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.

This paper — Benkei

Thanks. It indeed appears to be a good place to start with Kane's theory of ultimate responsibility and SFAs ("self forming actions"), in order to learn how he's dealing with the problem from luck.

To base free will on the mere fact that not all processes are predictable is even a worse case of not understanding what we're talking about in my view. — Benkei

Sure, but who does that?

My first red flag with Robert Kane is therefore his equivocation of indeterminism and chance. That means he appears to be firmly in the territory of epistemological indeterminism which simply isn't interesting for the reason above. I'll read his full paper later but that's just a first few remarks to clarify my position based on his first two pages. — Benkei

Tell me what paper you're reading first. I'm not an advocate of Kane's libertarian conception of free will, myself, but as I've suggested, there are some good point that he is making while he is addressing the problem from luck. So, I'd be interested to hear your objections to his take. They may even coincide with mine.

My argument is distinct from the luck argument I guess or Robert Kane misrepresents it in his paper. — Benkei

What paper? Kane has written dozens of papers and, maybe, half a dozen books on this topic.

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?

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).

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.

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.)

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.

So long as people can't wrap their head around the idea that saying "indeterminism is necessary for free will to exist", really means that everything they decide is totally random as a result, we'll continue to have these discussions. — Benkei

The idea that lack of causal determination of actions (by laws of nature and prior events and/or states of affair) entails mere randomness is generally acknowledged as the luck objection to libertarianism. The problem of luck is well known and acknowledged by contemporary incompatibilist libertarians. Robert Kane, for instance, has a fairly sophisticated response to it, which, albeit not being entirely successful, on my view, has some good positive features.

On the other hand, although compatibilist accounts of free will don't have to deal with the luck objection specifically, they have problems of their own. They must respond to the main arguments for incompatibilism such as Jaegwon Kim's causal exclusion argument or Peter van Inwagen's consequence argument. (On my view, dispositionalist accounts of compatibilist free will also suffer from a third problem which is that they tend to narrow the scope of freedom more narrowly than is required to account for our normal ascriptions of personal responsibility to mature human agents). So, it's not sufficient to acknowledge that some amount of causal determination is required for dealing with the problem of luck. In order to secure a satisfactory compatibilist account of free will, one also has to show how autonomous rational and moral agency isn't threatened by determinism just as much (although in different ways) as it would be by indeterminism.

Making the generic cause to be about the impossibility of placing a ball with arbitrary accuracy on an apex is both another way of saying the same thing, but not quite as strong a version as focusing on the impossibility of eliminating triggering fluctuations. — apokrisis

I rather like the idea of a generic cause of the symmetry breaking mechanism. The generic cause, in this case, is the (practically inobservable) fluctuation background that serves as the explanation of the emergence of the indeterministic law that governs the (practically) observable events of symmetry breaking. There being such a generic cause doesn't entail that there is a law on account of which a contrastive explanation can be given as to why the ball fell in one rather than another direction, or fell immediately rather than at a slightly later time. There may be no such law, and no such contrastive cause. (There may however be a emergent law specifying the half-life of a ball's staying poised before starting to fall).

That is my view. The ball fell because when it was released by whatever was holding it on the apex, its centre of mass was not exactly above the point of contact with the dome, so it started falling. — andrewk

Have you looked at the paper linked to in the OP, though? The case has been specially contrived such that even if the ball is placed exactly at the apex, and with no initial velocity at all, then, consistently with Newton's laws of motion, it could either remain stationary or start falling towards an arbitrary radial direction with a distance from the apex: r(t) = (1/144) (t – T)^4 (where T is the time when it would spontaneously start moving in the absence of any net force at that time).

Do you buy his story? — apokrisis

Not really, because it assumes metaphysical realism: the idea that there might conceivably be an external God's eye view of the world that amounts to a complete description of it, including its laws and causal relations. I think Kant has shown this not to a be possible account of our (or of any conceivable) empirical world. But this case remains an instructive mathematical possibility. I'll say a bit more later on.

