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Old 1st December 2008, 07:24 AM   #1
Paul C. Anagnostopoulos
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The dome: Nonprobabilistic indeterminism

John Norton, a philosopher at the University of Pittsburgh, presents a violation of determinism in Newtonian mechanics:

http://www.pitt.edu/~jdnorton/Goodies/Dome/index.html

The basic idea is that a ball sits on top of a specially-shaped, frictionless dome in an unchanging environment. Spontaneously, the ball rolls down the dome. One cannot assign a probability distribution to the time at which the ball rolls. In particular, there is no first instant of motion and therefore no cause of the motion. So we have an indeterminism that is nonprobabilistic.

I have at least one question about this. Is it a real possibility or just a thought experiment? In particular, there is no such thing as a frictionless surface.

Your thoughts?

~~ Paul
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Old 1st December 2008, 08:28 AM   #2
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We can put aside non-physical aspects, such as the dome having to exactly match a particular functional form, and there being no friction, and the ball being a point mass. After all, none of them are any more "non-physical" than the fact that we are using Newton's laws rather than Quantum mechanics.

What really makes this purely an abstract construct, with no physical implications whatsoever, is that it just amounts to assuming that all solutions to the differential equations used to describe Newton's laws, are actually consistent with Newtonian mechanics.

There are actually all sorts of examples in physics where the differential equations that describe the behavior of objects give many solutions, only some of which are physically realizable. What you have to remember is that the differential equation is just a mathematical representation of the natural laws that govern the behavior of an object. This is just an example of the general situation where one solution actually corresponds to Newtonian mechanics, and the others do not.

The author tries to gloss over this by saying that this unusual solution also "satisfies Newton's Laws", but in fact what he shows that it satisfies is the differential equation. He then goes on to interpret a common phrasing of Newton's laws in a way such that his solution also satisfies that, but ignores the fact that this particular phrasing is not in line with Newtonian Mechanics, which are, of course, completely deterministic.

It is also interesting to note that the types of shapes that need to be used to generate these strange solutions are ones that are not "smooth". That is, they involve singularities in their high-order derivatives. When one constructs thought experiments using such configurations, all sorts of strange things can happen.


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Old 1st December 2008, 08:44 AM   #3
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What is he talking about when he says that there is no first instant in which the mass moves? Wouldn't his argument apply to any object that was stationary and then started moving? It seems like this is related to Zeno's paradox.

~~ Paul
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Old 1st December 2008, 09:48 AM   #4
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I think there are some huge assumptions on that web page, mainly that there is not going to be a probability distribution for the point after the setting of the ball that it begins motion.

It is a thought experiment but it seems he just asserting that there is no probability distribution:
Factors that might cause the ball to move are not 'spontaneous', even if the author just defines them that way. The ball is being acted upon by a constant gravitational field.
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Old 1st December 2008, 10:06 AM   #5
Paul C. Anagnostopoulos
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I was discussing this with Michael Strevens, a philosopher at New York University.
Originally Posted by me
Yes, I guess that is what you are saying. But is Norton suggesting
that such a dome and ball can actually exist and behave this way?
For example, he says the ball is frictionless, but there is no such
thing.
Originally Posted by Strevens
Two different questions:
1. Could this happen in the actual world?
2. Is this a coherent physics for a world? Is such a world
metaphysically possible?
It's (2) that matters for our purposes, that is, for deciding whether
it is metaphysically possible that there exists a non-deterministic
but non-probabilistic universe.

To (1), the answer is surely no. To (2) -- the best way to make a case
against would be to argue that the dome world violates some
metaphysically necessary principle of causality. (People have tried
this.) But it's not totally easy to say what the principle is (in an
illuminating way, i.e., not just "dome worlds are impossible"). And
even if you articulate such a principle, it is hard to see how to
argue that it is true, let alone metaphysically necessary.
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Old 1st December 2008, 10:09 AM   #6
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I'm missing something.

Why, in the absence of any for besides perfectly normal gravity, would the ball begin to roll at all? I seem to recall something about "A body continues to maintain its state of rest or of uniform motion unless acted upon by an external unbalanced force."

