 'specially plead' against the conclusion that his dome provided an example of indeterminism within the strict framework of Newtonian mechanics.

One thing that struck me, though, is that Norton seems to be making an unnecessary concession to his critics while discussing one specific feature of the ideality of his thought experiment. What he is conceding is that the indeterminism that arises from the state where the ball is initially at rest at the apex of the dome only arises at the limit where the peculiar mathematical shape of the some is perfectly realized on an infinitesimal scale, and hence can't be realized in practice owing to the granular structure of real matter.

It rather seems to me that this indeterminism is an emergent feature that is already manifest under imperfect realizations of the dome. Whether or not it is manifested depends on how the ideal limit is being approached. One way to approach it, which seems to be the only way that Norton and his critics consider, is to assume that the ball is being located, at rest, precisely at the apex of the dome, and to realize the shape of the dome ever more precisely in the neighborhood of the apex. Only when the curvature at the apex blows up, will the ball's "excitation" (as Norton call's the spontaneous beginning of the motion from a state of rest) become physically possible.

But there is another way to approach (or approximate) the peculiar indeterministic nature of the dome, and to probe the corresponding bifurcation in phase space that characterizes it). We can stick with a merely approximate realization of the shape of the dome, where the curvature remains finite within a neighborhood of radius R from the apex, and the ball is being initially located (or sent sliding up) in the vicinity of the apex with some error distribution of commensurate size. We can compare, side by side, two experiments where the infinitesimal limit is being approached, one using an hemispherical dome, say, and the other one using Norton's dome. In the first case, under successive iterations of the experiment where the ball is placed (or sent) with an ever narrowing error spread towards the apex, and where the apex is materially shaped ever more closely to an ideal hemispherical shape, the time being spent by the ball in the neighborhood of the apex will tend towards infinity. In the case of Norton's dome, the time will tend towards zero (while the time required to move a fixed distance D away from the apex will remain roughly the same). As we move towards the ideal limit (with an ever smaller error spread, and an ever larger curvature within the narrowing neighborhood), the ball will not only become more sensitive to microscopic disturbances (which it will be both in the hemisphere and in the dome cases) but the cumulative effect of those triggering disturbances, as well as the small errors in initially setting up the ball at the apex, will be continuously amplified from the microscopic realm to the macroscopic realm (in a fixed time) in such a way as to make manifest the bifurcation in phase space as a truly emergent macroscopic phenomenon lacking a counterpart in the microphysical realm.

We can compare, side by side, two experiments where the infinitesimal limit is being approached, one using an hemispherical dome, say, and the other one using Norton's dome. — Pierre-Normand

If things can converge, then they can diverge. In one direction, the ultraviolet catastrophe. In the other, its matching infrared catastrophe.

So in terms of my metaphysical interests here, the dichotomous nature of any ideal limit is not a surprise. It would be a prediction. If you have fluctuations, as you do in quantum physics, then you are always going to be stuck between the two perils of everything adding up to infinity, or everything cancelling to zero.

Now those two perils are mathematically nicely-behaved but also observationally non-physical. The Universe actually exists in a way that suggests a finite cut-off before we can arrive at either two ideal limits to processes of convergence or divergence.

So that was something implicit in the OP. We need to explain finitude. There has to be an emergent scale of fluctuations that becomes too small to make a difference. Or indeed, to big to make a difference.

And here is where I would call on the holism and semiosis of hierarchy theory. In hierarchy theory, small scale fluctuations eventually become just a solid blur - from a middle ground perspective of them. And likewise, large scale fluctuations eventually become so large in spatiotemporal terms that they completely fill the available field of view. Change can no longer be seen as it is change that stretches wider than the visible world itself.

This is the usual contrast between blackholes and de Sitter spaces. Looking in one direction, fluctuations tend to a Planck scale quantum blur. Looking in the other, we encounter the large scale event horizon cut-off imposed by the speed of light.

So yes. There is always a dichotomy in play if there is any action at all. If there is a convergent limit, there is a divergent one to match it. And then tracking the physics of such limits with fluctuations also makes sense. But that then is nudging you towards this kind of hierarchical semiotics, this triadic story of being inside limits because of some kind of finitude-constructing mechanism, some kind of cut-off creating effect.

Again, the mathematical imagination is quick to believe that the infinite and the infinitesimal are in some sense achievable. But I'm thinking no. Finitude must arise somehow in the actually physical universe. And we don't have a lot of good tools for modelling that.

