Why do you think that? You asked a question about the future and I replied about the future. I didn't say anything about the past in that latest reply because the question was not about that.

The definition above prevents bifurcation in the past by means of the subset W. As I said earlier, one needs to be more cautious when talking about bifurcation in the past. Generally, for any given observation, no matter how ordinary, there are multiple past scenarios that could have led to it (example of ball at bottom of inverted dome), and distinguishing between them needs information about more particles than just the ball and the dome. But that's not bifurcation in the sense we've been discussing. It's just a lack of complete information.

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.

I am assuming that the system consists in the dome, the ball bearing, the ambient gravitational field, and nothing else. — Pierre-Normand

In that case it is impossible for the ball to have rolled up the dome, because there is nothing in the system that could have given 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 be derived from the 2nd law alone. The 1st law is not needed.

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.

Les us assume that the ball has been shot up with a canon, or hit with a cue stick, if you like. — Pierre-Normand

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.

The second point has a more general application than the first. If we truncate our histories before we get back to the firing of the cannon, so that it starts with the ball already in motion, the momentum of the dome will differ when the ball is at the top between the case where it was always there and the case where it rolled up.

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.

Having reflected a bit, I don't think I'm motivated to defend Newton's laws against suggestions that they don't always allow unambiguous projection into the past (retrojection?). I have always felt rather dubious about that idea, and would be not at all upset to find that it cannot hold in all situations.

I don't know why I've been arguing in its defence. I suppose I just got caught up in the momentum, and the challenge of coming up with arguments against a well-directed set of challenges.

I'm thinking of reverting to a version of the first law that only removes ambiguity (bifurcations) going forward, not backward, as that is what my intuitive feel for Newton's Laws is.

But I'll sleep on it first, in case I change my mind again.

There are always forces acting on the ball bearing such that - the initial condition is false. Forces like gravity, electromagnetism, heat, noise etc, are acting on the ball bearing all the time - such that it was never really "balanced." That's human perception. In physics, there is no stationary other than at absolute zero.

Gravity.
— creativesoul

What about it? — Pierre-Normand

Where's it being accounted for here?

I suppose I just got caught up in the momentum, and the challenge of coming up with arguments against a well-directed set of challenges — andrewk

Gotta love it...

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.

Newtonian gravity then...

Newtonian gravity then... — creativesoul

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

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

I mean, when taking gravity into consideration...

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.

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?

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.

Doesn't the net force change alongside molecular decay?

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. — Pierre-Normand

This part in particular. The sine of the slope changes with molecular decay(in the 'right' places), right?

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.

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 solve the conceptual issue since, according to Newton's laws of motion, the "fall" (or initiation of the movement) is possible even if there is no initial perturbation at all. — Pierre-Normand

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.

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.

...it still doesn't address the initial problem regarding Newton's laws: namely, that they allow the ball to start moving even in the case where there is no such initial departure from symmetry. — Pierre-Normand

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

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).

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). — Pierre-Normand

Hmmm... Sounds eerily similar to Zeno. It may be that the solution lies in what's being neglected by the problem itself(how it's being framed is the problem). Molecular decay disallows perfect spheres and perfect domes...

It may be that the solution lies in what's being neglected by the problem itself — creativesoul

Two different questions are being confused here.

The OP was not intended to be about Norton's dome and its claims of Newtonian indeterminism due to a latent jounce concealed in the initial conditions. The OP was about how we would think about an initiating cause when it comes to spontaneous symmetry breaking. The usual natural inclination is to finger one individual perturbation - the straw that broke the camel's back. But the alternative view is to point to the general impossibility of reaching the Platonic perfection modelled by a Newtonian set-up.

In that other view, Newtonianism is only arrived at via the suppression of environmental perturbations. And so, even if perturbations can be more or less completely suppressed - thus allowing Newtonian determinism to be a useful general description of an actual material system - they can also never be absolutely suppressed. Hence - generically - the environment still fluctuates, meaning there will always be a straw to break the camel's back.

"Molecular decay" would be just one more example of this generic inability to suppress all background fluctuations. Just the same as thermal jitter in the ball and dome. And if we get down to the quantum level, there just is always a probability of the ball quantum tunnelling across the threshold, moving sufficiently off-centre due to uncertainty.

In the real world, there would also be some residual frictional forces holding the ball bearing in place. The surfaces in contact would have some microscopic degree of roughness. So in reality, the whole frictionless dome set-up is unphysical.

@Pierre-Normand I've thought some more about why I have been defending The First Law's ability to prevent bifurcation when looking backwards in time. It looks like I started in response to your post quoted below.