But I can understand if such a simple physically demonstrable platitude is completely inapplicable to the high-flown world of mental experiments.
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Old 1st December 2008, 12:55 PM   #7
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Paul,

Quote:
What is he talking about when he says that there is no first instant in which the mass moves? Wouldn't his argument apply to any object that was stationary and then started moving? It seems like this is related to Zeno's paradox.
That's a tricky one to explain. If you consider the case where the dome is just a semi-sphere (for example), then you have to apply a force at some time T in order to get the ball moving. In such an example, the accelleration of the ball is zero for all times t<T, and non-zero for all times t>=T. Thus we can say that the ball begins moving at time t=T.

For the example he gives, the situation is different. In this case, the accelleration of the ball is zero for all times t<=T, and non-zero for all times t>T. It is a subtle difference, but it means that, as he says, there is no instant at which it begins moving. That is, there is no time x such that the velocity is not zero at x, but zero at all times t<x.


Dancing David,

Quote:
I think there are some huge assumptions on that web page, mainly that there is not going to be a probability distribution for the point after the setting of the ball that it begins motion.
It's not that he is assuming that there is no probability distribution, but rather that he is pointing out that Newton's laws do not provide a probability distribution. The implication being that if this scenario were physically possible, then Newton's laws would have to be incomplete. After all, if the scenario he described could happen, then one could perform the experiment, and empirically derive the distribution. So a distribution would necessarily have to exist. But Newton's laws do not provide it, so some additional natural laws would be required to do so.

Of course, since the scenario he describes can't happen, even in a hypothetical universe where Newton's laws apply, this implication of incompleteness does not obtain.

Quote:
Factors that might cause the ball to move are not 'spontaneous', even if the author just defines them that way. The ball is being acted upon by a constant gravitational field.
Incorrect. In the given scenario there is no net force on the ball until after it has begun moving. That is the whole point. What the author has done is present a non-physically-realizable scenario in which the equations describing Newtonian mechanics are consistent with non-deterministic solutions.


Neutrino Cannon,

Quote:
I'm missing something.

Why, in the absence of any for besides perfectly normal gravity, would the ball begin to roll at all? I seem to recall something about "A body continues to maintain its state of rest or of uniform motion unless acted upon by an external unbalanced force."
That is essentially the point I was making in my first reply. The mathematical equations the author uses in his derivation are a way of mathematically expressing the physical principle you just mentioned. What the author shows is that under very specially constructed conditions, one can show that these mathematical equations actually allow violations of the principle that they were invented to describe. He then essentially plays a game of semantics to try to make it sound like those original principles are also compatible with the strange scenario he describes.


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Old 1st December 2008, 01:30 PM   #8
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Thanks Stimpy, I still don't get it but will reread and think about it.
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Old 1st December 2008, 01:32 PM   #9
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How could we , really, tell the difference between the ball rolling:-
1. For no damn reason whatsoever.
2. Because a graviton from Alpha Centauri happened by.
3. Because god diddit.
4. Due to a Reiki energy malfunction.

Hidden variables, in other wor(l)ds.
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Old 1st December 2008, 02:32 PM   #10
Paul C. Anagnostopoulos
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Originally Posted by Stimpson
That's a tricky one to explain. If you consider the case where the dome is just a semi-sphere (for example), then you have to apply a force at some time T in order to get the ball moving. In such an example, the accelleration of the ball is zero for all times t<T, and non-zero for all times t>=T. Thus we can say that the ball begins moving at time t=T.

For the example he gives, the situation is different. In this case, the accelleration of the ball is zero for all times t<=T, and non-zero for all times t>T. It is a subtle difference, but it means that, as he says, there is no instant at which it begins moving. That is, there is no time x such that the velocity is not zero at x, but zero at all times t<x.
Oh my, I didn't realize that the exact shape was so important. Why does a semi-sphere require force, while this shape does not? Since it's a point-like mass, I wouldn't think the shape would matter, since there is only one point where the gravitational attactions cancel out, regardless of shape.

But I guess not. What's special about this apex?

~~ Paul
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Old 1st December 2008, 02:51 PM   #11
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The potential in question (i.e. the shape of the dome, h(r)) is non-analytic. In layman's terms, it is not smooth (its second derivative is infinite at r=0). Such potentials are not guaranteed to result in sensible physics, and this is a case where one doesn't.

Moreover, the solutions he considers are a set of measure zero. The full set of solutions to that problem contain a continuous parameter - the angular momentum - which is conserved by the motion. His solutions are for exactly zero angular momentum; any other value would avoid r=0 and have sensible and "deterministic" behavior. So there are at least two excellent reasons not to care about this.

