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.

The first and simplest reason is that we are able to discuss our intentional acts. If these acts were not involved in a causal chain leading to physical acts of speech and writing, we would be unable to discuss them. One could claim that intentional acts are physical, but doing so not only begs the question, it equivocates on the meaning of "physical" which refers to what is objective, rather than what is subjective. (See my several discussions of the Fundamental Abstraction on this forum, including the precis in my last post in this thread.) Further, if the causes within Kim's enclosure include any being we can discuss, the principle makes no meaningful claim, for it excludes nothing. — Dfpolis

This (and the rest of your post) is a very good response. I'll comment more fully shortly, within a day or two, hopefully.

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.

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.

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).

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.

However, primordial desire is nebulous, vague. For instance we feel thirst, a generic desire. This initial thirst may then be specifically satisfied with either water, coke, beer, pepsi, etc. Do you think this process from generic desires to specific fulfillment can accommodate some form of freedom of will? — TheMadFool

Yes, because the way in which we are making our decisions isn't merely a process of instrumental specification from generic or blind desires that we are passively being straddled with. The strength of our various desires and inclinations can both potentially help, or hinder, in various ways, the actualization our capacity for practical judgement.

We oftentimes act against our stronger raw inclinations when we judge that they ought not to be given voice in our practical deliberation in light of the rational or moral demands of the specific situation. It is a metaphysical prejudice to conclude that, whenever this occurs, and some of our raw inclinations are being silenced, it is because some other (and equally blind in point of rationality) raw inclination to do the contrary won out over them. The outcome of practical judgement (and hence what we decide to do) often is the outcome of our having concluded, on good rational and/or moral ground, that it is the desirable thing to do.

If our normal inclinations, and our characters, are in good order, then we are more inclined to do the right thing effortlessly. In that case, what is the right thing to do tends to align with what appears to be the most desirable thing to do. If they aren't in good order, then, doing the right thing may require more effort, stronger external incitatives, and we are more likely to fail to make a correct practical judgement.

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.

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).

Which I believe is the solution to the paradox. God can create the stone, but doesn't. — Michael

I rather like this purported solution, not because I am especially interested in saving the notion of an omnipotent god, but because it is a useful reminder of the general distinction between an agent (who may be an ordinary human being) lacking a power and her being in contingent conditions entailing that she will not exercise it. Failures to recognize this distinction often leads to some variations on the modal fallacy.

What stone? — Michael

Yes, it's true that if Her power to create such a stone remains unactualized, then, in that case, Her merely having this power doesn't entail a contradiction.

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.

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.

As I see no reason to give Kim his principle of causal closure, and many reasons to reject it, I am not bothered by the paradoxes that trouble physicalists. — Dfpolis

It is fine not to be bothered by problems that exercise proponents of dubious -isms (such as physicalism). I am not overly bothered by them either. But it's even better to provide a rationale as to why one is entitled not to be bothered by their specific objections to our non-physicalist views.

Incidentally, some quite smart non-physicalists (or anti-Humeans) about causation, such as Ruth Groff, counter Kim's causal exclusion argument by rejecting the principle of the physical closure of the physical. That seems to me to be a blunder. This principle is fine, although limited in scope. (Michel Bitbol argued that it is consistent with strong emergence, and the existence of systems that exhibit downward-causation). The faulty premise in Kim's argument, on my view, rather is the principle of the nomological character of causation (also famously endorsed by Donald Davidson).

How does this follow? — Michael

Because God thereby lacks the power to lift the stone.

So, to save the PSR all we need to do is say that the agent is the sufficient cause of his or her choice. One can deny this, but not on the ground of the PSR. One simply has to decide if agents can determine their own choices or not. If they can, they are sufficient to the task of making the choice. If they cannot, there is no free will. Either way, the PSR is unviolated. — Dfpolis

I rather agree with that, since I endorse a variety of agent-causation (and rational causation) myself. Many libertarian philosophers, and some compatibilist philosophers, endorse some sort of agent-causal view of the source and explanation of free human actions. The main challenge that is being presented to the compatibilist versions is the problem of dealing with causal overdetermination, or so called arguments from causal exclusion.