My OP illustrated one form of such a cut-off - the principle of indifference. If instead of having to count every tiniest, most infintesimal, fluctuation or contribution, we simply arrive at the generic point of not being able to suppress such contributions, then this is just such an internalist mechanism. The crucial property is not a sensitivity to the infinitesimal, but simply a loss of an ability to care about everything smaller in any particular sense. There is smaller shit happening just as there is also bigger shit happening in the other direction. It just isn't visible from our middle ground position due to a lack of the means to record that information. The holographic universe story in a nutshell.

In the first case, under successive iterations of the experiment where the ball is placed (or sent) with an ever narrowing error spread towards the apex, and where the apex is materially shaped ever more closely to an ideal hemispherical shape, the time being spent by the ball in the neighborhood of the apex will tend towards infinity. — Pierre-Normand

This may not be right. What I should have said (in the case of the hemispheric dome) is that the acceleration in the vicinity of the apex will be such that the total time from the moment when the ball will exit the shrinking neighborhood and travel to a predefined distance D from from the apex will tend towards infinity.

My OP illustrated one form of such a cut-off - the principle of indifference. If instead of having to count every tiniest, most infintesimal, fluctuation or contribution, we simply arrive at the generic point of not being able to suppress such contributions, then this is just such an internalist mechanism. The crucial property is not a sensitivity to the infinitesimal, but simply a loss of an ability to care about everything smaller in any particular sense. — apokrisis

This particular conclusion is convergent with my own. It seems interesting, to me, that the shape of Norton's dome creates a specific condition of instability such that the ability of the ball to move away from the equilibrium point, and further slide under the impetus of the tangential component of the gravitational force to a finite distance D from the apex in a finite time T, is insensitive to the magnitude of an initial perturbation from equilibrium. This condition of instability is somewhat independent of the condition under which the initial perturbation is enabled to arise (from thermal molecular agitation, or Heisenberg's uncertainty principle being applied to the initial state of the ball, or whatever).

Perhaps the defintion of cause itself needs to be looked at.

Anything could be a cause for the pencil tipping over (generic cause) BUT the actual event is caused by a breath of air or the table shaking (specific cause).

The world is full of causal effectors (fall). No one can deny that. However, to say that because this is true one can't ascribe a specific cause (push) would be foolish. Don't you think?

Yeah, it is not that I’m arguing that regular cause and effect explanations are wrong. They are useful descriptions of how things generally are in a world that has become cold and large, and so is acting like a collection of atomistic objects. But it is how we would think of causality as it applies to bifurcations or symmetry breakings. It is the causality that would apply to events such as the Big Bang.

So toppling pencils and rolling balls just serve as illustrations of the principles. And I’m arguing that while logic says there will always be some triggering cause, it also doesn’t make much sense to attribute anything much to that particular event - single it out as something uniquely significant and useful to know. The real cause of the change is the fact that triggering events couldn’t have been avoided. That generic fact of nature is what would be useful to know about and understand fully.

Specifically it's that no force (0 vector) is applied as an initial condition while the ball is at the apex that leaves room for the indeterminism.

Specifically it's that no force (0 vector) is applied as an initial condition while the ball is at the apex that leaves room for the indeterminism. — fdrake

Well, the fact that there is no force while the ball is initially at rest on the apex of Norton's dome enables it to remain stationary during some arbitrary length of time T. This corresponds to one possible trajectory in phase space, among many. But that would also be true of a ball resting on the apex of a sphere, or paraboloid. In those cases, though, the evolution would be deterministic since there would be no possibility for the ball ever to move off center any finite distance in a finite amount of time. That's not so in the case of Norton's dome. The ball can "fall off" (start moving away from the apex) at any time consistently with Newton's second law being obeyed at all times.

See the point. Perhaps I'm too poorly attuned to physics to see much of a distinction between a time symmetry and a radial one.

See the point. Perhaps I'm too poorly attuned to physics to see much of a distinction between a time symmetry and a radial one. — fdrake

I don't understand this comment. The dynamics, in this case, is indeterministic (branching out at the point in phase space representing the particle at rest at the apex) but it is also time symmetrical. The same branching out occurs in phase space towards the past.

Doesn't really matter what point I'm making for the purposes of the discussion, seeing as it's moved on. All I'm saying is that mapping t->t-T is a symmetry of the laws of motion here, but so is rotating the force vector; any path that the ball could take down the object would follow Newton's laws at all time, even though the arbitrary start time and arbitrary falling direction are unspecified. The major difference between the two in my reading is that the problem is 'set up' to be radially symmetric and so we're primed to think of the problem as of a single dimension (the radial parameter), but the time symmetry falls out of the equations and is surprising.