I'm not sure I understand your point. If the particle is perfectly balanced on top of the dome, then there it shall remain until some net force moves it. — LD Saunders

That's fairly intuitive, right? Yet, the mathematical analysis of the case contradicts this intuition. Look at the equation of motion of the ball bearing. At any time when it is at the apex, both the net force being applied on it, and its acceleration, are zero, as they should be, or else this equation would not be a solution to Newton's laws of motion. And yet, also, the ball bearing is radially being displaced away from the apex a finite distance after only a finite time. It needs not be the case that, as our intuition about causality seems to demand, that the cause (viz., the net force) begins operating before the effect (viz., the acceleration) starts being manifested. In this idealized Newtonian case, they both come into existence together (i.e. at any time t > T) without there being some other physical cause that accounts for the moment in time T when the ball starts moving away from the apex (with only an initial null acceleration at t=T, and at all t <= T).

All I can come up with is limited free will. We can't choose what we want but we can choose how we satisfy our wants. — TheMadFool

I don't quite see how one can consistently hold that view. If there is some generic end that you want to achieve, but that you can achieve in a variety of different ways, then you are going to do it in the way that you want to achieve it (after having pondered over the alternatives ways in which you can achieve it). But then, in that case, by your own premise, you will not be able to chose how (or in what way) to satisfy your generic want either.

But now if you time reverse the story, you still only can arrive infinitesimally close to the apex, not actually perched exactly on it. — apokrisis

No so. What you are saying would be true for any number of smooth convex domes, including spherical domes. But the particular shape being discussed in the paper that you linked to in the OP, referred to as "the dome" and defined radially as the surface with height h = -(2/3g) r^(3/2), where g is the vertical acceleration of gravity, is such that the equation of motion for a small spherical mass being perched at rest at the apex at t = 0 admits of two different sorts solutions. The first solution describes the mass remaining at rest at all times. The second class of solutions have the mass moving away from the apex with radial positon r(t) = (1/144) (t – T)^4, where T is an arbitrary time and t <= T; and r(t) = 0 at any time t before T.

So, in the time reversal scenario, when the ball is sent sliding up towards the apex with just the right speed, it doesn't slow down asymptotically as a function of time. It slows down to rest in a finite time and then (consistently with Newton's laws) remains at rest for an arbitrary amount of time at the apex before sliding back down in the same, or another arbitrary, radial direction.

If "the dome" didn't have this very specific shape, then, the equation of motion of the sphere instantaneously at rest at the apex, at a time, would (in most cases) be deterministic. It would necessarily remain at rest at all times. In the case of "the dome", indeterministic outcomes are consistent with Newton's laws of motion even in the idealized case where there is no initial disturbance at all away form an initial state of instantaneous rest! That's what makes "the dome" (Norton's) such an interesting shape.

How so? If the ball has mass, it has inertia. A push is required to set it moving. — apokrisis

Not so, as I've already explained to Bitter. The shape of the dome is such that, as the ball is getting infinitesimally close to the apex, the second derivatives of its horizontal motion tends towards zero; and hence, also, the horizontal component of the force.

If the net force acting on the ball bearing was zero, why would the ball move? On the other hand, can a ball sit on another ball without moving? Or must it eventually move as a chance event? — Bitter Crank

That's because of a specific queer mathematical property of the shape of the dome and how the system interacts with the vertical force of gravity. At the initial time when the ball gets rolling, its speed is zero. But the first derivative of its speed (its acceleration) also is zero. Hence, the force, at the instant in time when it is initially at rest, also is zero. This is why no force at all is required to set it in motion.

Another way to view it is to imagine the time reversal of the process where the ball is being sent rolling up the dome with just enough speed so that it will end up at rest at the apex, after a finite time. Thereafter -- and this is unmysterious -- it may remain at rest for an arbitrary period of time. If this is a valid solution to Newton's equations, then, so is the time reversal of this process where it remains at rest for some time and then "spontaneously" starts rolling (with an initial instantaneous null acceleration).

What's interesting about the dome is that the ball's starting from rest, and, after a finite time, rolling in an arbitrary direction, is a valid solution to Newton's laws of motion. So, in that case, there would be no need for as much as an infinitesimal nudge in order to set the ball in motion. It could start moving (consistently with Newton's laws) in any direction at a moment in time when the net force being applied on it is was null. Newton's laws of motion also leave it open when it would start moving, after having been at rest for an arbitrary length of time.