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. — Pierre-Normand

I have come to the tentative realisation that I don't have any intuitive sense that a model of the world needs to be satisfy that criterion. Key contributors to this feeling are:

• it seems entirely natural to me that there might have been more than one path by which we got to the situation we are now in. This contrasts with my feeling that a simple model like Newton's is more aesthetically pleasing without multiple possible future paths.
• I do not feel intuitively that tracing back into the past should be as easy as projecting into the future
• my sense of the purpose and meaning of Newton's Laws is that they are purely future directed. The observation that, in all physically possible cases, they still hold when we change the sign of the time coordinate, was made later by others, and I am not aware of any important physical theorems that rely on that observation.

With that in mind I have returned to favouring my original reformulation of Newton's First Law:

Where there is more than one future movement pattern (locus) 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 loci 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 loci. — andrewk

This is not just an artificial construction. It is my best attempt at stating formally what I believe intuitively to be the case for a Newtonian model.

It is carefully formulated so as to still allow movement where there is a time-varying external force on the particle, that is zero at time t when the particle is stationary. In that case, if the force starts to increase at time t (eg according to formula F(t') = t' - t), the object will commence movement at time t because it would contradict the 2nd law if it did not do so. But in the case of the dome there is no time-varying external force. The external force varies by position, not time.

I would be interested in comments on this position.

And another thing

Another way to avoid bifurcations, both future and past, might be to replace the 1st law by a new law saying that the relationship of any force to time must be analytic. Analytic functions are expressible as a power series, which implies, but is stronger than, the condition of being smooth. I have a strong intuitive sense that all force functions are analytic.

With this constraint, the dome could not be constructed, because its surface is not smooth, so it would require application of a non-smooth force to make it into that shape. I also doubt whether the ball could be positioned perfectly atop the dome by application of an analytic force.

For this thought too I would very much appreciate comments.

(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). — Pierre-Normand

The metaphysical reality of time reversal scenario's would also be in question here. If classical reality is actually the product of maximally constrained fluctuations - spontaneity is only ever suppressed - then time reversal can only be a locally effective story, not the generic metaphysical story.

Yes, I can drop your Ming vase on the floor and you will be able to gather up all its shards to glue it back together. Initial conditions can be recovered if no information is either created or lost in the time evolution of the event.

But thermodynamics would appear to say that is not the generic case, certainly at the Cosmic scale on which we are trying to write the laws of nature. So especially if we grant that the development of physical complexity erases past information - because hierarchical organisation acts downward to simplify the parts of which it is composed - then time irreversibility becomes the generic condition.

David Layzer makes this cosmological case - http://www.informationphilosopher.com/solutions/scientists/layzer/

The larger point I am course working towards is that when it comes to the big question - why does anything exist? - we mostly start from completely the wrong metaphysical place. The ball perched for all time on top of a dome is just an example to how far our most familiar models of causality are from the physical reality.

Layzer's approach shows how classicality exists as literally borrowed time. The Big Bang cosmos had to go through its phase change - gravitating mass had to condense out of the relativistic thermal fireball - to create "Newtonian" degrees of freedom. You suddenly had particles that could have a location because they could move at less than light speed and so inhabit a realm where there was emergently a space-like causal separation between events.

So when speculating about the beginnings of everything, we have built up a stock of false Newtonian intuitions. Newtonianism only describes a Cosmos that has got cold and large enough to have undergone a complete phase transition - one that takes it from a quantum description to a classical one. And if we try to time reverse that, how to do we cross the divide given it involves a massive loss/creation of information at the transition point? (A loss and a creation, depending on whether you are tracking the negentropic order created, or the entropic disorder lost.)

The quantum fireball was of course its own further crossing of a transition with its own symmetry-breaking creation story. But we can't say more about that until the lingering Newtonian determinism is completely dispensed with.

It may be that the solution lies in what's being neglected by the problem itself
— creativesoul

Two different questions are being confused here.

The OP was not intended to be about Norton's dome and its claims of Newtonian indeterminism due to a latent jounce concealed in the initial conditions. The OP was about how we would think about an initiating cause when it comes to spontaneous symmetry breaking. — apokrisis

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

Another way to avoid bifurcations, both future and past, might be to replace the 1st law by a new law saying that the relationship of any force to time must be analytic. — andrewk

There is no need to worry about trying to fix Newton's laws if you just accept them as effective descriptions. They are how reality would work in the limit. But reality can only approach such a limit.

It is defending Newtonianism as a literal truth that creates the problem.

And the metaphysical insight for philosophy of maths is that a third category beyond the discrete and the continuous needs to be recognised - that of the vague.

Limits in metaphysics can only be defined logically in dichotomous or complementary terms. The discrete and the continuous are ideals - opposing limitations on possibility defined by their formal reciprocality. That then leaves raw possibility - ie: vagueness or indeterminacy - sitting in the middle as the stuff that can approach one or other limit with arbitrary precision.