As for Streven's second question above.... it's obviously hard to say what is possible and what isn't. Quantum mechanics probably destroys this particular paradox, since it only affects one point (I'll have to think that over to be sure). So I think you'd need a world without quantum mechanics - but such a world would suffer from problems like the ultraviolet catastrophe. Perhaps it could exist, but it would be utterly different from ours.

ETA - out of curiosity I checked, and in QM the wavefunction near r=0 is only affected at third order even for zero angular momentum. So nothing weird happens.

Last edited by sol invictus; 1st December 2008 at 03:54 PM.
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Old 2nd December 2008, 02:30 AM   #12
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Paul,

Quote:
Oh my, I didn't realize that the exact shape was so important. Why does a semi-sphere require force, while this shape does not? Since it's a point-like mass, I wouldn't think the shape would matter, since there is only one point where the gravitational attactions cancel out, regardless of shape.
For a hemisphere, and many other types of shapes, the apex will still be an unstable equilibrium point. But that is not the issue. If you are exactly at the equilibrium point, then stable or not, you should not move.

The issue is that for this particular shape, you can get these strange additional solutions. For a shape like a hemisphere, you will only get one solution, which is the solution where the ball never moves. For this particular shape, and certain other shapes where the curve is non-analytic at the apex, you get additional solutions as well.

The author of that page sort of explains the special nature of this shape, with his time-reversal example. As he explains, for a hemispherical dome, if you start with the ball at the bottom, and give it exactly the right upward momentum in exactly the right direction, it would theoretically stop at the apex, but would take an infinite amount of time to reach it. For this special shape, it reaches it in a finite time. That is the source of the strange solutions.

And as I mentioned in my first reply, and Sol Invictus elaborated on, this is a result of the curve not being smooth at the peak. More to the point, it is non-analytic. As Sol mentioned, all sorts of strange things can happen when you start playing with non-analytic potentials. But those strange solutions are physically irrelevant, because they only obtain for those specific parameters. Tweak the potential at the peak by an arbitrarily small amount, and the strange solutions go away.

Incidentally, one need not consider anything as complicated as this to get such strange solutions. Simply construct a potential that is infinite, or discontinuous at some point, and you will see all sorts of strange things there too. The difference is that in those cases we tend to immediately reject them as "non-physical" due to the singularity. In this case it is again a singularity that causes the problem, but it is just less obvious because it appears in the second derivative of the potential, rather than in the potential itself or its first derivative. The deeper hidden the singularity is, the less obvious it is when looking at the curve. But it's still there, and it still plays havoc with any differential equations involving that potential.


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Old 2nd December 2008, 02:45 PM   #13
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Stimpson J. Cat,


What do you mean by "a potential that is infinite"? Or what do you mean a potential that is discontinuous at some point?

Also when you say singularity, I assume you do not mean a quantum singularity which is a black-hole right?


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Old 3rd December 2008, 05:56 AM   #14
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INRM,


Quote:
What do you mean by "a potential that is infinite"? Or what do you mean a potential that is discontinuous at some point?
I mean that the potential either diverges to infinity at some point, or undergoes a finite change at over an infinitesimal interval (in which case the derivative diverges to infinity).

Quote:
Also when you say singularity, I assume you do not mean a quantum singularity which is a black-hole right?
Correct. I mean "singularity" in the more general mathematical sense.
http://en.wikipedia.org/wiki/Mathematical_singularity


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Old 5th December 2008, 05:29 PM   #15
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Stimpson J. Cat,

Okay, I sort of understand what you mean.


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Old 2nd December 2018, 06:20 PM   #16
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I was going to start a new thread about Norton's Dome as it came up elsewhere, but I thought I would search for an existing one.

Turns out this came up over ten years ago, so I thought I would give it an epic bounce.

The problem seems to be that the shape of the dome is described by h=(2/3g)r^(3/2), in which case what is "g"?

From the context it is the Newtonian approximation for the acceleration due to gravity close to the surface of the Earth. But if this dome is supposed to be sitting on top of an Earth size mass this would differ infinitesimally from the top of the dome to the bottom.

For most purposes this does not matter since the difference is so small, but in this case it seems to make the conclusion invalid since the derived force from this shape would not be exact.

