@StreetlightX Although I hadn't found the time to comment, I had very much enjoyed the OP. I just now finished reading a paper by Sebastjan Vörös and Michel Bitbol -- Enacting Enaction: A Dialectic Between Knowing and Being -- recently published in the journal Constructivist Foundation. (I love the format of this journal which, like Behavioral and Brain Science, publishes target articles followed by several peer commentaries and then a response by the original author(s))

In the authors' response to the commentaries, they produced a quote from Varela that reminded me of your OP.

'As Varela himself pointed out, “ideas appear as movements of historical
networks in which individuals are formed, rather than vice versa,” and tracing a genesis of a given idea is like “making a fold in history where men and ideas live because we are points of accumulations among the social networks in which we live” (Varela 1996b: 408; our emphasis).'

Breifly, on this - Do you know if this is something that is in Daniel Everett's discussion of the Piraha language? — StreetlightX

I have Everett's book in my digital library but I am yet to read it too. The case of the Piraha and their (lack of) counting abilities had been a topic of discussion very many years ago on the now defunct (but still archived) Yahoo 'analytic' discussion group. From what I remember, there is an interpretive disagreement between the psychologist Peter Gordon, who also studied the Piraha (lack of) numerical abilities and Everett, who is more of a Chomskyan linguist and, hence, who is less inclined to take seriously the Whorf-Sapir hypothesis on the essential link between language and cognition. I'm siding more with Gordon's analysis than with Everett's, because of this issue, regarding this specific topic. (So, you might also be interested in digging up Gordon's relevant publications)

Let me add that, here on Earth, we have the Pirahas of the Amazon rainforest who don't have any use for natural numbers, not even the number 1, nor of the existential or universal quantifiers (which Hume, I think, argued were required for grounding the practice of counting).

Rovelli's argument is compelling to me. His view on natural numbers, in particular, meshes rather well with Frege's construal of them as second order functors that are predicated of first order functors (e.g. sortal concepts) where those first order concepts do the prior work of individuating the objects to be counted. If Jupiterians don't have a need for the sorts of first order functors that we make use of to individuate discrete persisting objects, then they wouldn't have much use for our concept of a natural number either. They maybe would have a use for a somewhat isomorphic concept, however, for purposes others than counting discrete persisting material entities. Some language games are structurally (almost) identical to other language games that have very different uses or pragmatic points.

At this stage, a Platonist mathematician might insist that the concept of a natural number latches on the structural invariant shared between two such language games. But I don't think such a defense would work since the variations in the pragmatic points of structurally similar language games being played with symbolic numerals would lead to variations in the way they are structured 'around the edge' as it were. They might not be axiomatized quite in the same way nor be projectable in the same manner to extended domains. (I'd have to conjure up some example that might be more compelling than Kripke's 'quus' example).

Mm, I'm of a know-thy-enemy type as well. You fight cancer by studying it rigorously and prodding it incessantly. The Sophist remains one of my favorite philosophical works. — StreetlightX

Great!

And the Plato I have in mind is more the Plato who valorizes eternity, who rejects becoming, and poses infantile questions. — StreetlightX

Fair enough. I have a tendency, myself, to try to pay attention to the best in the thought and hidden legacy of influential philosophers. This includes Descartes, whose influence I oppose relentlessly, but whose better insights have been aptly advocated by Daniel Robinson (who is nevertheless closely philosophically allied with the arch-Wittgensteinian and arch-anti-Cartesian P.M.S. Hacker!)

I'm not familiar with Goodman's new problem — csalisbury

It's not very new, though. The New Riddle of Induction is the fourth chapter in Goodman's book Fact, Fiction and Forecast, first published in 1955 and adapted from lectures given in 1953. It is related to Kripke's 'quus' alternative rule for addition discussed in his Wittgenstein on Rules and Private Language (1982).

Platonism is philosophical cancer. — StreetlightX

On the other hand, you are saluting the nominalistic proclivities of Sellars, Quine and Davidson, among others. Let me just focus on Sellars for now.

Sellars's nominalism (if we can call it that) operates through Kant, and pragmatizes it. It is especially indebted to Kant for its acknowledgement of the constitutive and ineliminable role of our conceptual powers in shaping up experience (while recognizing the equally indispensable reciprocal role of sensory intuitions, or receptivity, in the constitution of the power of judgement). It is epitomized, within Sellars' own neo-Kantian pragmatism, by its denunciation of the Myth of the (non-conceptual) Given. But it also jettisons the idea that the a priori forms of the understanding can be disclosed by means of pure armchair exercises of the power of the intellect.