It is an inertial frame. And I’m not claiming that there is no accelerating force. I argue that the necessary force ought to be considered generic rather than particular. The environment did it. Accidents happen because they can’t be suppressed. — apokrisis

That's a cop-out. You're just saying that if we don't have the capacity to determine the particular causes involved, we can just say it's a generic cause, "the environment did it". But that's an untruth, because it implies that the particular aspects which are the true causes, are acting together as a unified whole, called "the environment", when the claim of a concerted effort is unjustified. So your claim that "the environment" is an acting agent, is nonsense without some principles whereby "the environment" can be conceived as an acting, unified whole.

Doesn't really matter what point I'm making for the purposes of the discussion, seeing as it's moved on. — fdrake

It's fine to pick up again a sub-thread when something has been overlooked.

The major difference between the two in my reading is that the problem is 'set up' to be radially symmetric and so we're primed to think of the problem as of a single dimension (the radial parameter), but the time symmetry falls out of the equations and is surprising.

What is surprising? The indeterminism is uprising, but the time symmetry is expected since the laws of motion are time-symmetrical.

What is surprising? The indeterminism is uprising, but the time symmetry is expected since the laws of motion are time-symmetrical. — Pierre-Normand

I think we meant different things by indeterminism. In the paper's sense of 'a single past can be followed by many futures', the translational time symmetry of the non-zero solution is what facilitates that conclusion. If the ball decides to fall in a given direction, its behaviour is determined at every point on that path by the equations of motion (after redefining t-T=0).

Nevertheless, this collapses the issue to a one spatial-dimension one time-dimension problem- we have a radial direction and the time parameter, and the equations of motion are defined in terms of a radius which is a function of time. So it's pretty clear that the dynamics is radially symmetric.

What I was missing is that when we collapse down to a vertical cross section of the dome, then remove half of it (going from something that looks like /\ to something that looks like /), the one dimensional version of the problem exhibits the time symmetry.

What the radial symmetry highlights is 'the same future can be held by many pasts', where a future includes 'choosing' a direction to fall in, the time symmetry (specifically talking about the mapping t->t-T which shows up in the solution) highlights 'the same past can be held by many futures'. I was mixing up the two in my head.

So your claim that "the environment" is an acting agent, is nonsense without some principles whereby "the environment" can be conceived as an acting, unified whole. — Metaphysician Undercover

Apocrisis was talking about a generic force rather than a generic cause, or generic agent. I think is makes sense to speak of a general background condition that isn't happily conceived of as a cause of the events that they enable to occur (randomly, at some frequency). Causes ought to be explanatory. So, there may be events that are purely accidental and, hence, don't have a cause at all although they may be expected to arise with some definite probabilistic frequencies. Radioactive decay may be such an example. Consider also Aristotle's discussion of two friends accidentally meeting at a well. Even though each friend was caused to get there at that time (because she wanted to get water at that time, say), there need not be any cause for them to have both been there at the same time. Their meeting is a purely uncaused accident, although some background condition, such as there being only one well in the neighborhood, may have made it more likely.

I think we meant different things by indeterminism. In the paper's sense of 'a single past can be followed by many futures', the translational time symmetry of the non-zero solution is what facilitates that conclusion. If the ball decides to fall in a given direction, its behaviour is determined at every point on that path by the equations of motion (after redefining t-T=0). — fdrake

Yes, because the path of the system in phase space only is branching out at the point representing the particle being at rest at the apex. When the particle is already has acquired some momentum, some distance away from the apex, then its trajectory is fully determined in both time direction up to the point where it gets to (or came from) the bifurcation point (that is, to the apex, at rest).

By 'determinism', as predicated of a material system and its laws, I only mean that this system's state at a time (and the laws) uniquely determines its state at any other time. That a single past (either a single past instantaneous state, or a single past historical trajectory in phase space) leads to a unique future is a corollary.

Apokrisis: If it's an inertial frame, then how would that alone cause the point object to move? I don't see how it would.

You queried the ball being at absolute rest. We can presume for the sake of the thought experiment it is in an inertial frame. So that would be a reason the ball shouldn’t move. That source of possible acceleration has been removed for sake of argument.

The issue here is spontaneous symmetry breaking. So you’ve got to start with some plausible state of symmetry.