I suppose that it could refer to a dome being accelerated through empty space at an even 9.8 m/s^2, but then the force applied at the base of the dome would have to be altered slightly to compensate for the shifting weight due to the changing position of the point mass.

In order for the equations to hold, the change in force would have to happen instantaneously with the movement of the mass and so this would simply be a case of the mass moving because the force at the base was changed.
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Old 3rd December 2018, 01:55 AM   #17
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Originally Posted by Robin View Post
The problem seems to be that the shape of the dome is described by h=(2/3g)r^(3/2), in which case what is "g"?

From the context it is the Newtonian approximation for the acceleration due to gravity close to the surface of the Earth. But if this dome is supposed to be sitting on top of an Earth size mass this would differ infinitesimally from the top of the dome to the bottom.

For most purposes this does not matter since the difference is so small, but in this case it seems to make the conclusion invalid since the derived force from this shape would not be exact.
To a first approximation, the value for g is that of gravity at the apex of the dome, as this is the point at which the unstable equilibrium is defined. To a higher degree of approximation, it should in principle be possible to correct for the known variation in gravity by adjusting the shape of the dome accordingly. Since it's a thought experiment, we can assume that this calculation has been carried out correctly to recover the original scenario.

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Old 3rd December 2018, 03:05 AM   #18
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I don't see it. The height of the dome at a given point is defined in terms of the gravitational force at that height. How does that work?
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Old 3rd December 2018, 03:11 AM   #19
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So there's no air in this experiment? No molecular motion in the objects? It doesn't sound very practical to me. Besides that, many scenarios can demonstrate lack of causality in macro objects better than this. If you rig up a machine to perform an action upon the radioactive decay of a piece of matter, that's non-causal as the initial prompt is quantum and therefore unpredictable. I don't get what he's talking about.
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Old 3rd December 2018, 03:20 AM   #20
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Originally Posted by Robin View Post
I don't see it. The height of the dome at a given point is defined in terms of the gravitational force at that height. How does that work?
The equation of the dome profile is determined from a required acceleration profile of an object placed upon it, assuming constant g. If you want to account for variable g, then it would have to be re-derived based on the actual gravitational profile. It might be a very complex calculation, but it is in principle calculable.

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Me: So what you're saying is that, if the load carrying ability of the lower structure is reduced to the point where it can no longer support the load above it, it will collapse without a jolt, right?

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Old 3rd December 2018, 03:23 AM   #21
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Originally Posted by baron View Post
If you rig up a machine to perform an action upon the radioactive decay of a piece of matter, that's non-causal as the initial prompt is quantum and therefore unpredictable.
But that misses the entire point of the thought experiment, which is to argue that it is possible to construct a scenario strictly within Newtonian physics with a non-deterministic outcome.

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Old 3rd December 2018, 03:33 AM   #22
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Originally Posted by Dave Rogers View Post
But that misses the entire point of the thought experiment, which is to argue that it is possible to construct a scenario strictly within Newtonian physics with a non-deterministic outcome.

Dave
But it's not strictly within Newtonian physics (not that anything can be in any event). A perfectly smooth ball on a perfectly smooth surface will be subject to the quantum activity of the few particles that come into 'contact' with each other.
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Old 3rd December 2018, 04:41 AM   #23
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Originally Posted by baron View Post
But it's not strictly within Newtonian physics (not that anything can be in any event). A perfectly smooth ball on a perfectly smooth surface will be subject to the quantum activity of the few particles that come into 'contact' with each other.
True, but still missing the point of the argument. It's intended as a counter-argument to the claim that nondeterministic solutions can only arise from quantum effects, by constructing a hypothetical scenario purely within Newtonian physics that has no deterministic outcome.

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Old 3rd December 2018, 05:12 AM   #24
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Originally Posted by Dave Rogers View Post
True, but still missing the point of the argument. It's intended as a counter-argument to the claim that nondeterministic solutions can only arise from quantum effects, by constructing a hypothetical scenario purely within Newtonian physics that has no deterministic outcome.