So, Sellars's pragmatism is inimical to Plato, in that sense. But it still retains, from Plato (and through Kant), the idea that intellectual reflection can reveal to the intellect its own a priori forms. Those a priori forms, however, are conceived within Sellars's pragmatized neo-Kantianism rather more on the model of Wittgenstein's grammatical remarks, or hinge propositions, or the constitutive laws/rules of embodied, situated and historicized scientific practices, or in the way Strawson and Grice have conceived of "analytic" statements (comparable to Sellars's synthetic a priori statements) in their rejoinder to Quine's 1951 Two Dogmas of Empiricism (in In Defense of a Dogma, 1956).

I believe that an "event" is completely artificial, in the sense that "an event" only exists according to how it is individuated by the mind which individuates it. So the problem you refer to here is a function of this artificiality of any referred to event. It is a matter of removing something form its context, as if it could be an individual thing without being part of a larger whole. — Metaphysician Undercover

I agree with this. I am indeed stressing the fact that the event doesn't exist -- or can't be thought of, or referred to, as the sort of event that it is -- apart from its relational properties. And paramount among those constitutive relational properties are some of the intentional features of the 'mind' who is individuating the event, in accordance with her practical and/or theoretical interests, and embodied capacities.

(...) And when we see things this way we have to ask are any events really accidental or coincidental. it might just be a function of how they are individuated and removed from context, that makes them appear this way.

Yes, indeed. The context may be lost owing to a tendency to attempt reducing it to a description of its material constituent processes that abstracts away from the relevant relational and functional properties of the event (including constitutive relations to the interests and powers of the inquirer). But the very same reductionist tendency can lead one to assume that whenever a 'composite' event appears to be a mere accident there ought to be an underlying cause of its occurrence expressible in terms of the sufficient causal conditions of the constituent material processes purportedly making up this 'event'. Such causes may be wholly irrelevant to the explanation of the occurrence of the composite 'event', suitable described as the purported "meeting" of two human beings at a well, for instance.

There is a clear problem with this example, and this is the result of expecting that an event has only one cause. When we allow that events have multiple causes, then each of the two friends have reasons (cause) to be where they are, and these are the causes of their chance meeting. — Metaphysician Undercover

It's true, in a sense, that 'events' have multiple causes. Recent work on the contrastive characters of causation and of explanation highlight this. But what it highlights, and what Aristotelian considerations also highlight, is somewhat obscured by the tendency in modern philosophy to individuate 'events' (and hence, also, effects) in an atomic manner as if they were local occurrences in space and in time that are what they are independently from their causes, or from the character of the agents, and of their powers, that cause them to occur. This modern tendency is encouraged by broadly Humean considerations on causation, and the metaphysical realism of modern reductionist science, and of scientific materialism.

If we don't endorse metaphysical realism, then we must acknowledge that the event consisting in the two acquaintances meeting at the well can't be identified merely with 'what happens there and then' quite appart from our interest in the non-accidental features of this event that we have specifically picked up as at topic of inquiry. Hence, the event consisting in the two individuals' meeting can't be exhaustively decomposed into two separate component events each one consisting in the arrival of one individual at the well at that specific time. The obvious trouble with this attempted decomposition is that a complete causal explanation of each one of the 'component events' might do nothing to explain the non-accidental nature of the meeting, in the case where this meeting indeed wouldn't be accidental. In the case where it is, then, one might acknowledge, following Aristotle, that the 'event' is purely an accident and doesn't have a cause under that description (that is, viewed as a meeting).

So the event, the chance meeting, is caused, but it has multiple causes which must all come together.

Well, either it's a chance encounter or it's a non-accidental meeting. Only in the later case might a cause be found that is constitutive of the event being a meeting (maybe willed by a third individual, or probabilistically caused by non-accidental features of the surrounding topography, etc.)

When we look for "the cause", in the sense of a single cause, for an event which was caused by multiple factors, we may well conclude that the event has no cause, because there is no such thing as "the cause" of the event, there is a multitude of necessary factors, causes.

Agreed. Those separate causes, though, may explain separately the different features of the so called 'event' without amounting to an explanation why the whole 'event', as such, came together, and hence fail to constitute a cause for it (let alone the cause).

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.

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.

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.