The point is about how we like to assign causality to particular triggering events, but if a triggering event is almost sure to happen, then the particular loses its hallowed explanatory status. — apokrisis

OK. I don't find assigning causality a productive exercise, so I'll leave the field to those that do.

OK. I don't find assigning causality a productive exercise, so I'll leave the field to those that do. — andrewk

That's strange. I would have thought assigning causality to relevant agents, events, or states of affairs, is quite productive (pragmatically) whenever something occurs, we don't know why it occurred, but we would be interested in seeing to it that such an event occurs again, or in preventing it from reoccurring, for instance.

Did you have any thoughts about a clearer account of Norton’s set up?

Comments suggest to me that the cause of the sudden spontaneous motion is a concealed fourth derivative jounce. So it is like the ball is set down on the apex in the middle of just being about to snap. The maths allow this because the maths is blind to the concealed action. The maths only concerns itself with the first and second derivatives being zero to say the ball is at rest. It can’t pick up a singularity, as when the ball would be briefly motionless apex of its trajectory when tossed in the air.

That would seem a rather trivial get out though. And I don’t yet see why this particular curvature is so special. Any explanation for why this curve is somehow poised in a way that allows for the claimed indeterminism?

This is another good little commentary I was looking at - https://theconfused.me/blog/is-newtons-first-law-merely-a-special-case-of-the-second/

I don't often read external sources. But I trust your recommendation, so I had a look. It's an interesting setup. What is not immediately apparent in the presentation is that the displacement vector of the object is not smooth (infinitely differentiable), and to allow non-smooth functions destroys the notion of causality because they will have unexplained jumps in higher derivatives.

The location function given has discontinuous Jounce (aka Snap), which is the fourth derivative of displacement (second derivative of acceleration) wrt time.
The Jerk (3rd deriv of displacement) is non-differentiable.

Jerk is (t-T)/6 for t>T and 0 otherwise.
Jounce is 1/6 for t>T and 0 otherwise.

So I don't think this case does what it at first seems to do, which is to generate breaking symmetry out of nothing. The breaking symmetry is always there in the discontinuous Jounce, which we have simply assumed. The plausible physical solution is that which has smooth displacement and all derivatives are always zero - ie symmetry doesn't break.

It does raise an interesting question for me though. There are functions called 'bump functions' that are smooth and yet are zero everywhere except on a compact interval. An example is f(x) = exp(-1/x) for x>0 and f(x)= 0 for x<=0. This is zero until it gets to x=0 and then it suddenly starts increasing 'for no reason', and asymptotically heads towards 1 from below..

I wonder if one could construct a dome of a shape that made the displacement vector a bump function. That would be mysterious because the function is infinitely-differentiable, and hence doesn't implicitly already deny causality.

The way one excludes bump functions, when one wants to do so, is to restrict ourselves to analytic functions, which can be expressed locally as a power series. There are no analytic bump functions.

I have always found bump functions very mysterious, and like to ponder them when I have nothing else to do. They are a truly beautiful case of nothing happening, then something suddenly starts to happen, without a discontinuity anywhere to be found.

I will muse over whether one could construct a dome shape that would make the displacement vector a bump function.

Any explanation for why this curve is somehow poised in a way that allows for the claimed indeterminism? — apokrisis

The curve is constructed so that the displacement function is a constant multiple of (t-T)^4 for t>=T. The same would work for a displacement function proportional to (t-T)^n for any n>=4. In that case the non-differentiability won't appear until the (n-1)th derivative of the displacement function. So as long as n>=4 the nondifferentiability will be out of sight and out of mind.

So I don't think this case does what it at first seems to do, which is to generate breaking symmetry out of nothing. The breaking symmetry is always there in the discontinuous Jounce, which we have simply assumed. The plausible physical solution is that which has smooth displacement and all derivatives are always zero - ie symmetry doesn't break. — andrewk

That's interesting. I hadn't thought about the implications of that. But I am unsure about the implications that it has for symmetry breaking understood as a nondeterministic bifurcation in phase space. Imagine the ball bearing being sent sliding up the slope of this surface with just the right velocity such that it ends up being at rest at the apex. Granted, there will occur a discontinuity in the jounce at the time when the ball comes to rest. How are this process, or its time reversal, not both plausible physical possibilities (in any arbitrary radial direction)? And if they all are plausible physical possibilities, then it would appear that there is a bifurcation in the phase space of this system. (Granted, in a 'real world' implementation, it's still vanishingly improbable that the ball would ever be placed precisely at the apex, and laid there completely at rest.)