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.

And this is solely as a result of the shape of the dome? — creativesoul

Yes. Although Norton's dome isn't the only shape that allows this, many shapes, such as a spherical dome, or a paraboloid, wouldn't allow it since it would take an infinite amount of time for a perfectly balanced ball to "fall off" from the apex. (Or, equivalently, in a time-reversed scenario, it would take an infinite amount of time for a ball sent sliding up to come to rest at the apex).

I'm not seeing the need for an initial perturbation either. The system of molecular decay can change the net force causing the bearing to begin being in motion all the while never appealing to a force outside the system, aside from gravity. The physical structure of molecules changes over time. This change alone is enough to account for the movement of the bearing after sufficient time without introducing another force. — creativesoul

So, you are envisioning a spontaneous change in the microscopic shape of the ball. This would break the initial symmetry and move the ball's center of gravity away from directly above the apex of the dome. Fair enough. But it still doesn't address the initial problem regarding Newton's laws: namely, that they allow for the ball to start moving towards some arbitrary radial direction even in the case where there is no such initial departure from symmetry from any cause whatsoever.

"Why would God, Who can do anything, bother doing something so incredibly stupid and pointless?" — Michael1981

What if God IS the stone? — gloaming

I had very much the same thought. I was thinking that God (or whoever thought about herself that she was God) would kick herself for having performed such a dumb and pointless act of creation. And then she would pause to contemplate the almighty stone that's now defeating her powers, and call it her God.

Doesn't the net force change alongside with molecular decay? — creativesoul

Not sure what molecular decay is. But if you're thinking of thermal molecular motion, yes. It would be a source of fluctuation of the net force, and then could be appealed to as the cause of the fall. But that doesn't address the original conceptual puzzle since, according to Newton's laws of motion, the "fall" (or initiation of the movement) of the ball from Norton's dome is physically possible even if there is no initial perturbation at all. It occurs even in the idealized case where the ball and the dome are ideal solids, perfectly smooth and perfectly rigid, in a total vacuum.

Is it? Gravity is never zero. Accompanied by a significant enough amount of molecular decay of either the bearing or the dome, and it will fall...

Right? — creativesoul

That's right, although the force at issue, here, is the net force. For sure, you can allege that there ought to be some random force from thermal molecular motion that kicks the ball out of balance. But the puzzle remains since the equation of motion that accounts for the ball "falling away" from the apex towards some arbitrary radial direction remains valid and strictly consistent with Newton's laws of motion even when there is no such perturbative force being posited.

I suppose my simple mind is struggling to see the relevant difference between being pushed or falling...

I mean, when taking gravity into consideration... — creativesoul

The source of the puzzle regarding causality is that the cause of the initial departure from a state of rest is usually (or intuitively) being identified with the existence of the net force being exerted on the mass at the moment of departure from rest. But, in this case, this force is exactly zero. The net force only starts to grow after the ball has already begun to move away from the apex. So, what was the cause of the beginning of its movement? That's the conceptual puzzle.

Newtonian gravity then... — creativesoul

Yes, from a far away planet, with the variable attraction from the dome itself being neglected.

Where's it being accounted for here? — creativesoul

The presence of a uniform and constant field of gravity g is assumed in the setup of the problem. It's the source of the weight, mg, of the ball bearing. It's thus, indirectly, the source of the radial (horizontal) component of the reaction force exerted by the surface on the ball. This reaction force vector is constrained (when summed up with the weight vector) to maintaining the acceleration vector along the tangent to the slope. The radial component of the reaction force is proportional to the sine of the slope at the point of contact with the ball, and hence null when the ball is located at the apex. Those assumptions, together with Newton's second law, allow the derivation of the equations of motion of the ball. Since there is a plurality of such physically possible equations of motion, the system is indeterministic.