Dave
But the non-determinism does arise from quantum effects, just not any that have been explicitly defined. It's just a question of what terminology is used. He's chosen to use phrases like 'frictionless' when he could equally well talk of interactions between individual particles. And, of course, no matter how sensitive your experimental equipment, you could never set up such a scenario, so it's misleading in that sense too.
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Old 3rd December 2018, 05:32 AM   #25
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Originally Posted by baron View Post
But the non-determinism does arise from quantum effects, just not any that have been explicitly defined. It's just a question of what terminology is used. He's chosen to use phrases like 'frictionless' when he could equally well talk of interactions between individual particles.
All the terms used have valid meanings in Newtonian physics independently of quantum effects, and the non-determinism arises from a purely Newtonian analysis. You're choosing to apply a more realistic set of physical laws to the problem, which is not the point of the thought experiment. It's not a statement about reality; it's a statement about the approximation to reality contained in Newton's laws.

Originally Posted by baron View Post
And, of course, no matter how sensitive your experimental equipment, you could never set up such a scenario, so it's misleading in that sense too.
It's not intended to be a real world experiment. It's a thought experiment that sets up an idealised hypothetical scenario that's entirely consistent with Newtonian physics, in order to demonstrate that Newton's laws don't exclude the possibility of non-deterministic behaviour.

Dave
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Old 3rd December 2018, 07:40 AM   #26
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Originally Posted by Dave Rogers View Post
All the terms used have valid meanings in Newtonian physics independently of quantum effects, and the non-determinism arises from a purely Newtonian analysis. You're choosing to apply a more realistic set of physical laws to the problem, which is not the point of the thought experiment. It's not a statement about reality; it's a statement about the approximation to reality contained in Newton's laws.



It's not intended to be a real world experiment. It's a thought experiment that sets up an idealised hypothetical scenario that's entirely consistent with Newtonian physics, in order to demonstrate that Newton's laws don't exclude the possibility of non-deterministic behaviour.
I just don't see the point in it. Newton's laws are intended to be applied to the physical world. They have no other use. This guy is imagining a world that not only doesn't exist but could not exist, even in theory. Why bother? I asked this question when I was about ten - if a ball was on top of a hill which way would it roll down?. I didn't know anything about Newton back then, it was just something that occurred to me. I guess I should have waited.
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Old 3rd December 2018, 03:04 PM   #27
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If the height at r is a function of the gravitational pull, then you have to know the value of g at r to get the height at r.

But you also have to know the height at r to get the value of g at r.

That might be derivable, for example you could have it as the solution to a set of non-linear equations, but then he would have to show that there was the same non-deterministic solution to the new equation.

If Norton is saying, 'suppose there was a force, similar to gravity, except that it is constant and not proportional to distance ...' then OK.

But that seems to take away from the claim that it is a demonstration of indeterminism in Newtonian physics.
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Old 3rd December 2018, 03:21 PM   #28
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Originally Posted by baron View Post
I just don't see the point in it. Newton's laws are intended to be applied to the physical world. They have no other use. This guy is imagining a world that not only doesn't exist but could not exist, even in theory. Why bother? I asked this question when I was about ten - if a ball was on top of a hill which way would it roll down?. I didn't know anything about Newton back then, it was just something that occurred to me. I guess I should have waited.
Most mathematics does not have any real world instantiation.

In this case the point is to demonstrate that an ostensibly deterministic system might not actually be deterministic. It is something that could potentially apply to any deterministic system.
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Old 3rd December 2018, 09:34 PM   #29
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Originally Posted by Robin View Post
If the height at r is a function of the gravitational pull, then you have to know the value of g at r to get the height at r.

But you also have to know the height at r to get the value of g at r.

For the purposes of this thought experiment the height as well as the values of r and g are not relevant to the outcome. The principle should stand for any values > 0.
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Old 3rd December 2018, 11:59 PM   #30
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Originally Posted by Paul C. Anagnostopoulos View Post
John Norton, a philosopher at the University of Pittsburgh, presents a violation of determinism in Newtonian mechanics:

http://www.pitt.edu/~jdnorton/Goodies/Dome/index.html

The basic idea is that a ball sits on top of a specially-shaped, frictionless dome in an unchanging environment. Spontaneously, the ball rolls down the dome. One cannot assign a probability distribution to the time at which the ball rolls. In particular, there is no first instant of motion and therefore no cause of the motion. So we have an indeterminism that is nonprobabilistic.

I have at least one question about this. Is it a real possibility or just a thought experiment? In particular, there is no such thing as a frictionless surface.

Your thoughts?

~~ Paul

The cause of the mass starting to move seems to be that the scenario says that it does.
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