I think quantum wave collapses are best viewed as events that lack a contrastive cause, just like this idealized Newtonian example exemplifies. But this idea of events that happen (in some sort of symmetry breaking way) without being caused to so happen also may be the best construal of events that are determined by microscopic (symmetry breaking) causes where those causes simply are irrelevant to the emergent macroscopic dynamics that we are interested in.

Latest news... — Proto

If there are credible claims by people who knew Ford and Kavanaugh that both of them lied under oath, they should both be held accountable for their lies. It would be unfair to deny Kavanaugh the opportunity to sit on the Supreme Court, and not penalize Dr Ford as well. She should also be denied the opportunity to sit on the Supreme Court.

I don't know what all they will investigate, but from the POV of the administration, the less of a fishing expedition the better. — Bitter Crank

Their move is certainly effective. In the wake of the news of the tight grip that the White House is determined to exert on the investigation, the shares for Kavanaugh's being confirmed, on the Predictit prediction market have jumped up 10 points (from about 61 cents to about 71 cents). The reason they still remain as low as 71 cents, I conjecture, is because there is an expectation that the ambivalent Senators (Flake, Collins and Murkowski) might still vote "no" if they will feel the FBI investigation to have been a sham despite Kavanaugh having been "cleared".

From reports I've been reading, the FBI investigation will be extremely half-assed. — Maw

'Half-assed' is an understatement. It will be hundredth-assed. The scope of the investigation is defined by the White House while "White House counsel Don McGhan, who is the administration's leading advocate for Kavanaugh's confirmation, is overseeing the probe for the President and working closely with Senate Republican leaders." -- CNN's Ariane de Vogue, Dana Bash, Evan Perez and Kevin Liptak.

They've already decided that Kavanaugh's drinking history isn't part of the probe at all. Senator Lindsey Graham expressed what he views the proper scope of the probe to be: "They're going to Mark Judge, did you ever see Brett Kavanaugh drug women or engage in gang rape. I think that's going to be the focus of it."

wouldn't Mark Judge who was watching, take the key in his pocket? Would he leave the key there, so that she could somehow escape? — Agustino

You are assuming that they locked the door to prevent her from escaping. They may have locked the door so that nobody would walk in on them unexpectedly. They were drunk and may have expected, or hoped, that she would go along with their game with just a minimum of duress. Taking the key out might have been an unnecessarily threatening gesture (assuming there was a key at all; it may have been an interior spring door lock that you merely twist or depress). According to Ford, they were laughing a lot. Not all rapes or attempted rapes are premeditated.

I don’t know if it’s something that every conservative would rule for. — Michael

That's interesting. But this article doesn't make clear if there is anything about Kavanaugh that could lead us to expect that he would decide such as case differently than any other conservative judge.

There’s also an upcoming court case that is concerned with double jeopardy and federal pardons (i.e can you subsequently be charged for the equivalent state crime). — Michael

Would that be a case where Trump doesn't necessarily need that it be Kavanaugh specifically, but he needs that a fifth conservative seat be filled ASAP?

It seems tactically and politically stupid from where I'm standing but perhaps I'm missing an angle I'm not considering here. — Benkei

Maybe the Republicans don't mind so much if it isn't Kavanaugh who gets the seat. But Trump minds very much since Kavanaugh is the only one who asserted that a sitting President can't be indicted and that he ought to be able to fire at will a prosecutor investigating him. And the Republicans are bound not to make Trump and his base too angry.

I do think it's odd that Kavanaugh is so hostile to the idea of an FBI probe into these supposedly false accusations. — Erik

When he was pressed on this issue, he and the other Republicans appeared almost schizophrenic. On the one hand, they were arguing that the Senate handling of the allegations against him amounted to nothing more than a devious McCartyan show trial, and that the politicized process was more painful to Kavanaugh than living Hell. But when the possibility of a FBI handling of those matters was raised, Kavanaugh was insisting that he was delighted to being subjected to this process before the Senate committee, and that the very wise and respectable committee members were much better equipped for handling those delicate matters than the bumbling and clueless FBI.

:gasp: Woah. — StreetlightX

To be fair, he's always been a very loose cannon.

The partisanship seems to be owned wholly by the republicans by refusing to call Mark Judge to testify or to have the FBI look into Brett's moral character in more depth. — Benkei

The American Bar Association is now calling for the nomination process to be put on hold, and for the FBI to investigate. And Alan Dershowitz, of all people, is seconding their motion.