In that case the path that involves the ball having always been at the top of the dome will not be consistent, under the 2nd law, with the current state of the cannon or the cue stick (eg heat, momentum) Also, the momentum of the dome will be different in both cases, as the ball transfers its horizontal momentum to the dome (3rd law) as it climbs to the top. — andrewk

I had always assumed that the equation of motion of the ball (away from the potential bifurcation point) was as given in Norton's paper. For this solution to be exact, the potential motion of the dome is neglected. It this had not been the case, the force that maintains the dome up against gravity would have to be specified, as well as the dome's mass, moment of inertia, etc. Those complications would seem to be quite beside the issue being discussed in the paper (or in this thread). The surface of the dome is better conceived as a strict restriction on the range of motion of the ball, providing a reaction force just as strong as needed to keep the sphere along this mathematically defined surface.

In that case it is impossible for the ball to roll up the dome, because there is nothing to give it the necessary upward impulse. So if we observe it sitting at the top of the dome, the only possible history is that it has always been there. This can all be derived from the 2nd law alone. The 1st law is not needed. — andrewk

I don't see any reason why a physical system can't have some of its components initially in a state of motion. Velocity is relative to an inertial referential frame anyway. If it was initially at rest in some inertial frame, then it was initially moving relative to another inertial frame. And the laws of classical mechanics are Galilean-invariant. Les us assume that the ball has been shot up with a canon, or hit with a cue stick, if you like. The laws of motion govern its state of motion, thereafter, from the time after it was shot (or hit) right up until the time when it reaches the top of the dome.

I am assuming that the dynamical equations, together with whatever supplementary laws might be posited, which govern the system determine the set of the physically possible histories of the system. I am assuming that the system consists in the dome, the ball bearing, the ambient gravitational field, and nothing else. The physically possible histories are being represented by trajectories in phase space. The system is deemed deterministic (in the time-asymmetrical sense) if the set of all the physically possible trajectories in phase space present no bifurcations. A backward looking bifurcation at T would consist in a case where two or more partial histories of the system before T would be consistent (in respect of physical possibility) with the same unique partial history after T. Your law seems to allow for this possibility.

Also, since the laws of classical mechanics are symmetrical with respect to time, it's deemed to be impossible to tell if a movie depicting a segment of the history of a mechanical system is running forwards or backwards. But if your law were governing a system, and a movie was shown of a ball rolling up a Norton dome and coming to rest at the top, then it would be possible to tell for sure that the movie is being run forwards since the time-reversal of this scenario would be physically impossible.

Yes. Say it reaches the apex at time t2. Then there is a path compatible with the 2nd law in which it remains there for the period [t2,t2+h) for any h>0. So that path must be what happens rather than a path in which it continues down the other side. — andrewk

Very well. In that case your law doesn't describe a deterministic system under the time-symmetrical definition of determinism. It allows bifurcations of paths in phase space towards the past. But you had meant to strengthen your law in order precisely to remove such backward looking bifurcations.

My law prohibits the ball rolling down (a solution not in U) because there exists a solution in U, ie in which it does not roll down, and the law requires that a solution in U be taken in preference to a solution outside it. — andrewk

But what happens in the case where the ball is being sent rolling up towards the apex with the requisite speed? Consider the situation at any time T, when it already had been rolling up for awhile, and hasn't reached the apex yet. In this case, we fully know the pattern of motion of the ball in the temporal vicinity of t = T. Is your law mandating that the ball will stand still indefinitely after it has reached the apex, or is it rather silent regarding what will happen next?

What do you think about this? — prothero

Before commenting, let me also point to this short video by Daniel Dennett, discussing this issue, and with which I am in broad agreement.

I'm not sure I understand the question. The above law would mandate that a ball sitting stationary exactly on top of the dome would not roll down. The second law does not mandate that. — andrewk

The case where the ball stays at rest on the apex during a finite time interval merely constitutes a subset of the set of the trajectories in phase space that are aimed at the potential bifurcation point. Also included into that set are all the trajectories whereby the ball is rolling up the surface towards the apex with just enough speed to reach it with zero velocity. In that case, both the second law, and your expanded law (if I understand it correctly) are silent regarding what happens next.