I suppose if we send it sliding up with exactly the correct initial velocity, and no touching it after we release it, all higher derivatives of displacement will be zero once it is on its way up. It follows that it will stop at the top rather than continuing down the other side, because it will have zero velocity and zero horizontal force on it at that time.

The higher derivatives would have to be nonzero for the ball to pass the cime and go down the other side. If it stops there, there are no discontinuities because Jounce and Jerk were already zero on the way up.

Comments suggest to me that the cause of the sudden spontaneous motion is a concealed fourth derivative jounce. So it is like the ball is set down on the apex in the middle of just being about to snap. — apokrisis

Well, what is 'concealed' (or worth paying attention to) is the discontinuity in the jounce. But I am usure that this discontinuity can be construed as the cause of the spontaneous motion of the ball after the time interval where it has been at rest and was subject to no net force. What may be instructive, and maybe dispel some of the weirdness of the case, is to consider the limiting case of the effect of a small horizontal momentum P being transferred to the ball bearing (such as the impact from a single molecule of air) where this initial momentum transfer P tends towards zero. The integrated time that it takes for the sphere to thereafter fall along the surface in the radial direction where it has initially been pushed will tend towards a finite time T as P tends towards zero. This is why, in a sense, we may say that the symmetry breaking event requires no initial perturbation at all.

I suppose if we send it sliding up with exactly the correct initial velocity, and no touching it after we release it, all higher derivatives of displacement will be zero once it is on its way up. It follows that it will stop at the top rather than continuing down the other side, because it will have zero velocity and zero horizontal force on it at that time. — andrewk

Yes, I wasn't picturing the ball to keep on going to the other side. Rather, it reaches a bifurcation point in phase space. It is equally physically possible (with undefined probabilities) that it will stay at rest, or immediately start moving towards an arbitrary direction.

The higher derivatives would have to be nonzero for the ball to pass the cime and go down the other side. If it stops there, there are no discontinuities because Jounce and Jerk were already zero on the way up.

The jounce wasn't zero on the way up. It was constant and equal to 1/6. It will drop to zero only if the ball thereafter remains at rest at the apex (which is only one physical possibility among others).

The jounce wasn't zero on the way up. It was constant and equal to 1/6. — Pierre-Normand

True.

I realised that what I wrote above, as powers of (t-T), is not actually the Jounce, Jerk etc but rather the higher derivatives of the radial coordinate r. It's a scalar rather than a vector. I doubt that helps but I need to get my perspective right before commenting further. At present, I'm finding the case of sliding the ball up more perplexing than it starting at the top.

I'm also wondering whether we need to incorporate the dome into the system we are analysing rather than treating it as a supplier of external forces, in order to make sense of the scenario.

Also - I just realised that the dome is not smooth. a geodesic over the top will have a nondifferentiable first derivative, because the second derivative of r^(3/2) blows up at 0+. That can explain the discontinuity in what we've been calling Jounce. That Jounce is a function of the dome's shape, and the shape is not smooth at the top, so it is reasonable for there to be a discontinuity in Jounce there, as you point out there is.

A new perspective on the whole problem just came to me though. Here it is:

The paper argues that the ball could move because there is a solution to equation 2 (d^2r/dt^2 = sqrt(r)) in which the ball moves, where that equation 2 is derived from Newton's Second Law. But that doesn't mean the solution is applicable. To test whether it's applicable, we need to substitute it back into all Newton's Laws and see if they still hold.

Newton's first law says that an item will remain in its state of motion (which is interpreted to mean its velocity does not change) unless acted upon by a net external force. So the ball in a perfect, stationary position at the top will remain in its state of motion, which is stationary. It will not roll down. Hence the solution is non-Newtonian and must be rejected. It satisfies the second but not the first law.

The same goes for when we slide the ball up. When it arrives at the top it is stationary and balanced. Newton's first law says it will remain in that state until pushed.

For me, that solves the puzzle. The solution in which spontaneous movement occurs only satisfies Newton's 2nd and 3rd laws, not his first.

Newton's first law says that an item will remain in its state of motion (which is interpreted to mean its velocity does not change) unless acted upon by a (net) external force. So the ball in a perfect, stationary position at the top will remain in its state of motion, which is stationary. It will not roll down. Hence the solution is non-Newtonian and must be rejected. It satisfies the second but not the first law. — andrewk

I think your construal of the first law might be too strong, or too literal, and may make it inconsistent with the second law. This first law often is seen as a special case of the second, where the net force is zero. Thus construed, the force being applied to a mass at T only is relevant to its state of motion at T. Thus, the fact that the net force being applied to the mass at T is zero ought not to entail that the state of motion will remain unchanged at a later time but only that the rate of change of its velocity is zero at T. This is quite obvious when the mass moves within a variable field of force. Granted, it is less obvious in the case where a mass is instantaneously at rest in a variable field of force at a point where the force vanishes. This is the sort of case that allows for bifurcations in phase space. But if an overly strict construal of Newton's first law might dictate that a particle would remain stuck at such a point of vanishing force when it arrives there with a null velocity (so as to conveniently remove the bifurcation in phase space) then this construal of Newton's first law would also have the very unfortunate consequence that it makes it inconsistent with the second law in other cases. How would you account, consistently with such a strict construal of the first law, for the fact that the point mass does not remain at rest, in the case where the field of force varies at that point as a function of time and hence is null only for an instant?

this construal of Newton's first law would also have the very unfortunate consequence that it makes it inconsistent with the third law in other cases. — Pierre-Normand

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?

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.

Ah, that makes more sense. I think I'd approach it this way:

First we observe that Newton's laws were aimed at explaining real phenomena and so did not use the pernickety precision needed to cover all boundary cases. The case being discussed here is not only practically impossible but has probability zero of occurring even in Newtonian theory, so is also 'theoretically impossible' if we make the fudge of equating probability zero with impossible.

So to try to apply Newton's Laws to such a bizarre scenario we'd first have to expand them so it was clear what they entailed in that situation. I think expansions could be made both that make bifurcation possible and that prevent it.

An expansion that allowed bifurcation would be to essentially remove the First Law, by expressing it in a way that is unambiguously contained within the second law. We could do that by re-writing it as

'The time derivative of the velocity of a massive object is zero at any time at which the net force on it is zero'.

That makes it a special case of the 2nd law, so it adds nothing to 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.

I'm thinking there might be some sort of analogy to Asimov's three Laws of Robotics, where each law can overrule those above (or is it below) it, but I'm not sure if that works.

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

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

Or perhaps the "deep truth" is that the actual event that results in a push is unpredictable, just as the moment when the ball starts moving is unpredictable. Your thought experiment concerns chaos and complexity, a subject that interests me greatly, but on which I have no expertise to speak of. :blush:

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

:up: :smile:

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

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

But even internal vibrations are often demonstrated to have an external source. When I heat my lunch with the microwave, it causes internal vibrations, but there is an external source.

If the external source of the internal "push" cannot be found and identified, this doesn't mean that we can exclude the possibility of an external source, saying that there is no push just because we can't see the push.

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.

Hmm. I think removing backward-looking bifurcations would necessitate a further lengthening of the law statement. Perhaps something like this:

'Let P be the case that an object O has location L in phase space at time t. Let the set of patterns of motion of the object that are consistent with P under the 2nd and 3rd laws be S. Let U be the subset of S such that the object's velocity remains constant for some open interval [t,t+h), where h may vary by pattern.
Let W be the subset of S such that the object's velocity remains constant for some open interval (t-h,t], where h may vary by pattern.
Let V be the intersection of U and W.
Then the actual pattern of motion (both pre and post t) is in V if that is nonempty, else in U if that is nonempty, else in W if that is nonempty.

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.

This seems to do as much as possible to remove both future and past bifurcations in a way that is consistent with our intuitions.

Gravity.

Gravity. — creativesoul

What about it?

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?

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

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.

both the second law, and your expanded law (if I understand it correctly) are silent regarding what happens next. — Pierre-Normand

My expanded first 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.

As regards what happened in the past, to project backwards we need more information than just the current phase location of the ball. We need the phase locations of all the other elements of the system, including whatever it was that fired the ball up the slope, if that is what happened. This is a requirement for backwards projection in any multi-particle system, not just those with potential bifurcations. Essentially we are asking 'how did the ball get there', and we can't work that out just by looking at where the ball currently is. Eg consider a ball sitting at the bottom of an inverted dome. It could have rolled down from any angle, or dropped directly down out of the sky. Our inability to say which happened reflects that we don't know the current phase locations of all relevant particles, not that a bifurcation may have happened.

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?

Is your law mandating that the ball will stand still indefinitely after it has reached the apex — Pierre-Normand

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.

I think my use of the word 'observation' in the above rule may have conjured up images that were not intended. I have changed the words so that now it just refers to what the location in phase space is at that time, regardless of whether it is observed. It could, for instance, have been predicted from an earlier observation. In the case of your latest post, we have an observation at time T, which enables us to unambiguously project the path up to time t2, using only the 2nd law. At time t2 the 2nd law on its own allows for bifurcation, so the 1st law steps in and requires that the path that it follows is the one for which the velocity remains constant at zero from that point onwards (until such time as the dome wobbles or a new force acts on the ball).

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.