What do you think about this? — prothero

Here is a link to the source.

This states that the solution must have locally constant velocity both looking backwards and forwards if that is compatible with laws 2&3, else locally constant future velocity if compatible with 2&3, else locally constant past velocity if compatible with laws 2&3. Otherwise the law is silent. — andrewk

If the expanded law is allowed to remain silent for the specific set of states of the system whereby its trajectory in phase space aims precisely at the potential bifurcation point, how is that law any different from an indeterministic law that allows for any of the bifurcations that are merely consistent with the second law?

Gravity. — creativesoul

What about it?

An expansion that prevents bifurcation could be:

'Where there is more than one future movement pattern of an object that is compatible with the 2nd and 3rd laws and the conditions in place at time t, and one or more of those patterns involves the object's velocity remaining constant for the period [t,t+h) for some h>0, the pattern that occurs will be one of those latter patterns'.

Very wordy, I know, but it has to be in order to deal with nonphysical cases like this without just disappearing into Law 2. Note also that it leaves open the possibility that there may still be bifurcations possible with this law - not the one discussed in the paper, which would be ruled out, but other ones in even more pathological cases. I suspect it may be possible to prove there cannot be, but that's just a hunch. — andrewk

Notice, though, that this proposed expansion only shaves off 'branching outs' from bifurcation point towards the future. Determinism is commonly defined as a property of a system whereby the state of this system at a time, in conjunction with the dynamical laws governing its evolution, uniquely determine its state at any other time (either past or future from this point in time). This is a time-symmetrical definition of determinism. Under that definition, if the laws are such that there remains bifurcation points in phase space that are branching out towards the past, then the system still is indeterministic. The system past or present states uniquely determine its future; but its future or present states don't always uniquely determine its past.

(I deleted my post because it wasn't well thought out)

I got a bit lost here. Newton's third law is that for every action there is an equal and opposite reaction. I can't see how that law is relevant to the questions being examined in this scenario. Can you outline what you had in mind here? — andrewk

Sorry. I got confused. (Can you imagine that I have an undergraduate degree in mathematical physics?) I was thinking of Newton's second law (F = dp/dt) and wrongly labelled it Newton's third law. I don't remember making this mistake before. Maybe I can blame the emotional impact of the Kavanaugh saga.

As I stated above, the luck objection seems to me different from what I meant and I personally don't find the actual answer all that interesting. — Benkei

You were implying that whoever defends an incompatibilist version of free will (such that it requires indeterminism) ought to acknowledge that what they really believes, then, is that "everything they decide is totally random as a result". How is it a problem, in your view, that free actions would be totally random? Of course, I agree that it would be a huge problem. We couldn't be able to claim authorship of our "free" actions, or responsibility for them, if they were merely the random outcomes of indeterministic processes. In that case, whether we would be acting well or badly, in accordance with our wishes or against them, would be a matter of chance rather than an expression of our will and character. But that is precisely what the luck objection to crude versions of libertarian free will amounts to. If your objection is completely different from that, then I have no idea what your objection is.

Most everyone when they think luck and change are relevant. It stems from an inability for most to properly understand QM theories, which, admittedly, I only understand at a limited conceptual level but enough to spot the mistake. Too many think QM theory is an example of ontological indeterminism. It isn't. — Benkei

The question whether QM is fundamentally indeterministic at a fundamental level isn't really relevant to appraising responses to the luck objection to libertarian free will. Those responses avert to facts about human psychology and the processes of decision making that are quite independent of whatever physicists will ultimately disclose about the fundamental theories of particle physics or how the disputes regarding the interpretations of quantum mechanics will be resolved.

Incidentally, I think Kane requires that the laws of physics be fundamentally indeterministic for his account of free will to work, but I disagree with his endorsement of this requirement for genuine freedom of action, and it isn't relevant to his response to the luck objection anyway. The luck objection also can be directed to theories that appeal to complexity, mere epistemic ignorance, and/or features of deterministic chaos.